OncoSimulR: forward genetic simulation in asexual populations with arbitrary epistatic interactions and a focus on modeling tumor progression.
Ramon Diaz-Uriarte, Sergio Sanchez-Carrillo, Juan Antonio Miguel-Gonzalez
Dept. Biochemistry, Universidad Autónoma de Madrid, Instituto de Investigaciones Biomédicas ‘Alberto Sols’ (UAM-CSIC), Madrid, Spain.
ramon.diaz@iib.uam.es, https://ligarto.org/rdiaz
2021-10-26. OncoSimulR version 3.2.0
1 Introduction
OncoSimulR is an individual- or clone-based forward-time genetic simulator for biallelic markers (wildtype vs. mutated) in asexually reproducing populations without spatial structure (perfect mixing). Its design emphasizes flexible specification of fitness and mutator effects.
OncoSimulR was originally developed to simulate tumor progression with emphasis on allowing users to set restrictions in the accumulation of mutations as specified, for example, by Oncogenetic Trees (OT: Desper et al. 1999; Szabo and Boucher 2008) or Conjunctive Bayesian Networks (CBN: Beerenwinkel, Eriksson, and Sturmfels 2007; Gerstung et al. 2009, 2011), with the possibility of adding passenger mutations to the simulations and allowing for several types of sampling.
Since then, OncoSimulR has been vastly extended to allow you to specify other types of restrictions in the accumulation of genes, such as the XOR models of Korsunsky et al. (2014) or the “semimonotone” model of Farahani and Lagergren (2013). Moreover, different fitness effects related to the order in which mutations appear can also be incorporated, involving arbitrary numbers of genes. This is very different from “restrictions in the order of accumulation of mutations”. With order effects, described in a recent cancer paper by Ortmann and collaborators (Ortmann et al. 2015), the effect of having both mutations “A” and “B” differs depending on whether “A” appeared before or after “B” (the actual case involves genes JAK2 and TET2).
More generally, OncoSimulR now also allows you to specify arbitrary epistatic interactions between arbitrary collections of genes and to model, for example, synthetic mortality or synthetic viability (again, involving an arbitrary number of genes, some of which might also depend on other genes, or show order effects with other genes). Moreover, it is possible to specify the above interactions in terms of modules, not genes. This idea is discussed in, for example, Raphael and Vandin (2015) and Gerstung et al. (2011): the restrictions encoded in, say, CBNs or OT can be considered to apply not to genes, but to modules, where each module is a set of genes (and the intersection between modules is the empty set) that performs a specific biological function. Modules, then, play the role of a “union operation” over the set of genes in a module. In addition, arbitrary numbers of genes without interactions (and with fitness effects coming from any distribution you might want) are also possible.
You can also directly specify the mapping between genotypes and fitness and, thus, you can simulate on fitness landscapes of arbitrary complexity.
It is now (released initially in this repo as the freq-dep-fitness branch on February 2019) also possible to simulate scenarios with frequency-dependent fitness, where the fitness of one or more genotypes depends on the relative or absolute frequencies of other genotypes, as in game theory and adaptive dynamics. This makes it possible to model predation and parasitism, cooperation and mutualism, and commensalism. It also allows to model therapeutic interventions (where fitness changes at specified time points or as a function of the total populations size).
Simulations can start from arbitrary initial population compositions and it is also possible to simulate multiple species. Thus, simulations that involve both ecological and evolutionary processes are possible.
Mutator/antimutator genes, genes that alter the mutation rate of other genes (Gerrish et al. 2007; Tomlinson, Novelli, and Bodmer 1996), can also be simulated with OncoSimulR and specified with most of the mechanisms above (you can have, for instance, interactions between mutator genes). And, regardless of the presence or not of other mutator/antimutator genes, different genes can have different mutation rates.
Simulations can be stopped as a function of total population size, number of mutated driver genes, or number of time periods. Simulations can also be stopped with a stochastic detection mechanism where the probability of detecting a tumor increases with total population size. Simulations return the number of cells of every genotype/clone at each of the sampling periods and we can take samples from the former with single-cell or whole- tumor resolution, adding noise if we want. If we ask for them, simulations also store and return the genealogical relationships of all clones generated during the simulation.
The models so far implemented are all continuous time models, which are simulated using the BNB algorithm of Mather, Hasty, and Tsimring (2012). The core of the code is implemented in C++, providing for fast execution. To help with simulation studies, code to simulate random graphs of the kind often seen in CBNs, OTs, etc, is also available. Finally, OncoSimulR also allows for the generation of random fitness landscapes and the representation of fitness landscapes and provides statistics of evolutionary predictability.
1.1 Key features of OncoSimulR
As mentioned above, OncoSimulR is now a very general package for forward genetic simulation, with applicability well beyond tumor progression. This is a summary of some of its key features:
- You can specify arbitrary interactions between genes, with
arbitrary fitness effects, with explicit support for:
Restrictions in the accumulations of mutations, as specified by Oncogenetic Trees (OTs), Conjunctive Bayesian Networks (CBNs), semimonotone progression networks, and XOR relationships.
- Epistatic interactions including, but not limited to, synthetic viability and synthetic lethality.
Order effects.
- You can add passenger mutations.
- You can add mutator/antimutator effects.
- Fitness and mutation rates can be gene-specific.
You can add arbitrary numbers of non-interacting genes with arbitrary fitness effects.
you can allow for deviations from the OT, CBN, semimonotone, and XOR models, specifying a penalty for such deviations (the \(s_h\) parameter).
You can conduct multiple simulations, and sample from them with different temporal schemes and using both whole tumor or single cell sampling.
You can stop the simulations using a flexible combination of conditions: final time, number of drivers, population size, fixation of certain genotypes, and a stochastic stopping mechanism that depends on population size.
Right now, three different models are available, two that lead to exponential growth, one of them loosely based on Bozic et al. (2010), and another that leads to logistic-like growth, based on McFarland et al. (2013).
You can use large numbers of genes (e.g., see an example of 50000 in section 6.5.3).
Simulations are generally very fast: I use C++ to implement the BNB algorithm (see sections 15.5 and 15.6 for more detailed comments on the usage of this algorithm).
You can obtain the true sequence of events and the phylogenetic relationships between clones (see section 15.1 for the details of what we mean by “clone”).
You can generate random fitness landscapes (under the House of Cards, Rough Mount Fuji, or additive models, or combinations of the former and under the NK model) and use those landscapes as input to the simulation functions.
You can plot fitness landscapes.
You can obtain statistics of evolutionary predictability from the simulations.
You can now also use simulations with frequency-dependent fitness: fitness (birth rate) is not fixed for a genotype, but can be a function of the frequecies of the clones (see section 10). We can therefore use OncoSimulR to examine, via simulations, results from game theory and adaptive dynamics and study complex scenarios that are not amenable to analytical solutions. More generally, we can model predation and parasitism, cooperation and mutualism, and commensalism.
It is possible to start the simulation with arbitrary initial composition (section 6.7) and to simulate multiple species (section 6.8). You can thus run simulations that involve both ecological and evolutionary processes involving inter-species relationships plus genetic restrictions in evolution.
It is possible to simulate many different therapeutic interventions. Section 12 shows examples of interventions where certain genotypes change fitness (because of chemotherapy) at specified times. More generally, since fitness (birth rates) can be made a function of total populations sizes and/or frequencies (see section 10), many different arbitrary intervention schemes can be simulated. Possible models are, of course, not limited to cancer chemotherapy, but could include antibiotic treatment of bacteria, antiviral therapy, etc.
The table below, modified from the table at the Genetics Simulation Resources (GSR) page, provides a summary of the key features of OncoSimulR. (An explanation of the meaning of terms specific to the GSR table is available from https://popmodels.cancercontrol.cancer.gov/gsr/search/ or from the Genetics Simulation Resources table itself, by moving the mouse over each term).
| Attribute Category | Attribute |
|---|---|
| Target | |
| Type of Simulated Data | Haploid DNA Sequence |
| Variations | Biallelic Marker, Genotype or Sequencing Error |
| Simulation Method | Forward-time |
| Type of Dynamical Model | Continuous time |
| Entities Tracked | Clones (see 15.2) |
| Input | Program specific (R data frames and matrices specifying genotypes’ fitness, gene effects, and starting genotype) |
| Output | |
| Data Type | Genotype or Sequence, Individual Relationship (complete parent-child relationships between clones), Demographic (populations sizes of all clones at sampling times), Diversity Measures (LOD, POM, diversity of genotypes), Fitness |
| Sample Type | Random or Independent, Longitudinal, Other (proportional to population size) |
| Evolutionary Features | |
| Mating Scheme | Asexual Reproduction |
| Demographic | |
| Population Size Changes | Exponential (two models), Logistic (McFarland et al., 2013) |
| Fitness Components | |
| Birth Rate | Individually Determined from Genotype (models “Exp”, “McFL”, “McFLD”). Frequency-Dependently Determined from Genotype (models “Exp”, “McFL”, “McFLD”) |
| Death Rate | Individually Determined from Genotype (model “Bozic”), Influenced by Environment —population size (models “McFL” and “McFLD”) |
| Natural Selection | |
| Determinant | Single and Multi-locus, Fitness of Offspring, Environmental Factors (population size, genotype frequencies) |
| Models | Directional Selection, Multi-locus models, Epistasis, Random Fitness Effects, Frequency-Dependent |
| Mutation Models | Two-allele Mutation Model (wildtype, mutant), without back mutation |
| Events Allowed | Varying Genetic Features: change of individual mutation rates (mutator/antimutator genes) |
| Spatial Structure | No Spatial Structure (perfectly mixed and no migration) |
Further details about the original motivation for wanting to simulate data this way in the context of tumor progression can be found in Diaz-Uriarte (2015), where additional comments about model parameters and caveats are discussed.
Are there similar programs? The Java program by Reiter et al. (2013), TTP, offers somewhat similar functionality to the previous version of OncoSimulR, but it is restricted to at most four drivers (whereas v.1 of OncoSimulR allowed for up to 64), you cannot use arbitrary CBNs or OTs (or XORs or semimonotone graphs) to specify restrictions, there is no allowance for passengers, and a single type of model (a discrete time Galton-Watson process) is implemented. The current functionality of OncoSimulR goes well beyond the the previous version (and, thus, also the TPT of Reiter et al. (2013)). We now allow you to specify all types of fitness effects in other general forward genetic simulators such as FFPopSim (Zanini and Neher 2012), and some that, to our knowledge (e.g., order effects) are not available from any genetics simulator. In addition, the “Lego system” to flexibly combine different fitness specifications is also unique; by “Lego system” I mean that we can combine different pieces and blocks, similarly to what we do with Lego bricks. (I find this an intuitive and very graphical analogy, which I have copied from Hothorn et al. (2006) and Hothorn et al. (2008)). In a nutshell, salient features of OncoSimulR compared to other simulators are the unparalleled flexibility to specify fitness and mutator effects, with modules and order effects as particularly unique, and the options for sampling and stopping the simulations, particularly convenient in cancer evolution models. Also unique in this type of software is the addition of functions for simulating fitness landscapes and assessing evolutionary predictability.
1.2 What kinds of questions is OncoSimulR suited for?
OncoSimulR can be used to address questions that span from the effect of mutator genes in cancer to the interplay between fitness landscapes and mutation rates. The main types of questions that OncoSimulR can help address involve combinations of:
- Simulating asexual evolution (the
oncoSimul*functions) where:- Fitness is:
- A function of specific epistatic effects between genes
- A function of order effects
- A function of epistatic effects specified using
DAGs/posets where these DAGs/posets:
- Are user-specified
- Generated randomly (
simOGraph)
- Any mapping between genotypes and fitness where this mapping is:
- User-specified
- Generated randomly from families of random fitness landscapes (
rfitness)
- A function of the frequency of other genotypes (i.e., frequency-dependent fitness), such as in adaptive dynamics (see section 10 for more details). This also allows you to model competition, cooperation and mutualism, parasitism and predation, and commensalism between clones.
- Mutation rates can:
- Vary between genes
- Be affected by other genes
- Fitness is:
Examining times to evolutionarily or biomedically relevant events (fixation of genotypes, reaching a minimal size, acquiring a minimal number of driver genes, etc —specified with the stopping conditions to the
oncoSimul*functions).- Using different sampling schemes (
samplePop) that are related to:- Assessing genotypes from single-cell vs. whole tumor (or whole
population) with the
typeSampleargument - Genotyping error (
propErrorargument) - Timing of samples (
timeSampleargument) - … and assessing the consequences of those on the observed
genotypes and their diversity (
sampledGenotypes) and any other inferences that depend on the observational process. - (OncoSimulR returns the abundances of all genotypes at each of
the sampling points, so you are not restricted by what the
samplePopfunction provides.)
- Assessing genotypes from single-cell vs. whole tumor (or whole
population) with the
Tracking the genealogical relationships of clones (
plotClonePhylog) and assessing evolutionary predictability (LOD,POM).
Some specific questions that you can address with the help of OncoSimulR are discussed in section 1.3.
A quick overview of the main functions and their relationships is shown in Figure 1.1, where we use italics for the type/class of R object and courier font for the name of the functions.
Figure 1.1: Relationships between the main functions in OncoSimulR.
1.3 Examples of questions that can be addressed with OncoSimulR
Most of the examples in the rest of this vignette, starting with those in 1.7, focus on the mechanics. Here, we will illustrate some problems in cancer genomics and evolutionary genetics where OncoSimulR could be of help. This section does not try to provide an answer to any of these questions (those would be full papers by themselves). Instead, this section simply tries to illustrate some kinds of questions where you can use OncoSimulR; of course, the possible uses of OncoSimulR are only limited by your ingenuity. Here, I will only use short snippets of working code as we are limited by time of execution; for real work you would want to use many more scenarios and many more simulations, you would use appropriate statistical methods to compare the output of runs, etc, etc, etc.
## Load the package
library(OncoSimulR) 1.3.1 Recovering restrictions in the order of accumulation of mutations
This is a question that was addressed, for instance, in Diaz-Uriarte (2015): do methods that try to infer restrictions in the order of accumulation of mutations (e.g., Szabo and Boucher 2008; Gerstung et al. 2009; Ramazzotti et al. 2015) work well under different evolutionary models and with different sampling schemes?
A possible way to examine that question would involve:
- generating random DAGs that encode restrictions;
- simulating cancer evolution using those DAGs;
- sampling the data and adding different levels of noise to the sampled data;
- running the inferential method;
- comparing the inferred DAG with the original, true, one.
## For reproducibility
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
## Simulate a DAG
g1 <- simOGraph(4, out = "rT")
## Simulate 10 evolutionary trajectories
s1 <- oncoSimulPop(10, allFitnessEffects(g1, drvNames = 1:4),
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility of vignette
## Sample those data uniformly, and add noise
d1 <- samplePop(s1, timeSample = "unif", propError = 0.1)
##
## Subjects by Genes matrix of 10 subjects and 4 genes.
## You would now run the appropriate inferential method and
## compare observed and true. For example
## require(Oncotree)
## fit1 <- oncotree.fit(d1)
## Now, you'd compare fitted and original. This is well beyond
## the scope of this document (and OncoSimulR itself).1.3.2 Sign epistasis and probability of crossing fitness valleys
This question, and the question in the next section (1.3.3), encompass a wide range of issues that have been addressed in evolutionary genetics studies and which include from detailed analysis of simple models with a few uphill paths and valleys as in Weissman et al. (2009) or Ochs and Desai (2015), to questions that refer to larger, more complex fitness landscapes as in Szendro, Franke, et al. (2013) or Franke et al. (2011) or Krug (2019) (see below).
Using as an example Ochs and Desai (2015) (we will see this example again in section
5.3, where we cover different ways of specifying fitness), we
could specify the fitness landscape and run simulations until fixation
(with argument fixation to oncoSimulPop —see more details in section
6.3.3 and 6.3.4, again with this example). We would
then examine the proportion of genotypes fixed under different
scenarios. And we can extend this example by adding mutator genes:
## For reproducibility
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
## Specify fitness effects.
## Numeric values arbitrary, but set the intermediate genotype en
## route to ui as mildly deleterious so there is a valley.
## As in Ochs and Desai, the ui and uv genotypes
## can never appear.
u <- 0.2; i <- -0.02; vi <- 0.6; ui <- uv <- -Inf
od <- allFitnessEffects(
epistasis = c("u" = u, "u:i" = ui,
"u:v" = uv, "i" = i,
"v:-i" = -Inf, "v:i" = vi))
## For the sake of extending this example, also turn i into a
## mutator gene
odm <- allMutatorEffects(noIntGenes = c("i" = 50))
## How do mutation and fitness look like for each genotype?
evalAllGenotypesFitAndMut(od, odm, addwt = TRUE)
## Genotype Fitness MutatorFactor
## 1 WT 1.000 1
## 2 i 0.980 50
## 3 u 1.200 1
## 4 v 0.000 1
## 5 i, u 0.000 50
## 6 i, v 1.568 50
## 7 u, v 0.000 1
## 8 i, u, v 0.000 50Ochs and Desai explicitly say “Each simulated population was evolved
until either the uphill genotype or valley-crossing genotype fixed.”
So we will use fixation.
## Set a small initSize, as o.w. unlikely to pass the valley
initS <- 10
## The number of replicates is tiny, for the sake of speed
## of creation of the vignette. Even fewer in Windows, since we run on a single
## core
if(.Platform$OS.type == "windows") {
nruns <- 4
} else {
nruns <- 10
}
od_sim <- oncoSimulPop(nruns, od, muEF = odm,
fixation = c("u", "i, v"), initSize = initS,
model = "McFL",
mu = 1e-4, detectionDrivers = NA,
finalTime = NA,
detectionSize = NA, detectionProb = NA,
onlyCancer = TRUE,
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility
## What is the frequency of each final genotype?
sampledGenotypes(samplePop(od_sim))
##
## Subjects by Genes matrix of 10 subjects and 3 genes.
## Genotype Freq
## 1 i, v 4
## 2 u 6
##
## Shannon's diversity (entropy) of sampled genotypes: 0.67301171.3.3 Predictability of evolution in complex fitness landscapes
Focusing now on predictability in more general fitness landscapes, we would run simulations under random fitness landscapes with varied ruggedness, and would then examine the evolutionary predictability of the trajectories with measures such as “Lines of Descent” and “Path of the Maximum” (Szendro, Franke, et al. 2013) and the diversity of the sampled genotypes under different sampling regimes (see details in section 13). (See also related comments in section 1.5).
## For reproducibility
set.seed(7)
RNGkind("L'Ecuyer-CMRG")
## Repeat the following loop for different combinations of whatever
## interests you, such as number of genes, or distribution of the
## c and sd (which affect how rugged the landscape is), or
## reference genotype, or evolutionary model, or stopping criterion,
## or sampling procedure, or ...
## Generate a random fitness landscape, from the Rough Mount
## Fuji model, with g genes, and c ("slope" constant) and
## reference chosen randomly (reference is random by default and
## thus not specified below). Require a minimal number of
## accessible genotypes
g <- 6
c <- runif(1, 1/5, 5)
rl <- rfitness(g, c = c, min_accessible_genotypes = g)
## Plot it if you want; commented here as it takes long for a
## vignette
## plot(rl)
## Obtain landscape measures from MAGELLAN. Export to MAGELLAN and
## call your own copy of MAGELLAN's binary
## to_Magellan(rl, file = "rl1.txt") ## (Commented out here to avoid writing files)
## or use the binary copy provided with OncoSimulR
## see also below.
Magellan_stats(rl) ## (Commented out here to avoid writing files)
## ngeno npeaks nsinks gamma gamma. r.s
## 64.000 2.000 1.000 0.769 0.854 0.372
## nchains nsteps nori depth magn sign
## 1.000 5.000 4.000 2.000 0.863 0.129
## rsign f.1. X.2. f.3.. mode_f outD_m
## 0.008 0.916 0.010 0.074 1.000 0.346
## outD_v steps_m reach_m fitG_m opt_i mProbOpt_0
## 1.587 3.052 13.597 32.452 60.000 0.128
## opt_i.1 mProbOpt_1
## 63.000 0.872
## Simulate evolution in that landscape many times (here just 10)
simulrl <- oncoSimulPop(10, allFitnessEffects(genotFitness = rl),
keepPhylog = TRUE, keepEvery = 1,
initSize = 4000,
seed = NULL, ## for reproducibility
mc.cores = 2) ## adapt to your hardware
## Obtain measures of evolutionary predictability
diversityLOD(LOD(simulrl))
## [1] 1.418484
diversityPOM(POM(simulrl))
## [1] 1.418484
sampledGenotypes(samplePop(simulrl, typeSample = "whole"))
##
## Subjects by Genes matrix of 10 subjects and 6 genes.
## Genotype Freq
## 1 A 1
## 2 B 1
## 3 D, F 1
## 4 E 4
## 5 F 3
##
## Shannon's diversity (entropy) of sampled genotypes: 1.4184841.3.4 Mutator and antimutator genes
The effects of mutator and antimutator genes have been examined both in cancer genetics (Nowak 2006; Tomlinson, Novelli, and Bodmer 1996) and in evolutionary genetics (Gerrish et al. 2007), and are related to wider issues such as Muller’s ratchet and the evolution of sex. There are, thus, a large range of questions related to mutator and antimutator genes.
One question addressed in Tomlinson, Novelli, and Bodmer (1996) concerns under what circumstances mutator genes are likely to play a role in cancer progression. For instance, Tomlinson, Novelli, and Bodmer (1996) find that an increased mutation rate is more likely to matter if the number of required mutations in driver genes needed to reach cancer is large and if the mutator effect is large.
We might want to ask, then, how long it takes before to reach cancer under
different scenarios. Time to reach cancer is stored in the component
FinalTime of the output. We would specify different numbers and effects
of mutator genes (argument muEF). We would also change the criteria for
reaching cancer and in our case we can easily do that by specifying
different numbers in detectionDrivers. Of course, we would also want to
examine the effects of varying numbers of mutators, drivers, and possibly
fitness consequences of mutators. Below we assume mutators are neutral and
we assume there are no additional genes with deleterious mutations, but
this need not be so, of course (see also Tomlinson, Novelli, and Bodmer 1996; Gerrish et al. 2007; Christopher McFarland, Mirny, and Korolev 2014).
Let us run an example. For the sake of simplicity, we assume no epistatic interactions.
sd <- 0.1 ## fitness effect of drivers
sm <- 0 ## fitness effect of mutator
nd <- 20 ## number of drivers
nm <- 5 ## number of mutators
mut <- 10 ## mutator effect
fitnessGenesVector <- c(rep(sd, nd), rep(sm, nm))
names(fitnessGenesVector) <- 1:(nd + nm)
mutatorGenesVector <- rep(mut, nm)
names(mutatorGenesVector) <- (nd + 1):(nd + nm)
ft <- allFitnessEffects(noIntGenes = fitnessGenesVector,
drvNames = 1:nd)
mt <- allMutatorEffects(noIntGenes = mutatorGenesVector)Now, simulate using the fitness and mutator specification. We fix
the number of drivers to cancer, and we stop when those numbers of
drivers are reached. Since we only care about the time it takes to
reach cancer, not the actual trajectories, we set keepEvery = NA:
## For reproducibility
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
ddr <- 4
st <- oncoSimulPop(4, ft, muEF = mt,
detectionDrivers = ddr,
finalTime = NA,
detectionSize = NA,
detectionProb = NA,
onlyCancer = TRUE,
keepEvery = NA,
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility
## How long did it take to reach cancer?
unlist(lapply(st, function(x) x$FinalTime))
## [1] 370 141 1793 282(Incidentally, notice that it is easy to get OncoSimulR to throw an exception if you accidentally specify a huge mutation rate when all mutator genes are mutated: see section 15.7.)
1.3.5 Epistatic interactions between drivers and passengers in cancer and the consequences of order effects
1.3.5.1 Epistatic interactions between drivers and passengers
Bauer, Siebert, and Traulsen (2014) have examined the effects of epistatic relationships between drivers and passengers in cancer initiation. We could use their model as a starting point, and examine how likely cancer is to develop under different variations of their model and different evolutionary scenarios (e.g., initial sample size, mutation rates, evolutionary model, etc).
There are several ways to specify their model, as we discuss in section 5.1. We will use one based on DAGs here:
K <- 4
sp <- 1e-5
sdp <- 0.015
sdplus <- 0.05
sdminus <- 0.1
cnt <- (1 + sdplus)/(1 + sdminus)
prod_cnt <- cnt - 1
bauer <- data.frame(parent = c("Root", rep("D", K)),
child = c("D", paste0("s", 1:K)),
s = c(prod_cnt, rep(sdp, K)),
sh = c(0, rep(sp, K)),
typeDep = "MN")
fbauer <- allFitnessEffects(bauer)
(b1 <- evalAllGenotypes(fbauer, order = FALSE, addwt = TRUE))
## Genotype Fitness
## 1 WT 1.0000000
## 2 D 0.9545455
## 3 s1 1.0000100
## 4 s2 1.0000100
## 5 s3 1.0000100
## 6 s4 1.0000100
## 7 D, s1 0.9688636
## 8 D, s2 0.9688636
## 9 D, s3 0.9688636
## 10 D, s4 0.9688636
## 11 s1, s2 1.0000200
## 12 s1, s3 1.0000200
## 13 s1, s4 1.0000200
## 14 s2, s3 1.0000200
## 15 s2, s4 1.0000200
## 16 s3, s4 1.0000200
## 17 D, s1, s2 0.9833966
## 18 D, s1, s3 0.9833966
## 19 D, s1, s4 0.9833966
## 20 D, s2, s3 0.9833966
## 21 D, s2, s4 0.9833966
## 22 D, s3, s4 0.9833966
## 23 s1, s2, s3 1.0000300
## 24 s1, s2, s4 1.0000300
## 25 s1, s3, s4 1.0000300
## 26 s2, s3, s4 1.0000300
## 27 D, s1, s2, s3 0.9981475
## 28 D, s1, s2, s4 0.9981475
## 29 D, s1, s3, s4 0.9981475
## 30 D, s2, s3, s4 0.9981475
## 31 s1, s2, s3, s4 1.0000400
## 32 D, s1, s2, s3, s4 1.0131198
## How does the fitness landscape look like?
plot(b1, use_ggrepel = TRUE) ## avoid overlapping labels
## Warning: ggrepel: 12 unlabeled data points (too many overlaps).
## Consider increasing max.overlaps
Now run simulations and examine how frequently the runs end up with population sizes larger than a pre-specified threshold; for instance, below we look at increasing population size 4x in the default maximum number of 2281 time periods (for real, you would of course increase the number of total populations, the range of initial population sizes, model, mutation rate, required population size or number of drivers, etc):
## For reproducibility
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
totalpops <- 5
initSize <- 100
sb1 <- oncoSimulPop(totalpops, fbauer, model = "Exp",
initSize = initSize,
onlyCancer = FALSE,
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility
## What proportion of the simulations reach 4x initSize?
sum(summary(sb1)[, "TotalPopSize"] > (4 * initSize))/totalpops
## [1] 0.2Alternatively, to examine how long it takes to reach cancer for a
pre-specified size, you could look at the value of FinalTime as we
did above (section 1.3.4) after running simulations
with onlyCancer = TRUE and detectionSize set to some reasonable value:
totalpops <- 5
initSize <- 100
sb2 <- oncoSimulPop(totalpops, fbauer, model = "Exp",
initSize = initSize,
onlyCancer = TRUE,
detectionSize = 10 * initSize,
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility
## How long did it take to reach cancer?
unlist(lapply(sb2, function(x) x$FinalTime))
## [1] 416 354 339 445 2151.3.5.2 Consequences of order effects for cancer initiation
Instead of focusing on different models for epistatic interactions, you might want to examine the consequences of order effects (Ortmann et al. 2015). You would proceed as above, but using models that differ by, say, the presence or absence of order effects. Details on their specification are provided in section 3.6. Here is one particular model (you would, of course, want to compare this to models without order effects or with other magnitudes and types of order effects):
## Order effects involving three genes.
## Genotype "D, M" has different fitness effects
## depending on whether M or D mutated first.
## Ditto for genotype "F, D, M".
## Meaning of specification: X > Y means
## that X is mutated before Y.
o3 <- allFitnessEffects(orderEffects = c(
"F > D > M" = -0.3,
"D > F > M" = 0.4,
"D > M > F" = 0.2,
"D > M" = 0.1,
"M > D" = 0.5))
## With the above specification, let's double check
## the fitness of the possible genotypes
(oeag <- evalAllGenotypes(o3, addwt = TRUE, order = TRUE))
## Genotype Fitness
## 1 WT 1.00
## 2 D 1.00
## 3 F 1.00
## 4 M 1.00
## 5 D > F 1.00
## 6 D > M 1.10
## 7 F > D 1.00
## 8 F > M 1.00
## 9 M > D 1.50
## 10 M > F 1.00
## 11 D > F > M 1.54
## 12 D > M > F 1.32
## 13 F > D > M 0.77
## 14 F > M > D 1.50
## 15 M > D > F 1.50
## 16 M > F > D 1.50Now, run simulations and examine how frequently the runs do not end up in extinction. As above, for real, you would of course increase the number of total populations, the range of initial population sizes, mutation rate, etc:
## For reproducibility
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
totalpops <- 5
soe1 <- oncoSimulPop(totalpops, o3, model = "Exp",
initSize = 500,
onlyCancer = FALSE,
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility
## What proportion of the simulations do not end up extinct?
sum(summary(soe1)[, "TotalPopSize"] > 0)/totalpops
## [1] 0.4As we just said, alternatively, to examine how long it takes to
reach cancer you could run simulations with onlyCancer = TRUE and
look at the value of FinalTime as we did above (section
1.3.4).
1.3.6 Simulating evolution with frequency-dependent fitness
The new frequency-dependent fitness funcionality allows users to run simulations in a different way, defining fitness (birth rates) as functions of clone’s frequencies. We can thus model frequency-dependent selection, as well as predation and parasitism, cooperation and mutualism, and commensalism. See section 10 for further details and examples.
1.4 Trade-offs and what is OncoSimulR not well suited for
OncoSimulR is designed for complex fitness specifications and selection scenarios and uses forward-time simulations; the types of questions where OncoSimulR can be of help are discussed in sections 1.2 and 1.3 and running time and space consumption of OncoSimulR are addressed in section 2. You should be aware that coalescent simulations, sometimes also called backward-time simulations, are much more efficient for simulating neutral data as well as some special selection scenarios (Yuan et al. 2012; Carvajal-Rodriguez 2010; Hoban, Bertorelle, and Gaggiotti 2011).
In addition, since OncoSimulR allows you to specify fitness with arbitrary epistatic and order effects, as well as mutator effects, you need to learn the syntax of how to specify those effects and you might be paying a performance penalty if your scenario does not require this complexity. For instance, in the model of Beerenwinkel et al. (2007), the fitness of a genotype depends only on the total number of drivers mutated, but not on which drivers are mutated (and, thus, not on the epistatic interactions nor the order of accumulation of the drivers). This means that the syntax for specifying that model could probably be a lot simpler (e.g., specify \(s\) per driver).
But it also means that code written for just that case could probably run much faster. First, because fitness evaluation is easier. Second, and possibly much more important, because what we need to keep track of leads to much simpler and economic structures: we do not need to keep track of clones (where two cells are regarded as different clones if they differ anywhere in their genotype), but only of clone types or clone classes as defined by the number of mutated drivers, and keeping track of clones can be expensive —see sections 2 and 15.2.
So for those cases where you do not need the full flexibility of OncoSimulR, special purpose software might be easier to use and faster to run. Of course, for some types of problems this special purpose software might not be available, though.
1.5 Random fitness landscapes, clonal competition, predictability, and the strong selection weak mutation (SSWM) regime
Many studies about evolutionary predictability (among other topics) focus on the strong selection, weak mutation regime, SSWM (Gillespie 1984; Orr 2002) (see overview in Krug (2019)). In this regime, mutations are rare (much smaller than the mutation rate times the population size) and selection is strong (much larger than 1/population size), so that the population consists of a single clone most of the time, and evolution proceeds by complete, successive clonal expansions of advantageous mutations.
We can easily simulate variations around these scenarios with OncoSimulR, moving away from the SSWM by increasing the population size, or changing the size of the fitness differences.
The examples below, not run for the sake of speed, play with population size and fitness differences. To make sure we use a similar fitness landscape, we use the same simulated fitness landscape, scaled differently, so that the differences in fitness between mutants are increased or decreased while keeping their ranking identical (and, thus, having the same set of accessible and inaccessible genotypes and paths over the landscape).
If you run the code, you will see that as we increase population size we move further away from the SSWM: the population is no longer composed of a single clone most of the time.
Before running the examples, and to show the effects quantitatively, we define a simple wrapper to compute a few statistics.
## oncoSimul object -> measures of clonal interference
## they are not averaged over time. One value for sampled time
clonal_interf_per_time <- function(x) {
x <- x$pops.by.time
y <- x[, -1, drop = FALSE]
shannon <- apply(y, 1, OncoSimulR:::shannonI)
tot <- rowSums(y)
half_tot <- tot * 0.5
five_p_tot <- tot * 0.05
freq_most_freq <- apply(y/tot, 1, max)
single_more_half <- rowSums(y > half_tot)
## whether more than 1 clone with more than 5% pop.
how_many_gt_5p <- rowSums(y > five_p_tot)
several_gt_5p <- (how_many_gt_5p > 1)
return(cbind(shannon, ## Diversity of clones
freq_most_freq, ## Frequency of the most freq. clone
single_more_half, ## Any clone with a frequency > 50%?
several_gt_5p, ## Are there more than 1 clones with
## frequency > 5%?
how_many_gt_5p ## How many clones are there with
## frequency > 5%
))
}set.seed(1)
r7b <- rfitness(7, scale = c(1.2, 0, 1))
## Large pop sizes: clonal interference
(sr7b <- oncoSimulIndiv(allFitnessEffects(genotFitness = r7b),
model = "McFL",
mu = 1e-6,
onlyCancer = FALSE,
finalTime = 400,
initSize = 1e7,
keepEvery = 4,
detectionSize = 1e10))
plot(sr7b, show = "genotypes")
colMeans(clonal_interf_per_time(sr7b))## Small pop sizes: a single clone most of the time
(sr7c <- oncoSimulIndiv(allFitnessEffects(genotFitness = r7b),
model = "McFL",
mu = 1e-6,
onlyCancer = FALSE,
finalTime = 60000,
initSize = 1e3,
keepEvery = 4,
detectionSize = 1e10))
plot(sr7c, show = "genotypes")
colMeans(clonal_interf_per_time(sr7c))
## Even smaller fitness differences, but large pop. sizes
set.seed(1); r7b2 <- rfitness(7, scale = c(1.05, 0, 1))
(sr7b2 <- oncoSimulIndiv(allFitnessEffects(genotFitness = r7b2),
model = "McFL",
mu = 1e-6,
onlyCancer = FALSE,
finalTime = 3500,
initSize = 1e7,
keepEvery = 4,
detectionSize = 1e10))
sr7b2
plot(sr7b2, show = "genotypes")
colMeans(clonal_interf_per_time(sr7b2))
## Increase pop size further
(sr7b3 <- oncoSimulIndiv(allFitnessEffects(genotFitness = r7b2),
model = "McFL",
mu = 1e-6,
onlyCancer = FALSE,
finalTime = 1500,
initSize = 1e8,
keepEvery = 4,
detectionSize = 1e10))
sr7b3
plot(sr7b3, show = "genotypes")
colMeans(clonal_interf_per_time(sr7b3))
1.6 Steps for using OncoSimulR
Using this package will often involve the following steps:
Simulate cancer progression: section 6. You can simulate for a single individual or subject or for a set of subjects. You will need to:
Decide on a model. This basically amounts to choosing a model with exponential growth (“Exp” or “Bozic”) or a model with carrying capacity (“McFL”). If exponential growth, you can choose whether the the effects of mutations operate on the death rate (“Bozic”) or the birth rate (“Exp”)1.
Specify other parameters of the simulation. In particular, decide when to stop the simulation, mutation rates, etc.
Of course, at least for initial playing around, you can use the defaults.
Sample from the simulated data and do something with those simulated data (e.g., fit an OT model to them, examine diversity or time until cancer, etc). Most of what you do with the data, however, is outside the scope of this package and this vignette.
Before anything else, let us load the package in case it was not yet loaded. We also explicitly load graph and igraph for the vignette to work (you do not need that for your usual interactive work). And I set the default color for vertices in igraph.
library(OncoSimulR)
library(graph)
library(igraph)
igraph_options(vertex.color = "SkyBlue2")To be explicit, what version are we running?
packageVersion("OncoSimulR")
## [1] '3.2.0'1.7 Two quick examples of fitness specifications
Following 1.6 we will run two very minimal examples. First a model with a few genes and epistasis:
## 1. Fitness effects: here we specify an
## epistatic model with modules.
sa <- 0.1
sb <- -0.2
sab <- 0.25
sac <- -0.1
sbc <- 0.25
sv2 <- allFitnessEffects(epistasis = c("-A : B" = sb,
"A : -B" = sa,
"A : C" = sac,
"A:B" = sab,
"-A:B:C" = sbc),
geneToModule = c(
"A" = "a1, a2",
"B" = "b",
"C" = "c"),
drvNames = c("a1", "a2", "b", "c"))
evalAllGenotypes(sv2, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.000
## 2 a1 1.100
## 3 a2 1.100
## 4 b 0.800
## 5 c 1.000
## 6 a1, a2 1.100
## 7 a1, b 1.250
## 8 a1, c 0.990
## 9 a2, b 1.250
## 10 a2, c 0.990
## 11 b, c 1.000
## 12 a1, a2, b 1.250
## 13 a1, a2, c 0.990
## 14 a1, b, c 1.125
## 15 a2, b, c 1.125
## 16 a1, a2, b, c 1.125
## 2. Simulate the data. Here we use the "McFL" model and set
## explicitly parameters for mutation rate, initial size, size
## of the population that will end the simulations, etc
RNGkind("Mersenne-Twister")
set.seed(983)
ep1 <- oncoSimulIndiv(sv2, model = "McFL",
mu = 5e-6,
sampleEvery = 0.025,
keepEvery = 0.5,
initSize = 2000,
finalTime = 3000,
onlyCancer = FALSE)## 3. We will not analyze those data any further. We will only plot
## them. For the sake of a small plot, we thin the data.
plot(ep1, show = "drivers", xlim = c(0, 1500),
thinData = TRUE, thinData.keep = 0.5)
Figure 1.2: Plot of drivers of an epistasis simulation.
As a second example, we will use a model where we specify restrictions in the order of accumulation of mutations using a DAG with the pancreatic cancer poset in Gerstung et al. (2011) (see more details in section 5.5):
## 1. Fitness effects:
pancr <- allFitnessEffects(
data.frame(parent = c("Root", rep("KRAS", 4),
"SMAD4", "CDNK2A",
"TP53", "TP53", "MLL3"),
child = c("KRAS","SMAD4", "CDNK2A",
"TP53", "MLL3",
rep("PXDN", 3), rep("TGFBR2", 2)),
s = 0.1,
sh = -0.9,
typeDep = "MN"),
drvNames = c("KRAS", "SMAD4", "CDNK2A", "TP53",
"MLL3", "TGFBR2", "PXDN"))## Plot the DAG of the fitnessEffects object
plot(pancr)
Figure 1.3: Plot of DAG corresponding to fitnessEffects object.
## 2. Simulate from it. We change several possible options.
set.seed(1) ## Fix the seed, so we can repeat it
## We set a small finalTime to speed up the vignette
ep2 <- oncoSimulIndiv(pancr, model = "McFL",
mu = 1e-6,
sampleEvery = 0.02,
keepEvery = 1,
initSize = 1000,
finalTime = 20000,
detectionDrivers = 3,
onlyCancer = FALSE)## 3. What genotypes and drivers we get? And play with limits
## to show only parts of the data. We also aggressively thin
## the data.
par(cex = 0.7)
plot(ep2, show = "genotypes", xlim = c(500, 1800),
ylim = c(0, 2400),
thinData = TRUE, thinData.keep = 0.3)
Figure 1.4: Plot of genotypes of a simulation from a DAG.
The rest of this vignette explores all of those functions and arguments in much more detail.
1.8 Citing OncoSimulR and other documentation
In R, you can do
citation("OncoSimulR")
##
## If you use OncoSimulR, please cite the OncoSimulR
## Bioinformatics paper. OncoSimulR has been used in three
## large comparative studies of methods to infer restrictions
## in the order of accumulation of mutations (cancer
## progression models) published in PLoS Computational Biology,
## Bioinformatics and BMC Bioinformatics; you might want to
## cite those too, if appropriate, such as when referring to
## using evolutionary simulations to assess oncogenetic
## tree/cancer progression methods performance.
##
## R Diaz-Uriarte. OncoSimulR: genetic simulation with
## arbitrary epistasis and mutator genes in asexual
## populations. 2017. Bioinformatics, 33, 1898--1899.
## https://doi.org/10.1093/bioinformatics/btx077.
##
## R Diaz-Uriarte and C. Vasallo. Every which way? On
## predicting tumor evolution using cancer progression models
## 2019 PLoS Computational Biology
## https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1007246
##
## R Diaz-Uriarte. Cancer progression models and fitness
## landscapes: a many-to-many relationship 2017
## Bioinformatics.
## https://academic.oup.com/bioinformatics/advance-article/doi/10.1093/bioinformatics/btx663/
##
## R Diaz-Uriarte. Identifying restrictions in the order of
## accumulation of mutations during tumor progression:
## effects of passengers, evolutionary models, and sampling
## 2015. BMC Bioinformatics, 16(41).
##
## To see these entries in BibTeX format, use
## 'print(<citation>, bibtex=TRUE)', 'toBibtex(.)', or set
## 'options(citation.bibtex.max=999)'.which will tell you how to cite the package. Please, do cite the Bionformatics paper if you use the package in publications.
This is the URL for the Bioinformatics paper: https://doi.org/10.1093/bioinformatics/btx077 (there is also an early preprint at bioRxiv, but it should now point to the Bioinformatics paper).
1.8.1 HTML and PDF versions of the vignette
A PDF version of this vignette is available from https://rdiaz02.github.io/OncoSimul/pdfs/OncoSimulR.pdf. And an HTML version from https://rdiaz02.github.io/OncoSimul/OncoSimulR.html. These files should correspond to the most recent, GitHub version, of the package (i.e., they might include changes not yet available from the BioConductor package). Beware that the PDF might have figures and R code that do not fit on the page, etc.
1.9 Testing, code coverage, and other examples
OncoSimulR includes more than 2000 tests that are run at every check cycle. These tests provide a code coverage of more than 90% including both the C++ and R code. Another set of over 500 long-running (several hours) tests can be run on demand (see directory ‘/tests/manual’). In addition to serving as test cases, some of that code also provides further examples of usage.
1.10 Versions
In this vignette and the documentation I often refer to version 1 (v.1) and version 2 of OncoSimulR. Version 1 is the version available up to, and including, BioConductor v. 3.1. Version 2 of OncoSimulR is available starting from BioConductor 3.2 (and, of course, available too from development versions of BioC). So, if you are using the current stable or development version of BioConductor, or you grab the sources from GitHub (https://github.com/rdiaz02/OncoSimul) you are using what we call version 2. The functionality of version has been removed.
2 Running time and space consumption of OncoSimulR
Time to complete the simulations and size of returned objects (space consumption) depend on several, interacting factors. The usual rule of “experiment before launching a large number of simulations” applies, but here we will walk through several cases to get a feeling for the major factors that affect speed and size. Many of the comments on this section need to use ideas discussed in other places of this document; if you read this section first, you might want to come back after reading the relevant parts.
Speed will depend on:
- Your hardware, of course.
- The evolutionary model.
- The granularity of how often you keep data (
keepEveryargument). Note that the default, which is to keep as often as you sample (so that we preserve all history) can lead to slow execution times. - The mutation rate, because higher mutation rates lead to more clones, and more clones means we need to iterate over, well, more clones, and keep larger data structures.
- The fitness specification: more complex fitness specifications tend to be slightly slower but specially different fitness specifications can have radically different effects on the evolutionary trajectories, accessibility of fast growing genotypes and, generally, the evolutionary dynamics.
- The stopping conditions (
detectionProb,detectionDrivers,detectionSizearguments) and whether or not simulations are run until cancer is reached (onlyCancerargument). - Most of the above factors can interact in complex ways.
Size of returned objects will depend on:
- Any factor that affects the number of clones tracked/returned, in
particular: initial sizes and stopping conditions, mutation rate,
and how often you keep data (the
keepEveryargument can make a huge difference here). - Whether or not you keep the complete genealogy of clones (this affects slightly the size of returned object, not speed).
In the sections that follow, we go over several cases to understand some of the main settings that affect running time (or execution time) and space consumption (the size of returned objects). It should be understood, however, that many of the examples shown below do not represent typical use cases of OncoSimulR and are used only to identify what and how affects running time and space consumption. As we will see in most examples in this vignette, typical use cases of OncoSimulR involve hundreds to thousands of genes on population sizes up to \(10^5\) to \(10^7\).
Note that most of the code in this section is not executed during the building of the vignette to keep vignette build time reasonable and prevent using huge amounts of RAM. All of the code, ready to be sourced and run, is available from the ‘inst/miscell’ directory (and the summary output from some of the benchmarks is available from the ‘miscell-files/vignette_bench_Rout’ directory of the main OncoSimul repository at https://github.com/rdiaz02/OncoSimul).
2.1 Exp and McFL with “detectionProb” and pancreas example
To get familiar with some of they factors that affect time and size,
we will use the fitness specification from section
1.7, with the detectionProb stopping mechanism
(see 6.3.2). We will use the two main growth models
(exponential and McFarland). Each model will be run with two
settings of keepEvery. With keepEvery = 1 (runs exp1 and
mc1), population samples are stored at time intervals of 1 (even
if most of the clones in those samples later become extinct). With
keepEvery = NA (runs exp2 and mc2) no intermediate population
samples are stored, so clones that become extinct at any sampling
period are pruned and only the existing clones at the end of the
simulation are returned (see details in 15.2.1).
Will run 100 simulations. The results
I show are for a laptop with an 8-core Intel Xeon E3-1505M CPU,
running Debian GNU/Linux (the results from these benchmarks are
available as data(benchmark_1)).
## Specify fitness
pancr <- allFitnessEffects(
data.frame(parent = c("Root", rep("KRAS", 4),
"SMAD4", "CDNK2A",
"TP53", "TP53", "MLL3"),
child = c("KRAS","SMAD4", "CDNK2A",
"TP53", "MLL3",
rep("PXDN", 3), rep("TGFBR2", 2)),
s = 0.1,
sh = -0.9,
typeDep = "MN"),
drvNames = c("KRAS", "SMAD4", "CDNK2A", "TP53",
"MLL3", "TGFBR2", "PXDN"))
Nindiv <- 100 ## Number of simulations run.
## Increase this number to decrease sampling variation
## keepEvery = 1
t_exp1 <- system.time(
exp1 <- oncoSimulPop(Nindiv, pancr,
detectionProb = "default",
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = 1,
model = "Exp",
mc.cores = 1))["elapsed"]/Nindiv
t_mc1 <- system.time(
mc1 <- oncoSimulPop(Nindiv, pancr,
detectionProb = "default",
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = 1,
model = "McFL",
mc.cores = 1))["elapsed"]/Nindiv
## keepEvery = NA
t_exp2 <- system.time(
exp2 <- oncoSimulPop(Nindiv, pancr,
detectionProb = "default",
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = NA,
model = "Exp",
mc.cores = 1))["elapsed"]/Nindiv
t_mc2 <- system.time(
mc2 <- oncoSimulPop(Nindiv, pancr,
detectionProb = "default",
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = NA,
model = "McFL",
mc.cores = 1))["elapsed"]/Nindiv
We can obtain times, sizes of objects, and summaries of numbers of clones, iterations, and final times doing, for instance:
cat("\n\n\n t_exp1 = ", t_exp1, "\n")
object.size(exp1)/(Nindiv * 1024^2)
cat("\n\n")
summary(unlist(lapply(exp1, "[[", "NumClones")))
summary(unlist(lapply(exp1, "[[", "NumIter")))
summary(unlist(lapply(exp1, "[[", "FinalTime")))
summary(unlist(lapply(exp1, "[[", "TotalPopSize")))The above runs yield the following:
| Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, max. | keepEvery | |
|---|---|---|---|---|---|---|---|---|
| exp1 | 0 | 0.04 | 2 | 254 | 252 | 1,058 | 11,046 | 1 |
| mc1 | 0.74 | 3.9 | 12 | 816,331 | 20,406 | 696 | 979 | 1 |
| exp2 | 0 | 0.01 | 1 | 296 | 294 | 1,021 | 21,884 | NA |
| mc2 | 0.7 | 0.01 | 1 | 694,716 | 17,366 | 692 | 888 | NA |
The above table shows that a naive comparison (looking simply at execution
time) might conclude that the McFL model is much, much slower than the Exp
model. But that is not the complete story: using the detectionProb
stopping mechanism (see 6.3.2) will lead to stopping the
simulations very quickly in the exponential model because as soon as a
clone with fitness \(>1\) appears it starts growing exponentially. In fact,
we can see that the number of iterations and the final time are much
smaller in the Exp than in the McFL model. We will elaborate on this
point below (section 2.2.1), when we discuss the setting for
checkSizePEvery (here left at its default value of 20): checking the
exiting condition more often (smaller checkSizePEvery) would probably be
justified here (notice also the very large final times) and would lead to
a sharp decrease in number of iterations and, thus, running time.
This table also shows that the keepEvery = NA setting, which was
in effect in simulations exp2 and mc2, can make a difference
especially for the McFL models, as seen by the median number of
clones and the size of the returned object. Models exp2 and mc2
do not store any intermediate population samples so clones that
become extinct at any sampling period are pruned and only the
existing clones at the end of the simulation are returned. In
contrast, models exp1 and mc1 store population samples at time
intervals of 1 (keepEvery = 1), even if many of those clones
eventually become extinct. We will return to this issue below as
execution time and object size depend strongly on the number of
clones tracked.
We can run the exponential model again modifying the arguments of the
detectionProb mechanism; in two of the models below (exp3 and exp4) no
detection can take place unless populations are at least 100 times larger
than the initial population size, and probability of detection is 0.1 with a
population size 1,000 times larger than the initial one (PDBaseline = 5e4,
n2 = 5e5). In the other two models (exp5 and exp6), no detection can
take place unless populations are at least 1,000 times larger than the
initial population size, and probability of detection is 0.1 with a
population size 100,000 times larger than the initial one (PDBaseline = 5e5, n2 = 5e7)2. In runs exp3 and exp5 we set keepEvery = 1 and in
runs exp4 and exp6 we set keepEvery = NA.
t_exp3 <- system.time(
exp3 <- oncoSimulPop(Nindiv, pancr,
detectionProb = c(PDBaseline = 5e4,
p2 = 0.1, n2 = 5e5,
checkSizePEvery = 20),
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = 1,
model = "Exp",
mc.cores = 1))["elapsed"]/Nindiv
t_exp4 <- system.time(
exp4 <- oncoSimulPop(Nindiv, pancr,
detectionProb = c(PDBaseline = 5e4,
p2 = 0.1, n2 = 5e5,
checkSizePEvery = 20),
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = NA,
model = "Exp",
mc.cores = 1))["elapsed"]/Nindiv
t_exp5 <- system.time(
exp5 <- oncoSimulPop(Nindiv, pancr,
detectionProb = c(PDBaseline = 5e5,
p2 = 0.1, n2 = 5e7),
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = 1,
model = "Exp",
mc.cores = 1))["elapsed"]/Nindiv
t_exp6 <- system.time(
exp6 <- oncoSimulPop(Nindiv, pancr,
detectionProb = c(PDBaseline = 5e5,
p2 = 0.1, n2 = 5e7),
detectionSize = NA,
detectionDrivers = NA,
finalTime = NA,
keepEvery = NA,
model = "Exp",
mc.cores = 1))["elapsed"]/Nindiv
| Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, max. | keepEvery | PDBaseline | n2 | |
|---|---|---|---|---|---|---|---|---|---|---|
| exp3 | 0.01 | 0.41 | 14 | 2,754 | 1,890 | 6,798,358 | 2.7e+08 | 1 | 50,000 | 5e+05 |
| exp4 | 0.01 | 0.02 | 8 | 2,730 | 2,090 | 7,443,812 | 1.7e+08 | NA | 50,000 | 5e+05 |
| exp5 | 0.84 | 0.91 | 34 | 54,332 | 2,026 | 1.4e+09 | 4.2e+10 | 1 | 5e+05 | 5e+07 |
| exp6 | 0.54 | 0.02 | 27 | 44,288 | 2,026 | 1.2e+09 | 3.3e+10 | NA | 5e+05 | 5e+07 |
As above, keepEvery = NA (in exp4 and exp6) leads to much
smaller object sizes and slightly smaller numbers of clones and
execution times. Changing the exiting conditions (by changing
detectionProb arguments) leads to large increases in number of
iterations (in this case by factors of about 15x to 25x) and a
corresponding increase in execution time as well as much larger
population sizes (in some cases \(>10^{10}\)).
In some of the runs of exp5 and exp6 we get the (recoverable)
exception message from the C++ code: Recoverable exception ti set to DBL_MIN. Rerunning, which is related to those simulations reaching total
population sizes \(>10^{10}\); we return to this below (section
2.4). You might also wonder why total and median population
sizes are so large in these two runs, given the exiting conditions. One of
the reasons is that we are using the default checkSizePEvery = 20, so
the interval between successive checks of the exiting condition is large;
this is discussed at greater length in section 2.2.1.
All the runs above used the default value onlyCancer = TRUE. This means
that simulations will be repeated until the exiting conditions are reached
(see details in section 6.3) and, therefore, any simulation
that ends up in extinction will be repeated. This setting can thus have a
large effect on the exponential models, because when the initial
population size is not very large and we start from the wildtype, it is
not uncommon for simulations to become extinct (when birth and death
rates are equal and the population size is small, it is easy to reach
extinction before a mutation in a gene that increases fitness occurs). But
this is rarely the case in the McFarland model (unless we use really tiny
initial population sizes) because of the dependency of death rate on total
population size (see section 3.2.1).
The number of attempts until cancer was reached in the above
models is shown in Table 2.3 (the values can be obtained from
any of the above runs doing, for instance, median(unlist(lapply(exp1, function(x) x$other$attemptsUsed))) ):
| Attempts until Cancer, median | Attempts until Cancer, mean | Attempts until Cancer, max. | PDBaseline | n2 | |
|---|---|---|---|---|---|
| exp1 | 1 | 1.9 | 7 | 600 | 1,000 |
| mc1 | 1 | 1 | 1 | 600 | 1,000 |
| exp2 | 2 | 2.2 | 16 | 600 | 1,000 |
| mc2 | 1 | 1 | 1 | 600 | 1,000 |
| exp3 | 6 | 7.7 | 40 | 50,000 | 5e+05 |
| exp4 | 6 | 8 | 39 | 50,000 | 5e+05 |
| exp5 | 5 | 8.3 | 41 | 5e+05 | 5e+07 |
| exp6 | 5 | 7.2 | 30 | 5e+05 | 5e+07 |
The McFL models finish in a single attempt. The exponential model
simulations where we can exit with small population sizes (exp1, exp2)
need many fewer attempts to reach cancer than those where large population
sizes are required (exp3 to exp6). There is no relevant different
among those last four, which is what we would expect: a population that has
already reached a size of 50,000 cells from an initial population size of
500 is obviously a growing population where there is at least one mutant
with positive fitness; thus, it unlikely to go extinct and therefore
having to grow up to at least 500,000 will not significantly increase the
risk of extinction.
We will now rerun all of the above models with argument onlyCancer = FALSE. The results are shown in Table 2.4 (note that the
differences between this table and Table 2.1 for the McFL
models are due only to sampling variation).
| Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, mean | Total Population Size, max. | keepEvery | PDBaseline | n2 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| exp1_noc | 0.001 | 0.041 | 1.5 | 394 | 393 | 0 | 708 | 18,188 | 1 | 600 | 1,000 |
| mc1_noc | 0.69 | 3.9 | 12 | 673,910 | 16,846 | 692 | 700 | 983 | 1 | 600 | 1,000 |
| exp2_noc | 0.001 | 0.012 | 1 | 320 | 319 | 726 | 870 | 26,023 | NA | 600 | 1,000 |
| mc2_noc | 0.65 | 0.014 | 1 | 628,683 | 15,716 | 694 | 704 | 910 | NA | 600 | 1,000 |
| exp3_noc | 0.002 | 0.15 | 2 | 718 | 694 | 0 | 2,229,519 | 5.7e+07 | 1 | 50,000 | 5e+05 |
| exp4_noc | 0.002 | 0.013 | 0 | 600 | 599 | 0 | 3,122,765 | 1.3e+08 | NA | 50,000 | 5e+05 |
| exp5_noc | 0.17 | 0.22 | 3 | 848 | 777 | 0 | 5.9e+08 | 1.5e+10 | 1 | 5e+05 | 5e+07 |
| exp6_noc | 0.068 | 0.013 | 0 | 784 | 716 | 0 | 4.1e+08 | 1.3e+10 | NA | 5e+05 | 5e+07 |
Now most simulations under the exponential model end up in extinction, as
seen by the median population size of 0 (but not all, as the mean and
max. population size are clearly away from zero). Consequently,
simulations under the exponential model are now faster (and the size of
the average returned object is smaller). Of course, whether one should run
simulations with onlyCancer = TRUE or onlyCancer = FALSE will depend
on the question being asked (see, for example, section 1.3.5 for
a question where we will naturally want to use onlyCancer = FALSE).
To make it easier to compare results with those of the next section, Table 2.5 shows all the runs so far.
| Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, mean | Total Population Size, max. | keepEvery | PDBaseline | n2 | onlyCancer | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| exp1 | 0.001 | 0.037 | 2 | 254 | 252 | 1,058 | 1,277 | 11,046 | 1 | 600 | 1,000 | TRUE |
| mc1 | 0.74 | 3.9 | 12 | 816,331 | 20,406 | 696 | 702 | 979 | 1 | 600 | 1,000 | TRUE |
| exp2 | 0.001 | 0.012 | 1 | 296 | 294 | 1,021 | 1,392 | 21,884 | NA | 600 | 1,000 | TRUE |
| mc2 | 0.7 | 0.014 | 1 | 694,716 | 17,366 | 692 | 698 | 888 | NA | 600 | 1,000 | TRUE |
| exp3 | 0.01 | 0.41 | 14 | 2,754 | 1,890 | 6,798,358 | 1.7e+07 | 2.7e+08 | 1 | 50,000 | 5e+05 | TRUE |
| exp4 | 0.009 | 0.016 | 8 | 2,730 | 2,090 | 7,443,812 | 1.5e+07 | 1.7e+08 | NA | 50,000 | 5e+05 | TRUE |
| exp5 | 0.84 | 0.91 | 34 | 54,332 | 2,026 | 1.4e+09 | 3.5e+09 | 4.2e+10 | 1 | 5e+05 | 5e+07 | TRUE |
| exp6 | 0.54 | 0.021 | 27 | 44,288 | 2,026 | 1.2e+09 | 3.2e+09 | 3.3e+10 | NA | 5e+05 | 5e+07 | TRUE |
| exp1_noc | 0.001 | 0.041 | 1.5 | 394 | 393 | 0 | 708 | 18,188 | 1 | 600 | 1,000 | FALSE |
| mc1_noc | 0.69 | 3.9 | 12 | 673,910 | 16,846 | 692 | 700 | 983 | 1 | 600 | 1,000 | FALSE |
| exp2_noc | 0.001 | 0.012 | 1 | 320 | 319 | 726 | 870 | 26,023 | NA | 600 | 1,000 | FALSE |
| mc2_noc | 0.65 | 0.014 | 1 | 628,683 | 15,716 | 694 | 704 | 910 | NA | 600 | 1,000 | FALSE |
| exp3_noc | 0.002 | 0.15 | 2 | 718 | 694 | 0 | 2,229,519 | 5.7e+07 | 1 | 50,000 | 5e+05 | FALSE |
| exp4_noc | 0.002 | 0.013 | 0 | 600 | 599 | 0 | 3,122,765 | 1.3e+08 | NA | 50,000 | 5e+05 | FALSE |
| exp5_noc | 0.17 | 0.22 | 3 | 848 | 777 | 0 | 5.9e+08 | 1.5e+10 | 1 | 5e+05 | 5e+07 | FALSE |
| exp6_noc | 0.068 | 0.013 | 0 | 784 | 716 | 0 | 4.1e+08 | 1.3e+10 | NA | 5e+05 | 5e+07 | FALSE |
2.1.1 Changing fitness: \(s=0.1\) and \(s=0.05\)
In the above fitness specification the fitness effect of each gene
(when its restrictions are satisfied) is \(s = 0.1\) (see section
3.2 for details). Here we rerun all the above benchmarks
using \(s= 0.05\) (the results from these benchmarks are available as
data(benchmark_1_0.05)) and results are shown below in Table
2.6.
| Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, mean | Total Population Size, max. | keepEvery | PDBaseline | n2 | onlyCancer | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| exp1 | 0.002 | 0.043 | 2 | 316 | 315 | 1,104 | 1,181 | 3,176 | 1 | 600 | 1,000 | TRUE |
| mc1 | 1.7 | 11 | 17 | 2e+06 | 50,696 | 644 | 647 | 761 | 1 | 600 | 1,000 | TRUE |
| exp2 | 0.001 | 0.012 | 1 | 274 | 273 | 1,129 | 1,281 | 7,608 | NA | 600 | 1,000 | TRUE |
| mc2 | 1.6 | 0.016 | 1 | 1,615,197 | 40,376 | 644 | 651 | 772 | NA | 600 | 1,000 | TRUE |
| exp3 | 0.012 | 0.63 | 15 | 3,995 | 2,919 | 3,798,540 | 5,892,376 | 4.5e+07 | 1 | 50,000 | 5e+05 | TRUE |
| exp4 | 0.011 | 0.017 | 9 | 4,288 | 3,276 | 4,528,072 | 6,551,319 | 3.2e+07 | NA | 50,000 | 5e+05 | TRUE |
| exp5 | 0.3 | 1.2 | 34 | 68,410 | 2,751 | 6.8e+08 | 1e+09 | 8.2e+09 | 1 | 5e+05 | 5e+07 | TRUE |
| exp6 | 0.26 | 0.022 | 23 | 44,876 | 2,499 | 4.3e+08 | 8.9e+08 | 7.3e+09 | NA | 5e+05 | 5e+07 | TRUE |
| exp1_noc | 0.001 | 0.039 | 2 | 310 | 308 | 0 | 522 | 2,239 | 1 | 600 | 1,000 | FALSE |
| mc1_noc | 1.6 | 11 | 17 | 2e+06 | 50,776 | 638 | 643 | 757 | 1 | 600 | 1,000 | FALSE |
| exp2_noc | 0.001 | 0.012 | 0 | 340 | 336 | 0 | 599 | 3,994 | NA | 600 | 1,000 | FALSE |
| mc2_noc | 1.7 | 0.017 | 1 | 2,102,439 | 52,556 | 645 | 650 | 740 | NA | 600 | 1,000 | FALSE |
| exp3_noc | 0.002 | 0.11 | 2 | 618 | 615 | 0 | 150,978 | 6,093,498 | 1 | 50,000 | 5e+05 | FALSE |
| exp4_noc | 0.002 | 0.013 | 0 | 813 | 812 | 0 | 558,225 | 2.3e+07 | NA | 50,000 | 5e+05 | FALSE |
| exp5_noc | 0.031 | 0.23 | 3 | 917 | 914 | 0 | 1.1e+08 | 3.7e+09 | 1 | 5e+05 | 5e+07 | FALSE |
| exp6_noc | 0.046 | 0.013 | 0 | 628 | 610 | 0 | 1.7e+08 | 5.1e+09 | NA | 5e+05 | 5e+07 | FALSE |
As expected, having a smaller \(s\) leads to slower processes in most cases, since it takes longer to reach the exiting conditions sooner. Particularly noticeable are the runs for the McFL models (notice the increases in population size and number of iterations —see also below).
That is not the case, however, for exp5 and exp6 (and exp5_noc and
exp6_noc). When running with \(s=0.05\) the simulations exit at a later
time (see column “Final Time”) but they exit with smaller population
sizes. Here we have an interaction between sampling frequency, speed of
growth of the population, mutation events and number of clones. In
populations that grow much faster mutation events will happen more often
(which will trigger further iterations of the algorithm); in addition,
more new clones will be created, even if they only exist for short times
and become extinct by the following sampling period (so they are not
reflected in the pops.by.time matrix). These differences are
proportionally larger the larger the rate of growth of the
population. Thus, they are larger between, say, the exp5 at \(s=0.1\) and
\(s=0.05\) than between the exp4 at the two different \(s\): the exp5 exit
conditions can only be satisfied at much larger population sizes so at
populations sizes when growth is much faster (recall we are dealing with
exponential growth).
Recall also that with the default settings in detectionProb, we
assess the exiting condition every 20 time periods (argument
checkSizePEvery); this means that for fast growing populations,
the increase in population size between successive checks of the
exit conditions will be much larger (this phenomenon is also discussed in
section 2.2.1).
Thus, what is happening in the exp5 and exp6 with \(s=0.1\) is
that close to the time the exit conditions could be satisfied, they
are growing very fast, accumulating mutants, and incurring in
additional iterations. They exit sooner in terms of time periods,
but they do much more work before arriving there.
The setting of checkSizePEvery is also having a huge effect on the McFL
model simulations (the number of iterations is \(>10^6\)). Even more than in
the previous section, checking the exiting condition more often (smaller
checkSizePEvery) would probably be justified here (notice also the very
large final times) and would lead to a sharp decrease in number of
iterations and, thus, running time.
The moral here is that in complex simulations like this (and most simulations are complex), the effects of some parameters (\(s\) in this case) might look counter-intuitive at first. Thus the need to “experiment before launching a large number of simulations”.
2.2 Several “common use cases” runs
Let us now execute some simulations under more usual conditions. We will use seven different fitness specifications: the pancreas example, two random fitness landscapes, and four sets of independent genes (200 to 4000 genes) with fitness effects randomly drawn from exponential distributions:
pancr <- allFitnessEffects(
data.frame(parent = c("Root", rep("KRAS", 4),
"SMAD4", "CDNK2A",
"TP53", "TP53", "MLL3"),
child = c("KRAS","SMAD4", "CDNK2A",
"TP53", "MLL3",
rep("PXDN", 3), rep("TGFBR2", 2)),
s = 0.1,
sh = -0.9,
typeDep = "MN"),
drvNames = c("KRAS", "SMAD4", "CDNK2A", "TP53",
"MLL3", "TGFBR2", "PXDN"))
## Random fitness landscape with 6 genes
## At least 50 accessible genotypes
rfl6 <- rfitness(6, min_accessible_genotypes = 50)
attributes(rfl6)$accessible_genotypes ## How many accessible
rf6 <- allFitnessEffects(genotFitness = rfl6)
## Random fitness landscape with 12 genes
## At least 200 accessible genotypes
rfl12 <- rfitness(12, min_accessible_genotypes = 200)
attributes(rfl12)$accessible_genotypes ## How many accessible
rf12 <- allFitnessEffects(genotFitness = rfl12)
## Independent genes; positive fitness from exponential distribution
## with mean around 0.1, and negative from exponential with mean
## around -0.02. Half of genes positive fitness effects, half
## negative.
ng <- 200 re_200 <- allFitnessEffects(noIntGenes = c(rexp(ng/2, 10),
-rexp(ng/2, 50)))
ng <- 500
re_500 <- allFitnessEffects(noIntGenes = c(rexp(ng/2, 10),
-rexp(ng/2, 50)))
ng <- 2000
re_2000 <- allFitnessEffects(noIntGenes = c(rexp(ng/2, 10),
-rexp(ng/2, 50)))
ng <- 4000
re_4000 <- allFitnessEffects(noIntGenes = c(rexp(ng/2, 10),
-rexp(ng/2, 50)))
2.2.1 Common use cases, set 1.
We will use the Exp and the McFL models, run with different parameters. The
script is provided as ‘benchmark_2.R’, under ‘/inst/miscell’, with output in
the ‘miscell-files/vignette_bench_Rout’ directory of the main OncoSimul
repository at https://github.com/rdiaz02/OncoSimul. The data are available
as data(benchmark_2).
For the Exp model the call will be
oncoSimulPop(Nindiv,
fitness,
detectionProb = NA,
detectionSize = 1e6,
initSize = 500,
detectionDrivers = NA,
keepPhylog = TRUE,
model = "Exp",
errorHitWallTime = FALSE,
errorHitMaxTries = FALSE,
finalTime = 5000,
onlyCancer = FALSE,
mc.cores = 1,
sampleEvery = 0.5,
keepEvery = 1)And for McFL:
initSize <- 1000
oncoSimulPop(Nindiv,
fitness,
detectionProb = c(
PDBaseline = 1.4 * initSize,
n2 = 2 * initSize,
p2 = 0.1,
checkSizePEvery = 4),
initSize = initSize,
detectionSize = NA,
detectionDrivers = NA,
keepPhylog = TRUE,
model = "McFL",
errorHitWallTime = FALSE,
errorHitMaxTries = FALSE,
finalTime = 5000,
max.wall.time = 10,
onlyCancer = FALSE,
mc.cores = 1,
keepEvery = 1)
For the exponential model we will stop simulations when populations
have \(>10^6\) cells (simulations start from 500 cells). For the McFarland
model we will use the detectionProb mechanism (see section
6.3.2 for details); we could have used as stopping mechanism
detectionSize = 2 * initSize (which would be basically equivalent to
reaching cancer, as argued in (McFarland et al. 2013)) but we want to provide
further examples under the detectionProb mechanism. We will start from 1000
cells, not 500 (starting from 1000 we almost always reach cancer in a
single run).
Why not use the detectionProb mechanism with the Exp models? Because
it can be hard to intuitively understand what are reasonable settings for
the parameters of the detectionProb mechanism when used in a population
that is growing exponentially, especially if different genes have very
different effects on fitness. Moreover, we are using fitness
specifications that are very different (compare the fitness landscape of
six genes, the pancreas specification, and the fitness specification with
4000 genes with fitness effects drawn from an exponential distribution
—re_4000). In contrast, the detectionProb mechanism might be simpler
to reason about in a population that is growing under a model of carrying
capacity with possibly large periods of stasis. Let us emphasize that it
is not that the detectionProb mechanism does not make sense with the Exp
model; it is simply that the parameters might need finer adjustment for
them to make sense, and in these benchmarks we are dealing with widely
different fitness specifications.
Note also that we specify checkSizePEvery = 4 (instead of the default,
which is 20). Why? Because the fitness specifications where fitness
effects are drawn from exponential distributions (re_200 to re_4000
above) include many genes (well, up to 4000) some of them with possibly
very large effects. In these conditions, simulations can run very fast in
the sense of “units of time”. If we check exiting conditions every 20
units the population could have increased its size several orders of
magnitude in between checks (this is also discussed in sections
2.1.1 and 6.3.2). You can verify this by running the
script with other settings for checkSizePEvery (and being aware that
large settings might require you to wait for a long time). To ensure that
populations have really grown, we have increased the setting of
PDBaseline so that no simulation can be considered for stopping unless
its size is 1.4 times larger than initSize.
In all cases we use keepEvery = 1 and keepPhylog = TRUE (so we store
the population sizes of all clones every 1 time unit and we keep the
complete genealogy of clones). Finally, we run all models with
errorHitWallTime = FALSE and errorHitMaxTries = FALSE so that we can
see results even if stopping conditions are not met.
The results of the benchmarks, using 100 individual simulations, are shown in Table 2.7.
| Model | Fitness | Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, mean | Total Population Size, max. |
|---|---|---|---|---|---|---|---|---|---|
| Exp | pancr | 0.002 | 0.12 | 3 | 1,397 | 697 | 0 | 164,222 | 1,053,299 |
| McFL | pancr | 0.12 | 0.56 | 8 | 2e+05 | 5,000 | 1,037 | 1,144 | 1,938 |
| Exp | rf6 | 0.002 | 0.064 | 6 | 783 | 391 | 1e+06 | 594,899 | 1,309,497 |
| McFL | rf6 | 0.019 | 0.071 | 3 | 23,297 | 582 | 1,884 | 1,975 | 4,636 |
| Exp | rf12 | 0.01 | 0.13 | 4 | 1,178 | 542 | 0 | 287,669 | 1,059,141 |
| McFL | rf12 | 0.14 | 0.82 | 18 | 2e+05 | 5,000 | 1,252 | 1,295 | 1,695 |
| Exp | re_200 | 0.013 | 0.67 | 230 | 1,185 | 223 | 1,060,944 | 859,606 | 1,536,242 |
| McFL | re_200 | 0.018 | 0.22 | 47 | 9,679 | 240 | 2,166 | 2,973 | 29,301 |
| Exp | re_500 | 0.09 | 2.7 | 771 | 2,732 | 152 | 1,068,732 | 959,026 | 1,285,522 |
| McFL | re_500 | 0.024 | 0.44 | 91 | 7,056 | 172 | 2,148 | 2,578 | 8,234 |
| Exp | re_2000 | 0.91 | 29 | 3,376 | 7,412 | 70 | 1,163,990 | 1,143,041 | 1,741,492 |
| McFL | re_2000 | 0.031 | 1.9 | 186 | 3,546 | 80 | 2,870 | 3,704 | 13,248 |
| Exp | re_4000 | 3.3 | 113 | 7,088 | 12,216 | 52 | 1,217,568 | 1,309,185 | 2,713,200 |
| McFL | re_4000 | 0.063 | 6.5 | 326 | 2,731 | 52 | 4,592 | 13,601 | 729,611 |
In most cases, simulations run reasonably fast (under 0.1 seconds per individual simulation) and the returned objects are small. I will only focus on a few cases.
The McFL model with random fitness landscape rf12 and with pancr does
not satisfy the conditions of detectionProb in most cases: its median
final time is 5000, which was the maximum final time specified. This
suggests that the fitness landscape is such that it is unlikely that we
will reach population sizes \(> 1400\) (remember we the setting for
PDBaseline) before 5000 time units. There is nothing particular about
using a fitness landscape of 12 genes and other runs in other 12-gene
random fitness landscapes do not show this pattern. However, complex
fitness landscapes might be such that genotypes of high fitness (those
that allow reaching a large population size quickly) are not easily
accessible3 so reaching them might take a long time. This does not
affect the exponential model in the same way because, well, because there
is exponential growth in that model: any genotype with fitness \(>1\) will
grow exponentially (of course, at possibly very different rates). You
might want to play with the script and modify the call to rfitness
(using different values of reference and c, for instance) to have
simpler paths to a maximum or modify the call to oncoSimulPop (with,
say, finalTime to much larger values). Some of these issues are related
to more general questions about fitness landscapes and accessibility (see
section 1.3.2 and references therein).
You could also set onlyCancer = TRUE. This might make sense if you are
interested in only seeing simulations that “reach cancer” (where “reach
cancer” means reaching a state you define as a function of population size
or drivers). However, if you are exploring fitness landscapes, onlyCancer = TRUE might not always be reasonable as reaching a particular population
size, for instance, might just not be possible under some fitness
landscapes (this phenomenon is of course not restricted to random fitness
landscapes —see also section 2.3.3).
As we anticipated above, the detectionProb mechanism has to be used with
care: some of the simulations run in very short “time units”, such as
those for the fitness specifications with 2000 and 4000 genes. Having used
a checkSizePEvery = 20 probably would not have made sense.
Finally, it is interesting that in the cases examined here, the two
slowest running simulations are from “Exp”, with fitnesses re_2000
and re_4000 (and the third slowest is also Exp, under
re_500). These are also the cases with the largest number of
clones. Why? In the “Exp” model there is no competition, and fitness
specifications re_2000 and re_4000 have genomes with many genes
with positive fitness contributions. It is thus very easy to obtain,
from the wildtype ancestor, a large number of clones all of which
have birth rates \(>1\) and, thus, clones that are unlikely to become
extinct.
2.2.2 Common use cases, set 2.
We will now rerun the simulations above changing the following:
finalTimeset to 25000.onlyCancerset to TRUE.- The “Exp” models will stop when population size \(> 10^5\).
This is in script ‘benchmark_3.R’, under ‘/inst/miscell’, with
output in the ‘miscell-files/vignette_bench_Rout’ directory of the
main OncoSimul repository at https://github.com/rdiaz02/OncoSimul.
The data are available as data(benchmark_3).
| Model | Fitness | Elapsed Time, average per simulation (s) | Object Size, average per simulation (MB) | Number of Clones, median | Number of Iterations, median | Final Time, median | Total Population Size, median | Total Population Size, mean | Total Population Size, max. |
|---|---|---|---|---|---|---|---|---|---|
| Exp | pancr | 0.012 | 0.32 | 10 | 3,480 | 1,718 | 1e+05 | 1e+05 | 108,805 |
| McFL | pancr | 0.41 | 1.7 | 14 | 4e+05 | 9,955 | 1,561 | 1,555 | 1,772 |
| Exp | rf6 | 0.003 | 0.058 | 4 | 866 | 430 | 107,492 | 109,774 | 135,257 |
| McFL | rf6 | 0.033 | 0.12 | 4 | 35,216 | 880 | 2,003 | 2,010 | 3,299 |
| Exp | rf12 | 0.012 | 0.098 | 9 | 1,138 | 561 | 1e+05 | 1e+05 | 112,038 |
| McFL | rf12 | 0.17 | 0.76 | 16 | 1e+05 | 2,511 | 1,486 | 1,512 | 1,732 |
| Exp | re_200 | 0.004 | 0.39 | 106 | 723 | 252 | 1e+05 | 105,586 | 122,338 |
| McFL | re_200 | 0.026 | 0.33 | 61 | 13,484 | 335 | 1,830 | 2,049 | 3,702 |
| Exp | re_500 | 0.007 | 0.61 | 168 | 490 | 117 | 110,311 | 112,675 | 134,860 |
| McFL | re_500 | 0.018 | 0.33 | 70 | 5,157 | 126 | 2,524 | 3,455 | 19,899 |
| Exp | re_2000 | 0.046 | 5.7 | 651 | 1,078 | 68 | 106,340 | 109,081 | 153,146 |
| McFL | re_2000 | 0.029 | 1.8 | 186 | 3,444 | 80 | 2,837 | 4,009 | 37,863 |
| Exp | re_4000 | 0.1 | 19 | 1,140 | 1,722 | 51 | 111,256 | 113,499 | 168,958 |
| McFL | re_4000 | 0.057 | 6.7 | 325 | 3,081 | 60 | 3,955 | 8,892 | 265,183 |
Since we increased the maximum final time and forced runs to “reach
cancer” the McFL run with the pancreas fitness specification takes a bit
longer because it also has to do a larger number of
iterations. Interestingly, notice that the median final time is close to
10000, so the runs in 2.2.1 with maximum final time of 5000 would
have had a hard time finishing with onlyCancer = TRUE.
Forcing simulations to “reach cancer” and just random differences between
the random fitness landscape also affect the McFL run under rf12: final
time is below 5000 and the median number of iterations is about half of
what was above.
Finally, by stopping the Exp simulations at \(10^5\), simulations with
re_2000 and re_4000 finish now in much shorter times (but they still
take longer than their McFL counterparts) and the number of clones created
is much smaller.
2.3 Can we use a large number of genes?
Yes. In fact, in OncoSimulR there is no pre-set limit on genome size. However, large numbers of genes can lead to unacceptably large returned object sizes and/or running time. We discuss several examples next that illustrate some of the major issues to consider. Another example with 50,000 genes is shown in section 6.5.3.
We have seen in 2.1 and 2.2.1 that for the Exp model,
benchmark results using detectionProb require a lot of care and can be
misleading. Here, we will fix initial population sizes (to 500) and all
final population sizes will be set to \(\geq 10^6\). In addition, to avoid
the confounding factor of the onlyCancer = TRUE argument, we will set it
to FALSE, so we measure directly the time of individual runs.
2.3.1 Exponential model with 10,000 and 50,000 genes
2.3.1.1 Exponential, 10,000 genes, example 1
We will start with 10000 genes and an exponential model, where we stop when the population grows over \(10^6\) individuals:
ng <- 10000
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng/2),
rep(-0.1, ng/2)))
t_e_10000 <- system.time(
e_10000 <- oncoSimulPop(5, u, model = "Exp", mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
mutationPropGrowth = TRUE,
mc.cores = 1))t_e_10000
## user system elapsed
## 4.368 0.196 4.566
summary(e_10000)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 5017 1180528 415116 143 7547
## 2 3726 1052061 603612 131 5746
## 3 4532 1100721 259510 132 6674
## 4 4150 1283115 829728 99 6646
## 5 4430 1139185 545958 146 6748
print(object.size(e_10000), units = "MB")
## 863.9 Mb
Each simulation takes about 1 second but note that the number of clones for most simulations is already over 4000 and that the size of the returned object is close to 1 GB (a more detailed explanation of where this 1 GB comes from is deferred until section 2.3.1.6).
2.3.1.2 Exponential, 10,000 genes, example 2
We can decrease the size of the returned object if we use the keepEvery = NA argument (this setting was explained in detail in section
2.1):
t_e_10000b <- system.time(
e_10000b <- oncoSimulPop(5,
u,
model = "Exp",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = TRUE,
mc.cores = 1
))
t_e_10000b
## user system elapsed
## 5.484 0.100 5.585
summary(e_10000b)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 2465 1305094 727989 91 6447
## 2 2362 1070225 400329 204 8345
## 3 2530 1121164 436721 135 8697
## 4 2593 1206293 664494 125 8149
## 5 2655 1186994 327835 191 8572
print(object.size(e_10000b), units = "MB")
## 488.3 Mb
2.3.1.3 Exponential, 50,000 genes, example 1
Let’s use 50,000 genes. To keep object sizes reasonable we use
keepEvery = NA. For now, we also set mutationPropGrowth = FALSE
so that the mutation rate does not become really large in clones
with many mutations but, of course, whether or not this is a
reasonable decision depends on the problem; see also below.
ng <- 50000
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng/2),
rep(-0.1, ng/2)))
t_e_50000 <- system.time(
e_50000 <- oncoSimulPop(5,
u,
model = "Exp",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_e_50000
## user system elapsed
## 44.192 1.684 45.891
summary(e_50000)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 7367 1009949 335455 75.00 18214
## 2 8123 1302324 488469 63.65 17379
## 3 8408 1127261 270690 72.57 21144
## 4 8274 1138513 318152 80.59 20994
## 5 7520 1073131 690814 70.00 18569
print(object.size(e_50000), units = "MB")
## 7598.6 MbOf course, simulations now take longer and the size of the returned object is over 7 GB (we are keeping more than 7,000 clones, even if when we prune all those that went extinct).
2.3.1.4 Exponential, 50,000 genes, example 2
What if we had not pruned?
ng <- 50000
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng/2),
rep(-0.1, ng/2)))
t_e_50000np <- system.time(
e_50000np <- oncoSimulPop(5,
u,
model = "Exp",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = 1,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_e_50000np
## user system elapsed
## 42.316 2.764 45.079
summary(e_50000np)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 13406 1027949 410074 71.97 19469
## 2 12469 1071325 291852 66.00 17834
## 3 11821 1089834 245720 90.00 16711
## 4 14008 1165168 505607 77.61 19675
## 5 14759 1074621 205954 87.68 20597
print(object.size(e_50000np), units = "MB")
## 12748.4 Mb
The main effect is not on execution time but on object size (it has grown by 5 GB). We are tracking more than 10,000 clones.
2.3.1.5 Exponential, 50,000 genes, example 3
What about the mutationPropGrowth setting? We will rerun the example in
2.3.1.3 leaving keepEvery = NA but with the default
mutationPropGrowth:
ng <- 50000
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng/2),
rep(-0.1, ng/2)))
t_e_50000c <- system.time(
e_50000c <- oncoSimulPop(5,
u,
model = "Exp",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = TRUE,
mc.cores = 1
))
t_e_50000c
## user system elapsed
## 84.228 2.416 86.665
summary(e_50000c)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 11178 1241970 344479 84.74 27137
## 2 12820 1307086 203544 91.94 33448
## 3 10592 1126091 161057 83.81 26064
## 4 11883 1351114 148986 65.68 25396
## 5 10518 1101392 253523 99.79 26082
print(object.size(e_50000c), units = "MB")
## 10904.9 Mb
As expected (because the mutation rate per unit time is increasing in the fastest growing clones), we have many more clones, larger objects, and longer times of execution here: we almost double the time and the size of the object increases by almost 3 GB.
What about larger population sizes or larger mutation rates? The number of clones starts growing fast, which means much slower execution times and much larger returned objects (see also the examples below).
2.3.1.6 Interlude: where is that 1 GB coming from?
In section 2.3.1.1 we have seen an apparently innocuous simulation producing a returned object of almost 1 GB. Where is that coming from? It means that each simulation produced almost 200 MB of output.
Let us look at one simulation in more detail:
r1 <- oncoSimulIndiv(u,
model = "Exp",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
mutationPropGrowth = TRUE
)
summary(r1)[c(1, 8)]
## NumClones FinalTime
## 1 3887 345
print(object.size(r1), units = "MB")
## 160 Mb
## Size of the two largest objects inside:
sizes <- lapply(r1, function(x) object.size(x)/(1024^2))
sort(unlist(sizes), decreasing = TRUE)[1:2]
## Genotypes pops.by.time
## 148.28 10.26
dim(r1$Genotypes)
## [1] 10000 3887The above shows the reason: the Genotypes matrix is a 10,000 by 3,887
integer matrix (with a 0 and 1 indicating not-mutated/mutated for each
gene in each genotype) and in R integers use 4 bytes each. The
pops.by.time matrix is 346 by 3,888 (the 1 in \(346 = 345 + 1\) comes from
starting at 0 and going up to the final time, both included; the 1 in
\(3888 = 3887 + 1\) is from the column of time) double matrix and doubles
use 8 bytes4.
2.3.2 McFarland model with 50,000 genes; the effect of keepEvery
We show an example of McFarland’s model with 50,000 genes in section 6.5.3. We will show here a few more examples with those many genes but with a different fitness specification and changing several other settings.
2.3.2.1 McFarland, 50,000 genes, example 1
Let’s start with mutationPropGrowth = FALSE and keepEvery = NA. Simulations end when population size \(\geq 10^6\).
ng <- 50000
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng/2),
rep(-0.1, ng/2)))
t_mc_50000_nmpg <- system.time(
mc_50000_nmpg <- oncoSimulPop(5,
u,
model = "McFL",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg
## user system elapsed
## 30.46 0.54 31.01
summary(mc_50000_nmpg)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 1902 1002528 582752 284.2 31137
## 2 2159 1002679 404858 274.8 36905
## 3 2247 1002722 185678 334.5 42429
## 4 2038 1009606 493574 218.4 32519
## 5 2222 1004661 162628 291.0 38470
print(object.size(mc_50000_nmpg), units = "MB")
## 2057.6 Mb
We are already dealing with 2000 clones.
2.3.2.2 McFarland, 50,000 genes, example 2
Setting keepEvery = 1 (i.e., keeping track of clones with an
interval of 1):
t_mc_50000_nmpg_k <- system.time(
mc_50000_nmpg_k <- oncoSimulPop(5,
u,
model = "McFL",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = 1,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg_k
## user system elapsed
## 30.000 1.712 31.714
summary(mc_50000_nmpg_k)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 8779 1000223 136453 306.7 38102
## 2 7442 1006563 428150 345.3 35139
## 3 8710 1003509 224543 252.3 35659
## 4 8554 1002537 103889 273.7 36783
## 5 8233 1003171 263005 301.8 35236
print(object.size(mc_50000_nmpg_k), units = "MB")
## 8101.4 MbComputing time increases slightly but the major effect is seen on the size of the returned object, that increases by a factor of about 4x, up to 8 GB, corresponding to the increase in about 4x in the number of clones being tracked (see details of where the size of this object comes from in section 2.3.1.6).
2.3.2.3 McFarland, 50,000 genes, example 3
We will set keepEvery = NA again, but we will now increase
detection size by a factor of 3 (so we stop when total population
size becomes \(\geq 3 * 10^6\)).
ng <- 50000
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng/2),
rep(-0.1, ng/2)))
t_mc_50000_nmpg_3e6 <- system.time(
mc_50000_nmpg_3e6 <- oncoSimulPop(5,
u,
model = "McFL",
mu = 1e-7,
detectionSize = 3e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg_3e6
## user system elapsed
## 77.240 1.064 78.308
summary(mc_50000_nmpg_3e6)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 5487 3019083 836793 304.5 65121
## 2 4812 3011816 789146 286.3 53087
## 3 4463 3016896 1970957 236.6 45918
## 4 5045 3028142 956026 360.3 63464
## 5 4791 3029720 916692 358.1 55012
print(object.size(mc_50000_nmpg_3e6), units = "MB")
## 4759.3 Mb
Compared with the first run (2.3.2.1) we have approximately doubled computing time, number of iterations, number of clones, and object size.
2.3.2.4 McFarland, 50,000 genes, example 4
Let us use the same detectionSize = 1e6 as in the first example
(2.3.2.1), but with 5x the mutation rate:
t_mc_50000_nmpg_5mu <- system.time(
mc_50000_nmpg_5mu <- oncoSimulPop(5,
u,
model = "McFL",
mu = 5e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg_5mu
## user system elapsed
## 167.332 1.796 169.167
summary(mc_50000_nmpg_5mu)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 7963 1004415 408352 99.03 57548
## 2 8905 1010751 120155 130.30 74738
## 3 8194 1005465 274661 96.98 58546
## 4 9053 1014049 119943 112.23 75379
## 5 8982 1011817 95047 99.95 76757
print(object.size(mc_50000_nmpg_5mu), units = "MB")
## 8314.4 MbThe number of clones we are tracking is about 4x the number of clones of the first example (2.3.2.1), and roughly similar to the number of clones of the second example (2.3.2.2), and size of the returned object is similar to that of the second example. But computing time has increased by a factor of about 5x and iterations have increased by a factor of about 2x. Iterations increase because mutation is more frequent; in addition, at each sampling period each iteration needs to do more work as it needs to loop over a larger number of clones and this larger number includes clones that are not shown here, because they are pruned (they are extinct by the time we exit the simulation —again, pruning is discussed with further details in 15.2.1).
2.3.2.5 McFarland, 50,000 genes, example 5
Now let’s run the above example but with keepEvery = 1:
t_mc_50000_nmpg_5mu_k <- system.time(
mc_50000_nmpg_5mu_k <- oncoSimulPop(5,
u,
model = "McFL",
mu = 5e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = 1,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg_5mu_k
## user system elapsed
## 174.404 5.068 179.481
summary(mc_50000_nmpg_5mu_k)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 25294 1001597 102766 123.4 74524
## 2 23766 1006679 223010 124.3 71808
## 3 21755 1001379 203638 114.8 62609
## 4 24889 1012103 161003 119.3 75031
## 5 21844 1002927 255388 108.8 64556
print(object.size(mc_50000_nmpg_5mu_k), units = "MB")
## 22645.8 Mb
We have already seen these effects before in section
2.3.2.2: using keepEvery = 1 leads to a slight increase
in execution time. What is really affected is the size of the
returned object which increases by a factor of about 3x (and is now
over 20GB). That 3x corresponds, of course, to the increase in the
number of clones being tracked (now over 20,000). This, by the way,
also allows us to understand the comment above, where we said that
in these two cases (where we have increased mutation rate) at each
iteration we need to do more work as at every update of the
population the algorithm needs to loop over a much larger number of
clones (even if many of those are eventually pruned).
2.3.2.6 McFarland, 50,000 genes, example 6
Finally, we will run the example in section 2.3.2.1 with the
default of mutationPropGrowth = TRUE:
t_mc_50000 <- system.time(
mc_50000 <- oncoSimulPop(5,
u,
model = "McFL",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = TRUE,
mc.cores = 1
))
t_mc_50000
## user system elapsed
## 303.352 2.808 306.223
summary(mc_50000)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 13928 1010815 219814 210.9 91255
## 2 12243 1003267 214189 178.1 67673
## 3 13880 1014131 124354 161.4 88322
## 4 14104 1012941 75521 205.7 98583
## 5 12428 1005594 232603 167.4 70359
print(object.size(mc_50000), units = "MB")
## 12816.6 Mb
Note the huge increase in computing time (related of course to the huge increase in number of iterations) and in the size of the returned object: we have gone from having to track about 2000 clones to tracking over 12000 clones even when we prune all clones without descendants.
2.3.3 Examples with \(s = 0.05\)
A script with the above runs but using \(s=0.05\) instead of \(s=0.1\) is available from the repository (‘miscell-files/vignette_bench_Rout/large_num_genes_0.05.Rout’). I will single out a couple of cases here.
First, we repeat the run shown in section 2.3.2.5:
t_mc_50000_nmpg_5mu_k <- system.time(
mc_50000_nmpg_5mu_k <- oncoSimulPop(2,
u,
model = "McFL",
mu = 5e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = 1,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg_5mu_k
## user system elapsed
## 305.512 5.164 310.711
summary(mc_50000_nmpg_5mu_k)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 61737 1003273 104460 295.8731 204214
## 2 65072 1000540 133068 296.6243 210231
print(object.size(mc_50000_nmpg_5mu_k), units = "MB")
## 24663.6 Mb
Note we use only two replicates, since those two already lead to a 24 GB returned object as we are tracking more than 60,000 clones, more than twice those with \(s=0.1\). The reason for the difference in number of clones and iterations is of course the change from \(s=0.1\) to \(s=0.05\): under the McFarland model to reach population sizes of \(10^6\) starting from an equilibrium population of 500 we need about 43 mutations (whereas only about 22 are needed if \(s=0.1\)5).
Next, let us rerun 2.3.2.1:
t_mc_50000_nmpg <- system.time(
mc_50000_nmpg <- oncoSimulPop(5,
u,
model = "McFL",
mu = 1e-7,
detectionSize = 1e6,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
keepEvery = NA,
mutationPropGrowth = FALSE,
mc.cores = 1
))
t_mc_50000_nmpg
## user system elapsed
## 111.236 0.596 111.834
summary(mc_50000_nmpg)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 2646 1000700 217188 734.475 108566
## 2 2581 1001626 209873 806.500 107296
## 3 2903 1001409 125148 841.700 120859
## 4 2310 1000146 473948 906.300 91519
## 5 2704 1001290 448409 838.800 103556
print(object.size(mc_50000_nmpg), units = "MB")
## 2638.3 Mb
Using \(s=0.05\) leads to a large increase in final time and number of
iterations. However, as we are using the keepEvery = NA setting,
the increase in number of clones tracked and in size of returned
object is relatively small.
2.3.4 The different consequences of keepEvery = NA in the Exp and McFL models
We have seen that keepEvery = NA often leads to much smaller returned
objects when using the McFarland model than when using the Exp model. Why?
Because in the McFarland model there is strong competition and there can
be complete clonal sweeps so that in extreme cases a single clone might be
all that is left after some time. This is not the case in the exponential
models.
Of course, the details depend on the difference in fitness effects
between different genotypes (or clones). In particular, we have seen
several examples where even with keepEvery=NA there are a lot of
clones in the McFL models. In those examples many clones had
identical fitness (the fitness effects of all genes with positive
fitness was the same, and ditto for the genes with negative fitness
effects), so no clone ends up displacing all the others.
2.3.5 Are we keeping the complete history (genealogy) of the clones?
Yes we are if we run with keepPhylog = TRUE, regardless of the
setting for keepEvery. As explained in section 15.2,
OncoSimulR prunes clones that never had a population size larger
than zero at any sampling period (so they are not reflected in the
pops.by.time matrix in the output). And when we set keepEvery = NA we are telling OncoSimulR to discard all sampling periods except
the very last one (i.e., the pops.by.time matrix contains only the
clones with 1 or more cells at the end of the simulation).
keepPhylog operates differently: it records the exact time at
which a clone appeared and the clone that gave rise to it. This
information is kept regardless of whether or not those clones appear
in the pops.by.time matrix.
Keeping the complete genealogy might be of limited use if the
pops.by.time matrix only contains the very last period. However,
you can use plotClonePhylog and ask to be shown only clones that
exist in the very last period (while of course showing all of their
ancestors, even if those are now extinct —i.e., regardless of
their abundance).
For instance, in run 2.3.1.3 we could have looked at the information stored about the genealogy of clones by doing (we look at the first “individual” of the simulation, of the five “individuals” we simulated):
head(e_50000[[1]]$other$PhylogDF)
## parent child time
## 1 3679 0.8402
## 2 4754 1.1815
## 3 20617 1.4543
## 4 15482 2.3064
## 5 4431 3.7130
## 6 41915 4.0628
tail(e_50000[[1]]$other$PhylogDF)
## parent child time
## 20672 3679, 20282 3679, 20282, 22359 75.0
## 20673 3679, 17922, 22346 3679, 17922, 22346, 35811 75.0
## 20674 2142, 3679 2142, 3679, 25838 75.0
## 20675 3679, 17922, 19561 3679, 17922, 19561, 43777 75.0
## 20676 3679, 15928, 19190, 20282 3679, 15928, 19190, 20282, 49686 75.0
## 20677 2142, 3679, 16275 2142, 3679, 16275, 24201 75.0where each row corresponds to one event of appearance of a new clone, the column labeled “parent” are the mutated genes in the parent, and the column labeled “child” are the mutated genes in the child.
And we could plot the genealogical relationships of clones that have a population size of at least one in the last period (again, while of course showing all of their ancestors, even if those are now extinct —i.e., regardless of their current numbers) doing:
plotClonePhylog(e_50000[[1]]) ## plot not shown
What is the cost of keep the clone genealogies? In terms of time it
is minor. In terms of space, and as shown in the example above, we
can end up storing a data frame with tends of thousands of rows and
three columns (two factors, one float). In the example above the
size of that data frame is approximately 2 MB for a single
simulation. This is much smaller than the pops.by.time or
Genotypes matrices, but it can quickly build up if you routinely
launch, say, 1000 simulations via oncoSimulPop. That is why the
default is keepPhylog = FALSE as this information is not needed as
often as that in the other two matrices (pops.by.time and
Genotypes).
2.4 Population sizes \(\geq 10^{10}\)
We have already seen examples where population sizes reach \(10^{8}\) to \(10^{10}\), as in Tables 2.2, 2.4, 2.6. What about even larger population sizes?
The C++ code will unconditionally alert if population sizes exceed
\(4*10^{15}\) as in those cases loosing precision (as we are using
doubles) would be unavoidable, and we would also run into problems
with the generation of binomial random variates (code that
illustrates and discusses this problem is available in file
“example-binom-problems.cpp”, in directory
“/inst/miscell”). However, well before we reach \(4*10^{15}\) we loose
precision from other sources. One of the most noticeable ones is
that when we reach population sizes around \(10^{11}\) the C++ code
will often alert us by throwing exceptions with the message
Recoverable exception ti set to DBL_MIN. Rerunning. I throw this
exception because \(t_i\), the random variable for time to next
mutation, is less than DBL_MIN, the minimum representable
floating-point number. This happens because, unless we use really
tiny mutation rates, the time to a mutation starts getting closer to
zero as population sizes grow very large. It might be possible to
ameliorate these problems somewhat by using long doubles (instead of
doubles) or special purpose libraries that provide more
precision. However, this would make it harder to run the same code
in different operating systems and would likely decrease execution
speed on the rest of the common scenarios for which OncoSimulR has
been designed.
The following code shows some examples where we use population sizes
of \(10^{10}\) or larger. Since we do not want simulations in the
exponential model to end because of extinction, I use a fitness
specification where all genes have a positive fitness effect and we
start all simulations from a large population (to make it unlikely
that the population will become extinct before cells mutate and
start increasing in numbers). We set the maximum running time to 10
minutes. We keep the genealogy of the clones and use keepEvery = 1.
ng <- 50
u <- allFitnessEffects(noIntGenes = c(rep(0.1, ng)))t_mc_k_50_1e11 <- system.time(
mc_k_50_1e11 <- oncoSimulPop(5,
u,
model = "McFL",
mu = 1e-7,
detectionSize = 1e11,
initSize = 1e5,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
mutationPropGrowth = FALSE,
keepEvery = 1,
finalTime = 5000,
mc.cores = 1,
max.wall.time = 600
))
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
t_mc_k_50_1e11
## user system elapsed
## 613.612 0.040 613.664
summary(mc_k_50_1e11)[, c(1:3, 8, 9)]
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 5491 100328847809 44397848771 1019.950 942764
## 2 3194 100048090441 34834178374 789.675 888819
## 3 5745 100054219162 24412502660 927.950 929231
## 4 4017 101641197799 60932177160 750.725 480938
## 5 5393 100168156804 41659212367 846.250 898245
## print(object.size(mc_k_50_1e11), units = "MB")
## 177.8 Mb
We get to \(10^{11}\). But notice the exception with the warning about \(t_i\). Notice also that this takes a long time and we run a very large number of iterations (getting close to one million in some cases).
Now the exponential model with detectionSize = 1e11:
t_exp_k_50_1e11 <- system.time(
exp_k_50_1e11 <- oncoSimulPop(5,
u,
model = "Exp",
mu = 1e-7,
detectionSize = 1e11,
initSize = 1e5,
detectionDrivers = NA,
detectionProb = NA,
keepPhylog = TRUE,
onlyCancer = FALSE,
mutationPropGrowth = FALSE,
keepEvery = 1,
finalTime = 5000,
mc.cores = 1,
max.wall.time = 600,
errorHitWallTime = FALSE,
errorHitMaxTries = FALSE
))
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Hitted wall time. Exiting.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Hitted wall time. Exiting.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Recoverable exception ti set to DBL_MIN. Rerunning.
## Hitted wall time. Exiting.
## Hitted wall time. Exiting.
t_exp_k_50_1e11
## user system elapsed
## 2959.068 0.128 2959.556
try(summary(exp_k_50_1e11)[, c(1:3, 8, 9)])
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 6078 65172752616 16529682757 235.7590 1883438
## 2 5370 106476643712 24662446729 232.0000 2516675
## 3 2711 21911284363 17945303353 224.8608 543698
## 4 2838 13241462284 2944300245 216.8091 372298
## 5 7289 76166784312 10941729810 240.0217 1999489
print(object.size(exp_k_50_1e11), units = "MB")
## 53.5 Mb
Note that we almost reached max.wall.time (600 * 5 = 3000). What if we
wanted to go up to \(10^{12}\)? We would not be able to do it in 10
minutes. We could set max.wall.time to a value larger than 600 to allow
us to reach larger sizes but then we would be waiting for a possibly
unacceptable time for simulations to finish. Moreover, this would
eventually fail as simulations would keep hitting the \(t_i\) exception
without ever being able to complete. Finally, even if we were very
patient, hitting that \(t_i\) exception should make us worry about possible
biases in the samples.
2.5 A summary of some determinants of running time and space consumption
To summarize this section, we have seen:
Both McFL and Exp can be run in short times over a range of sizes for the
detectionProbanddetectionSizemechanisms using a complex fitness specification with moderate numbers of genes. These are the typical or common use cases of OncoSimulR.The
keepEveryargument can have a large effect on time in the McFL models and specially on object sizes. If only the end result of the simulation is to be used, you should setkeepEvery = NA.The distribution of fitness effects and the fitness landscape can have large effects on running times. Sometimes these are intuitive and simple to reason about, sometimes they are not as they interact with other factors (e.g., stopping mechanism, numbers of clones, etc). In general, there can be complex interactions between different settings, from mutation rate to fitness effects to initial size. As usual, test before launching a massive simulation.
Simulations start to slow down and lead to a very large object size when we keep track of around 6000 to 10000 clones. Anything that leads to these patterns will slow down the simulations.
OncoSimulR needs to keep track of genotypes (or clones), not just numbers of drivers and passengers, because it allows you to use complex fitness and mutation specifications that depend on specific genotypes. The
keepEvery = NAis an approach to store only the minimal information needed, but it is unavoidable that during the simulations we might be forced to deal with many thousands of different clones.
3 Specifying fitness effects
OncoSimulR uses a standard continuous time model, where individual cells divide, die, and mutate with rates that can depend on genotype and population size; over time the abundance of the different genotypes changes by the action of selection (due to differences in net growth rates among genotypes), drift, and mutation. As a result of a mutation in a pre-existing clone new clones arise, and the birth rate of a newly arisen clone is determined at the time of its emergence as a function of its genotype. Simulations can use an use exponential growth model or a model with carrying capacity that follows McFarland et al. (2013). For the exponential growth model, the death rate is fixed at one whereas in the model with carrying capacity death rate increases with population size. In both cases, therefore, fitness differences among genotypes in a given population at a given time are due to differences in the mapping between genotype and birth rate. There is second exponential model (called “Bozic”) where birth rate is fixed at one, and genotype determines death rate instead of birth rate (see details in 3.2). So when we discuss specifying fitness effects or the effects of genes on fitness, we are actually referring to specifying effects on birth (or death) rates, which then translate into differences in fitness (since the other rate, death or birth, is either fixed, as in the Exp and Bozic models, or depends on the population size). This is also shown in Table 1.1, in the rows for “Fitness components”, under “Evolutionary Features”.
In the case of frequency-dependent fitness simulations (10), the
fitness effects must be reevaluated frequently so that birth rate, death
rate, or both, depending the model used, are updated. To do this it is
necessary to use a short step to reevaluate fitness; this is done using a
small value for sampleEvery parameter in oncoSimulindv (see
15.8 for more details), as is the case when using
McFarland model.
Incidentally, notice that with OncoSimulR we do not directly specify fitness itself (even if, for the sake of simplicity, we often refer to fitness in the documentation) as fitness is, arguably, a derived quantity (Doebeli, Ispolatov, and Simon 2017). Rather, we specify how birth and/or death rates, which are the actual mechanistic drivers of evolutionary dynamics, are related to genotypes (or to the frequencies of the different genotypes).
3.1 Introduction to the specification of fitness effects
With OncoSimulR you can specify different types of effects on fitness:
A special type of epistatic effect that is particularly amenable to be represented as a graph (a DAG). In this graph having, say, “B” be a child of “A” means that a mutation in B can only accumulate if a mutation in A is already present. This is what OT (Desper et al. 1999; Szabo and Boucher 2008), CBN (Beerenwinkel, Eriksson, and Sturmfels 2007; Gerstung et al. 2009, 2011), progression networks (Farahani and Lagergren 2013), and other similar models (Korsunsky et al. 2014) generally mean. Details are provided in section 3.4. Note that this is not an order effect (discussed below): the fitness of a genotype from this DAGs is a function of whether or not the restrictions in the graph are satisfied, not the historical sequence of how they were satisfied.
Effects where the order in which mutations are acquired matters, as illustrated in section 3.6. There is, in fact, empirical evidence of these effects (Ortmann et al. 2015). For instance, the fitness of genotype “A, B” would differ depending on whether A or B was acquired first (or, as in the actual example in (Ortmann et al. 2015), the fitness of the mutant with JAK2 and TET2 mutated will depend on which of the genes was mutated first).
General epistatic effects (e.g., section 3.7), including synthetic viability (e.g., section 3.9) and synthetic lethality/mortality (e.g., section 3.10).
Genes that have independent effects on fitness (section 3.3).
Modules (see section 3.5) allow you to specify any of the above effects (except those for genes without interactions, as it would not make sense there) in terms of modules (sets of genes), not individual genes. We will introduce them right after 3.4, and we will continue using them thereafter.
A guiding design principle of OncoSimulR is to try to make the specification of those effects as simple as possible but also as flexible as possible. Thus, there are two main ways of specifying fitness effects:
Combining different types of effects in a single specification. For instance, you can combine epistasis with order effects with no interaction genes with modules. What you would do here is specify the effects that different mutations (or their combinations) have on fitness (the fitness effects) and then have OncoSimulR take care of combining them as if each of these were lego pieces. We will refer to this as the lego system of fitness effects. (As explained above, I find this an intuitive and very graphical analogy, which I have copied from Hothorn et al. (2006) and Hothorn et al. (2008)).
Explicitly passing to OncoSimulR a mapping of genotypes to fitness. Here you specify the fitness of each genotype. We will refer to this as the explicit mapping of genotypes to fitness.
Both approaches have advantages and disadvantages. Here I emphasize some relevant differences.
With the lego system you can specify huge genomes with an enormous variety of interactions, since the possible genotypes are not constructed in advance. You would not be able to do this with the explicit mapping of genotypes to fitness if you wanted to, say, construct that mapping for a modest genotype of 500 genes (you’d have more genotypes than particles in the observable Universe).
For many models/data you often intuitively start with the fitness of the genotypes, not the fitness consequences of the different mutations. In these cases, you’d need to do the math to specify the terms you want if you used the lego system so you’ll probably use the specification with the direct mapping genotype \(\rightarrow\) fitness.
Likewise, sometimes you already have a moderate size genotype \(\rightarrow\) fitness mapping and you certainly do not want to do the math by hand: here the lego system would be painful to use.
But sometimes we do think in terms of “the effects on fitness of such and such mutations are” and that immediately calls for the lego system, where you focus on the effects, and let OncoSimulR take care of doing the math of combining.
If you want to use order effects, you must use the lego system (at least for now).
If you want to specify modules, you must use the lego system (the explicit mapping of genotypes is, by its very nature, ill-suited for this).
The lego system might help you see what your model really means: in many cases, you can obtain fairly succinct specifications of complex fitness models with just a few terms. Similarly, depending on what your emphasis is, you can often specify the same fitness landscape in several different ways.
Regardless of the route, you need to get that information into
OncoSimulR’s functions. The main function we will use is
allFitnessEffects: this is the function in charge of reading
the fitness specifications. We also need to discuss how, what, and where
you have to pass to allFitnessEffects.
3.1.1 Explicit mapping of genotypes to fitness
Conceptually, the simplest way to specify fitness is to specify the mapping of all genotypes to fitness explicitly. An example will make this clear. Let’s suppose you have a simple two-gene scenario, so a total of four genotypes, and you have a data frame with genotypes and fitness, where genoytpes are specified as character vectors, with mutated genes separated by commas:
m4 <- data.frame(G = c("WT", "A", "B", "A, B"), F = c(1, 2, 3, 4))Now, let’s give that to the allFitnessEffects function:
fem4 <- allFitnessEffects(genotFitness = m4)
## Column names of object not Genotype and Fitness. Renaming them assuming that is what you wanted(The message is just telling you what the program guessed you wanted.)
That’s it. You can try to plot that fitnessEffects object
try(plot(fem4))
## Error in plot.fitnessEffects(fem4) :
## This fitnessEffects object can not be ploted this way. It is probably one with fitness landscape specification, so you might want to plot the fitness landscape instead.In this case, you probably want to plot the fitness landscape.
plotFitnessLandscape(evalAllGenotypes(fem4))
You can also check what OncoSimulR thinks the fitnesses are, with the
evalAllGenotypes function that we will use repeatedly below
(of course, here we should see the same fitnesses we entered):
evalAllGenotypes(fem4, addwt = TRUE)
## Genotype Fitness
## 1 WT 1
## 2 A 2
## 3 B 3
## 4 A, B 4And you can plot the fitness landscape:
plotFitnessLandscape(evalAllGenotypes(fem4))
To specify the mapping you can also use a matrix (or data frame) with \(g + 1\) columns; each of the first \(g\) columns contains a 1 or a 0 indicating that the gene of that column is mutated or not. Column \(g+ 1\) contains the fitness values. And you do not even need to specify all the genotypes: the missing genotypes are assigned a fitness 0 —except for the WT genotype which, if missing, is assigned a fitness of 1:
m6 <- cbind(c(1, 1), c(1, 0), c(2, 3))
fem6 <- allFitnessEffects(genotFitness = m6)
## No column names: assigning gene names from LETTERS
## Warning in to_genotFitness_std(genotFitness,
## frequencyDependentFitness = FALSE, : No wildtype in the fitness
## landscape!!! Adding it with fitness 1.
evalAllGenotypes(fem6, addwt = TRUE)
## Genotype Fitness
## 1 WT 1
## 2 A 3
## 3 B 0
## 4 A, B 2
## plot(fem6)plotFitnessLandscape(evalAllGenotypes(fem6))
This way of giving a fitness specification to OncoSimulR might be ideal if you directly generate random mappings of genotypes to fitness (or random fitness landscapes), as we will do in section 9. Specially when the fitness landscape contains many non-viable genotypes (which are considered those with fitness —birth rate— \(<1e-9\)) this can result in considerable savings as we only need to check the fitness of the viable genotypes in a table (a C++ map). Note, however, that using the Bozic model with the fitness landscape specification is not tested. In addition, for speed, missing genotypes from the fitness landscape specification are taken to be non-viable genotypes (beware!! this is a breaking change relative to versions < 2.9.1)6.
In the case of frequency-dependent fitness situations, the only way
to specify fitness effects is using genoFitnes as we have shown
before, but now you need to set frequencyDependentFitness = TRUE
in allFitnessEffects. The fundamental difference is the Fitness
column in genoFitnes. Now this column must be a character vector
and each element (character also) is a function whose variables are
the relative frequencies of the clones in the population. You must
specify the variables like f_, for frequency of wild type, f_1 or
f_A for frequency of mutant A or position 1, f_1_2 or f_A_B for
double mutant, and so on. Mathematical operations and symbols
allowed are described in the documentation of C++ library ExprTk
(http://www.partow.net/programming/exprtk/). ExprTk is the library
used to parse and evaluate the fitness equations. The numeric vector
spPopSizes is only necesary to evaluate genotypes through
evalGenotype or evalAllGenotypes functions because population
sizes are needed to calculate the clone’s frequencies.
r <- data.frame(Genotype = c("WT", "A", "B", "A, B"),
Fitness = c("10 * f_",
"10 * f_1",
"50 * f_2",
"200 * (f_1 + f_2) + 50 * f_1_2"))
afe <- allFitnessEffects(genotFitness = r,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
plotFitnessLandscape(evalAllGenotypes(afe,
spPopSizes = c(WT = 2500, A = 2000,
B = 5500, "A, B" = 700)))
The above example is simple enough in terms of genes and genotypes
that using f_1 is OK. But it will be better, as examples get more
complex, to use:
r <- data.frame(Genotype = c("WT", "A", "B", "A, B"),
Fitness = c("10 * f_",
"10 * f_A",
"50 * f_B",
"200 * (f_A + f_B) + 50 * f_A_B"))which makes explicit what depends on what (i.e., you do not need to
keep in mind the mapping of letters to numbers). In other words, we
write f_genotype expressed as combination of gene names, with the
gene names we are actually using. And those f_something_other,
will match the genotypes given in Genotype (there will a
something, other genotype).
3.1.2 How to specify fitness effects with the lego system
An alternative general approach followed in many genetic simulators is to specify how particular combinations of alleles modify the wildtype genotype or the genotype that contains the individual effects of the interacting genes (e.g., see equation 1 in the supplementary material for FFPopSim (Zanini and Neher 2012)). For example, if we specify that a mutation in “A” contributes 0.04, a mutation in “B” contributes 0.03, and the double mutation “A:B” contributes 0.1, that means that the fitness of the “A, B” genotype (the genotype with A and B mutated) is that of the wildtype (1, by default), plus (actually, times —see section 3.2— but plus on the log scale) the effects of having A mutated, plus (times) the effects of having B mutated, plus (times) the effects of “A:B” both being mutated.
We will see below that with the “lego system” it is possible to do something very similar to the explicit mapping of section 3.1.1. But this will sometimes require a more cumbersome notation (and sometimes also will require your doing some math). We will see examples in sections 3.7.1, 3.7.2 and 3.7.3 or the example in 5.4.2. But then, if we can be explicit about (at least some of) the mappings \(genotype \rightarrow fitness\), how are these procedures different? When you use the “lego system” you can combine both a partial explicit mapping of genotypes to fitness with arbitrary fitness effects of other genes/modules. In other words, with the “lego system” OncoSimulR makes it simple to be explicit about the mapping of specific genotypes, while also using the “how this specific effects modifies previous effects” logic, leading to a flexible specification. This also means that in many cases the same fitness effects can be specified in several different ways.
Most of the rest of this section is devoted to explaining how to combine those pieces. Before that, however, we need to discuss the fitness model we use.
3.2 Numeric values of fitness effects
We evaluate fitness using the usual (Zanini and Neher 2012; Gillespie 1993; Beerenwinkel, Eriksson, and Sturmfels 2007; Datta et al. 2013) multiplicative model: fitness is \(\prod (1 + s_i)\) where \(s_i\) is the fitness effect of gene (or gene interaction) \(i\). In all models except Bozic, this fitness refers to the growth rate (the death rate being fixed to 17). The original model of McFarland et al. (2013) has a slightly different parameterization, but you can go easily from one to the other (see section 3.2.1).
For the Bozic model (Bozic et al. 2010), however, the birth rate is set to 1, and the death rate then becomes \(\prod (1 - s_i)\).
3.2.1 McFarland parameterization
In the original model of McFarland et al. (2013), the effects of drivers contribute to the numerator of the birth rate, and those of the (deleterious) passengers to the denominator as: \(\frac{(1 + s)^d}{(1 + s_p)^p}\), where \(d\) and \(p\) are, respectively, the total number of drivers and passengers in a genotype, and here the fitness effects of all drivers is the same (\(s\)) and that of all passengers the same too (\(s_p\)). Note that, as written above, and as explicitly said in McFarland et al. (2013) (see p. 2911) and CD McFarland (2014) (see p. 9), “(…) \(s_p\) is the fitness disadvantage conferred by a passenger”. In other words, the larger the \(s_p\) the more deleterious the passenger.
This is obvious, but I make it explicit because in our parameterization a positive \(s\) means fitness advantage, whereas fitness disadvantages are associated with negative \(s\). Of course, if you rewrite the above expression as \(\frac{(1 + s)^d}{(1 - s_p)^p}\) then we are back to the “positive means fitness advantage and negative means fitness disadvantage”.
As CD McFarland (2014) explains (see p. 9, bottom), we can rewrite the above expression so that there are no terms in the denominator. McFarland writes it as (I copy verbatim from the fourth and fifth lines from the bottom on his p. 9) \((1 + s_d)^{n_d} (1 - s_p^{'})^{n_p}\) where \(s_p^{'} = s_p/(1 + s_p)\).
However, if we want to express everything as products (no ratios) and use the “positive s means advantage and negative s means disadvantage” rule, we want to write the above expression as \((1 + s_d)^{n_d} (1 + s_{pp})^{n_p}\) where \(s_{pp} = -s_p/(1 + s_p)\). And this is actually what we do in v.2. There is an example, for instance, in section 6.5.2 where you will see:
sp <- 1e-3
spp <- -sp/(1 + sp)so we are going from the “(…) \(s_p\) is the fitness disadvantage conferred by a passenger” in McFarland et al. (2013) (p. 2911) and CD McFarland (2014) (p. 9) to the expression where we have a product \(\prod (1 + s_i)\), with the “positive s means advantage and negative s means disadvantage” rule. This reparameterization applies to v.2. In v.1 we used the same parameterization as in the original one in McFarland et al. (2013), but with the “positive s means advantage and negative s means disadvantage” rule (so we are using expression \(\frac{(1 + s)^d}{(1 - s_p)^p}\)).
3.2.1.1 Death rate under the McFarland model
For death rate, we use the expression that McFarland et al. (2013) (see their p. 2911) use “(…) for large cancers (grown to \(10^6\) cells)”: \(D(N) = \log(1 + N/K)\) where \(K\) is the initial equilibrium population size. As the authors explain, for large N/K the above expression “(…) recapitulates Gompertzian dynamics observed experimentally for large tumors”.
By default, OncoSimulR uses a value of \(K=initSize/(e^{1} - 1)\) so that the starting population is at equilibrium.
A consequence of the above expression for death rate is that if the population size decreases the death rate decreases. This is not relevant in most cases (as mutations, or some mutations, will inexorably lead to population size increases). And this prevents the McFL model from resulting in extinction even with very small population sizes as long as birth rate \(\ge\) death rate. (For small population sizes, it is likely that the population will become extinct if birth rate = death rate; you can try this with the exponential model).
But this is not what we want in some other models, such as
frequency-dependent ones, where modeling population collapse (which
will happen if birth rate < death rate) can be important (as in the
example in 10.3). Here, it makes sense to set \(D(N) = \max(1, \log(1 + N/K))\) so that the death rate never decreases
below 1. (Using 1 is reasonable if we consider the equilibrium birth
rate in the absence of any mutants to be 1). You can specify this
behaviour using model McFLD (a shorthand for McFarlandLogD).
3.2.2 No viability of clones and types of models
For all models where fitness affects directly the birth rate (all except Bozic), if you specify that some event (say, mutating gene A) has \(s_A \le -1\), if that event happens then birth rate becomes zero. This is taken to indicate that the clone is not even viable and thus disappears immediately without any chance for mutation8.
Models based on Bozic, however, have a birth rate of 1 and mutations affect the death rate. In this case, a death rate larger than birth rate, per se, does not signal immediate extinction and, moreover, even for death rates that are a few times larger than birth rates, the clone could mutate before becoming extinct9.
In general, if you want to identify some mutations or some combinations of mutations as leading to immediate extinction (i.e., no viability), of the affected clone, set it to \(-\infty\) as this would work even if how birth rates of 0 are handled changes. Most examples below evaluate fitness by its effects on the birth rate. You can see one where we do it both ways in Section 3.11.1.
3.3 Genes without interactions
This is a simple scenario. Each gene \(i\) has a fitness effect \(s_i\) if mutated. The \(s_i\) can come from any distribution you want. As an example let’s use three genes. We know there are no order effects, but we will also see what happens if we examine genotypes as ordered.
ai1 <- evalAllGenotypes(allFitnessEffects(
noIntGenes = c(0.05, -.2, .1)), order = FALSE)We can easily verify the first results:
ai1
## Genotype Fitness
## 1 1 1.050
## 2 2 0.800
## 3 3 1.100
## 4 1, 2 0.840
## 5 1, 3 1.155
## 6 2, 3 0.880
## 7 1, 2, 3 0.924all(ai1[, "Fitness"] == c( (1 + .05), (1 - .2), (1 + .1),
(1 + .05) * (1 - .2),
(1 + .05) * (1 + .1),
(1 - .2) * (1 + .1),
(1 + .05) * (1 - .2) * (1 + .1)))
## [1] TRUEAnd we can see that considering the order of mutations (see section 3.6) makes no difference:
(ai2 <- evalAllGenotypes(allFitnessEffects(
noIntGenes = c(0.05, -.2, .1)), order = TRUE,
addwt = TRUE))
## Genotype Fitness
## 1 WT 1.000
## 2 1 1.050
## 3 2 0.800
## 4 3 1.100
## 5 1 > 2 0.840
## 6 1 > 3 1.155
## 7 2 > 1 0.840
## 8 2 > 3 0.880
## 9 3 > 1 1.155
## 10 3 > 2 0.880
## 11 1 > 2 > 3 0.924
## 12 1 > 3 > 2 0.924
## 13 2 > 1 > 3 0.924
## 14 2 > 3 > 1 0.924
## 15 3 > 1 > 2 0.924
## 16 3 > 2 > 1 0.924(The meaning of the notation in the output table is as follows: “WT” denotes the wild-type, or non-mutated clone. The notation \(x > y\) means that a mutation in “x” happened before a mutation in “y”. A genotype \(x > y\ \_\ z\) means that a mutation in “x” happened before a mutation in “y”; there is also a mutation in “z”, but that is a gene for which order does not matter).
And what if I want genes without interactions but I want modules (see section 3.5)? Go to section 3.8.
3.4 Using DAGs: Restrictions in the order of mutations as extended posets
3.4.1 AND, OR, XOR relationships
The literature on Oncogenetic trees, CBNs, etc, has used graphs as a way of showing the restrictions in the order in which mutations can accumulate. The meaning of “convergent arrows” in these graphs, however, differs. In Figure 1 of Korsunsky et al. (2014) we are shown a simple diagram that illustrates the three basic different meanings of convergent arrows using two parental nodes. We will illustrate it here with three. Suppose we focus on node “g” in the following figure (we will create it shortly)
data(examplesFitnessEffects)
plot(examplesFitnessEffects[["cbn1"]])
In relationships of the type used in Conjunctive Bayesian Networks (CBN) (e.g., Gerstung et al. 2009), we are modeling an AND relationship, also called CMPN by Korsunsky et al. (2014) or monotone relationship by Farahani and Lagergren (2013). If the relationship in the graph is fully respected, then “g” will only appear if all of “c”, “d”, and “e” are already mutated.
Semimonotone relationships sensu Farahani and Lagergren (2013) or DMPN sensu Korsunsky et al. (2014) are OR relationships: “g” will appear if one or more of “c”, “d”, or “e” are already mutated.
XMPN relationships (Korsunsky et al. 2014) are XOR relationships: “g” will be present only if exactly one of “c”, “d”, or “e” is present.
Note that Oncogenetic trees (Desper et al. 1999; Szabo and Boucher 2008) need not deal with the above distinctions, since the DAGs are trees: no node has more than one incoming connection or more than one parent10.
To have a flexible way of specifying all of these restrictions, we will want to be able to say what kind of dependency each child node has on its parents.
3.4.2 Fitness effects
Those DAGs specify dependencies and, as explained in
Diaz-Uriarte (2015), it is simple to map them to a simple evolutionary
model: any set of mutations that does not conform to the restrictions
encoded in the graph will have a fitness of 0. However, we might not want
to require absolute compliance with the DAG. This means we might want to
allow deviations from the DAG with a corresponding penalization that is,
however, not identical to setting fitness to 0 (again, see Diaz-Uriarte 2015). This we can do by being explicit about the
fitness effects of the deviations from the restrictions encoded in the
DAG. We will use below a column of s for the fitness effect when
the restrictions are satisfied and a column of sh when they are
not. (See also 3.2 for the details about the meaning of the
fitness effects).
That way of specifying fitness effects makes it also trivial to use the
model in Hjelm, Höglund, and Lagergren (2006) where all mutations might be allowed to occur,
but the presence of some mutations increases the probability of occurrence
of other mutations. For example, the values of sh could be all
small positive ones (or for mildly deleterious effects, small negative
numbers), while the values of s are much larger positive numbers.
3.4.3 Extended posets
In version 1 of this package we used posets in the sense of
Beerenwinkel, Eriksson, and Sturmfels (2007) and Gerstung et al. (2009), as explained in
the help for poset. The
functionality for simulating directly from such two column matrices
has been removed. Instead, we use what we call extended posets.
With the extended posets, we continue using two columns, that specify parents and children, but we add columns for the specific values of fitness effects (both s and sh —i.e., fitness effects for what happens when restrictions are and are not satisfied) and for the type of dependency as explained in section 3.4.1.
We can now illustrate the specification of different fitness effects using DAGs.
3.4.4 DAGs: A first conjunction (AND) example
cs <- data.frame(parent = c(rep("Root", 4), "a", "b", "d", "e", "c"),
child = c("a", "b", "d", "e", "c", "c", rep("g", 3)),
s = 0.1,
sh = -0.9,
typeDep = "MN")
cbn1 <- allFitnessEffects(cs)(We skip one letter, just to show that names need not be consecutive or have any particular order.)
We can get a graphical representation using the default “graphNEL”
plot(cbn1)
or one using “igraph”:
plot(cbn1, "igraph")
Since we have a parent and children, the reingold.tilford layout is probably the best here, so you might want to use that:
library(igraph) ## to make the reingold.tilford layout available
plot(cbn1, "igraph", layout = layout.reingold.tilford)
And what is the fitness of all genotypes?
gfs <- evalAllGenotypes(cbn1, order = FALSE, addwt = TRUE)
gfs[1:15, ]
## Genotype Fitness
## 1 WT 1.00
## 2 a 1.10
## 3 b 1.10
## 4 c 0.10
## 5 d 1.10
## 6 e 1.10
## 7 g 0.10
## 8 a, b 1.21
## 9 a, c 0.11
## 10 a, d 1.21
## 11 a, e 1.21
## 12 a, g 0.11
## 13 b, c 0.11
## 14 b, d 1.21
## 15 b, e 1.21You can verify that for each genotype, if a mutation is present without all of its dependencies present, you get a \((1 - 0.9)\) multiplier, and you get a \((1 + 0.1)\) multiplier for all the rest with its direct parents satisfied. For example, genotypes “a”, or “b”, or “d”, or “e” have fitness \((1 + 0.1)\), genotype “a, b, c” has fitness \((1 + 0.1)^3\), but genotype “a, c” has fitness \((1 + 0.1) (1 - 0.9) = 0.11\).
3.4.5 DAGs: A second conjunction example
Let’s try a first attempt at a somewhat more complex example, where the fitness consequences of different genes differ.
c1 <- data.frame(parent = c(rep("Root", 4), "a", "b", "d", "e", "c"),
child = c("a", "b", "d", "e", "c", "c", rep("g", 3)),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, rep(0.2, 3)),
sh = c(rep(0, 4), c(-.1, -.2), c(-.05, -.06, -.07)),
typeDep = "MN")
try(fc1 <- allFitnessEffects(c1))
## Error in FUN(X[[i]], ...) : Not all sh identical within a childIf you try this, you’ll get an error. There is an error because the “sh” varies within a child, and we do not allow that for a poset-type specification, as it is ambiguous. If you need arbitrary fitness values for arbitrary combinations of genotypes, you can specify them using epistatic effects as in section 3.7 and order effects as in section 3.6.
Why do we need to specify as many “s” and “sh” as there are rows (or a single one, that gets expanded to those many) when the “s” and “sh” are properties of the child node, not of the edges? Because, for ease, we use a data.frame.
We fix the error in our specification. Notice that the “sh” is not set to \(-1\) in these examples. If you want strict compliance with the poset restrictions, you should set \(sh = -1\) or, better yet, \(sh = -\infty\) (see section 3.2.2), but having an \(sh > -1\) will lead to fitnesses that are \(> 0\) and, thus, is a way of modeling small deviations from the poset (see discussion in Diaz-Uriarte 2015).
Note that for those nodes that depend only on “Root” the type of dependency is irrelevant.
c1 <- data.frame(parent = c(rep("Root", 4), "a", "b", "d", "e", "c"),
child = c("a", "b", "d", "e", "c", "c", rep("g", 3)),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, rep(0.2, 3)),
sh = c(rep(0, 4), c(-.9, -.9), rep(-.95, 3)),
typeDep = "MN")
cbn2 <- allFitnessEffects(c1)We could get graphical representations but the figures would be the same as in the example in section 3.4.4, since the structure has not changed, only the numeric values.
What is the fitness of all possible genotypes? Here, order of events per se does not matter, beyond that considered in the poset. In other words, the fitness of genotype “a, b, c” is the same no matter how we got to “a, b, c”. What matters is whether or not the genes on which each of “a”, “b”, and “c” depend are present or not (I only show the first 10 genotypes)
gcbn2 <- evalAllGenotypes(cbn2, order = FALSE)
gcbn2[1:10, ]
## Genotype Fitness
## 1 a 1.0100
## 2 b 1.0200
## 3 c 0.1000
## 4 d 1.0300
## 5 e 1.0400
## 6 g 0.0500
## 7 a, b 1.0302
## 8 a, c 0.1010
## 9 a, d 1.0403
## 10 a, e 1.0504Of course, if we were to look at genotypes but taking into account order of occurrence of mutations, we would see no differences
gcbn2o <- evalAllGenotypes(cbn2, order = TRUE, max = 1956)
gcbn2o[1:10, ]
## Genotype Fitness
## 1 a 1.0100
## 2 b 1.0200
## 3 c 0.1000
## 4 d 1.0300
## 5 e 1.0400
## 6 g 0.0500
## 7 a > b 1.0302
## 8 a > c 0.1010
## 9 a > d 1.0403
## 10 a > e 1.0504(The \(max = 1956\) is there so that we show all the genotypes, even if they are more than 256, the default.)
You can check the output and verify things are as they should. For instance:
all.equal(
gcbn2[c(1:21, 22, 28, 41, 44, 56, 63 ) , "Fitness"],
c(1.01, 1.02, 0.1, 1.03, 1.04, 0.05,
1.01 * c(1.02, 0.1, 1.03, 1.04, 0.05),
1.02 * c(0.10, 1.03, 1.04, 0.05),
0.1 * c(1.03, 1.04, 0.05),
1.03 * c(1.04, 0.05),
1.04 * 0.05,
1.01 * 1.02 * 1.1,
1.01 * 0.1 * 0.05,
1.03 * 1.04 * 0.05,
1.01 * 1.02 * 1.1 * 0.05,
1.03 * 1.04 * 1.2 * 0.1, ## notice this
1.01 * 1.02 * 1.03 * 1.04 * 1.1 * 1.2
))
## [1] TRUEA particular one that is important to understand is genotype with mutated genes “c, d, e, g”:
gcbn2[56, ]
## Genotype Fitness
## 56 c, d, e, g 0.128544
all.equal(gcbn2[56, "Fitness"], 1.03 * 1.04 * 1.2 * 0.10)
## [1] TRUEwhere “g” is taken as if its dependencies are satisfied (as “c”, “d”, and “e” are present) even when the dependencies of “c” are not satisfied (and that is why the term for “c” is 0.9).
3.4.6 DAGs: A semimonotone or “OR” example
We will reuse the above example, changing the type of relationship:
s1 <- data.frame(parent = c(rep("Root", 4), "a", "b", "d", "e", "c"),
child = c("a", "b", "d", "e", "c", "c", rep("g", 3)),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, rep(0.2, 3)),
sh = c(rep(0, 4), c(-.9, -.9), rep(-.95, 3)),
typeDep = "SM")
smn1 <- allFitnessEffects(s1)It looks like this (where edges are shown in blue to denote the semimonotone relationship):
plot(smn1)
gsmn1 <- evalAllGenotypes(smn1, order = FALSE)Having just one parental dependency satisfied is now enough, in contrast to what happened before. For instance:
gcbn2[c(8, 12, 22), ]
## Genotype Fitness
## 8 a, c 0.10100
## 12 b, c 0.10200
## 22 a, b, c 1.13322
gsmn1[c(8, 12, 22), ]
## Genotype Fitness
## 8 a, c 1.11100
## 12 b, c 1.12200
## 22 a, b, c 1.13322
gcbn2[c(20:21, 28), ]
## Genotype Fitness
## 20 d, g 0.05150
## 21 e, g 0.05200
## 28 a, c, g 0.00505
gsmn1[c(20:21, 28), ]
## Genotype Fitness
## 20 d, g 1.2360
## 21 e, g 1.2480
## 28 a, c, g 1.33323.4.7 An “XMPN” or “XOR” example
Again, we reuse the example above, changing the type of relationship:
x1 <- data.frame(parent = c(rep("Root", 4), "a", "b", "d", "e", "c"),
child = c("a", "b", "d", "e", "c", "c", rep("g", 3)),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, rep(0.2, 3)),
sh = c(rep(0, 4), c(-.9, -.9), rep(-.95, 3)),
typeDep = "XMPN")
xor1 <- allFitnessEffects(x1)It looks like this (edges in red to denote the “XOR” relationship):
plot(xor1)
gxor1 <- evalAllGenotypes(xor1, order = FALSE)Whenever “c” is present with both “a” and “b”, the fitness component for “c” will be \((1 - 0.1)\). Similarly for “g” (if more than one of “d”, “e”, or “c” is present, it will show as \((1 - 0.05)\)). For example:
gxor1[c(22, 41), ]
## Genotype Fitness
## 22 a, b, c 0.10302
## 41 d, e, g 0.05356
c(1.01 * 1.02 * 0.1, 1.03 * 1.04 * 0.05)
## [1] 0.10302 0.05356However, having just both “a” and “b” is identical to the case with CBN and the monotone relationship (see sections 3.4.5 and 3.4.6). If you want the joint presence of “a” and “b” to result in different fitness than the product of the individual terms, without considering the presence of “c”, you can specify that using general epistatic effects (section 3.7).
We also see a very different pattern compared to CBN (section 3.4.5) here:
gxor1[28, ]
## Genotype Fitness
## 28 a, c, g 1.3332
1.01 * 1.1 * 1.2
## [1] 1.3332as exactly one of the dependencies for both “c” and “g” are satisfied.
But
gxor1[44, ]
## Genotype Fitness
## 44 a, b, c, g 0.123624
1.01 * 1.02 * 0.1 * 1.2
## [1] 0.123624is the result of a \(0.1\) for “c” (and a \(1.2\) for “g” that has exactly one of its dependencies satisfied).
3.4.8 Posets: the three types of relationships
p3 <- data.frame(
parent = c(rep("Root", 4), "a", "b", "d", "e", "c", "f"),
child = c("a", "b", "d", "e", "c", "c", "f", "f", "g", "g"),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, 0.2, 0.2, 0.3, 0.3),
sh = c(rep(0, 4), c(-.9, -.9), c(-.95, -.95), c(-.99, -.99)),
typeDep = c(rep("--", 4),
"XMPN", "XMPN", "MN", "MN", "SM", "SM"))
fp3 <- allFitnessEffects(p3)This is how it looks like:
plot(fp3)
We can also use “igraph”:
plot(fp3, "igraph", layout.reingold.tilford)
gfp3 <- evalAllGenotypes(fp3, order = FALSE)Let’s look at a few:
gfp3[c(9, 24, 29, 59, 60, 66, 119, 120, 126, 127), ]
## Genotype Fitness
## 9 a, c 1.1110000
## 24 d, f 0.0515000
## 29 a, b, c 0.1030200
## 59 c, f, g 0.0065000
## 60 d, e, f 1.2854400
## 66 a, b, c, f 0.0051510
## 119 c, d, e, f, g 0.1671072
## 120 a, b, c, d, e, f 0.1324260
## 126 b, c, d, e, f, g 1.8749428
## 127 a, b, c, d, e, f, g 0.1721538
c(1.01 * 1.1, 1.03 * .05, 1.01 * 1.02 * 0.1, 0.1 * 0.05 * 1.3,
1.03 * 1.04 * 1.2, 1.01 * 1.02 * 0.1 * 0.05,
0.1 * 1.03 * 1.04 * 1.2 * 1.3,
1.01 * 1.02 * 0.1 * 1.03 * 1.04 * 1.2,
1.02 * 1.1 * 1.03 * 1.04 * 1.2 * 1.3,
1.01 * 1.02 * 1.03 * 1.04 * 0.1 * 1.2 * 1.3)
## [1] 1.1110000 0.0515000 0.1030200 0.0065000 1.2854400 0.0051510
## [7] 0.1671072 0.1324260 1.8749428 0.1721538As before, looking at the order of mutations makes no difference (look at the test directory to see a test that verifies this assertion).
3.5 Modules
As already mentioned, we can think of all the effects of fitness in terms not of individual genes but, rather, modules. This idea is discussed in, for example, Raphael and Vandin (2015), Gerstung et al. (2011): the restrictions encoded in, say, the DAGs can be considered to apply not to genes, but to modules, where each module is a set of genes (and the intersection between modules is the empty set). Modules, then, play the role of a “union operation” over sets of genes. Of course, if we can use modules for the restrictions in the DAGs we should also be able to use them for epistasis and order effects, as we will see later (e.g., 3.6.2).
3.5.1 What does a module provide
Modules can provide very compact ways of specifying relationships when you want to, well, model the existence of modules. For simplicity suppose there is a module, “A”, made of genes “a1” and “a2”, and a module “B”, made of a single gene “b1”. Module “B” can mutate if module “A” is mutated, but mutating both “a1” and “a2” provides no additional fitness advantage compared to mutating only a single one of them. We can specify this as:
s <- 0.2
sboth <- (1/(1 + s)) - 1
m0 <- allFitnessEffects(data.frame(
parent = c("Root", "Root", "a1", "a2"),
child = c("a1", "a2", "b", "b"),
s = s,
sh = -1,
typeDep = "OR"),
epistasis = c("a1:a2" = sboth))
evalAllGenotypes(m0, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 1.20
## 3 a2 1.20
## 4 b 0.00
## 5 a1, a2 1.20
## 6 a1, b 1.44
## 7 a2, b 1.44
## 8 a1, a2, b 1.44Note that we need to add an epistasis term, with value “sboth” to capture the idea of “mutating both”a1" and “a2” provides no additional fitness advantage compared to mutating only a single one of them"; see details in section 3.7.
Now, specify it using modules:
s <- 0.2
m1 <- allFitnessEffects(data.frame(
parent = c("Root", "A"),
child = c("A", "B"),
s = s,
sh = -1,
typeDep = "OR"),
geneToModule = c("Root" = "Root",
"A" = "a1, a2",
"B" = "b1"))
evalAllGenotypes(m1, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 1.20
## 3 a2 1.20
## 4 b1 0.00
## 5 a1, a2 1.20
## 6 a1, b1 1.44
## 7 a2, b1 1.44
## 8 a1, a2, b1 1.44This captures the ideas directly. The typing savings here are small, but they can be large with modules with many genes.
3.5.2 Specifying modules
How do you specify modules? The general procedure is simple: you pass a vector that makes explicit the mapping from modules to sets of genes. We just saw an example. There are several additional examples such as 3.5.3, 3.6.2, 3.7.4.
It is important to note that, once you specify modules, we expect all of the relationships (except those that involve the non interacting genes) to be specified as modules. Thus, all elements of the epistasis, posets (the DAGs) and order effects components should be specified in terms of modules. But you can, of course, specify a module as containing a single gene (and a single gene with the same name as the module).
What about the “Root” node? If you use a “restriction table”, that
restriction table (that DAG) must have a node named “Root” and in the
mapping of genes to module there must be a first entry that has a
module and gene named “Root”, as we saw above with
geneToModule = c("Root" = "Root", ....
We force you to do this to be explicit about
the “Root” node. This is not needed (thought it does not hurt) with
other fitness specifications. For instance, if we have a model with two
modules, one of them with two genes (see details in section
3.8) we do not need to pass a “Root” as in
fnme <- allFitnessEffects(epistasis = c("A" = 0.1,
"B" = 0.2),
geneToModule = c("A" = "a1, a2",
"B" = "b1"))
evalAllGenotypes(fnme, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 1.10
## 3 a2 1.10
## 4 b1 1.20
## 5 a1, a2 1.10
## 6 a1, b1 1.32
## 7 a2, b1 1.32
## 8 a1, a2, b1 1.32but it is also OK to have a “Root” in the geneToModule:
fnme2 <- allFitnessEffects(epistasis = c("A" = 0.1,
"B" = 0.2),
geneToModule = c(
"Root" = "Root",
"A" = "a1, a2",
"B" = "b1"))
evalAllGenotypes(fnme, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 1.10
## 3 a2 1.10
## 4 b1 1.20
## 5 a1, a2 1.10
## 6 a1, b1 1.32
## 7 a2, b1 1.32
## 8 a1, a2, b1 1.323.5.3 Modules and posets again: the three types of relationships and modules
We use the same specification of poset, but add modules. To keep it manageable, we only add a few genes for some modules, and have some modules with a single gene. Beware that the number of genotypes is starting to grow quite fast, though. We capitalize to differentiate modules (capital letters) from genes (lowercase with a number), but this is not needed.
p4 <- data.frame(
parent = c(rep("Root", 4), "A", "B", "D", "E", "C", "F"),
child = c("A", "B", "D", "E", "C", "C", "F", "F", "G", "G"),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, 0.2, 0.2, 0.3, 0.3),
sh = c(rep(0, 4), c(-.9, -.9), c(-.95, -.95), c(-.99, -.99)),
typeDep = c(rep("--", 4),
"XMPN", "XMPN", "MN", "MN", "SM", "SM"))
fp4m <- allFitnessEffects(
p4,
geneToModule = c("Root" = "Root", "A" = "a1",
"B" = "b1, b2", "C" = "c1",
"D" = "d1, d2", "E" = "e1",
"F" = "f1, f2", "G" = "g1"))By default, plotting shows the modules:
plot(fp4m)
but we can show the gene names instead of the module names:
plot(fp4m, expandModules = TRUE)
or
plot(fp4m, "igraph", layout = layout.reingold.tilford,
expandModules = TRUE)
We obtain the fitness of all genotypes in the usual way:
gfp4 <- evalAllGenotypes(fp4m, order = FALSE, max = 1024)Let’s look at a few of those:
gfp4[c(12, 20, 21, 40, 41, 46, 50, 55, 64, 92,
155, 157, 163, 372, 632, 828), ]
## Genotype Fitness
## 12 a1, b2 1.030200
## 20 b1, b2 1.020000
## 21 b1, c1 1.122000
## 40 c1, g1 0.130000
## 41 d1, d2 1.030000
## 46 d2, e1 1.071200
## 50 e1, f1 0.052000
## 55 f2, g1 0.065000
## 64 a1, b2, c1 0.103020
## 92 b1, b2, c1 1.122000
## 155 c1, f2, g1 0.006500
## 157 d1, d2, f1 0.051500
## 163 d1, f1, f2 0.051500
## 372 d1, d2, e1, f2 1.285440
## 632 d1, d2, e1, f1, f2 1.285440
## 828 b2, c1, d1, e1, f2, g1 1.874943
c(1.01 * 1.02, 1.02, 1.02 * 1.1, 0.1 * 1.3, 1.03,
1.03 * 1.04, 1.04 * 0.05, 0.05 * 1.3,
1.01 * 1.02 * 0.1, 1.02 * 1.1, 0.1 * 0.05 * 1.3,
1.03 * 0.05, 1.03 * 0.05, 1.03 * 1.04 * 1.2, 1.03 * 1.04 * 1.2,
1.02 * 1.1 * 1.03 * 1.04 * 1.2 * 1.3)
## [1] 1.030200 1.020000 1.122000 0.130000 1.030000 1.071200 0.052000
## [8] 0.065000 0.103020 1.122000 0.006500 0.051500 0.051500 1.285440
## [15] 1.285440 1.8749433.6 Order effects
As explained in the introduction (section 1), by order effects we mean a phenomenon such as the one shown empirically by Ortmann et al. (2015): the fitness of a double mutant “A”, “B” is different depending on whether “A” was acquired before “B” or “B” before “A”. This, of course, can be generalized to more than two genes.
Note that order effects are different from the restrictions in the order of accumulation of mutations discussed in section 3.4. With restrictions in the order of accumulation of mutations we might say that acquiring “B” depends or is facilitated by having “A” mutated (and, unless we allowed for multiple mutations, having “A” mutated means having “A” mutated before “B”). However, once you have the genotype “A, B”, its fitness does not depend on the order in which “A” and “B” appeared.
3.6.1 Order effects: three-gene orders
Consider this case, where three specific three-gene orders and two two-gene orders (one of them a subset of one of the three) lead to different fitness compared to the wild-type. We add also modules, to show its usage (but just limit ourselves to using one gene per module here).
Order effects are specified using a \(x > y\), which means that that order effect is satisfied when module \(x\) is mutated before module \(y\).
o3 <- allFitnessEffects(orderEffects = c(
"F > D > M" = -0.3,
"D > F > M" = 0.4,
"D > M > F" = 0.2,
"D > M" = 0.1,
"M > D" = 0.5),
geneToModule =
c("M" = "m",
"F" = "f",
"D" = "d") )
(ag <- evalAllGenotypes(o3, addwt = TRUE, order = TRUE))
## Genotype Fitness
## 1 WT 1.00
## 2 d 1.00
## 3 f 1.00
## 4 m 1.00
## 5 d > f 1.00
## 6 d > m 1.10
## 7 f > d 1.00
## 8 f > m 1.00
## 9 m > d 1.50
## 10 m > f 1.00
## 11 d > f > m 1.54
## 12 d > m > f 1.32
## 13 f > d > m 0.77
## 14 f > m > d 1.50
## 15 m > d > f 1.50
## 16 m > f > d 1.50(The meaning of the notation in the output table is as follows: “WT” denotes the wild-type, or non-mutated clone. The notation \(x > y\) means that a mutation in “x” happened before a mutation in “y”. A genotype \(x > y\ \_\ z\) means that a mutation in “x” happened before a mutation in “y”; there is also a mutation in “z”, but that is a gene for which order does not matter).
The values for the first nine genotypes come directly from the fitness specifications. The 10th genotype matches \(D > F > M\) (\(= (1 + 0.4)\)) but also \(D > M\) (\((1 + 0.1)\)). The 11th matches \(D > M > F\) and \(D > M\). The 12th matches \(F > D > M\) but also \(D > M\). Etc.
3.6.2 Order effects and modules with multiple genes
Consider the following case:
ofe1 <- allFitnessEffects(
orderEffects = c("F > D" = -0.3, "D > F" = 0.4),
geneToModule =
c("F" = "f1, f2",
"D" = "d1, d2") )
ag <- evalAllGenotypes(ofe1, order = TRUE)There are four genes, \(d1, d2, f1, f2\), where each \(d\) belongs to module \(D\) and each \(f\) belongs to module \(F\).
What to expect for cases such as \(d1 > f1\) or \(f1 > d1\) is clear, as shown in
ag[5:16,]
## Genotype Fitness
## 5 d1 > d2 1.0
## 6 d1 > f1 1.4
## 7 d1 > f2 1.4
## 8 d2 > d1 1.0
## 9 d2 > f1 1.4
## 10 d2 > f2 1.4
## 11 f1 > d1 0.7
## 12 f1 > d2 0.7
## 13 f1 > f2 1.0
## 14 f2 > d1 0.7
## 15 f2 > d2 0.7
## 16 f2 > f1 1.0Likewise, cases such as \(d1 > d2 > f1\) or \(f2 > f1 > d1\) are clear, because in terms of modules they map to $ D > F$ or \(F > D\): the observed order of mutation \(d1 > d2 > f1\) means that module \(D\) was mutated first and module \(F\) was mutated second. Similar for \(d1 > f1 > f2\) or \(f1 > d1 > d2\): those map to \(D > F\) and \(F > D\). We can see the fitness of those four case in:
ag[c(17, 39, 19, 29), ]
## Genotype Fitness
## 17 d1 > d2 > f1 1.4
## 39 f2 > f1 > d1 0.7
## 19 d1 > f1 > d2 1.4
## 29 f1 > d1 > d2 0.7and they correspond to the values of those order effects, where \(F > D = (1 - 0.3)\) and \(D > F = (1 + 0.4)\):
ag[c(17, 39, 19, 29), "Fitness"] == c(1.4, 0.7, 1.4, 0.7)
## [1] TRUE TRUE TRUE TRUEWhat if we match several patterns? For example, \(d1 > f1 > d2 > f2\) and \(d1 > f1 > f2 > d2\)? The first maps to \(D > F > D > F\) and the second to \(D > F > D\). But since we are concerned with which one happened first and which happened second we should expect those two to correspond to the same fitness, that of pattern \(D > F\), as is the case:
ag[c(43, 44),]
## Genotype Fitness
## 43 d1 > f1 > d2 > f2 1.4
## 44 d1 > f1 > f2 > d2 1.4
ag[c(43, 44), "Fitness"] == c(1.4, 1.4)
## [1] TRUE TRUEMore generally, that applies to all the patterns that start with one of the “d” genes:
all(ag[41:52, "Fitness"] == 1.4)
## [1] TRUESimilar arguments apply to the opposite pattern, \(F > D\), which apply to all the possible gene mutation orders that start with one of the “f” genes. For example:
all(ag[53:64, "Fitness"] == 0.7)
## [1] TRUE3.6.3 Order and modules with 325 genotypes
We can of course have more than two genes per module. This just repeats
the above, with five genes (there are 325 genotypes, and that is why we
pass the “max” argument to evalAllGenotypes, to allow for
more than the default 256).
ofe2 <- allFitnessEffects(
orderEffects = c("F > D" = -0.3, "D > F" = 0.4),
geneToModule =
c("F" = "f1, f2, f3",
"D" = "d1, d2") )
ag2 <- evalAllGenotypes(ofe2, max = 325, order = TRUE)We can verify that any combination that starts with a “d” gene and then contains at least one “f” gene will have a fitness of \(1+0.4\). And any combination that starts with an “f” gene and contains at least one “d” genes will have a fitness of \(1 - 0.3\). All other genotypes have a fitness of 1:
all(ag2[grep("^d.*f.*", ag2[, 1]), "Fitness"] == 1.4)
## [1] TRUE
all(ag2[grep("^f.*d.*", ag2[, 1]), "Fitness"] == 0.7)
## [1] TRUE
oe <- c(grep("^f.*d.*", ag2[, 1]), grep("^d.*f.*", ag2[, 1]))
all(ag2[-oe, "Fitness"] == 1)
## [1] TRUE3.6.4 Order effects and genes without interactions
We will now look at both order effects and interactions. To make things more interesting, we name genes so that the ordered names do split nicely between those with and those without order effects (this, thus, also serves as a test of messy orders of names).
foi1 <- allFitnessEffects(
orderEffects = c("D>B" = -0.2, "B > D" = 0.3),
noIntGenes = c("A" = 0.05, "C" = -.2, "E" = .1))You can get a verbose view of what the gene names and modules are (and their automatically created numeric codes) by:
foi1[c("geneModule", "long.geneNoInt")]
## $geneModule
## Gene Module GeneNumID ModuleNumID
## 1 Root Root 0 0
## 2 B B 1 1
## 3 D D 2 2
##
## $long.geneNoInt
## Gene GeneNumID s
## A A 3 0.05
## C C 4 -0.20
## E E 5 0.10We can get the fitness of all genotypes (we set \(max = 325\) because that is the number of possible genotypes):
agoi1 <- evalAllGenotypes(foi1, max = 325, order = TRUE)
head(agoi1)
## Genotype Fitness
## 1 B 1.00
## 2 D 1.00
## 3 A 1.05
## 4 C 0.80
## 5 E 1.10
## 6 B > D 1.30Now:
rn <- 1:nrow(agoi1)
names(rn) <- agoi1[, 1]
agoi1[rn[LETTERS[1:5]], "Fitness"] == c(1.05, 1, 0.8, 1, 1.1)
## [1] TRUE TRUE TRUE TRUE TRUEAccording to the fitness effects we have specified, we also know that any genotype with only two mutations, one of which is either “A”, “C” “E” and the other is “B” or “D” will have the fitness corresponding to “A”, “C” or “E”, respectively:
agoi1[grep("^A > [BD]$", names(rn)), "Fitness"] == 1.05
## [1] TRUE TRUE
agoi1[grep("^C > [BD]$", names(rn)), "Fitness"] == 0.8
## [1] TRUE TRUE
agoi1[grep("^E > [BD]$", names(rn)), "Fitness"] == 1.1
## [1] TRUE TRUE
agoi1[grep("^[BD] > A$", names(rn)), "Fitness"] == 1.05
## [1] TRUE TRUE
agoi1[grep("^[BD] > C$", names(rn)), "Fitness"] == 0.8
## [1] TRUE TRUE
agoi1[grep("^[BD] > E$", names(rn)), "Fitness"] == 1.1
## [1] TRUE TRUEWe will not be playing many additional games with regular expressions, but let us check those that start with “D” and have all the other mutations, which occupy rows 230 to 253; fitness should be equal (within numerical error, because of floating point arithmetic) to the order effect of “D” before “B” times the other effects \((1 - 0.3) * 1.05 * 0.8 * 1.1 = 0.7392\)
all.equal(agoi1[230:253, "Fitness"] ,
rep((1 - 0.2) * 1.05 * 0.8 * 1.1, 24))
## [1] TRUEand that will also be the value of any genotype with the five mutations where “D” comes before “B” such as those in rows 260 to 265, 277, or 322 and 323, but it will be equal to \((1 + 0.3) * 1.05 * 0.8 * 1.1 = 1.2012\) in those where “B” comes before “D”. Analogous arguments apply to four, three, and two mutation genotypes.
3.7 Epistasis
3.7.1 Epistasis: two alternative specifications
We want the following mapping of genotypes to fitness:
| A | B | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | \(1 + s_b\) |
| M | wt | \(1 + s_a\) |
| M | M | \(1 + s_{ab}\) |
Suppose that the actual numerical values are \(s_a = 0.2, s_b = 0.3, s_{ab} = 0.7\).
We specify the above as follows:
sa <- 0.2
sb <- 0.3
sab <- 0.7
e2 <- allFitnessEffects(epistasis =
c("A: -B" = sa,
"-A:B" = sb,
"A : B" = sab))
evalAllGenotypes(e2, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.2
## 3 B 1.3
## 4 A, B 1.7That uses the “-” specification, so we explicitly exclude some patterns: with “A:-B” we say “A when there is no B”.
But we can also use a specification where we do not use the “-”. That requires a different numerical value of the interaction, because now, as we are rewriting the interaction term as genotype “A is mutant, B is mutant” the double mutant will incorporate the effects of “A mutant”, “B mutant” and “both A and B mutants”. We can define a new \(s_2\) that satisfies \((1 + s_{ab}) = (1 + s_a) (1 + s_b) (1 + s_2)\) so \((1 + s_2) = (1 + s_{ab})/((1 + s_a) (1 + s_b))\) and therefore specify as:
s2 <- ((1 + sab)/((1 + sa) * (1 + sb))) - 1
e3 <- allFitnessEffects(epistasis =
c("A" = sa,
"B" = sb,
"A : B" = s2))
evalAllGenotypes(e3, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.2
## 3 B 1.3
## 4 A, B 1.7Note that this is the way you would specify effects with FFPopsim (Zanini and Neher 2012). Whether this specification or the previous one with “-” is simpler will depend on the model. For synthetic mortality and viability, I think the one using “-” is simpler to map genotype tables to fitness effects. See also section 3.7.2 and 3.7.3 and the example in section 5.4.2.
Finally, note that we can also specify some of these effects by combining the graph and the epistasis, as shown in section 5.2.1 or 5.4.2.
3.7.2 Epistasis with three genes and two alternative specifications
Suppose we have
| A | B | C | Fitness |
|---|---|---|---|
| M | wt | wt | \(1 + s_a\) |
| wt | M | wt | \(1 + s_b\) |
| wt | wt | M | \(1 + s_c\) |
| M | M | wt | \(1 + s_{ab}\) |
| wt | M | M | \(1 + s_{bc}\) |
| M | wt | M | \((1 + s_a) (1 + s_c)\) |
| M | M | M | \(1 + s_{abc}\) |
where missing rows have a fitness of 1 (they have been deleted for conciseness). Note that the mutant for exactly A and C has a fitness that is the product of the individual terms (so there is no epistasis in that case).
sa <- 0.1
sb <- 0.15
sc <- 0.2
sab <- 0.3
sbc <- -0.25
sabc <- 0.4
sac <- (1 + sa) * (1 + sc) - 1
E3A <- allFitnessEffects(epistasis =
c("A:-B:-C" = sa,
"-A:B:-C" = sb,
"-A:-B:C" = sc,
"A:B:-C" = sab,
"-A:B:C" = sbc,
"A:-B:C" = sac,
"A : B : C" = sabc)
)
evalAllGenotypes(E3A, order = FALSE, addwt = FALSE)
## Genotype Fitness
## 1 A 1.10
## 2 B 1.15
## 3 C 1.20
## 4 A, B 1.30
## 5 A, C 1.32
## 6 B, C 0.75
## 7 A, B, C 1.40We needed to pass the \(s_{ac}\) coefficient explicitly, even if it that term was just the product. We can try to avoid using the “-”, however (but we will need to do other calculations). For simplicity, I use capital “S” in what follows where the letters differ from the previous specification:
sa <- 0.1
sb <- 0.15
sc <- 0.2
sab <- 0.3
Sab <- ( (1 + sab)/((1 + sa) * (1 + sb))) - 1
Sbc <- ( (1 + sbc)/((1 + sb) * (1 + sc))) - 1
Sabc <- ( (1 + sabc)/
( (1 + sa) * (1 + sb) * (1 + sc) *
(1 + Sab) * (1 + Sbc) ) ) - 1
E3B <- allFitnessEffects(epistasis =
c("A" = sa,
"B" = sb,
"C" = sc,
"A:B" = Sab,
"B:C" = Sbc,
## "A:C" = sac, ## not needed now
"A : B : C" = Sabc)
)
evalAllGenotypes(E3B, order = FALSE, addwt = FALSE)
## Genotype Fitness
## 1 A 1.10
## 2 B 1.15
## 3 C 1.20
## 4 A, B 1.30
## 5 A, C 1.32
## 6 B, C 0.75
## 7 A, B, C 1.40The above two are, of course, identical:
all(evalAllGenotypes(E3A, order = FALSE, addwt = FALSE) ==
evalAllGenotypes(E3B, order = FALSE, addwt = FALSE))
## [1] TRUEWe avoid specifying the “A:C”, as it just follows from the individual “A” and “C” terms, but given a specified genotype table, we need to do a little bit of addition and multiplication to get the coefficients.
3.7.3 Why can we specify some effects with a “-”?
Let’s suppose we want to specify the synthetic viability example seen before:
| A | B | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | 0 |
| M | wt | 0 |
| M | M | (1 + s) |
where “wt” denotes wild type and “M” denotes mutant.
If you want to directly map the above table to the fitness table for the program, to specify the genotype “A is wt, B is a mutant” you can specify it as “-A,B”, not just as “B”. Why? Because just the presence of a “B” is also compatible with genotype “A is mutant and B is mutant”. If you use “-” you are explicitly saying what should not be there so that “-A,B” is NOT compatible with “A, B”. Otherwise, you need to carefully add coefficients. Depending on what you are trying to model, different specifications might be simpler. See the examples in section 3.7.1 and 3.7.2. You have both options.
3.7.4 Epistasis: modules
There is nothing conceptually new, but we will show an example here:
sa <- 0.2
sb <- 0.3
sab <- 0.7
em <- allFitnessEffects(epistasis =
c("A: -B" = sa,
"-A:B" = sb,
"A : B" = sab),
geneToModule = c("A" = "a1, a2",
"B" = "b1, b2"))
evalAllGenotypes(em, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 a1 1.2
## 3 a2 1.2
## 4 b1 1.3
## 5 b2 1.3
## 6 a1, a2 1.2
## 7 a1, b1 1.7
## 8 a1, b2 1.7
## 9 a2, b1 1.7
## 10 a2, b2 1.7
## 11 b1, b2 1.3
## 12 a1, a2, b1 1.7
## 13 a1, a2, b2 1.7
## 14 a1, b1, b2 1.7
## 15 a2, b1, b2 1.7
## 16 a1, a2, b1, b2 1.7Of course, we can do the same thing without using the “-”, as in section 3.7.1:
s2 <- ((1 + sab)/((1 + sa) * (1 + sb))) - 1
em2 <- allFitnessEffects(epistasis =
c("A" = sa,
"B" = sb,
"A : B" = s2),
geneToModule = c("A" = "a1, a2",
"B" = "b1, b2")
)
evalAllGenotypes(em2, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 a1 1.2
## 3 a2 1.2
## 4 b1 1.3
## 5 b2 1.3
## 6 a1, a2 1.2
## 7 a1, b1 1.7
## 8 a1, b2 1.7
## 9 a2, b1 1.7
## 10 a2, b2 1.7
## 11 b1, b2 1.3
## 12 a1, a2, b1 1.7
## 13 a1, a2, b2 1.7
## 14 a1, b1, b2 1.7
## 15 a2, b1, b2 1.7
## 16 a1, a2, b1, b2 1.73.8 I do not want epistasis, but I want modules!
Sometimes you might want something like having several modules, say “A” and “B”, each with a number of genes, but with “A” and “B” showing no interaction.
It is a terminological issue whether we should allow noIntGenes
(no interaction genes), as explained in section 3.3 to actually be
modules. The reasoning for not allowing them is that the situation
depicted above (several genes in module A, for example) actually is one of
interaction: the members of “A” are combined using an “OR” operator
(i.e., the fitness consequences of having one or more genes of A mutated
are the same), not just simply multiplying their fitness; similarly for
“B”. This is why no interaction genes also mean no modules allowed.
So how do you get what you want in this case? Enter the names of
the modules in the epistasis component but have no term for “:”
(the colon). Let’s see an example:
fnme <- allFitnessEffects(epistasis = c("A" = 0.1,
"B" = 0.2),
geneToModule = c("A" = "a1, a2",
"B" = "b1, b2, b3"))
evalAllGenotypes(fnme, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 1.10
## 3 a2 1.10
## 4 b1 1.20
## 5 b2 1.20
## 6 b3 1.20
## 7 a1, a2 1.10
## 8 a1, b1 1.32
## 9 a1, b2 1.32
## 10 a1, b3 1.32
## 11 a2, b1 1.32
## 12 a2, b2 1.32
## 13 a2, b3 1.32
## 14 b1, b2 1.20
## 15 b1, b3 1.20
## 16 b2, b3 1.20
## 17 a1, a2, b1 1.32
## 18 a1, a2, b2 1.32
## 19 a1, a2, b3 1.32
## 20 a1, b1, b2 1.32
## 21 a1, b1, b3 1.32
## 22 a1, b2, b3 1.32
## 23 a2, b1, b2 1.32
## 24 a2, b1, b3 1.32
## 25 a2, b2, b3 1.32
## 26 b1, b2, b3 1.20
## 27 a1, a2, b1, b2 1.32
## 28 a1, a2, b1, b3 1.32
## 29 a1, a2, b2, b3 1.32
## 30 a1, b1, b2, b3 1.32
## 31 a2, b1, b2, b3 1.32
## 32 a1, a2, b1, b2, b3 1.32In previous versions these was possible using the longer, still accepted
way of specifying a : with a value of 0, but this is no longer
needed:
fnme <- allFitnessEffects(epistasis = c("A" = 0.1,
"B" = 0.2,
"A : B" = 0.0),
geneToModule = c("A" = "a1, a2",
"B" = "b1, b2, b3"))
evalAllGenotypes(fnme, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 1.10
## 3 a2 1.10
## 4 b1 1.20
## 5 b2 1.20
## 6 b3 1.20
## 7 a1, a2 1.10
## 8 a1, b1 1.32
## 9 a1, b2 1.32
## 10 a1, b3 1.32
## 11 a2, b1 1.32
## 12 a2, b2 1.32
## 13 a2, b3 1.32
## 14 b1, b2 1.20
## 15 b1, b3 1.20
## 16 b2, b3 1.20
## 17 a1, a2, b1 1.32
## 18 a1, a2, b2 1.32
## 19 a1, a2, b3 1.32
## 20 a1, b1, b2 1.32
## 21 a1, b1, b3 1.32
## 22 a1, b2, b3 1.32
## 23 a2, b1, b2 1.32
## 24 a2, b1, b3 1.32
## 25 a2, b2, b3 1.32
## 26 b1, b2, b3 1.20
## 27 a1, a2, b1, b2 1.32
## 28 a1, a2, b1, b3 1.32
## 29 a1, a2, b2, b3 1.32
## 30 a1, b1, b2, b3 1.32
## 31 a2, b1, b2, b3 1.32
## 32 a1, a2, b1, b2, b3 1.32This can, of course, be extended to more modules.
3.9 Synthetic viability
Synthetic viability and synthetic lethality (e.g., Ashworth, Lord, and Reis-Filho 2011; Hartman, Garvik, and Hartwell 2001) are just special cases of epistasis (section 3.7) but we deal with them here separately.
3.9.1 A simple synthetic viability example
A simple and extreme example of synthetic viability is shown in the following table, where the joint mutant has fitness larger than the wild type, but each single mutant is lethal.
| A | B | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | 0 |
| M | wt | 0 |
| M | M | (1 + s) |
where “wt” denotes wild type and “M” denotes mutant.
We can specify this (setting \(s = 0.2\)) as (I play around with spaces, to show there is a certain flexibility with them):
s <- 0.2
sv <- allFitnessEffects(epistasis = c("-A : B" = -1,
"A : -B" = -1,
"A:B" = s))Now, let’s look at all the genotypes (we use “addwt” to also get the wt, which by decree has fitness of 1), and disregard order:
(asv <- evalAllGenotypes(sv, order = FALSE, addwt = TRUE))
## Genotype Fitness
## 1 WT 1.0
## 2 A 0.0
## 3 B 0.0
## 4 A, B 1.2Asking the program to consider the order of mutations of course makes no difference:
evalAllGenotypes(sv, order = TRUE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 0.0
## 3 B 0.0
## 4 A > B 1.2
## 5 B > A 1.2Another example of synthetic viability is shown in section 5.2.2.
Of course, if multiple simultaneous mutations are not possible in the simulations, it is not possible to go from the wildtype to the double mutant in this model where the single mutants are not viable.
3.9.2 Synthetic viability, non-zero fitness, and modules
This is a slightly more elaborate case, where there is one module and the single mutants have different fitness between themselves, which is non-zero. Without the modules, this is the same as in Misra, Szczurek, and Vingron (2014), Figure 1b, which we go over in section 5.2.
| A | B | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | \(1 + s_b\) |
| M | wt | \(1 + s_a\) |
| M | M | \(1 + s_{ab}\) |
where \(s_a, s_b < 0\) but \(s_{ab} > 0\).
sa <- -0.1
sb <- -0.2
sab <- 0.25
sv2 <- allFitnessEffects(epistasis = c("-A : B" = sb,
"A : -B" = sa,
"A:B" = sab),
geneToModule = c(
"A" = "a1, a2",
"B" = "b"))
evalAllGenotypes(sv2, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 0.90
## 3 a2 0.90
## 4 b 0.80
## 5 a1, a2 0.90
## 6 a1, b 1.25
## 7 a2, b 1.25
## 8 a1, a2, b 1.25And if we look at order, of course it makes no difference:
evalAllGenotypes(sv2, order = TRUE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 a1 0.90
## 3 a2 0.90
## 4 b 0.80
## 5 a1 > a2 0.90
## 6 a1 > b 1.25
## 7 a2 > a1 0.90
## 8 a2 > b 1.25
## 9 b > a1 1.25
## 10 b > a2 1.25
## 11 a1 > a2 > b 1.25
## 12 a1 > b > a2 1.25
## 13 a2 > a1 > b 1.25
## 14 a2 > b > a1 1.25
## 15 b > a1 > a2 1.25
## 16 b > a2 > a1 1.253.10 Synthetic mortality or synthetic lethality
In contrast to section 3.9, here the joint mutant has decreased viability:
| A | B | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | \(1 + s_b\) |
| M | wt | \(1 + s_a\) |
| M | M | \(1 + s_{ab}\) |
where \(s_a, s_b > 0\) but \(s_{ab} < 0\).
sa <- 0.1
sb <- 0.2
sab <- -0.8
sm1 <- allFitnessEffects(epistasis = c("-A : B" = sb,
"A : -B" = sa,
"A:B" = sab))
evalAllGenotypes(sm1, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.1
## 3 B 1.2
## 4 A, B 0.2And if we look at order, of course it makes no difference:
evalAllGenotypes(sm1, order = TRUE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.1
## 3 B 1.2
## 4 A > B 0.2
## 5 B > A 0.23.11 Possible issues with Bozic model
3.11.1 Synthetic viability using Bozic model
If we were to use the above specification with Bozic’s models, we might not get what we think we should get:
evalAllGenotypes(sv, order = FALSE, addwt = TRUE, model = "Bozic")
## Genotype Death_rate
## 1 WT 1.0
## 2 A 2.0
## 3 B 2.0
## 4 A, B 0.8What gives here? The simulation code would alert you of this (see section 3.11.2) in this particular case because there are “-1”, which might indicate that this is not what you want. The problem is that you probably want the Death rate to be infinity (the birth rate was 0, so no clone viability, when we used birth rates —section 3.2.2).
Let us say so explicitly:
s <- 0.2
svB <- allFitnessEffects(epistasis = c("-A : B" = -Inf,
"A : -B" = -Inf,
"A:B" = s))
evalAllGenotypes(svB, order = FALSE, addwt = TRUE, model = "Bozic")
## Genotype Death_rate
## 1 WT 1.0
## 2 A Inf
## 3 B Inf
## 4 A, B 0.8Likewise, values of \(s\) larger than one have no effect beyond setting \(s = 1\) (a single term of \((1 - 1)\) will drive the product to 0, and as we cannot allow negative death rates negative values are set to 0):
s <- 1
svB1 <- allFitnessEffects(epistasis = c("-A : B" = -Inf,
"A : -B" = -Inf,
"A:B" = s))
evalAllGenotypes(svB1, order = FALSE, addwt = TRUE, model = "Bozic")
## Genotype Death_rate
## 1 WT 1
## 2 A Inf
## 3 B Inf
## 4 A, B 0
s <- 3
svB3 <- allFitnessEffects(epistasis = c("-A : B" = -Inf,
"A : -B" = -Inf,
"A:B" = s))
evalAllGenotypes(svB3, order = FALSE, addwt = TRUE, model = "Bozic")
## Genotype Death_rate
## 1 WT 1
## 2 A Inf
## 3 B Inf
## 4 A, B 0Of course, death rates of 0.0 are likely to lead to trouble down the road, when we actually conduct simulations (see section 3.11.2).
3.11.2 Numerical issues with death rates of 0 in Bozic model
As we mentioned above (section 3.11.1) death rates of 0 can lead to trouble when using Bozic’s model:
i1 <- allFitnessEffects(noIntGenes = c(1, 0.5))
evalAllGenotypes(i1, order = FALSE, addwt = TRUE,
model = "Bozic")
## Genotype Death_rate
## 1 WT 1.0
## 2 1 0.0
## 3 2 0.5
## 4 1, 2 0.0
i1_b <- oncoSimulIndiv(i1, model = "Bozic")
## Warning in nr_oncoSimul.internal(rFE = fp, birth = birth, death
## = death, : You are using a Bozic model with the new restriction
## specification, and you have at least one s of 1. If that gene is
## mutated, this will lead to a death rate of 0 and the simulations
## will abort when you get a non finite value.
##
## DEBUG2: Value of rnb = nan
##
## DEBUG2: Value of m = 1
##
## DEBUG2: Value of pe = 0
##
## DEBUG2: Value of pm = 1
##
## this is spP
##
## popSize = 1
## birth = 1
## death = 0
## W = 1
## R = 1
## mutation = 1e-10
## timeLastUpdate = 533.709
## absfitness = -inf
## numMutablePos = 0
##
## Unrecoverable exception: Algo 2: retval not finite. Aborting.Of course, there is no problem in using the above with other models:
evalAllGenotypes(i1, order = FALSE, addwt = TRUE,
model = "Exp")
## Genotype Fitness
## 1 WT 1.0
## 2 1 2.0
## 3 2 1.5
## 4 1, 2 3.0
i1_e <- oncoSimulIndiv(i1, model = "Exp")
summary(i1_e)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 3 200196519 200110030 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 803.7087 1204
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE NA 5e+05 5e+05
## OccurringDrivers
## 13.12 A longer example: Poset, epistasis, synthetic mortality and viability, order effects and genes without interactions, with some modules
We will now put together a complex example. We will use the poset from section 3.5.3 but will also add:
- Order effects that involve genes in the poset. In this case, if C happens before F, fitness decreases by \(1 - 0.1\). If it happens the other way around, there is no effect on fitness beyond their individual contributions.
- Order effects that involve two new modules, “H” and “I” (with genes “h1, h2” and “i1”, respectively), so that if H happens before I fitness increases by \(1 + 0.12\).
- Synthetic mortality between modules “I” (already present in the epistatic interaction) and “J” (with genes “j1” and “j2”): the joint presence of these modules leads to cell death (fitness of 0).
- Synthetic viability between modules “K” and “M” (with genes “k1”, “k2” and “m1”, respectively), so that their joint presence is viable but adds nothing to fitness (i.e., mutation of both has fitness \(1\)), whereas each single mutant has a fitness of \(1 - 0.5\).
- A set of 5 driver genes (\(n1, \ldots, n5\)) with fitness that comes from an exponential distribution with rate of 10.
As we are specifying many different things, we will start by writing each set of effects separately:
p4 <- data.frame(
parent = c(rep("Root", 4), "A", "B", "D", "E", "C", "F"),
child = c("A", "B", "D", "E", "C", "C", "F", "F", "G", "G"),
s = c(0.01, 0.02, 0.03, 0.04, 0.1, 0.1, 0.2, 0.2, 0.3, 0.3),
sh = c(rep(0, 4), c(-.9, -.9), c(-.95, -.95), c(-.99, -.99)),
typeDep = c(rep("--", 4),
"XMPN", "XMPN", "MN", "MN", "SM", "SM"))
oe <- c("C > F" = -0.1, "H > I" = 0.12)
sm <- c("I:J" = -1)
sv <- c("-K:M" = -.5, "K:-M" = -.5)
epist <- c(sm, sv)
modules <- c("Root" = "Root", "A" = "a1",
"B" = "b1, b2", "C" = "c1",
"D" = "d1, d2", "E" = "e1",
"F" = "f1, f2", "G" = "g1",
"H" = "h1, h2", "I" = "i1",
"J" = "j1, j2", "K" = "k1, k2", "M" = "m1")
set.seed(1) ## for reproducibility
noint <- rexp(5, 10)
names(noint) <- paste0("n", 1:5)
fea <- allFitnessEffects(rT = p4, epistasis = epist,
orderEffects = oe,
noIntGenes = noint,
geneToModule = modules)How does it look?
plot(fea)
or
plot(fea, "igraph")
We can, if we want, expand the modules using a “graphNEL” graph
plot(fea, expandModules = TRUE)
or an “igraph” one
plot(fea, "igraph", expandModules = TRUE)
We will not evaluate the fitness of all genotypes, since the number of all ordered genotypes is \(> 7*10^{22}\). We will look at some specific genotypes:
evalGenotype("k1 > i1 > h2", fea) ## 0.5
## [1] 0.5
evalGenotype("k1 > h1 > i1", fea) ## 0.5 * 1.12
## [1] 0.56
evalGenotype("k2 > m1 > h1 > i1", fea) ## 1.12
## [1] 1.12
evalGenotype("k2 > m1 > h1 > i1 > c1 > n3 > f2", fea)
## [1] 0.005113436
## 1.12 * 0.1 * (1 + noint[3]) * 0.05 * 0.9Finally, let’s generate some ordered genotypes randomly:
randomGenotype <- function(fe, ns = NULL) {
gn <- setdiff(c(fe$geneModule$Gene,
fe$long.geneNoInt$Gene), "Root")
if(is.null(ns)) ns <- sample(length(gn), 1)
return(paste(sample(gn, ns), collapse = " > "))
}
set.seed(2) ## for reproducibility
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: j2 > d2 > n4 > f1 > k2 > n1 > h2 > i1 > f2 > b1 > h1 > a1 > b2 > n3 > j1 > k1 > e1 > m1 > g1 > c1 > n2
## Individual s terms are : 0.0755182 0.118164 0.0145707 0.0139795 0.01 0.02 -0.9 0.03 0.04 0.2 0.3 -1 0.12
## Fitness: 0
## [1] 0
## Genotype: k2 > i1 > c1 > n1 > m1
## Individual s terms are : 0.0755182 -0.9
## Fitness: 0.107552
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: f2 > j1 > f1 > k1 > i1 > n4
## Individual s terms are : 0.0139795 -0.95 -1 -0.5
## Fitness: 0
## [1] 0
## Genotype: n2 > h1 > h2
## Individual s terms are : 0.118164
## Fitness: 1.11816
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: d2 > n1 > f2 > f1 > i1 > n5 > b1 > e1 > k2 > b2 > c1 > j1 > a1 > k1 > n3 > d1
## Individual s terms are : 0.0755182 0.0145707 0.0436069 0.01 0.02 -0.9 0.03 0.04 0.2 -1 -0.5
## Fitness: 0
## [1] 0
## Genotype: d2 > k2 > c1 > f2 > n4 > m1 > n3 > f1 > b1 > g1 > n5 > h1 > j2
## Individual s terms are : 0.0145707 0.0139795 0.0436069 0.02 0.1 0.03 -0.95 0.3 -0.1
## Fitness: 0.0725829
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: a1 > m1 > f1 > c1 > i1 > d1 > b1 > n4 > d2 > n1 > e1 > k2 > j2 > n2 > g1
## Individual s terms are : 0.0755182 0.118164 0.0139795 0.01 0.02 -0.9 0.03 0.04 0.2 0.3 -1
## Fitness: 0
## [1] 0
## Genotype: h2 > c1 > f1 > n2 > b2 > a1 > n1 > i1
## Individual s terms are : 0.0755182 0.118164 0.01 0.02 -0.9 -0.95 -0.1 0.12
## Fitness: 0.00624418
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: g1 > j2 > m1 > d2 > n1 > n4 > i1 > b2 > f1
## Individual s terms are : 0.0755182 0.0139795 0.02 0.03 -0.95 0.3 -1 -0.5
## Fitness: 0
## [1] 0
## Genotype: h2 > j1 > m1 > d2 > i1 > b2 > k2 > d1 > b1 > n3 > n1 > g1 > h1 > c1 > k1 > e1 > a1 > f1 > n5 > f2
## Individual s terms are : 0.0755182 0.0145707 0.0436069 0.01 0.02 -0.9 0.03 0.04 0.2 0.3 -1 -0.1 0.12
## Fitness: 0
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: i1 > k2 > d1 > d2 > n4 > f2 > j2 > c1 > a1 > j1 > n1 > n3 > h1 > m1 > h2 > b2 > n5 > k1 > e1 > n2 > b1 > g1
## Individual s terms are : 0.0755182 0.118164 0.0145707 0.0139795 0.0436069 0.01 0.02 -0.9 0.03 0.04 0.2 0.3 -1
## Fitness: 0
## [1] 0
## Genotype: n1 > m1 > n3 > i1 > j1 > n5 > k1
## Individual s terms are : 0.0755182 0.0145707 0.0436069 -1
## Fitness: 0
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: n2 > a1 > c1 > d2 > j1 > e1 > k1 > b2 > d1 > n3 > j2 > f2 > i1 > g1 > k2 > h2 > n4 > n5 > m1 > f1 > h1 > n1 > b1
## Individual s terms are : 0.0755182 0.118164 0.0145707 0.0139795 0.0436069 0.01 0.02 -0.9 0.03 0.04 0.2 0.3 -1 -0.1
## Fitness: 0
## [1] 0
## Genotype: d2 > n1 > g1 > f1 > f2 > c1 > b1 > d1 > k1 > a1 > b2 > i1 > n4 > h2 > n2
## Individual s terms are : 0.0755182 0.118164 0.0139795 0.01 0.02 -0.9 0.03 -0.95 0.3 -0.5
## Fitness: 0.00420528
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: d2 > a1 > h2
## Individual s terms are : 0.01 0.03
## Fitness: 1.0403
## [1] 1.0403
## Genotype: j1 > f1 > j2 > a1 > n4 > c1 > n3 > k1 > d1 > h1
## Individual s terms are : 0.0145707 0.0139795 0.01 0.1 0.03 -0.95 -0.5
## Fitness: 0.0294308
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: n2 > f2
## Individual s terms are : 0.118164 -0.95
## Fitness: 0.05590821
## [1] 0.05590821
## Genotype: n5 > f2 > f1 > h2 > n4 > c1 > n3 > b1
## Individual s terms are : 0.0145707 0.0139795 0.0436069 0.02 0.1 -0.95
## Fitness: 0.0602298
evalGenotype(randomGenotype(fea), fea, echo = TRUE, verbose = TRUE)
## Genotype: n5 > n1 > d1 > f1 > c1 > b1 > n2 > i1 > a1 > n3 > n4 > e1 > k2 > b2 > h1 > m1 > j2
## Individual s terms are : 0.0755182 0.118164 0.0145707 0.0139795 0.0436069 0.01 0.02 -0.9 0.03 0.04 0.2 -1
## Fitness: 0
## [1] 0
## Genotype: h1 > d1 > f2
## Individual s terms are : 0.03 -0.95
## Fitness: 0.0515
3.13 Homozygosity, heterozygosity, oncogenes, tumor suppressors
We are using what is conceptually a single linear chromosome. However, you can use it to model scenarios where the numbers of copies affected matter, by properly duplicating the genes.
Suppose we have a tumor suppressor gene, G, with two copies, one from Mom and one from Dad. We can have a table like:
| \(O_M\) | \(O_D\) | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | 1 |
| M | wt | 1 |
| M | M | \((1 + s)\) |
where \(s > 0\), meaning that you need two hits, one in each copy, to trigger the clonal expansion.
What about oncogenes? A simple model is that one single hit leads to clonal expansion and additional hits lead to no additional changes, as in this table for gene O, where again the M or D subscript denotes the copy from Mom or from Dad:
| \(O_M\) | \(O_D\) | Fitness |
|---|---|---|
| wt | wt | 1 |
| wt | M | \((1 + s)\) |
| M | wt | \((1 + s)\) |
| M | M | \((1 + s)\) |
If you have multiple copies you can proceed similarly. As you can see, these are nothing but special cases of synthetic mortality (3.10), synthetic viability (3.9) and epistasis (3.7).
3.14 Gene-specific mutation rates
You can specify gene-specific mutation rates. Instead of passing a scalar
value for mu, you pass a named vector. (This does not work with
the old v. 1 format, though; yet another reason to stop using that
format). This is a simple example (many more are available in the tests,
see file ./tests/testthat/test.per-gene-mutation-rates.R).
muvar2 <- c("U" = 1e-6, "z" = 5e-5, "e" = 5e-4, "m" = 5e-3,
"D" = 1e-4)
ni1 <- rep(0, 5)
names(ni1) <- names(muvar2) ## We use the same names, of course
fe1 <- allFitnessEffects(noIntGenes = ni1)
bb <- oncoSimulIndiv(fe1,
mu = muvar2, onlyCancer = FALSE,
initSize = 1e5,
finalTime = 25,
seed =NULL)3.15 Mutator genes
You can specify mutator/antimutator genes (e.g. Gerrish et al. 2007; Tomlinson, Novelli, and Bodmer 1996). These are genes that, when mutated, lead to an increase/decrease in the mutation rate all over the genome (similar to what happens with, say, mutations in mismatch-repair genes or microsatellite instability in cancer).
The specification is very similar to that for fitness effects, except we do not (at least for now) allow the use of DAGs nor of order effects (we have seen no reference in the literature to suggest any of these would be relevant). You can, however, specify epistasis and use modules. Note that the mutator genes must be a subset of the genes in the fitness effects; if you want to have mutator genes that have no direct fitness effects, give them a fitness effect of 0.
This first is a very simple example with simple fitness effects and modules for mutators. We will specify the fitness and mutator effects and evaluate the fitness and mutator effects:
fe2 <- allFitnessEffects(noIntGenes =
c(a1 = 0.1, a2 = 0.2,
b1 = 0.01, b2 = 0.3, b3 = 0.2,
c1 = 0.3, c2 = -0.2))
fm2 <- allMutatorEffects(epistasis = c("A" = 5,
"B" = 10,
"C" = 3),
geneToModule = c("A" = "a1, a2",
"B" = "b1, b2, b3",
"C" = "c1, c2"))
## Show the fitness effect of a specific genotype
evalGenotype("a1, c2", fe2, verbose = TRUE)
##
## Individual s terms are : 0.1 -0.2
## [1] 0.88
## Show the mutator effect of a specific genotype
evalGenotypeMut("a1, c2", fm2, verbose = TRUE)
##
## Individual mutator product terms are : 5 3
## [1] 15
## Fitness and mutator of a specific genotype
evalGenotypeFitAndMut("a1, c2", fe2, fm2, verbose = TRUE)
##
## Individual s terms are : 0.1 -0.2
##
## Individual mutator product terms are : 5 3
## [1] 0.88 15.00You can also use the evalAll functions. We do not show the output
here to avoid cluttering the vignette:
## Show only all the fitness effects
evalAllGenotypes(fe2, order = FALSE)
## Show only all mutator effects
evalAllGenotypesMut(fm2)
## Show all fitness and mutator
evalAllGenotypesFitAndMut(fe2, fm2, order = FALSE)Building upon the above, the next is an example where we have a bunch of no interaction genes that affect fitness, and a small set of genes that affect the mutation rate (but have no fitness effects).
set.seed(1) ## for reproducibility
## 17 genes, 7 with no direct fitness effects
ni <- c(rep(0, 7), runif(10, min = -0.01, max = 0.1))
names(ni) <- c("a1", "a2", "b1", "b2", "b3", "c1", "c2",
paste0("g", 1:10))
fe3 <- allFitnessEffects(noIntGenes = ni)
fm3 <- allMutatorEffects(epistasis = c("A" = 5,
"B" = 10,
"C" = 3,
"A:C" = 70),
geneToModule = c("A" = "a1, a2",
"B" = "b1, b2, b3",
"C" = "c1, c2"))Let us check what the effects are of a few genotypes:
## These only affect mutation, not fitness
evalGenotypeFitAndMut("a1, a2", fe3, fm3, verbose = TRUE)
##
## Individual s terms are : 0 0
##
## Individual mutator product terms are : 5
## [1] 1 5
evalGenotypeFitAndMut("a1, b3", fe3, fm3, verbose = TRUE)
##
## Individual s terms are : 0 0
##
## Individual mutator product terms are : 5 10
## [1] 1 50
## These only affect fitness: the mutator multiplier is 1
evalGenotypeFitAndMut("g1", fe3, fm3, verbose = TRUE)
##
## Individual s terms are : 0.019206
## [1] 1.019206 1.000000
evalGenotypeFitAndMut("g3, g9", fe3, fm3, verbose = TRUE)
##
## Individual s terms are : 0.0530139 0.0592025
## [1] 1.115355 1.000000
## These affect both
evalGenotypeFitAndMut("g3, g9, a2, b3", fe3, fm3, verbose = TRUE)
##
## Individual s terms are : 0 0 0.0530139 0.0592025
##
## Individual mutator product terms are : 5 10
## [1] 1.115355 50.000000Finally, we will do a simulation with those data
set.seed(1) ## so that it is easy to reproduce
mue1 <- oncoSimulIndiv(fe3, muEF = fm3,
mu = 1e-6,
initSize = 1e5,
model = "McFL",
detectionSize = 5e6,
finalTime = 500,
onlyCancer = FALSE)## We do not show this in the vignette to avoid cluttering it
## with output
mue1Of course, it is up to you to keep things reasonable: mutator effects are multiplicative, so if you specify, say, 20 genes (without modules), or 20 modules, each with a mutator effect of 50, the overall mutation rate can be increased by a factor of \(50^{20}\) and that is unlikely to be what you really want (see also section 15.7).
You can play with the following case (an extension of the example
above), where a clone with a mutator phenotype and some fitness
enhancing mutations starts giving rise to many other clones, some
with additional mutator effects, and thus leading to the number of
clones blowing up (as some also accumulate additional
fitness-enhancing mutations). Things start getting out of hand
shortly after time 250. The code below takes a few minutes to run
and is not executed here, but you can run it to get an idea of the
increase in the number of clones and their relationships (the usage
of plotClonePhylog is explained in section 8).
set.seed(1) ## for reproducibility
## 17 genes, 7 with no direct fitness effects
ni <- c(rep(0, 7), runif(10, min = -0.01, max = 0.1))
names(ni) <- c("a1", "a2", "b1", "b2", "b3", "c1", "c2",
paste0("g", 1:10))
## Next is for nicer figure labeling.
## Consider as drivers genes with s >0
gp <- which(ni > 0)
fe3 <- allFitnessEffects(noIntGenes = ni,
drvNames = names(ni)[gp])
set.seed(12)
mue1 <- oncoSimulIndiv(fe3, muEF = fm3,
mu = 1e-6,
initSize = 1e5,
model = "McFL",
detectionSize = 5e6,
finalTime = 270,
keepPhylog = TRUE,
onlyCancer = FALSE)
mue1
## If you decrease N even further it gets even more cluttered
op <- par(ask = TRUE)
plotClonePhylog(mue1, N = 10, timeEvents = TRUE)
plot(mue1, plotDrivers = TRUE, addtot = TRUE,
plotDiversity = TRUE)
## The stacked plot is slow; be patient
## Most clones have tiny population sizes, and their lines
## are piled on top of each other
plot(mue1, addtot = TRUE,
plotDiversity = TRUE, type = "stacked")
par(op)4 Plotting fitness landscapes
The evalAllGenotypes and related functions allow you to obtain
tables of the genotype to fitness mappings. It might be more convenient to
actually plot that, allowing us to quickly identify local minima and
maxima and get an idea of how the fitness landscape looks.
In plotFitnessLandscape I have blatantly and shamelessly copied most
of the looks of the plots of MAGELLAN (Brouillet et al. 2015) (see
also http://wwwabi.snv.jussieu.fr/public/Magellan/), a very nice
web-based tool for fitness landscape plotting and analysis (MAGELLAN
provides some other extra functionality and epistasis statistics not
provided here).
As an example, let us show the example of Weissman et al. we saw in 5.4:
d1 <- -0.05 ## single mutant fitness 0.95
d2 <- -0.08 ## double mutant fitness 0.92
d3 <- 0.2 ## triple mutant fitness 1.2
s2 <- ((1 + d2)/(1 + d1)^2) - 1
s3 <- ( (1 + d3)/((1 + d1)^3 * (1 + s2)^3) ) - 1
wb <- allFitnessEffects(
epistasis = c(
"A" = d1,
"B" = d1,
"C" = d1,
"A:B" = s2,
"A:C" = s2,
"B:C" = s2,
"A:B:C" = s3))plotFitnessLandscape(wb, use_ggrepel = TRUE) 
We have set use_ggrepel = TRUE to avoid overlap of labels.
For some types of objects, directly invoking plot will give
you the fitness landscape plot:
(ewb <- evalAllGenotypes(wb, order = FALSE))
## Genotype Fitness
## 1 A 0.95
## 2 B 0.95
## 3 C 0.95
## 4 A, B 0.92
## 5 A, C 0.92
## 6 B, C 0.92
## 7 A, B, C 1.20
plot(ewb, use_ggrepel = TRUE) 
This is example (section 5.5) will give a very busy plot:
par(cex = 0.7)
pancr <- allFitnessEffects(
data.frame(parent = c("Root", rep("KRAS", 4),
"SMAD4", "CDNK2A",
"TP53", "TP53", "MLL3"),
child = c("KRAS","SMAD4", "CDNK2A",
"TP53", "MLL3",
rep("PXDN", 3), rep("TGFBR2", 2)),
s = 0.1,
sh = -0.9,
typeDep = "MN"))
plot(evalAllGenotypes(pancr, order = FALSE), use_ggrepel = TRUE)
## Warning: ggrepel: 103 unlabeled data points (too many overlaps).
## Consider increasing max.overlaps
## Warning: ggrepel: 7 unlabeled data points (too many overlaps).
## Consider increasing max.overlaps
5 Specifying fitness effects: some examples from the literature
5.1 Bauer et al., 2014
In the model of Bauer and collaborators (Bauer, Siebert, and Traulsen 2014, 54) we have “For cells without the primary driver mutation, each secondary driver mutation leads to a change in the cell’s fitness by \(s_P\). For cells with the primary driver mutation, the fitness advantage obtained with each secondary driver mutation is \(s_{DP}\).”
The proliferation probability is given as:
- \(\frac{1}{2}(1 + s_p)^k\) when there are \(k\) secondary drivers mutated and no primary diver;
- \(\frac{1}{2}\frac{1+S_D^+}{1+S_D^-} (1 + S_{DP})^k\) when the primary driver is mutated;
apoptosis is one minus the proliferation rate.
5.1.1 Using a DAG
We cannot find a simple mapping from their expressions to our fitness parameterization, but we can get fairly close by using a DAG; in this one, note the unusual feature of having one of the “s” terms (that for the driver dependency on root) be negative. Using the parameters given in the legend of their Figure 3 for \(s_p, S_D^+, S_D^-, S_{DP}\) and obtaining that negative value for the dependency of the driver on root we can do:
K <- 4
sp <- 1e-5
sdp <- 0.015
sdplus <- 0.05
sdminus <- 0.1
cnt <- (1 + sdplus)/(1 + sdminus)
prod_cnt <- cnt - 1
bauer <- data.frame(parent = c("Root", rep("D", K)),
child = c("D", paste0("s", 1:K)),
s = c(prod_cnt, rep(sdp, K)),
sh = c(0, rep(sp, K)),
typeDep = "MN")
fbauer <- allFitnessEffects(bauer)
(b1 <- evalAllGenotypes(fbauer, order = FALSE, addwt = TRUE))
## Genotype Fitness
## 1 WT 1.0000000
## 2 D 0.9545455
## 3 s1 1.0000100
## 4 s2 1.0000100
## 5 s3 1.0000100
## 6 s4 1.0000100
## 7 D, s1 0.9688636
## 8 D, s2 0.9688636
## 9 D, s3 0.9688636
## 10 D, s4 0.9688636
## 11 s1, s2 1.0000200
## 12 s1, s3 1.0000200
## 13 s1, s4 1.0000200
## 14 s2, s3 1.0000200
## 15 s2, s4 1.0000200
## 16 s3, s4 1.0000200
## 17 D, s1, s2 0.9833966
## 18 D, s1, s3 0.9833966
## 19 D, s1, s4 0.9833966
## 20 D, s2, s3 0.9833966
## 21 D, s2, s4 0.9833966
## 22 D, s3, s4 0.9833966
## 23 s1, s2, s3 1.0000300
## 24 s1, s2, s4 1.0000300
## 25 s1, s3, s4 1.0000300
## 26 s2, s3, s4 1.0000300
## 27 D, s1, s2, s3 0.9981475
## 28 D, s1, s2, s4 0.9981475
## 29 D, s1, s3, s4 0.9981475
## 30 D, s2, s3, s4 0.9981475
## 31 s1, s2, s3, s4 1.0000400
## 32 D, s1, s2, s3, s4 1.0131198(We use “D” for “driver” or “primary driver”, as is it is called in the original paper, and “s” for secondary drivers, somewhat similar to passengers).
Note that what we specify as “typeDep” is irrelevant (MN, SMN, or XMPN make no difference).
This is the DAG:
plot(fbauer)
And if you compare the tabular output of evalAllGenotypes you can
see that the values of fitness reproduces the fitness landscape that they
show in their Figure 1. We can also use our plot for fitness landscapes:
plot(b1, use_ggrepel = TRUE)
## Warning: ggrepel: 20 unlabeled data points (too many overlaps).
## Consider increasing max.overlaps
5.1.2 Specifying fitness of genotypes directly
An alternative approach to specify the fitness, if the number of
genotypes is reasonably small, is to directly evaluate fitness as
given by their expressions. Then, use the genotFitness argument to
allFitnessEffects.
We will create all possible genotypes; then we will write a function
that gives the fitness of each genotype according to their
expression; finally, we will call this function on the data frame of
genotypes, and pass this data frame to allFitnessEffects.
m1 <- expand.grid(D = c(1, 0), s1 = c(1, 0), s2 = c(1, 0),
s3 = c(1, 0), s4 = c(1, 0))
fitness_bauer <- function(D, s1, s2, s3, s4,
sp = 1e-5, sdp = 0.015, sdplus = 0.05,
sdminus = 0.1) {
if(!D) {
b <- 0.5 * ( (1 + sp)^(sum(c(s1, s2, s3, s4))))
} else {
b <- 0.5 *
(((1 + sdplus)/(1 + sdminus) *
(1 + sdp)^(sum(c(s1, s2, s3, s4)))))
}
fitness <- b - (1 - b)
our_fitness <- 1 + fitness ## prevent negative fitness and
## make wt fitness = 1
return(our_fitness)
}
m1$Fitness <-
apply(m1, 1, function(x) do.call(fitness_bauer, as.list(x)))
bauer2 <- allFitnessEffects(genotFitness = m1)Now, show the fitness of all genotypes:
evalAllGenotypes(bauer2, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0000000
## 2 D 0.9545455
## 3 s1 1.0000100
## 4 s2 1.0000100
## 5 s3 1.0000100
## 6 s4 1.0000100
## 7 D, s1 0.9688636
## 8 D, s2 0.9688636
## 9 D, s3 0.9688636
## 10 D, s4 0.9688636
## 11 s1, s2 1.0000200
## 12 s1, s3 1.0000200
## 13 s1, s4 1.0000200
## 14 s2, s3 1.0000200
## 15 s2, s4 1.0000200
## 16 s3, s4 1.0000200
## 17 D, s1, s2 0.9833966
## 18 D, s1, s3 0.9833966
## 19 D, s1, s4 0.9833966
## 20 D, s2, s3 0.9833966
## 21 D, s2, s4 0.9833966
## 22 D, s3, s4 0.9833966
## 23 s1, s2, s3 1.0000300
## 24 s1, s2, s4 1.0000300
## 25 s1, s3, s4 1.0000300
## 26 s2, s3, s4 1.0000300
## 27 D, s1, s2, s3 0.9981475
## 28 D, s1, s2, s4 0.9981475
## 29 D, s1, s3, s4 0.9981475
## 30 D, s2, s3, s4 0.9981475
## 31 s1, s2, s3, s4 1.0000400
## 32 D, s1, s2, s3, s4 1.0131198Can we use modules in this example, if we use the “lego system”? Sure, as in any other case.
5.2 Misra et al., 2014
Figure 1 of Misra, Szczurek, and Vingron (2014) presents three scenarios which are different types of epistasis.
5.2.1 Example 1.a
In that figure it is evident that the fitness effect of “A” and “B” are the same. There are two different models depending on whether “AB” is just the product of both, or there is epistasis. In the first case probably the simplest is:
s <- 0.1 ## or whatever number
m1a1 <- allFitnessEffects(data.frame(parent = c("Root", "Root"),
child = c("A", "B"),
s = s,
sh = 0,
typeDep = "MN"))
evalAllGenotypes(m1a1, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 A 1.10
## 3 B 1.10
## 4 A, B 1.21If the double mutant shows epistasis, as we saw before (section 3.7.1) we have a range of options. For example:
s <- 0.1
sab <- 0.3
m1a2 <- allFitnessEffects(epistasis = c("A:-B" = s,
"-A:B" = s,
"A:B" = sab))
evalAllGenotypes(m1a2, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.1
## 3 B 1.1
## 4 A, B 1.3But we could also modify the graph dependency structure, and we have to change the value of the coefficient, since that is what multiplies each of the terms for “A” and “B”: \((1 + s_{AB}) = (1 + s)^2(1 + s_{AB3})\)
sab3 <- ((1 + sab)/((1 + s)^2)) - 1
m1a3 <- allFitnessEffects(data.frame(parent = c("Root", "Root"),
child = c("A", "B"),
s = s,
sh = 0,
typeDep = "MN"),
epistasis = c("A:B" = sab3))
evalAllGenotypes(m1a3, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.1
## 3 B 1.1
## 4 A, B 1.3And, obviously
all.equal(evalAllGenotypes(m1a2, order = FALSE, addwt = TRUE),
evalAllGenotypes(m1a3, order = FALSE, addwt = TRUE))
## [1] TRUE5.2.2 Example 1.b
This is a specific case of synthetic viability (see also section 3.9):
Here, \(S_A, S_B < 0\), \(S_B < 0\), \(S_{AB} > 0\) and \((1 + S_{AB}) (1 + S_A) (1 + S_B) > 1\).
As before, we can specify this in several different ways. The simplest is to specify all genotypes:
sa <- -0.6
sb <- -0.7
sab <- 0.3
m1b1 <- allFitnessEffects(epistasis = c("A:-B" = sa,
"-A:B" = sb,
"A:B" = sab))
evalAllGenotypes(m1b1, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 0.4
## 3 B 0.3
## 4 A, B 1.3We could also use a tree and modify the “sab” for the epistasis, as before (5.2.1).
5.2.3 Example 1.c
The final case, in figure 1.c of Misra et al., is just epistasis, where a mutation in one of the genes is deleterious (possibly only mildly), in the other is beneficial, and the double mutation has fitness larger than any of the other two.
Here we have that \(s_A > 0\), \(s_B < 0\), \((1 + s_{AB}) (1 + s_A) (1 + s_B) > (1 + s_{AB})\) so \(s_{AB} > \frac{-s_B}{1 + s_B}\)
As before, we can specify this in several different ways. The simplest is to specify all genotypes:
sa <- 0.2
sb <- -0.3
sab <- 0.5
m1c1 <- allFitnessEffects(epistasis = c("A:-B" = sa,
"-A:B" = sb,
"A:B" = sab))
evalAllGenotypes(m1c1, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.2
## 3 B 0.7
## 4 A, B 1.5We could also use a tree and modify the “sab” for the epistasis, as before (5.2.1).
5.3 Ochs and Desai, 2015
In Ochs and Desai (2015) the authors present a model shown graphically as (the actual numerical values are arbitrarily set by me):
In their model, \(s_u > 0\), \(s_v > s_u\), \(s_i < 0\), we can only arrive at \(v\) from \(i\), and the mutants “ui” and “uv” can never appear as their fitness is 0, or \(-\infty\), so \(s_{ui} = s_{uv} = -1\) (or \(-\infty\)).
We can specify this combining a graph and epistasis specifications:
su <- 0.1
si <- -0.05
fvi <- 1.2 ## the fitness of the vi mutant
sv <- (fvi/(1 + si)) - 1
sui <- suv <- -1
od <- allFitnessEffects(
data.frame(parent = c("Root", "Root", "i"),
child = c("u", "i", "v"),
s = c(su, si, sv),
sh = -1,
typeDep = "MN"),
epistasis = c(
"u:i" = sui,
"u:v" = suv))A figure showing that model is
plot(od)
And the fitness of all genotype is
evalAllGenotypes(od, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 i 0.95
## 3 u 1.10
## 4 v 0.00
## 5 i, u 0.00
## 6 i, v 1.20
## 7 u, v 0.00
## 8 i, u, v 0.00We could alternatively have specified fitness either directly specifying the fitness of each genotype or specifying epistatic effects. Let us use the second approach:
%% this was wrong %% u <- 0.2; i <- -0.02; vi <- 0.6; ui <- uv <- -Inf
u <- 0.1; i <- -0.05; vi <- (1.2/0.95) - 1; ui <- uv <- -Inf
od2 <- allFitnessEffects(
epistasis = c("u" = u, "u:i" = ui,
"u:v" = uv, "i" = i,
"v:-i" = -Inf, "v:i" = vi))
evalAllGenotypes(od2, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 i 0.95
## 3 u 1.10
## 4 v 0.00
## 5 i, u 0.00
## 6 i, v 1.20
## 7 u, v 0.00
## 8 i, u, v 0.00We will return to this model when we explain the usage of fixation
for stopping the simulations (see 6.3.3 and 6.3.4).
5.4 Weissman et al., 2009
In their figure 1a, Weissman et al. (2009) present this model (actual numeric values are set arbitrarily)
5.4.1 Figure 1.a
where the “1” and “2” in the figure refer to the total number of mutations in two different loci. This is, therefore, very similar to the example in section 5.2.2. Here we have, in their notation, \(\delta_1 < 0\), fitness of single “A” or single “B” = \(1 + \delta_1\), \(S_{AB} > 0\), \((1 + S_{AB})(1 + \delta_1)^2 > 1\).
5.4.2 Figure 1.b
In their figure 1b they show
Where, as before, 1, 2, 3, denote the total number of mutations over three different loci and \(\delta_1 < 0\), \(\delta_2 < 0\), fitness of single mutant is \((1 + \delta_1)\), of double mutant is \((1 + \delta_2)\) so that \((1 + \delta_2) = (1 + \delta_1)^2 (1 + s_2)\) and of triple mutant is \((1 + \delta_3)\), so that \((1 + \delta_3) = (1 + \delta_1)^3 (1 + s_2)^3 (1 + s_3)\).
We can specify this combining a graph with epistasis:
d1 <- -0.05 ## single mutant fitness 0.95
d2 <- -0.08 ## double mutant fitness 0.92
d3 <- 0.2 ## triple mutant fitness 1.2
s2 <- ((1 + d2)/(1 + d1)^2) - 1
s3 <- ( (1 + d3)/((1 + d1)^3 * (1 + s2)^3) ) - 1
w <- allFitnessEffects(
data.frame(parent = c("Root", "Root", "Root"),
child = c("A", "B", "C"),
s = d1,
sh = -1,
typeDep = "MN"),
epistasis = c(
"A:B" = s2,
"A:C" = s2,
"B:C" = s2,
"A:B:C" = s3))The model can be shown graphically as:
plot(w)
And fitness of all genotypes is:
evalAllGenotypes(w, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 A 0.95
## 3 B 0.95
## 4 C 0.95
## 5 A, B 0.92
## 6 A, C 0.92
## 7 B, C 0.92
## 8 A, B, C 1.20Alternatively, we can directly specify what each genotype adds to the fitness, given the included genotype. This is basically replacing the graph by giving each of “A”, “B”, and “C” directly:
wb <- allFitnessEffects(
epistasis = c(
"A" = d1,
"B" = d1,
"C" = d1,
"A:B" = s2,
"A:C" = s2,
"B:C" = s2,
"A:B:C" = s3))
evalAllGenotypes(wb, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 A 0.95
## 3 B 0.95
## 4 C 0.95
## 5 A, B 0.92
## 6 A, C 0.92
## 7 B, C 0.92
## 8 A, B, C 1.20The plot, of course, is not very revealing and we cannot show that there is a three-way interaction (only all three two-way interactions):
plot(wb)
As we have seen several times already (sections 3.7.1, 3.7.2, 3.7.3) we can also give the genotypes directly and, consequently, the fitness of each genotype (not the added contribution):
wc <- allFitnessEffects(
epistasis = c(
"A:-B:-C" = d1,
"B:-C:-A" = d1,
"C:-A:-B" = d1,
"A:B:-C" = d2,
"A:C:-B" = d2,
"B:C:-A" = d2,
"A:B:C" = d3))
evalAllGenotypes(wc, order = FALSE, addwt = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 A 0.95
## 3 B 0.95
## 4 C 0.95
## 5 A, B 0.92
## 6 A, C 0.92
## 7 B, C 0.92
## 8 A, B, C 1.205.5 Gerstung et al., 2011, pancreatic cancer poset
We can specify the pancreatic cancer poset in Gerstung et al. (2011) (their figure 2B, left). We use directly the names of the genes, since that is immediately supported by the new version.
pancr <- allFitnessEffects(
data.frame(parent = c("Root", rep("KRAS", 4),
"SMAD4", "CDNK2A",
"TP53", "TP53", "MLL3"),
child = c("KRAS","SMAD4", "CDNK2A",
"TP53", "MLL3",
rep("PXDN", 3), rep("TGFBR2", 2)),
s = 0.1,
sh = -0.9,
typeDep = "MN"))
plot(pancr)
Of course the “s” and “sh” are set arbitrarily here.
5.6 Raphael and Vandin’s 2014 modules
In Raphael and Vandin (2015), the authors show several progression models in terms of modules. We can code the extended poset for the colorectal cancer model in their Figure 4.a is (s and sh are arbitrary):
rv1 <- allFitnessEffects(data.frame(parent = c("Root", "A", "KRAS"),
child = c("A", "KRAS", "FBXW7"),
s = 0.1,
sh = -0.01,
typeDep = "MN"),
geneToModule = c("Root" = "Root",
"A" = "EVC2, PIK3CA, TP53",
"KRAS" = "KRAS",
"FBXW7" = "FBXW7"))
plot(rv1, expandModules = TRUE, autofit = TRUE)
We have used the (experimental) autofit option to fit the labels to the
edges. Note how we can use the same name for genes and modules, but we
need to specify all the modules.
Their Figure 5b is
rv2 <- allFitnessEffects(
data.frame(parent = c("Root", "1", "2", "3", "4"),
child = c("1", "2", "3", "4", "ELF3"),
s = 0.1,
sh = -0.01,
typeDep = "MN"),
geneToModule = c("Root" = "Root",
"1" = "APC, FBXW7",
"2" = "ATM, FAM123B, PIK3CA, TP53",
"3" = "BRAF, KRAS, NRAS",
"4" = "SMAD2, SMAD4, SOX9",
"ELF3" = "ELF3"))
plot(rv2, expandModules = TRUE, autofit = TRUE)
6 Running and plotting the simulations: starting, ending, and examples
6.1 Starting and ending
After you have decided the specifics of the fitness effects and the model, you need to decide:
Where will you start your simulation from. This involves deciding the initial population size (argument
initSize) and, possibly, the genotype of the initial population; the later is covered in section 6.2.When will you stop it: how long to run it, and whether or not to require simulations to reach cancer (under some definition of what it means to reach cancer). This is covered in 6.3.
6.2 Can I start the simulation from a specific mutant?
You bet. In version 2 you can specify the genotype for the
initial mutant with the same flexibility as in evalGenotype. Here we show
a couple of examples (we use the representation of the parent-child
relationships —discussed in section 8— of the clones so that
you can see which clones appear, and from which, and check that we are not
making mistakes).
o3init <- allFitnessEffects(orderEffects = c(
"M > D > F" = 0.99,
"D > M > F" = 0.2,
"D > M" = 0.1,
"M > D" = 0.9),
noIntGenes = c("u" = 0.01,
"v" = 0.01,
"w" = 0.001,
"x" = 0.0001,
"y" = -0.0001,
"z" = -0.001),
geneToModule =
c("M" = "m",
"F" = "f",
"D" = "d") )
oneI <- oncoSimulIndiv(o3init, model = "McFL",
mu = 5e-5, finalTime = 200,
detectionDrivers = 3,
onlyCancer = FALSE,
initSize = 1000,
keepPhylog = TRUE,
initMutant = c("m > u > d")
)
plotClonePhylog(oneI, N = 0)
## Note we also disable the stopping stochastically as a function of size
## to allow the population to grow large and generate may different
## clones.
## For speed, we set a small finalTime and we fix the seed
## for reproducilibity. Beware: since finalTime is short, sometimes
## we do not reach cancer
set.seed(1)
RNGkind("L'Ecuyer-CMRG")
ospI <- oncoSimulPop(2,
o3init, model = "Exp",
mu = 5e-5, finalTime = 200,
detectionDrivers = 3,
onlyCancer = TRUE,
initSize = 10,
keepPhylog = TRUE,
initMutant = c("d > m > z"),
mc.cores = 2,
seed = NULL
)
## Show just one example
## op <- par(mar = rep(0, 4), mfrow = c(1, 2))
plotClonePhylog(ospI[[1]])
## plotClonePhylog(ospI[[2]])
## par(op)
set.seed(1)
RNGkind("L'Ecuyer-CMRG")
ossI <- oncoSimulSample(2,
o3init, model = "Exp",
mu = 5e-5, finalTime = 200,
detectionDrivers = 2,
onlyCancer = TRUE,
initSize = 10,
initMutant = c("z > d"),
## check presence of initMutant:
thresholdWhole = 1,
seed = NULL
)
## Successfully sampled 2 individuals
##
## Subjects by Genes matrix of 2 subjects and 9 genes.
## No phylogeny is kept with oncoSimulSample, but look at the
## OcurringDrivers and the sample
ossI$popSample
## d f m u v w x y z
## [1,] 1 0 1 0 0 0 0 0 1
## [2,] 1 0 1 0 0 0 0 0 1
ossI$popSummary[, "OccurringDrivers", drop = FALSE]
## OccurringDrivers
## 1
## 2Since version 2.21.994, it is possible to start the simulations from arbitrary initial configurations: this uses multiple initial mutants (see section 6.7) and allows for multispecies simulations (section 6.8).
6.3 Ending the simulations
OncoSimulR provides very flexible ways to decide when to stop a simulation. Here we focus on a single simulation; see further options with multiple simulations in 7.
6.3.1 Ending the simulations: conditions
onlyCancer = TRUE. A simulation will be repeated until any one of the “reach cancer” conditions is met, if this happens before the simulation reachesfinalTime11. These conditions are:- Total population size becomes larger than
detectionSize. - The number of drivers in any one genotype or clone becomes equal
to, or larger than,
detectionDrivers; note that this allows you to stop the simulation as soon as a specific genotype is found, by using exactly and only the genes that make that genotype as the drivers. This is not allowed by the moment in frequency-dependent fitness simulations. - A gene or gene combination among those listed in
fixationbecomes fixed in the population (i.e., has a frequency is 1) (see details in (6.3.3 and 6.3.4). - The tumor is detected according to a stochastic detection
mechanism, where the probability of “detecting the tumor” increases
with population size; this is explained below (6.3.2) and
is controlled by argument
detectionProb.
As we exit as soon as any of the exiting conditions is reached, if you only care about one condition, set the other to
NA(see also section 6.3.2.1).- Total population size becomes larger than
onlyCancer = FALSE. A simulation will run only once, and will exit as soon as any of the above conditions are met or as soon as the total population size becomes zero or we reachfinalTime.
As an example of onlyCancer = TRUE, focusing on the first two
mechanisms, suppose you give detectionSize = 1e4 and detectionDrivers =3 (and you have detectionProb = NA). A simulation will exit as soon
as it reaches a total population size of \(10^4\) or any clone has four
drivers, whichever comes first (if any of these happen before
finalTime).
In the onlyCancer = TRUE case, what happens if we reach
finalTime (or the population size becomes zero) before any of the
“reach cancer” conditions have been fulfilled? The simulation will be
repeated again, within the following limits:
max.wall.time: the total wall time we allow an individual simulation to run;max.num.tries: the maximum number of times we allow a simulation to be repeated to reach cancer;max.wall.time.totalandmax.num.tries.total, similar to the above but over a set of simulations in functiononcoSimulSample.
Incidentally, we keep track of the number of attempts used (the component
other$attemptsUsed$) before we reach cancer, so you can estimate
(as from a negative binomial sampling) the probability of reaching your
desired end point under different scenarios.
The onlyCancer = FALSE case might be what you want to do when you
examine general population genetics scenarios without focusing on possible
sampling issues. To do this, set finalTime to the value you want
and set onlyCancer = FALSE; in addition, set
detectionProb to “NA” and detectionDrivers and
detectionSize to “NA” or to huge numbers12. In this
scenario you simply collect the simulation output at the end of the run,
regardless of what happened with the population (it became extinct, it did
not reach a large size, it did not accumulate drivers, etc).
6.3.2 Stochastic detection mechanism: “detectionProb”
This is the process that is controlled by the argument
detectionProb. Here the probability of tumor detection increases
with the total population size. This is biologically a reasonable
assumption: the larger the tumor, the more likely it is it will be
detected.
At regularly spaced times during the simulation, we compute the probability of detection as a function of size and determine (by comparing against a random uniform number) if the simulation should finish. For simplicity, and to make sure the probability is bounded between 0 and 1, we use the function
\[\begin{equation} P(N) = \begin{cases} 1 - e^{ -c ( (N - B)/B)} & \text{if } N > B \\ 0 & \text{if } N \leq B \end{cases} \label{eq:2} \end{equation}\]
where \(P(N)\) is the probability that a tumor with a population size \(N\)
will be detected, and \(c\) (argument \(cPDetect\) in the oncoSimul*
functions) controls how fast \(P(N)\) increases with increasing population
size relative to a baseline, \(B\) (\(PDBaseline\) in the oncoSimul*
functions); with \(B\) we both control the minimal population size at which
this mechanism stats operating (because we will rarely want detection
unless there is some meaningful increase of population size over
initSize) and we model the increase in \(P(N)\) as a function of relative
differences with respect to \(B\). (Note that this is a major change in
version 2.9.9. Before version 2.9.9, the expression used was
\(P(N) = 1 - e^{ -c ( N - B)}\), so we did not make the increase relative to
\(B\); of course, you can choose an appropriate \(c\) to make different models
comparable, but the expression used before 2.9.9 made it much harder to
compare simulations with very different initial population sizes, as
baselines are often naturall a function of initial population sizes.)
The \(P(N)\) refers to the probability of detection at each one of the
occasions when we assess the probability of exiting. When, or how often,
do we do that? When we assess probability of exiting is controlled by
checkSizePEvery, which will often be much larger than
sampleEvery13. Biologically, a way to think of
checkSizePEvery is “time between doctor appointments”.
An important warning, though: for populations that are growing
very, very fast or where some genes might have very large effects on
fitness even a moderate checkSizePEvery of, say, 10, might be
inappropriate, since populations could have increased by several
orders of magnitude between successive checks. This issue is also
discussed in section 2.1.1 and 2.2.
Finally, you can specify \(c\) (\(cPDetect\)) directly (you will need to set
n2 and p2 to NA). However, it might be more intuitive to specify the
pair n2, p2, such that \(P(n2) = p2\) (and from that pair we solve for
the value of \(cPDetect\)).
You can get a feeling for the effects of these arguments by playing with the following code, that we do not execute here for the sake of speed. Here no mutation has any effect, but there is a non-zero probability of exiting as soon as the total population size becomes larger than the initial population size. So, eventually, all simulations will exit and, as we are using the McFarland model, population size will vary slightly around the initial population size.
gi2 <- rep(0, 5)
names(gi2) <- letters[1:5]
oi2 <- allFitnessEffects(noIntGenes = gi2)
s5 <- oncoSimulPop(200,
oi2,
model = "McFL",
initSize = 1000,
detectionProb = c(p2 = 0.1,
n2 = 2000,
PDBaseline = 1000,
checkSizePEvery = 2),
detectionSize = NA,
finalTime = NA,
keepEvery = NA,
detectionDrivers = NA)
s5
hist(unlist(lapply(s5, function(x) x$FinalTime)))As you decrease checkSizePEvery the distribution of “FinalTime”
will resemble more and more an exponential distribution.
In this vignette, there are some further examples of using this mechanism in 6.5.4 and 6.5.2, with the default arguments.
6.3.2.1 Stochastic detection mechanism and minimum number of drivers
We said above that we exit as soon as any of the conditions is
reached (i.e., we use an OR operation over the exit
conditions). There is a special exception to this procedure: if you
set AND_DrvProbExit = TRUE, both the number of drivers and the
detectionProb mechanism condition must fulfilled. This means that
the detectionProb mechanism not assessed unless the
detectionDrivers condition is. Using AND_DrvProbExit = TRUE
allows to run simulations and ensure that all of the returned
simulations will have at least some cells with the number of drivers
as specified by detectionDrivers. Note, though, that this does not
guarantee that when you sample the population, all those drivers
will be detected (as this depends on the actual proportion of cells
with the drivers and the settings of samplePop).
6.3.3 Fixation of genes/gene combinations
In some cases we might be interested in running simulations until a particular set of genes, or gene combinations, reaches fixation. This exit condition might be more relevant than some of the above in many non-cancer-related evolutionary genetics scenarios.
Simulations will stop as soon as any of the genes or gene combinations in
the vector (or list) fixation reaches a frequency of 1. These gene
combinations might have non-zero intersection (i.e., they might share
genes), and those genes need not be drivers. If we want simulations to
only stop when fixation of those genes/gene combinations is reached, we
will set all other stopping conditions to NA. It is, of course, up to
you to ensure that those stopping conditions are reasonable (that they can
be reached) and to use, or not, finalTime; otherwise, simulations will
eventually abort (e.g., when max.wall.time or max.num.tries are
reached). Since we are asking for fixation, the Exp or Bozic models
will often not be appropriate here; instead, models with competition such
as McFL are more appropriate.
We return here to the example from section 5.3.
u <- 0.2; i <- -0.02; vi <- 0.6; ui <- uv <- -Inf
od2 <- allFitnessEffects(
epistasis = c("u" = u, "u:i" = ui,
"u:v" = uv, "i" = i,
"v:-i" = -Inf, "v:i" = vi))Ochs and Desai explain that “Each simulated population was evolved
until either the uphill genotype or valley-crossing genotype fixed.”
(see Ochs and Desai (2015), p.2, section “Simulations”). We will do the same
here. We specify that we want to end the simulation when either the
“u” or the “v, i” genotypes have reached fixation, by passing those
genotype combinations as the fixation argument (in this example
using fixation = c("u", "v") would have been equivalent, since the
“v” genotype by itself has fitness of 0).
We want to be explicit that fixation will be the one and only
condition for ending the simulations, and thus we set arguments
detectionDrivers, finalTime, detectionSize and detectionProb
explicitly to NA. (We set the number of repetitions only to 10 for
the sake of speed when creating the vignette).
initS <- 20
## We use only a small number of repetitions for the sake
## of speed. Even fewer in Windows, since we run on a single
## core
if(.Platform$OS.type == "windows") {
nruns <- 4
} else {
nruns <- 10
}
od100 <- oncoSimulPop(nruns, od2,
fixation = c("u", "v, i"),
model = "McFL",
mu = 1e-4,
detectionDrivers = NA,
finalTime = NA,
detectionSize = NA,
detectionProb = NA,
onlyCancer = TRUE,
initSize = initS,
mc.cores = 2)What is the frequency of each genotype among the simulations? (or, what is the frequency of fixation of each genotype?)
sampledGenotypes(samplePop(od100))
##
## Subjects by Genes matrix of 10 subjects and 3 genes.
## Genotype Freq
## 1 i, v 2
## 2 u 8
##
## Shannon's diversity (entropy) of sampled genotypes: 0.5004024Note the very large variability in reaching fixation
head(summary(od100)[, c(1:3, 8:9)])
## NumClones TotalPopSize LargestClone FinalTime NumIter
## 1 3 20 20 3792.850 151732
## 2 4 35 35 3225.750 129047
## 3 3 18 18 1756.975 70288
## 4 3 28 28 1295.300 51822
## 5 4 25 25 1866.075 74658
## 6 3 23 23 2332.900 933246.3.4 Fixation of genotypes
Section 6.3.3 deals with the fixation of gene/gene combinations. What if you want fixation on specific genotypes? To give an example, suppose we have a five loci genotype and suppose that you want to stop the simulations only if genotypes “A”, “B, E”, or “A, B, C, D, E” reach fixation. You do not want to stop it if, say, genotype “A, B, E” reaches fixation. To specify genotypes, you prepend the genotype combinations with a "_,", and that tells OncoSimulR that you want fixation of genotypes, not just gene combinations.
An example of the differences between the mechanisms can be seen from this code:
## Create a simple fitness landscape
rl1 <- matrix(0, ncol = 6, nrow = 9)
colnames(rl1) <- c(LETTERS[1:5], "Fitness")
rl1[1, 6] <- 1
rl1[cbind((2:4), c(1:3))] <- 1
rl1[2, 6] <- 1.4
rl1[3, 6] <- 1.32
rl1[4, 6] <- 1.32
rl1[5, ] <- c(0, 1, 0, 0, 1, 1.5)
rl1[6, ] <- c(0, 0, 1, 1, 0, 1.54)
rl1[7, ] <- c(1, 0, 1, 1, 0, 1.65)
rl1[8, ] <- c(1, 1, 1, 1, 0, 1.75)
rl1[9, ] <- c(1, 1, 1, 1, 1, 1.85)
class(rl1) <- c("matrix", "genotype_fitness_matrix")
## plot(rl1) ## to see the fitness landscape
## Gene combinations
local_max_g <- c("A", "B, E", "A, B, C, D, E")
## Specify the genotypes
local_max <- paste0("_,", local_max_g)
fr1 <- allFitnessEffects(genotFitness = rl1, drvNames = LETTERS[1:5])
initS <- 2000
######## Stop on gene combinations #####
r1 <- oncoSimulPop(10,
fp = fr1,
model = "McFL",
initSize = initS,
mu = 1e-4,
detectionSize = NA,
sampleEvery = .03,
keepEvery = 1,
finalTime = 50000,
fixation = local_max_g,
detectionDrivers = NA,
detectionProb = NA,
onlyCancer = TRUE,
max.num.tries = 500,
max.wall.time = 20,
errorHitMaxTries = TRUE,
keepPhylog = FALSE,
mc.cores = 2)
sp1 <- samplePop(r1, "last", "singleCell")
##
## Subjects by Genes matrix of 10 subjects and 5 genes.
sgsp1 <- sampledGenotypes(sp1)
## often you will stop on gene combinations that
## are not local maxima in the fitness landscape
sgsp1
## Genotype Freq
## 1 A 1
## 2 A, B, C, D 4
## 3 A, C, D 2
## 4 B, E 3
##
## Shannon's diversity (entropy) of sampled genotypes: 1.279854
sgsp1$Genotype %in% local_max_g
## [1] TRUE FALSE FALSE TRUE
####### Stop on genotypes ####
r2 <- oncoSimulPop(10,
fp = fr1,
model = "McFL",
initSize = initS,
mu = 1e-4,
detectionSize = NA,
sampleEvery = .03,
keepEvery = 1,
finalTime = 50000,
fixation = local_max,
detectionDrivers = NA,
detectionProb = NA,
onlyCancer = TRUE,
max.num.tries = 500,
max.wall.time = 20,
errorHitMaxTries = TRUE,
keepPhylog = FALSE,
mc.cores = 2)
## All final genotypes should be local maxima
sp2 <- samplePop(r2, "last", "singleCell")
##
## Subjects by Genes matrix of 10 subjects and 5 genes.
sgsp2 <- sampledGenotypes(sp2)
sgsp2$Genotype %in% local_max_g
## [1] TRUE TRUE TRUE6.3.5 Fixation: tolerance, number of periods, minimal size
In particular if you specify stopping on genotypes, you might want to
think about three additional parameters: fixation_tolerance,
min_successive_fixation, and fixation_min_size.
fixation_tolerance: fixation is considered to have happened if the
genotype/gene combinations specified as genotypes/gene combinations for
fixation have reached a frequency \(> 1 - fixation\_tolerance\). (The
default is 0, so we ask for genotypes/gene combinations with a frequency
of 1, which might not be what you want with large mutation rates and
complex fitness landscape with genotypes of similar fitness.)
min_successive_fixation: during how many successive sampling periods the
conditions of fixation need to be fulfilled before declaring
fixation. These must be successive sampling periods without interruptions
(i.e., a single period when the condition is not fulfilled will set the
counter to 0). This can help to exclude short, transitional, local maxima
that are quickly replaced by other genotypes. (The default is 50, but this
is probably too small for “real life” usage).
fixation_min_size: you might only want to consider fixation to have
happened if a minimal size has been reached (this can help weed out local
maxima that have fitness that is barely above that of the wild-type
genotype). (The default is 0).
An example of using those options:
## Create a simple fitness landscape
rl1 <- matrix(0, ncol = 6, nrow = 9)
colnames(rl1) <- c(LETTERS[1:5], "Fitness")
rl1[1, 6] <- 1
rl1[cbind((2:4), c(1:3))] <- 1
rl1[2, 6] <- 1.4
rl1[3, 6] <- 1.32
rl1[4, 6] <- 1.32
rl1[5, ] <- c(0, 1, 0, 0, 1, 1.5)
rl1[6, ] <- c(0, 0, 1, 1, 0, 1.54)
rl1[7, ] <- c(1, 0, 1, 1, 0, 1.65)
rl1[8, ] <- c(1, 1, 1, 1, 0, 1.75)
rl1[9, ] <- c(1, 1, 1, 1, 1, 1.85)
class(rl1) <- c("matrix", "genotype_fitness_matrix")
## plot(rl1) ## to see the fitness landscape
## The local fitness maxima are
## c("A", "B, E", "A, B, C, D, E")
fr1 <- allFitnessEffects(genotFitness = rl1, drvNames = LETTERS[1:5])
initS <- 2000
## Stop on genotypes
r3 <- oncoSimulPop(10,
fp = fr1,
model = "McFL",
initSize = initS,
mu = 1e-4,
detectionSize = NA,
sampleEvery = .03,
keepEvery = 1,
finalTime = 50000,
fixation = c(paste0("_,",
c("A", "B, E", "A, B, C, D, E")),
fixation_tolerance = 0.1,
min_successive_fixation = 200,
fixation_min_size = 3000),
detectionDrivers = NA,
detectionProb = NA,
onlyCancer = TRUE,
max.num.tries = 500,
max.wall.time = 20,
errorHitMaxTries = TRUE,
keepPhylog = FALSE,
mc.cores = 2)6.3.6 Mixing stopping on gene combinations and genotypes
This would probably be awfully confusing and is not tested formally (though it should work). Let me know if you think this is an important feature. (Pull requests with tests welcome.)
6.4 Plotting genotype/driver abundance over time; plotting the simulated trajectories
We have seen many of these plots already, starting with Figure
1.2 and Figure 1.4 and we will see many
more below, in the examples, starting with section 6.5.1
such as in figures 6.1 and 6.2. In a
nutshell, what we are plotting is the information contained in the
pops.by.time matrix, the matrix that contains the abundances of
all the clones (or genotypes) at each of the sampling periods.
The functions that do the work are called plot and these are actually
methods for objects of class “oncosimul” and “oncosimulpop”. You can access
the help by doing ?plot.oncosimul, for example.
What entities are shown in the plot? You can show the trajectories of:
(Of course, showing “drivers” requires that you have specified certain genes as drivers.)
What types of plots are available?
line plots;
stacked plots;
stream plots.
All those three are shown in both of Figure 6.1 and Figure 6.2.
If you run multiple simulations using oncoSimulPop you can plot
the trajectories of all of the simulations.
6.5 Several examples of simulations and plotting simulation trajectories
6.5.1 Bauer’s example again
We will use the model of Bauer, Siebert, and Traulsen (2014) that we saw in section 5.1.
K <- 5
sd <- 0.1
sdp <- 0.15
sp <- 0.05
bauer <- data.frame(parent = c("Root", rep("p", K)),
child = c("p", paste0("s", 1:K)),
s = c(sd, rep(sdp, K)),
sh = c(0, rep(sp, K)),
typeDep = "MN")
fbauer <- allFitnessEffects(bauer, drvNames = "p")
set.seed(1)
## Use fairly large mutation rate
b1 <- oncoSimulIndiv(fbauer, mu = 5e-5, initSize = 1000,
finalTime = NA,
onlyCancer = TRUE,
detectionProb = "default")We will now use a variety of plots
par(mfrow = c(3, 1))
## First, drivers
plot(b1, type = "line", addtot = TRUE)
plot(b1, type = "stacked")
plot(b1, type = "stream")
Figure 6.1: Three drivers’ plots of a simulation of Bauer’s model
par(mfrow = c(3, 1))
## Next, genotypes
plot(b1, show = "genotypes", type = "line")
plot(b1, show = "genotypes", type = "stacked")
plot(b1, show = "genotypes", type = "stream")
Figure 6.2: Three genotypes’ plots of a simulation of Bauer’s model
In this case, probably the stream plots are most helpful. Note, however, that (in contrast to some figures in the literature showing models of clonal expansion) the stream plot (or the stacked plot) does not try to explicitly show parent-descendant relationships, which would hardly be realistically possible in these plots (although the plots of phylogenies in section 8 could be of help).
6.5.2 McFarland model with 5000 passengers and 70 drivers
set.seed(678)
nd <- 70
np <- 5000
s <- 0.1
sp <- 1e-3
spp <- -sp/(1 + sp)
mcf1 <- allFitnessEffects(noIntGenes = c(rep(s, nd), rep(spp, np)),
drvNames = seq.int(nd))
mcf1s <- oncoSimulIndiv(mcf1,
model = "McFL",
mu = 1e-7,
detectionProb = "default",
detectionSize = NA,
detectionDrivers = NA,
sampleEvery = 0.025,
keepEvery = 8,
initSize = 2000,
finalTime = 4000,
onlyCancer = FALSE)
summary(mcf1s)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 973 2855 2412 3 2
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 2 11 2841.05 117409
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.01308528 1694.892 1972.387
## OccurringDrivers
## 1 7, 9, 10, 11, 16, 20, 22, 23, 33, 45, 59par(mfrow = c(2, 1))
## I use thinData to make figures smaller and faster
plot(mcf1s, addtot = TRUE, lwdClone = 0.9, log = "",
thinData = TRUE, thinData.keep = 0.5)
plot(mcf1s, show = "drivers", type = "stacked",
thinData = TRUE, thinData.keep = 0.3,
legend.ncols = 2)
With the above output (where we see there are over 500 different genotypes) trying to represent the genotypes makes no sense.
6.5.3 McFarland model with 50,000 passengers and 70 drivers: clonal competition
The next is too slow (takes a couple of minutes in an i5 laptop) and too big to run in a vignette, because we keep track of over 4000 different clones (which leads to a result object of over 800 MB):
set.seed(123)
nd <- 70
np <- 50000
s <- 0.1
sp <- 1e-4 ## as we have many more passengers
spp <- -sp/(1 + sp)
mcfL <- allFitnessEffects(noIntGenes = c(rep(s, nd), rep(spp, np)),
drvNames = seq.int(nd))
mcfLs <- oncoSimulIndiv(mcfL,
model = "McFL",
mu = 1e-7,
detectionSize = 1e8,
detectionDrivers = 100,
sampleEvery = 0.02,
keepEvery = 2,
initSize = 1000,
finalTime = 2000,
onlyCancer = FALSE)But you can access the pre-stored results and plot them (beware: this object has been trimmed by removing empty passenger rows in the Genotype matrix)
data(mcfLs)
plot(mcfLs, addtot = TRUE, lwdClone = 0.9, log = "",
thinData = TRUE, thinData.keep = 0.3,
plotDiversity = TRUE)
The argument plotDiversity = TRUE asks to show a small plot on top
with Shannon’s diversity index.
summary(mcfLs)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 4458 1718 253 3 3
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 3 70 2000 113759
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.01921737 184.1019 199.6085
## OccurringDrivers
## 1 13, 38, 40, 69
## number of passengers per clone
summary(colSums(mcfLs$Genotypes[-(1:70), ]))
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 4.000 6.000 5.673 7.750 13.000Note that we see clonal competition between clones with the same number of drivers (and with different drivers, of course). We will return to this (section 6.5.5).
A stacked plot might be better to show the extent of clonal competition (plotting takes some time —a stream plot reveals similar patterns and is also slower than the line plot). I will aggressively thin the data for this plot so it is faster and smaller (but we miss some of the fine grain, of course):
plot(mcfLs, type = "stacked", thinData = TRUE,
thinData.keep = 0.1,
plotDiversity = TRUE,
xlim = c(0, 1000))
6.5.4 Simulation with a conjunction example
We will use several of the previous examples. Most of them are in file
examplesFitnessEffects, where they are stored inside a list,
with named components (names the same as in the examples above):
data(examplesFitnessEffects)
names(examplesFitnessEffects)
## [1] "cbn1" "cbn2" "smn1" "xor1" "fp3" "fp4m" "o3"
## [8] "ofe1" "ofe2" "foi1" "sv" "svB" "svB1" "sv2"
## [15] "sm1" "e2" "E3A" "em" "fea" "fbauer" "w"
## [22] "pancr"set.seed(1)We will simulate using the simple CBN-like restrictions of section 3.4.4 with two different models.
data(examplesFitnessEffects)
evalAllGenotypes(examplesFitnessEffects$cbn1, order = FALSE)[1:10, ]
## Genotype Fitness
## 1 a 1.10
## 2 b 1.10
## 3 c 0.10
## 4 d 1.10
## 5 e 1.10
## 6 g 0.10
## 7 a, b 1.21
## 8 a, c 0.11
## 9 a, d 1.21
## 10 a, e 1.21
sm <- oncoSimulIndiv(examplesFitnessEffects$cbn1,
model = "McFL",
mu = 5e-7,
detectionSize = 1e8,
detectionDrivers = 2,
detectionProb = "default",
sampleEvery = 0.025,
keepEvery = 5,
initSize = 2000,
onlyCancer = TRUE)
summary(sm)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 4 2635 2014 2 2
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 2 3 372.5 14905
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.01265916 314478.5 333333.3
## OccurringDrivers
## 1 a, b, eset.seed(1234)
evalAllGenotypes(examplesFitnessEffects$cbn1, order = FALSE,
model = "Bozic")[1:10, ]
## Genotype Death_rate
## 1 a 0.90
## 2 b 0.90
## 3 c 1.90
## 4 d 0.90
## 5 e 0.90
## 6 g 1.90
## 7 a, b 0.81
## 8 a, c 1.71
## 9 a, d 0.81
## 10 a, e 0.81
sb <- oncoSimulIndiv(examplesFitnessEffects$cbn1,
model = "Bozic",
mu = 5e-6,
detectionProb = "default",
detectionSize = 1e8,
detectionDrivers = 4,
sampleEvery = 2,
initSize = 2000,
onlyCancer = TRUE)
summary(sb)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 12 26655 25030 2 2
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 1 6 550 310
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE NA 33333.33 33333.33
## OccurringDrivers
## 1 a, b, c, d, e, gAs usual, we will use several plots here.
## Show drivers, line plot
par(cex = 0.75, las = 1)
plot(sb,show = "drivers", type = "line", addtot = TRUE,
plotDiversity = TRUE)
## Drivers, stacked
par(cex = 0.75, las = 1)
plot(sb,show = "drivers", type = "stacked", plotDiversity = TRUE)
## Drivers, stream
par(cex = 0.75, las = 1)
plot(sb,show = "drivers", type = "stream", plotDiversity = TRUE)
## Genotypes, line plot
par(cex = 0.75, las = 1)
plot(sb,show = "genotypes", type = "line", plotDiversity = TRUE)
## Genotypes, stacked
par(cex = 0.75, las = 1)
plot(sb,show = "genotypes", type = "stacked", plotDiversity = TRUE)
## Genotypes, stream
par(cex = 0.75, las = 1)
plot(sb,show = "genotypes", type = "stream", plotDiversity = TRUE)
The above illustrates again that different types of plots can be useful to reveal different patterns in the data. For instance, here, because of the huge relative frequency of one of the clones/genotypes, the stacked and stream plots do not reveal the other clones/genotypes as we cannot use a log-transformed y-axis, even if there are other clones/genotypes present.
6.5.5 Simulation with order effects and McFL model
(We use a somewhat large mutation rate than usual, so that the simulation runs quickly.)
set.seed(4321)
tmp <- oncoSimulIndiv(examplesFitnessEffects[["o3"]],
model = "McFL",
mu = 5e-5,
detectionSize = 1e8,
detectionDrivers = 3,
sampleEvery = 0.025,
max.num.tries = 10,
keepEvery = 5,
initSize = 2000,
finalTime = 6000,
onlyCancer = FALSE) We show a stacked and a line plot of the drivers:
par(las = 1, cex = 0.85)
plot(tmp, addtot = TRUE, log = "", plotDiversity = TRUE,
thinData = TRUE, thinData.keep = 0.2)
par(las = 1, cex = 0.85)
plot(tmp, type = "stacked", plotDiversity = TRUE,
ylim = c(0, 5500), legend.ncols = 4,
thinData = TRUE, thinData.keep = 0.2)
In this example (and at least under Linux, with both GCC and clang —random number streams in C++, and thus simulations, can differ between combinations of operating system and compiler), we can see that the mutants with three drivers do not get established when we stop the simulation at time 6000. This is one case where the summary statistics about number of drivers says little of value, as fitness is very different for genotypes with the same number of mutations, and does not increase in a simple way with drivers:
evalAllGenotypes(examplesFitnessEffects[["o3"]], addwt = TRUE,
order = TRUE)
## Genotype Fitness
## 1 WT 1.00
## 2 d 1.00
## 3 f 1.00
## 4 m 1.00
## 5 d > f 1.00
## 6 d > m 1.10
## 7 f > d 1.00
## 8 f > m 1.00
## 9 m > d 1.50
## 10 m > f 1.00
## 11 d > f > m 1.54
## 12 d > m > f 1.32
## 13 f > d > m 0.77
## 14 f > m > d 1.50
## 15 m > d > f 1.50
## 16 m > f > d 1.50A few figures could help:
plot(tmp, show = "genotypes", ylim = c(0, 5500), legend.ncols = 3,
thinData = TRUE, thinData.keep = 0.5)
(When reading the figure legends, recall that genotype \(x > y\ \_\ z\) is one where a mutation in “x” happened before a mutation in “y”, and there is also a mutation in “z” for which order does not matter. Here, there are no genes for which order does not matter and thus there is nothing after the "_").
In this case, the clones with three drivers end up displacing those with two by the time we stop; moreover, notice how those with one driver never really grow to a large population size, so we basically go from a population with clones with zero drivers to a population made of clones with two or three drivers:
set.seed(15)
tmp <- oncoSimulIndiv(examplesFitnessEffects[["o3"]],
model = "McFL",
mu = 5e-5,
detectionSize = 1e8,
detectionDrivers = 3,
sampleEvery = 0.025,
max.num.tries = 10,
keepEvery = 5,
initSize = 2000,
finalTime = 20000,
onlyCancer = FALSE,
extraTime = 1500)
tmp
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = examplesFitnessEffects[["o3"]], model = "McFL",
## mu = 5e-05, detectionSize = 1e+08, detectionDrivers = 3,
## sampleEvery = 0.025, initSize = 2000, keepEvery = 5, extraTime = 1500,
## finalTime = 20000, onlyCancer = FALSE, max.num.tries = 10)
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 7 3984 3984 3 3
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 3 3 7178.375 288895
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.01343206 6253.799 6666.667
## OccurringDrivers
## 1 d, f, m
##
## Final population composition:
## Genotype N
## 1 _ 0
## 2 d _ 0
## 3 d > m _ 0
## 4 f _ 0
## 5 m _ 0
## 6 m > d _ 0
## 7 m > d > f _ 3984use a drivers plot:
par(las = 1, cex = 0.85)
plot(tmp, addtot = TRUE, log = "", plotDiversity = TRUE,
thinData = TRUE, thinData.keep = 0.5)
par(las = 1, cex = 0.85)
plot(tmp, type = "stacked", plotDiversity = TRUE,
legend.ncols = 4, ylim = c(0, 5200), xlim = c(3400, 5000),
thinData = TRUE, thinData.keep = 0.5)
Now show the genotypes explicitly:
## Improve telling apart the most abundant
## genotypes by sorting colors
## differently via breakSortColors
## Modify ncols of legend, so it is legible by not overlapping
## with plot
par(las = 1, cex = 0.85)
plot(tmp, show = "genotypes", breakSortColors = "distave",
plotDiversity = TRUE, legend.ncols = 4,
ylim = c(0, 5300), xlim = c(3400, 5000),
thinData = TRUE, thinData.keep = 0.5)
As before, the argument plotDiversity = TRUE asks to show a small
plot on top with Shannon’s diversity index. Here, as before, the quick
clonal expansion of the clone with two drivers leads to a sudden drop in
diversity (for a while, the population is made virtually of a single
clone). Note, however, that compared to section 6.5.3, we are
modeling here a scenario with very few genes, and correspondingly very few
possible genotypes, and thus it is not strange that we observe very little
diversity.
(We have used extraTime to continue the simulation well past the
point of detection, here specified as three drivers. Instead of specifying
extraTime we can set the detectionDrivers value to a
number larger than the number of existing possible drivers, and the
simulation will run until finalTime if onlyCancer = FALSE.)
6.6 Interactive graphics
It is possible to create interactive stacked area and stream plots using the streamgraph package, available from https://github.com/hrbrmstr/streamgraph. However, that package is not available as a CRAN or BioConductor package, and thus we cannot depend on it for this vignette (or this package). You can, however, paste the code below and make it run locally.
Before calling the streamgraph function, though, we need to
convert the data from the original format in which it is stored into
“long format”. A simple convenience function is provided as
OncoSimulWide2Long in OncoSimulR.
As an example, we will use the data we generated above for section 6.5.1.
## Convert the data
lb1 <- OncoSimulWide2Long(b1)
## Install the streamgraph package from GitHub and load
library(devtools)
devtools::install_github("hrbrmstr/streamgraph")
library(streamgraph)
## Stream plot for Genotypes
sg_legend(streamgraph(lb1, Genotype, Y, Time, scale = "continuous"),
show=TRUE, label="Genotype: ")
## Staked area plot and we use the pipe
streamgraph(lb1, Genotype, Y, Time, scale = "continuous",
offset = "zero") %>%
sg_legend(show=TRUE, label="Genotype: ")6.7 Multiple initial mutants: starting the simulation from arbitrary configurations
You can specify the population composition when you start the
simulation: in other words, you can use multiple initial
mutants. Simply pass a vector to initMutant and a vector of the
same length to initSize: the first are the genotypes/clones, the
second the population sizes of the corresponding genotypes/clones.
(It often makes no sense to start the simulation with genotypes with birth rate of 0: you can try it, but you will be told about it.)
Two examples.
r2 <- rfitness(6)
## Make sure these always viable for interesting stuff
r2[2, 7] <- 1 + runif(1) # A
r2[4, 7] <- 1 + runif(1) # C
r2[8, 7] <- 1 + runif(1) # A, B
o2 <- allFitnessEffects(genotFitness = r2)
ag <- evalAllGenotypes(o2)
out1 <- oncoSimulIndiv(o2, initMutant = c("A", "C"),
initSize = c(100, 200),
onlyCancer = FALSE,
finalTime = 200)
No WT, nor any other genotypes with a single mutation (except “A” and “C”) would thus be possible either (it is impossible to obtain, say, a “B” if there are no WT).
We can do something similar with the frequency-dependent functionality (section 10):
gffd0 <- data.frame(
Genotype = c(
"A", "A, B",
"C", "C, D", "C, E"),
Fitness = c(
"1.3",
"1.4",
"1.4",
"1.1 + 0.7*((f_A + f_A_B) > 0.3)",
"1.2 + sqrt(f_A + f_C + f_C_D)"))
afd0 <- allFitnessEffects(genotFitness = gffd0,
frequencyDependentFitness = TRUE)
## frequencyType set to 'auto'
sp <- 1:5
names(sp) <- c("A", "C", "A, B", "C, D", "C, E")
eag0 <- evalAllGenotypes(afd0, spPopSizes = sp)
os0 <- oncoSimulIndiv(afd0,
initMutant = c("A", "C"),
finalTime = 20, initSize = c(1e4, 1e5),
onlyCancer = FALSE, model = "McFLD")6.8 Multispecies simulations
Since we can use arbitrary initial populations to start the simulation (section 6.7) and we can use arbitrary fitness specifications, you can run multi-species simulations using a simple trick.
Suppose you want to use a two species simulation, where the first species has two loci and the second three loci. This is a possible procedure:
- Create a genotype of total length (1 + 2) + (1 + 3).
- The first locus in each set we will use as the “species indicator”.
- Thus, Species A will use loci 1, 2, 3 and Species B loci 4 to 7.
- Make all genotype combinations where loci from different species are mutated lethal (e.g., a genotype with loci 2 and 5 mutated is not viable).
- Now you can start the simulation from the “WT of each species”: the initMutants will have genotypes with only loci 1 or loci 4 mutated.
This trick is really only an approximation: mutation to the other species is actually death. So there is “leakage” from mutation to death, as in example 10.4: both species are leaking a small number of children via mutation to non-viable “hybrids”. Factor this into your equations for death rate, but this should be negligible if death rate \(\gg\) mutation rate. (You can ameliorate this problem slightly by making mutation to the “species indicator” locus very small, say \(10^{-10}\) —do not set it to 0, as you will get an error).
Of course, you can extend the scheme above to arbitrary numbers of species.
Let’s give several examples.
We use a capital letter for the “species indicator locus” and name each of the species-specific loci with the lower case and a number. We then ameliorate the leakage issue by making mutation to “A” or “B” tiny (though there is still leakage from, say, “A” to “A, b1”).
mspec <- data.frame(
Genotype = c("A",
"A, a1", "A, a2", "A, a1, a2",
"B",
"B, b1", "B, b2", "B, b3",
"B, b1, b2", "B, b1, b3", "B, b1, b2, b3"),
Fitness = 1 + runif(11)
)
fmspec <- allFitnessEffects(genotFitness = mspec)
## Warning in allGenotypes_to_matrix(x, frequencyDependentFitness): No
## WT genotype. Setting its fitness to 1.
afmspec <- evalAllGenotypes(fmspec)
## Show only viable ones
afmspec[afmspec$Fitness >= 1, ]
## Genotype Fitness
## 1 A 1.932927
## 2 B 1.455660
## 9 A, a1 1.693240
## 10 A, a2 1.990411
## 16 B, b1 1.872527
## 17 B, b2 1.650764
## 18 B, b3 1.799822
## 34 A, a1, a2 1.552544
## 51 B, b1, b2 1.355038
## 52 B, b1, b3 1.375106
## 93 B, b1, b2, b3 1.766015
muv <- c(1e-10, rep(1e-5, 2), 1e-10, rep(1e-5, 3))
names(muv) <- c("A", paste0("a", 1:2), "B", paste0("b", 1:3))
out1 <- oncoSimulIndiv(fmspec, initMutant = c("A", "B"),
initSize = c(100, 200),
mu = muv,
onlyCancer = FALSE,
finalTime = 200)We can do something similar with the frequency-dependent-fitness functionality. (We use a somewhat silly specification, so that checking equations is easy)
mspecF <- data.frame(
Genotype = c("A",
"A, a1", "A, a2", "A, a1, a2",
"B",
"B, b1", "B, b2", "B, b3",
"B, b1, b2", "B, b1, b3", "B, b1, b2, b3"),
Fitness = c("1 + f_A_a1",
"1 + f_A_a2",
"1 + f_A_a1_a2",
"1 + f_B",
"1 + f_B_b1",
"1 + f_B_b2",
"1 + f_B_b3",
"1 + f_B_b1_b2",
"1 + f_B_b1_b3",
"1 + f_B_b1_b2_b3",
"1 + f_A")
)
fmspecF <- allFitnessEffects(genotFitness = mspecF,
frequencyDependentFitness = TRUE)
## frequencyType set to 'auto'
## Remeber, spPopSizes correspond to the genotypes
## shown in
fmspecF$full_FDF_spec
## A B a1 a2 b1 b2 b3 Genotype_as_numbers Genotype_as_letters
## 1 1 0 0 0 0 0 0 1 A
## 2 0 1 0 0 0 0 0 2 B
## 3 1 0 1 0 0 0 0 1, 3 A, a1
## 4 1 0 0 1 0 0 0 1, 4 A, a2
## 5 0 1 0 0 1 0 0 2, 5 B, b1
## 6 0 1 0 0 0 1 0 2, 6 B, b2
## 7 0 1 0 0 0 0 1 2, 7 B, b3
## 8 1 0 1 1 0 0 0 1, 3, 4 A, a1, a2
## 9 0 1 0 0 1 1 0 2, 5, 6 B, b1, b2
## 10 0 1 0 0 1 0 1 2, 5, 7 B, b1, b3
## 11 0 1 0 0 1 1 1 2, 5, 6, 7 B, b1, b2, b3
## Genotype_as_fvars Fitness_as_fvars Fitness_as_letters
## 1 f_1 1 + f_1_3 1 + f_A_a1
## 2 f_2 1 + f_2_5 1 + f_B_b1
## 3 f_1_3 1 + f_1_4 1 + f_A_a2
## 4 f_1_4 1 + f_1_3_4 1 + f_A_a1_a2
## 5 f_2_5 1 + f_2_6 1 + f_B_b2
## 6 f_2_6 1 + f_2_7 1 + f_B_b3
## 7 f_2_7 1 + f_2_5_6 1 + f_B_b1_b2
## 8 f_1_3_4 1 + f_2 1 + f_B
## 9 f_2_5_6 1 + f_2_5_7 1 + f_B_b1_b3
## 10 f_2_5_7 1 + f_2_5_6_7 1 + f_B_b1_b2_b3
## 11 f_2_5_6_7 1 + f_1 1 + f_A
## in exactly that order if it is unnamed.
afmspecF <- evalAllGenotypes(fmspecF,
spPopSizes = 1:11)
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Alternatively, pass a named vector, which is the recommended approach
spp <- 1:11
names(spp) <- c("A","B",
"A, a1", "A, a2",
"B, b1", "B, b2", "B, b3",
"A, a1, a2",
"B, b1, b2", "B, b1, b3", "B, b1, b2, b3")
afmspecF <- evalAllGenotypes(fmspecF,
spPopSizes = spp)
## Show only viable ones
afmspecF[afmspecF$Fitness >= 1, ]
## Genotype Fitness
## 2 A 1.045455
## 3 B 1.075758
## 10 A, a1 1.060606
## 11 A, a2 1.121212
## 17 B, b1 1.090909
## 18 B, b2 1.106061
## 19 B, b3 1.136364
## 35 A, a1, a2 1.030303
## 52 B, b1, b2 1.151515
## 53 B, b1, b3 1.166667
## 94 B, b1, b2, b3 1.015152
## Expected values of fitness
exv <- 1 + c(3, 5, 4, 8, 6, 7, 9, 2, 10, 11, 1)/sum(1:11)
stopifnot(isTRUE(all.equal(exv, afmspecF[afmspecF$Fitness >= 1, ]$Fitness)))
muv <- c(1e-10, rep(1e-5, 2), 1e-10, rep(1e-5, 3))
names(muv) <- c("A", paste0("a", 1:2), "B", paste0("b", 1:3))
out1 <- oncoSimulIndiv(fmspecF, initMutant = c("A", "B"),
initSize = c(1e4, 1e5),
mu = muv,
finalTime = 20,
model = "McFLD",
onlyCancer = FALSE)
## Init Mutant with birth == 1.0
## Init Mutant with birth == 1.0Some further examples are given below, as in 10.4.2.
7 Sampling multiple simulations
Often, you will want to simulate multiple runs of the same scenario, and then obtain the matrix of runs by mutations (a matrix of individuals/samples by genes or, equivalently, a vector of “genotypes”), and do something with them. OncoSimulR offers several ways of doing this.
The key function here is samplePop, either called explicitly
after oncoSimulPop (or oncoSimulIndiv), or implicitly as
part of a call to oncoSimulSample. With samplePop you can
use single cell or whole tumor sampling (for details see the
help of samplePop). Depending on how the simulations were
conducted, you might also sample at different times, or as a
function of population sizes. A major difference between procedures
has to do with whether or not you want to keep the complete history
of the simulations.
You want to keep the complete history of population sizes of clones during the simulations. You will simulate using:
oncoSimulIndivrepeatedly (maybe withinmclapply, to parallelize the run).oncoSimulPop.oncoSimulPopis basically a thin wrapper aroundoncoSimulIndivthat usesmclapply.
In both cases, you specify the conditions for ending the simulations
(as explained in 6.3). Then, you use function
samplePop to obtain the matrix of samples by mutations.
You do not want to keep the complete history of population sizes of clones during the simulations. You will simulate using:
oncoSimulIndivrepeatedly, with argumentkeepEvery = NA.oncoSimulPop, with argumentkeepEvery = NA.In both cases you specify the conditions for ending the simulations (as explained in 6.3). Then, you use function
samplePop.oncoSimulSample, specifying the conditions for ending the simulations (as explained in 6.3). In this case, you will not usesamplePop, as that is implicitly called byoncoSimulSample. The output is directly the matrix (and a little bit of summary from each run), and during the simulation it only stores one time point.
Why the difference between the above cases? If you keep the complete
history of population sizes, you can take samples at any of the
times between the beginning and the end of the simulations. If you
do not keep the history, you can only sample at the time the
simulation exited (see section 15.2). Why would you
want to use the second route? If we are only interested in the final
matrix of individuals by mutations, keeping the complete history
above is wasteful because we store fully all of the simulations (for
example in the call to oncoSimulPop) and then sample (in the call
to samplePop).
Further criteria to use when choosing between sampling procedures is
whether you need detectionSize and detectionDrivers do differ
between simulations: if you use oncoSimulPop the arguments for
detectionSize and detectionDrivers must be the same for all
simulations but this is not the case for oncoSimulSample. See
further comments in 7.2. Finally, parallelized
execution is available for oncoSimulPop but, by design, not for
oncoSimulSample.
The following are a few examples. First we run oncoSimulPop to obtain 4
simulations and in the last line we sample from them:
pancrPop <- oncoSimulPop(4, pancr,
detectionSize = 1e7,
keepEvery = 10,
mc.cores = 2)
summary(pancrPop)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 10 10585920 10519876 0 0
## 2 10 10787256 10682097 0 0
## 3 17 10950556 8096187 0 0
## 4 13 11040862 10374965 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 364 1040
## 2 0 0 555 1231
## 3 0 0 2068 2580
## 4 0 0 253 937
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE NA 142857.1 142857.1
## 2 FALSE FALSE NA 142857.1 142857.1
## 3 FALSE FALSE NA 142857.1 142857.1
## 4 FALSE FALSE NA 142857.1 142857.1
## OccurringDrivers
## 1
## 2
## 3
## 4
samplePop(pancrPop)
##
## Subjects by Genes matrix of 4 subjects and 7 genes.
## CDNK2A KRAS MLL3 PXDN SMAD4 TGFBR2 TP53
## [1,] 0 1 0 0 0 0 0
## [2,] 0 1 0 0 0 0 0
## [3,] 0 1 0 0 1 0 0
## [4,] 0 1 0 0 0 0 0Now a simple multiple call to oncoSimulIndiv wrapped inside
mclapply; this is basically the same we just did above. We set
the class of the object to allow direct usage of
samplePop. (Note: in Windows mc.cores > 1 is not
supported, so for the vignette to run in Windows, Linux, and Mac we
explicitly set it here in the call to mclapply. For regular
usage, you will not need to do this; just use whatever is appropriate for
your operating system and number of cores. As well, we do not need any of
this with oncoSimulPop because the code inside
oncoSimulPop already takes care of setting mc.cores
to 1 in Windows).
library(parallel)
if(.Platform$OS.type == "windows") {
mc.cores <- 1
} else {
mc.cores <- 2
}
p2 <- mclapply(1:4, function(x) oncoSimulIndiv(pancr,
detectionSize = 1e7,
keepEvery = 10),
mc.cores = mc.cores)
class(p2) <- "oncosimulpop"
samplePop(p2)
##
## Subjects by Genes matrix of 4 subjects and 7 genes.
## CDNK2A KRAS MLL3 PXDN SMAD4 TGFBR2 TP53
## [1,] 0 1 0 0 0 0 0
## [2,] 0 1 0 0 1 0 0
## [3,] 0 1 0 0 0 0 0
## [4,] 0 1 0 0 0 0 0Above, we have kept the complete history of the simulations as you can check by doing, for instance
tail(pancrPop[[1]]$pops.by.time)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [33,] 320 1133 0 0 125656 0 0 0 0 0
## [34,] 330 1038 13 0 344578 3 0 0 1 0
## [35,] 340 919 260 0 951715 51 0 4 88 1
## [36,] 350 907 2086 0 2579255 379 0 3 1072 4
## [37,] 360 863 15724 0 7053271 3649 24 4 8680 5
## [38,] 364 907 36465 1 10519876 8126 86 9 20252 3
## [,11]
## [33,] 0
## [34,] 0
## [35,] 0
## [36,] 7
## [37,] 93
## [38,] 195If we were not interested in the complete history of simulations we could
have done instead (note the argument keepEvery = NA)
pancrPopNH <- oncoSimulPop(4, pancr,
detectionSize = 1e7,
keepEvery = NA,
mc.cores = 2)
summary(pancrPopNH)
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 10 10807980 10711808 0 0
## 2 10 10010137 9640132 0 0
## 3 10 10267869 10086812 0 0
## 4 13 10907229 6686739 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 786 1521
## 2 0 0 2133 2783
## 3 0 0 1823 2502
## 4 0 0 410 983
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE NA 142857.1 142857.1
## 2 FALSE FALSE NA 142857.1 142857.1
## 3 FALSE FALSE NA 142857.1 142857.1
## 4 FALSE FALSE NA 142857.1 142857.1
## OccurringDrivers
## 1
## 2
## 3
## 4
samplePop(pancrPopNH)
##
## Subjects by Genes matrix of 4 subjects and 7 genes.
## CDNK2A KRAS MLL3 PXDN SMAD4 TGFBR2 TP53
## [1,] 0 1 0 0 0 0 0
## [2,] 0 1 0 0 0 0 0
## [3,] 0 1 0 0 0 0 0
## [4,] 0 1 0 0 0 0 0which only keeps the very last sample:
pancrPopNH[[1]]$pops.by.time
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 786 1746 16776 2 10711808 65183 1 6 893 7 11558Or we could have used oncoSimulSample:
pancrSamp <- oncoSimulSample(4, pancr)
## Successfully sampled 4 individuals
##
## Subjects by Genes matrix of 4 subjects and 7 genes.
pancrSamp$popSamp
## CDNK2A KRAS MLL3 PXDN SMAD4 TGFBR2 TP53
## [1,] 0 1 0 0 0 0 0
## [2,] 0 1 0 0 0 0 0
## [3,] 0 1 1 0 0 0 0
## [4,] 0 1 0 0 1 0 0Again, why the above differences? If we are only interested in the
final matrix of populations by mutations, keeping the complete
history the above is wasteful, because we store fully all of the
simulations (in the call to oncoSimulPop) and then sample (in
the call to samplePop).
7.1 Whole-tumor and single-cell sampling, and do we always want to sample?
samplePop is designed to emulate the process of obtaining a sample
from a (set of) “patient(s)”. But there is no need to sample. The
history of the population, with a granularity that is controlled by
argument keepEvery, is kept in the matrix pops.by.time which
contains the number of cells of every clone at every sampling point
(see further details in 15.2). This is the information
used in the plots that show the trajectory of a simulation: the
plots that show the change in genotype or driver abundance over time
(see section 6.4 and examples mentioned there).
Regardless of whether and how you plot the information in pops.by.time,
you can also sample one or multiple simulations using samplePop. In
whole-tumor sampling the resolution is the whole tumor (or the whole
population). Thus, a key argument is thresholdWhole, the threshold for
detecting a mutation: a gene is considered mutated if it is altered in at
least “thresholdWhole” proportion of the cells in that simulation (at a
particular time point). This of course means that your “sampled genotype”
might not correspond to any existing genotype because we are summing over
all cells in the population. For instance, suppose that at the time we take
the sample there are only two clones in the population, one clone with a
frequency of 0.4 that has gene A mutated, and a second clone one with a
frequency of 0.6 that has gene B mutated. If you set thresholdWhole to
values \(\leq 0.4\) the sampled genotype will show both A and B
mutated. Single-cell sampling is provided as an option in contrast to
whole-tumor sampling. Here any sampled genotype will correspond to an
existing genotype as you are sampling with single-cell resolution.
When samplePop is run on a set of simulated data of, say, 100 simulated
trajectories (100 “subjects”), it will produce a matrix with 100 rows (100
“subjects”). But if it makes sense in the context of your problem (e.g.,
multiple samples per patient?) you can of course run samplePop repeatedly.
7.2 Differences between “samplePop” and “oncoSimulSample”
samplePop provides two sampling times: “last” and
“uniform”. It also allows you to sample at the first sample time(s) at
which the population(s) reaches a given size, which can be either the same
or different for each simulation (with argument popSizeSample).
“last” means to sample each individual in the very last time period of the
simulation. “uniform” means sampling each individual at a time chosen
uniformly from all the times recorded in the simulation between the time
when the first driver appeared and the final time period. “unif” means
that it is almost sure that different individuals will be sampled at
different times. “last” does not guarantee that different individuals will
be sampled at the same time unit, only that all will be sampled in the
last time unit of their simulation.
With oncoSimulSample we obtain samples that correspond to
timeSample = "last" in samplePop by specifying a
unique value for detectionSize and
detectionDrivers. The data from each simulation will
correspond to the time point at which those are reached (analogous to
timeSample = "last"). How about uniform sampling? We pass a
vector of detectionSize and detectionDrivers,
where each value of the vector comes from a uniform distribution. This is
not identical to the “uniform” sampling of oncoSimulSample,
as we are not sampling uniformly over all time periods, but are stopping
at uniformly distributed values over the stopping conditions. Arguably,
however, the procedure in samplePop might be closer to what we
mean with “uniformly sampled over the course of the disease” if that
course is measured in terms of drivers or size of tumor.
An advantage of oncoSimulSample is that we can specify
arbitrary sampling schemes, just by passing the appropriate vector
detectionSize and detectionDrivers. A disadvantage
is that if we change the stopping conditions we can not just resample the
data, but we need to run it again.
There is no difference between oncoSimulSample and
oncoSimulPop + samplePop in terms of the
typeSample argument (whole tumor or single cell).
Finally, there are some additional differences between the two
functions. oncoSimulPop can run parallelized (it uses
mclapply). This is not done with oncoSimulSample
because this function is designed for simulation experiments where you
want to examine many different scenarios simultaneously. Thus, we provide
additional stopping criteria (max.wall.time.total and
max.num.tries.total) to determine whether to continue running the
simulations, that bounds the total running time of all the simulations in
a call to oncoSimulSample. And, if you are running multiple
different scenarios, you might want to make multiple, separate,
independent calls (e.g., from different R processes) to
oncoSimulSample, instead of relying in mclapply,
since this is likely to lead to better usage of multiple cores/CPUs if you
are examining a large number of different scenarios.
8 Showing the genealogical relationships of clones
If you run simulations with keepPhylog = TRUE, the simulations keep track
of when every clone is generated, and that will allow us to see the
parent-child relationships between clones. (This is disabled by default).
Let us re-run a previous example:
set.seed(15)
tmp <- oncoSimulIndiv(examplesFitnessEffects[["o3"]],
model = "McFL",
mu = 5e-5,
detectionSize = 1e8,
detectionDrivers = 3,
sampleEvery = 0.025,
max.num.tries = 10,
keepEvery = 5,
initSize = 2000,
finalTime = 20000,
onlyCancer = FALSE,
extraTime = 1500,
keepPhylog = TRUE)
tmp
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = examplesFitnessEffects[["o3"]], model = "McFL",
## mu = 5e-05, detectionSize = 1e+08, detectionDrivers = 3,
## sampleEvery = 0.025, initSize = 2000, keepEvery = 5, extraTime = 1500,
## finalTime = 20000, onlyCancer = FALSE, keepPhylog = TRUE,
## max.num.tries = 10)
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 7 3984 3984 3 3
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 3 3 7178.375 288895
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.01343206 6253.799 6666.667
## OccurringDrivers
## 1 d, f, m
##
## Final population composition:
## Genotype N
## 1 _ 0
## 2 d _ 0
## 3 d > m _ 0
## 4 f _ 0
## 5 m _ 0
## 6 m > d _ 0
## 7 m > d > f _ 3984We can plot the parent-child relationships14 of every clone ever created (with fitness larger than 0 —clones without viability are never shown):
plotClonePhylog(tmp, N = 0)
However, we often only want to show clones that exist (have number of cells \(>0\)) at a certain time (while of course showing all of their ancestors, even if those are now extinct —i.e., regardless of their current numbers).
plotClonePhylog(tmp, N = 1)
If we set keepEvents = TRUE the arrows show how many times each
clone appeared:
(The next can take a while)
plotClonePhylog(tmp, N = 1, keepEvents = TRUE)
And we can show the plot so that the vertical axis is proportional to time (though you might see overlap of nodes if a child node appeared shortly after the parent):
plotClonePhylog(tmp, N = 1, timeEvents = TRUE)
We can obtain the adjacency matrix doing
get.adjacency(plotClonePhylog(tmp, N = 1, returnGraph = TRUE))
## 4 x 4 sparse Matrix of class "dgCMatrix"
## _ m _ m > d _ m > d > f _
## _ . 1 . .
## m _ . . 1 .
## m > d _ . . . 1
## m > d > f _ . . . .We can see another example here:
set.seed(456)
mcf1s <- oncoSimulIndiv(mcf1,
model = "McFL",
mu = 1e-7,
detectionSize = 1e8,
detectionDrivers = 100,
sampleEvery = 0.025,
keepEvery = 2,
initSize = 2000,
finalTime = 1000,
onlyCancer = FALSE,
keepPhylog = TRUE)Showing only clones that exist at the end of the simulation (and all their parents):
plotClonePhylog(mcf1s, N = 1)
Notice that the labels here do not have a "_", since there were no order effects in fitness. However, the labels show the genes that are mutated, just as before.
Similar, but with vertical axis proportional to time:
par(cex = 0.7)
plotClonePhylog(mcf1s, N = 1, timeEvents = TRUE)
What about those that existed in the last 200 time units?
par(cex = 0.7)
plotClonePhylog(mcf1s, N = 1, t = c(800, 1000))
And try now to show also when the clones appeared (we restrict the time to between 900 and 1000, to avoid too much clutter):
par(cex = 0.7)
plotClonePhylog(mcf1s, N = 1, t = c(900, 1000), timeEvents = TRUE)
(By playing with t, it should be possible to obtain animations of
the phylogeny. We will not pursue it here.)
If the previous graph seems cluttered, we can represent it in a different way by calling igraph directly after storing the graph and using the default layout:
g1 <- plotClonePhylog(mcf1s, N = 1, t = c(900, 1000),
returnGraph = TRUE)plot(g1)
which might be easier to show complex relationships or identify central or key clones.
It is of course quite possible that, especially if we consider few genes, the parent-child relationships will form a network, not a tree, as the same child node can have multiple parents. You can play with this example, modified from one we saw before (section 3.4.6):
op <- par(ask = TRUE)
while(TRUE) {
tmp <- oncoSimulIndiv(smn1, model = "McFL",
mu = 5e-5, finalTime = 500,
detectionDrivers = 3,
onlyCancer = FALSE,
initSize = 1000, keepPhylog = TRUE)
plotClonePhylog(tmp, N = 0)
}
par(op)8.1 Parent-child relationships from multiple runs
If you use oncoSimulPop you can store and plot the
“phylogenies” of the different runs:
oi <- allFitnessEffects(orderEffects =
c("F > D" = -0.3, "D > F" = 0.4),
noIntGenes = rexp(5, 10),
geneToModule =
c("F" = "f1, f2, f3",
"D" = "d1, d2") )
oiI1 <- oncoSimulIndiv(oi, model = "Exp")
oiP1 <- oncoSimulPop(4, oi,
keepEvery = 10,
mc.cores = 2,
keepPhylog = TRUE)We will plot the first two:
op <- par(mar = rep(0, 4), mfrow = c(2, 1))
plotClonePhylog(oiP1[[1]])
plotClonePhylog(oiP1[[2]])
par(op)This is so far disabled in function oncoSimulSample, since
that function is optimized for other uses. This might change in the future.
9 Generating random fitness landscapes
9.1 Random fitness landscapes from a Rough Mount Fuji model
In most of the examples seen above, we have fully specified the fitness of the different genotypes (either by providing directly the full mapping genotypes to fitness, or by providing that mapping by specifying the effects of the different gene combinations). In some cases, however, we might want to specify a particular model that generates the fitness landscape, and then have fitnesses be random variables obtained under this model. In other words, in this random fitness landscape the fitness of the genotypes is a random variable generated under some specific model. Random fitness landscapes are used extensively, for instance, to understand the evolutionary consequences of different types of epistatic interactions (e.g., Szendro, Schenk, et al. 2013; Franke et al. 2011) and there are especially developed tools for plotting and analyzing random fitness landscapes (e.g., Brouillet et al. 2015).
With OncoSimulR it is possible to generate mappings of genotype to
fitness using the function rfitness that allows you to use from a
pure House of Cards model to a purely additive model (see
9.2 for NK model). I have followed
Szendro, Schenk, et al. (2013) and Franke et al. (2011) and model
fitness as
\[\begin{equation} f_i = -c\ d(i, reference) + x_i \label{eq:1} \end{equation}\]
where \(d(i, j)\) is the Hamming distance between genotypes \(i\) and \(j\) (the
number of positions that differ), \(c\) is the decrease in fitness of a
genotype per each unit increase in Hamming distance from the reference
genotype, and \(x_i\) is a random variable (in this
case, a normal deviate of mean 0 and standard deviation \(sd\)). You can
change the reference genotype to any of the genotypes: for the
deterministic part, you make the fittest genotype be the one with all
positions mutated by setting reference = "max", or use the
wildtype by using a string of 0s, or randomly select a genotype as a
reference by using reference = "random" or reference = "random2".
And by changing \(c\) and \(sd\) you can flexibly modify the relative weight
of the purely House of Cards vs. additive component. The expression used
above is also very similar to the one on Greene and Crona (2014) if you use
rfitness with the argument reference = "max".
What can you do with these genotype to fitness mappings? You could plot
them, you could use them as input for oncoSimulIndiv and related
functions, or you could export them (to_Magellan) and plot them externally
(e.g., in MAGELLAN: http://wwwabi.snv.jussieu.fr/public/Magellan/,
Brouillet et al. (2015)).
## A small example
rfitness(3)
## A B C Fitness
## [1,] 0 0 0 1.0000000
## [2,] 1 0 0 1.3484510
## [3,] 0 1 0 1.0170041
## [4,] 0 0 1 1.0572110
## [5,] 1 1 0 2.3799363
## [6,] 1 0 1 0.9259539
## [7,] 0 1 1 2.1454996
## [8,] 1 1 1 0.9874939
## attr(,"class")
## [1] "matrix" "array"
## [3] "genotype_fitness_matrix"
## A 5-gene example, where the reference genotype is the
## one with all positions mutated, similar to Greene and Crona,
## 2014. We will plot the landscape and use it for simulations
## We downplay the random component with a sd = 0.5
r1 <- rfitness(5, reference = rep(1, 5), sd = 0.6)
plot(r1)
oncoSimulIndiv(allFitnessEffects(genotFitness = r1))
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = allFitnessEffects(genotFitness = r1))
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 10 212432090 154439733 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 333 1394
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE NA 85790.66 2e+05
## OccurringDrivers
## 1
##
## Final population composition:
## Genotype N
## 1 2070
## 2 A, C, D 473
## 3 A, D 240
## 4 B 0
## 5 B, C, D 553
## 6 B, D 238
## 7 C, D 154439733
## 8 C, D, E 326
## 9 D 57856440
## 10 D, E 1320179.2 Random fitness landscapes from Kauffman’s NK model
You can also use Kauffman’s NK model (e.g., Ferretti et al. 2016; Brouillet et al. 2015). We call the function fl_generate from
MAGELLAN (Brouillet et al. 2015).
rnk <- rfitness(5, K = 3, model = "NK")
plot(rnk)
oncoSimulIndiv(allFitnessEffects(genotFitness = rnk))
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = allFitnessEffects(genotFitness = rnk))
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 7 200041229 200036350 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 388.6968 1733
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE NA 103231 2e+05
## OccurringDrivers
## 1
##
## Final population composition:
## Genotype N
## 1 1019
## 2 A 0
## 3 A, C 2570
## 4 B, C 661
## 5 C 200036350
## 6 C, D 345
## 7 C, E 2849.3 Random fitness landscapes from an additive model
This model evaluates fitness with different contributions of each allele, which will be randomly generated.
Given a number a genes by the user, the code uses rnorm to generate random contribution for the mutated allele in each locus. Later, this constributions will be used in the generation of the matrix that gives the value of fitness for each combination of wild type/mutated alleles by addition of the values for each locus and combination.
radd <- rfitness(4, model = "Additive", mu = 0, sd = 0.5)
plot(radd)
9.4 Random fitness landscapes from Eggbox model
You can also use Eggbox model (e.g., Ferretti et al. 2016; Brouillet et al. 2015), where each locus is either high or low fitness
(depending on the “e” parameter value), with a systematic change between
each neighbor. We call the function fl_generate from MAGELLAN
(Brouillet et al. 2015) to generate these landscapes.
regg <- rfitness(4, model = "Eggbox", e = 1, E = 0.5)
plot(regg)
9.5 Random fitness landscapes from Ising model
In the Ising model (e.g., Ferretti et al. 2016; Brouillet et al. 2015), loci are arranged sequentially and each
locus interacts with its physical neighbors. For each pair of interacting
loci, there is a cost to (log)fitness if both alleles are not identical (and
therefore ‘compatible’); in this case, the cost for incompatibility i is
applied. The last and the first loci will interact only if ‘circular’ is
set. The implementation of this model is decribedin
(Brouillet et al. 2015), and we use a call to MAGELLAN code to
generate the landscape.
ris <- rfitness(g = 4, model = "Ising", i = 0.002, I = 0.45)
plot(ris)
9.6 Random fitness landscapes from Full models
MAGELLAN also offers the possibility to combine different models with their own parameters in order to generate a Full model. The models combined are:
- House of cards:
His the number of interacting genes. - Multiplicative (what we call additive):
sandSmean and SD for generating random fitnesses.dis a diminishing (negative) or increasing (positive) return as you approach the peak. - Kauffman NK: each locus interacts with any other
Kloci that can be chosen randomly pasing “r = TRUE” or among its neigbors. - RMF: explained at 9.1.
- Ising:
iandImean and SD for incompatibility. Ifcircularoption is provided, the last and first alleles can interact (circular arrangement). - Eggbox: In this model, each locus is considered as high or low fitness. From
one locus to another the fitness changes its sign so that one is benefficial
and its neigbor is detrimental.
eandEare fitness and noise for fitness. - Optimum: there is an optimum fitness contribution defined by
o\(mu\) andO\(sigma\) and every locus has a productionpandP(also mean and sd respectively).
All models can be taken into account for the fitness calculation. With
default parameters, neither of Ising, Eggbox or Optimum contribute to
fitness lanadcape generation as all i, e, o and p all == 0.
Also, as all parameters refering to standard deviations have value == -1, those are also have no effect unless changed. Further details can be
found in MAGELLAN’s webpage
http://wwwabi.snv.jussieu.fr/public/Magellan/ and
(Brouillet et al. 2015).
rnk <- rfitness(4, model = "Full", i = 0.5, I = 0.1,
e = 2, s = 0.3, S = 0.08)
plot(rnk)
9.7 Epistasis and fitness landscape statistics
We can call MAGELLAN’s (Brouillet et al. 2015) fl_statistics to
obtain fitness landscape statistics, including measures of sign and
reciprocal sign epistasis. See the help of Magellan_stats for further
details on output format. For example:
rnk1 <- rfitness(6, K = 1, model = "NK")
Magellan_stats(rnk1)
## ngeno npeaks nsinks gamma gamma. r.s
## 64.000 2.000 1.000 0.830 0.733 0.979
## nchains nsteps nori depth magn sign
## 2.000 7.000 4.000 2.000 0.292 0.200
## rsign f.1. X.2. f.3.. mode_f outD_m
## 0.033 0.731 0.269 0.000 1.000 2.030
## outD_v steps_m reach_m fitG_m opt_i mProbOpt_0
## 1.778 3.085 14.774 32.403 41.000 0.292
## opt_i.1 mProbOpt_1
## 59.000 0.708
rnk2 <- rfitness(6, K = 4, model = "NK")
Magellan_stats(rnk2)
## ngeno npeaks nsinks gamma gamma. r.s
## 64.000 8.000 7.000 0.206 0.154 3.568
## nchains nsteps nori depth magn sign
## 4.000 8.000 6.000 3.000 0.417 0.321
## rsign f.1. X.2. f.3.. mode_f outD_m
## 0.263 0.155 0.315 0.530 2.000 2.140
## outD_v steps_m reach_m fitG_m opt_i mProbOpt_0
## 3.524 2.095 12.732 34.875 5.000 0.038
## opt_i.1 mProbOpt_1 opt_i.2 mProbOpt_2 opt_i.3 mProbOpt_3
## 9.000 0.066 14.000 0.246 27.000 0.040
## opt_i.4 mProbOpt_4 opt_i.5 mProbOpt_5 opt_i.6 mProbOpt_6
## 36.000 0.195 42.000 0.180 49.000 0.108
## opt_i.7 mProbOpt_7
## 63.000 0.127(These fitness landscapes are, of course, frequency-independent fitness landscapes; with frequency-dependent fitness, as in section @ref{fdf} fitness landscapes as such are not defined.)
10 Frequency-dependent fitness
(Note that except for the example of 10.2, based on Hurlbut et al. (2018), the examples below are not used because of their biological realism, but rather to show some key features of the software)
With frequency-dependence fitness we can make fitness (actually, birth rate) depend on the frequency of other genotypes. We specify how the fitness (birth rate) of each genotype depends (or not) on other genotypes. Thus, this is similar to the explicit mapping of genotypes to fitness (see 3.1.1), but fitness can be a function of the abundance (relative or absolute) of other genotypes. Frequency-dependent fitness allows you to examine models from game theory and adaptive dynamics. Game theory has long tradition in evolutionary biology (Maynard Smith 1982) and has been widely used in cancer (see, for example, Tomlinson 1997; Basanta and Deutsch 2008; Archetti and Pienta 2019, for classical papers that cover from early uses to a very recent review).
Since birth rate can be an arbitrary function of the frequencies of
other clones, we can model competition, cooperation and mutualism,
parasitism and predation, and commensalism. (Recall that in the
“Exp” model death rate is constant and fixed to 1. In the “McFL” and
“McFLD” models, death rate is density-dependent —but not
frequency-dependent. You can thus model all those phenomena by, for
example, making the effects of clones \(i\), \(j\) on each other and on
their own be asymmetric on their birth rates). See examples in
section 10.4; as explained there, if you use the “Exp”
model, you might want to decrease the value of sampleEvery.
The procedure for working with the frequency-dependent functionality
is the general one with OncoSimulR. We first create a data frame with the mapping between genotypes
and their (frequency-dependent) fitness, similar to section
3.1.1. For example, a two-column data frame, where the
first column are the genotypes and the second column contains, as
strings, the expressions for the function that relate fitness to
frequencies of other genotypes. (We can also use a data frame with
\(g + 1\) columns; each of the first \(g\) columns contains a 1 or a 0
indicating that the gene of that column is mutated or not. Column
\(g+ 1\) contains the expressions for the fitness specifications; see
oncoSimulIndiv and allFitnessEffects for examples). Once this
data frame is created, we pass it to allFitnessEffects. From
there, simulation proceeds as usual.
How complex can the functions that specify fitness be? We use library ExprTk for the fitness specifications so the range of functions you can use is very large (http://www.partow.net/programming/exprtk/), including of course the usual arithmetic expressions, logical expressions (so you can model thresholds or jumps and use step functions), and a wide range of mathematical functions (so linear, non-linear, convex, concave, etc, functions can be used, including of course affine fitness functions as in Gerstung, Nakhoul, and Beerenwinkel 2011).
10.1 A first example with frequency-dependent fitness
The following is an arbitrary example. We will model birth rate of
some genotypes as a function of the relative frequencies of other
genotypes; we use f_1 to denote the relative frequency of the
genotype with the first gene mutated, f_1_2 to denote the relative
frequency of the genotype with the first and second genes mutated,
etc, and f_ to denote the frequency of the WT genotype —below,
in sections 10.3, 10.5, and 10.4, we
will use absolute number of cells instead of relative frequencies).
(As we have discussed already, instead of f_1 you can, and
probably should for any example except trivial ones, use f_A or
f_genotype expressed as combination of gene names).
As you can see below, the birth rate of genotype “A” \(= 1.2 + 1.5* f\_A\_B\) and that of the wildtype \(= 1 + 1.5* f\_A\_B\). Genotype
“A, B” in this example could be a genotype whose presence leads to
an increase in the growth of other genotypes (maybe via diffusible
factors, induction of angiogenesis, etc). Genotype “B” does not show
frequency-dependence. The birth rate of genotype “C” increases with
the frequency of f_A_B and increases (adding 0.7) with the
frequency of genotypes “A” and “B”, but only if the sum of the
frequencies of genotypes “A” and “B” is larger than 0.3. For
genotype “A, B” its fitness increases with the square root of the
sum of the frequencies of genotypes “A”, “B”, and “C”, but it
decreases (i.e., shows increased intra-clone competition) if its own
frequency is larger than 0.5. Genotypes not defined explicitly have
a fitness of 0.
## Define fitness of the different genotypes
gffd <- data.frame(
Genotype = c("WT", "A", "B", "C", "A, B"),
Fitness = c("1 + 1.5 * f_A_B",
"1.3 + 1.5 * f_A_B",
"1.4",
"1.1 + 0.7*((f_A + f_B) > 0.3) + f_A_B",
"1.2 + sqrt(f_1 + f_C + f_B) - 0.3 * (f_A_B > 0.5)"))(In the data frame creation, we use stringsAsFactors = FALSE
to avoid messages about conversions between factors and characters
in former versions of R).
You could also specify that as
## Define fitness of the different genotypes
gffdn <- data.frame(
Genotype = c("WT", "A", "B", "C", "A, B"),
Fitness = c("1 + 1.5 * f_1_2",
"1.3 + 1.5 * f_1_2",
"1.4",
"1.1 + 0.7*((f_1 + f_2) > 0.3) + f_1_2",
"1.2 + sqrt(f_1 + f_3 + f_2) - 0.3 * (f_1_2 > 0.5)"),
stringsAsFactors = FALSE)but it is strongly preferred to use explicit gene name letters (otherwise, you must keep in mind how R orders names of genes when making the mapping from letters to numbers).
Let us verify that we have specified what we think we have specified
using evalAllGenotypes (we have done this repeatedly in this
vignette, for example in 1.3.2 or 1.7 or
5.1. Because fitness can depend on population sizes of
different populations, we need to pass the
populations sizes at which we want fitness evaluated in
evalAllGenotypes.
Note that when calling allFitnessEffects we have to set the
paramenter frequencyDependentFitness to TRUE. Since we are using
relative frequencies, we can be explicit and specify freqType = "rel" (though it is not needed). We will see below
(10.3, 10.5, and 10.4) several examples
with absolute numbers.
When passing spPopSizes it is also strongly preferred to use a
named vector as that allows the code to run some checks. Otherwise,
the order of the population sizes must be identical to that in the
table with the fitness descriptions (component full_FDF_spec in
the fitness effects object).
evalAllGenotypes(allFitnessEffects(genotFitness = gffd,
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(WT = 100, A = 20, B = 20, C = 30, "A, B" = 0))
## Genotype Fitness
## 1 WT 1.000000
## 2 A 1.300000
## 3 B 1.400000
## 4 C 1.100000
## 5 A, B 1.841689
## 6 A, C 0.000000
## 7 B, C 0.000000
## 8 A, B, C 0.000000
## Notice the warning
evalAllGenotypes(allFitnessEffects(genotFitness = gffd,
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(100, 30, 40, 0, 10))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.083333
## 2 A 1.383333
## 3 B 1.400000
## 4 C 1.855556
## 5 A, B 1.823610
## 6 A, C 0.000000
## 7 B, C 0.000000
## 8 A, B, C 0.000000
evalAllGenotypes(allFitnessEffects(genotFitness = gffd,
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(100, 30, 40, 0, 100))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.555556
## 2 A 1.855556
## 3 B 1.400000
## 4 C 1.470370
## 5 A, B 1.709175
## 6 A, C 0.000000
## 7 B, C 0.000000
## 8 A, B, C 0.000000The numbered one gives the same results. Note as well that using
frequencyType is not needed (the default, auto, infers
the type)
evalAllGenotypes(allFitnessEffects(genotFitness = gffdn,
frequencyDependentFitness = TRUE),
spPopSizes = c(100, 20, 20, 30, 0))
## frequencyType set to 'auto'
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.000000
## 2 A 1.300000
## 3 B 1.400000
## 4 C 1.100000
## 5 A, B 1.841689
## 6 A, C 0.000000
## 7 B, C 0.000000
## 8 A, B, C 0.000000
evalAllGenotypes(allFitnessEffects(genotFitness = gffdn,
frequencyDependentFitness = TRUE),
spPopSizes = c(100, 30, 40, 0, 10))
## frequencyType set to 'auto'
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.083333
## 2 A 1.383333
## 3 B 1.400000
## 4 C 1.855556
## 5 A, B 1.823610
## 6 A, C 0.000000
## 7 B, C 0.000000
## 8 A, B, C 0.000000
evalAllGenotypes(allFitnessEffects(genotFitness = gffdn,
frequencyDependentFitness = TRUE),
spPopSizes = c(100, 30, 40, 0, 100))
## frequencyType set to 'auto'
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.555556
## 2 A 1.855556
## 3 B 1.400000
## 4 C 1.470370
## 5 A, B 1.709175
## 6 A, C 0.000000
## 7 B, C 0.000000
## 8 A, B, C 0.000000The fitness specification is correct. Let us now create the
allFitnessEffects object and simulate. We will use the McFL
model, so in addition to the frequency dependence in the birth
rates, there is also density dependence in the death rate (see
section 3).
afd <- allFitnessEffects(genotFitness = gffd,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(1) ## for reproducibility
sfd <- oncoSimulIndiv(afd,
model = "McFL",
onlyCancer = FALSE,
finalTime = 55, ## short, for speed
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(sfd, show = "genotypes")
There is no need to specify the fitness of all possible genotypes (and no need to always specify the fitness of a WT): those are taken to be 0. But no fitness expression can, thus, contain a function of the genotypes for which fitness is not specified (e.g., suppose you do not pass the fitness of genotype “B”: it will be taken as 0; but no genotype can have, in its fitness, a function such as “2 * f_B”).
10.2 Hurlbut et al., 2018: a four-cell example with angiogenesis and cytotoxicity
The following example is based on Hurlbut et al. (2018). As explained in p. 3 of that paper, “Stromal cancer cells (A-) [WT in the code below] have no particular benefit or cost unique to themselves, and they are considered a baseline neutral cell within the context of the model. In contrast, angiogenesis-factor producing cells (A+) [A in the code below] vascularize the local tumor area which consequently introduces a nutrient rich blood to the benefit of all interacting cells. Nutrient recruitment expands when A+ cells interact with one another. Cytotoxic cells (C) release a chemical compound which harms heterospecific cells and increases their rate of cell death. The cytotoxic cells benefit from the resulting disruption in competition caused by the interaction. For simplicity, our model presumes that cytotoxic cells are themselves immune to this class of agent. Finally, proliferative cells (P) possess a reproductive or metabolic advantage relative to the other cell types. In our model this advantage does not compound with the nutrient enrichment produced by vascularization when A+ cells are present; however, it does place the proliferative cell at a greater vulnerability to cytotoxins.”
They provide, in p. 4, the payoff matrix reproduced in Figure 10.1:
Figure 10.1: Payoff matrix from Table 2 of Hurlbut et al., 2018, ‘Game Theoretical Model of Cancer Dynamics withFour Cell Phenotypes’, Games, 9, 61; doi:10.3390/g9030061.
As explained in p. 6 (equation 1) of Hurlbut et al. (2018), we can write the fitness of the four types as f=Gu, where f and u are the vectors with the four fitnesses and frequencies (where each element of u corresponds to the relative frequencies of stromal, angiogenic, proliferative, and cytotoxic cells), and G is the payoff matrix in Figure 10.1.
To allow modelling scenarios with different values for the parameters in Figure 10.1 we will define a function to create the data frame of frequency-dependent fitnesses. First, we will assume that each one of types A+, P, and C, are all derived from WT by a single mutation in one of three genes, say, A, P, C, respectively.
create_fe <- function(a, b, c, d, e, f, g,
gt = c("WT", "A", "P", "C")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ",
d, " * f_A ",
"- ", c, " * f_C"),
paste0("1 - ", a,
" + ", d, " + ",
f, " * f_A ",
"- ", c, " * f_C"),
paste0("1 + ", g, " + ",
d, " * f_A ",
"- ", c, " * (1 + ",
g, ") * f_C"),
paste0("1 - ", b, " + ",
e, " * f_ + ",
"(", d, " + ",
e, ") * f_A + ",
e , " * f_P")))
}We can check we recover Figure 10.1:
create_fe("a", "b", "c", "d", "e", "f", "g")
## Genotype Fitness
## 1 WT 1 + d * f_A - c * f_C
## 2 A 1 - a + d + f * f_A - c * f_C
## 3 P 1 + g + d * f_A - c * (1 + g) * f_C
## 4 C 1 - b + e * f_ + (d + e) * f_A + e * f_PWe could model a different set of ancestor-dependent relationships:
## Different assumption about origins from mutation:
## WT -> P; P -> A,P; P -> C,P
create_fe2 <- function(a, b, c, d, e, f, g,
gt = c("WT", "A", "P", "C", "A, P", "A, C",
"C, P")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ",
d, " * f_A_P ",
"- ", c, " * f_P_C"),
"0",
paste0("1 + ", g, " + ",
d, " * f_A_P ",
"- ", c, " * (1 + ",
g, ") * f_P_C"),
"0",
paste0("1 - ", a, " + ",
d, " + ",
f, " * f_A_P ",
"- ", c, " * f_P_C"),
"0",
paste0("1 - ", b, " + ",
e, " * f_ + ",
"(", d, " + ",
e, ") * f_A_P + ",
e , " * f_P")),
stringsAsFactors = FALSE)
}
## And check:
create_fe2("a", "b", "c", "d", "e", "f", "g")
## Genotype Fitness
## 1 WT 1 + d * f_A_P - c * f_P_C
## 2 A 0
## 3 P 1 + g + d * f_A_P - c * (1 + g) * f_P_C
## 4 C 0
## 5 A, P 1 - a + d + f * f_A_P - c * f_P_C
## 6 A, C 0
## 7 C, P 1 - b + e * f_ + (d + e) * f_A_P + e * f_PNote: we are writing f_P_C: this is remapped internally to f_C_P
(which is the genotype name, with gene names reordered).
To show two examples, we will run the analyses Hurlbut et al. (2018) use for Figures 3a and 3b (p.8 of their paper):
## Figure 3a
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(0.02, 0.04, 0.08, 0.06,
0.15, 0.1, 0.06),
frequencyDependentFitness = TRUE)
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 160,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
## plot(s_3_a, show = "genotypes",
## xlim = c(40, 200),
## col = c("black", "green", "red", "blue"))
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue"),
thinData = TRUE)
## Figure 3b
afe_3_b <- allFitnessEffects(
genotFitness =
create_fe(0.02, 0.04, 0.08, 0.1,
0.15, 0.1, 0.05),
frequencyDependentFitness = TRUE)
set.seed(2)
## Use a short finalTime, for speed of vignette execution
s_3_b <- oncoSimulIndiv(afe_3_b,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100, ## 160,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
## plot(s_3_b, show = "genotypes",
## col = c("black", "green", "red", "blue"))
plot(s_3_b, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue"),
thinData = TRUE) ## thin for faster plotting
Of course, if we assume that the mutations leading to the different cell types are different, the results can change:
## Figure 3b. Now with WT -> P; P -> A,P; P -> C,P
## For speed, we set finalTiem = 100
afe_3_b_2 <- allFitnessEffects(
genotFitness =
create_fe2(0.02, 0.04, 0.08, 0.1,
0.15, 0.1, 0.05),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(2)
s_3_b_2 <- oncoSimulIndiv(afe_3_b_2,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100, ## 300
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_b_2, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue"),
thinData = TRUE)
(Examples for the remaining Figures 2 and 3 are provided, using also
oncoSimulPop, in file ‘inst/miscell/hurlbut-ex.R’)
10.3 An example with absolute numbers and population collapse
In the following example we use absolute numbers (thus the n_1,
etc, instead of the f_1 in the fitness definition). This is a toy
model where there is a change in fitness in two of the genotypes if
the other is above a specified threshold (this also shows again the
usage of a logical inequality). The genotype with gene B mutated has
birth rate less than 1, unless there are at least more than 10 cells
with genotype A mutated.
gffd3 <- data.frame(Genotype = c("WT", "A", "B"),
Fitness = c("1",
"1 + 0.2 * (n_B > 10)",
".9 + 0.4 * (n_A > 10)"
))
afd3 <- allFitnessEffects(genotFitness = gffd3,
frequencyDependentFitness = TRUE)As usual, let us verify that we have specified what we think we have
specified using evalAllGenotypes:
evalAllGenotypes(allFitnessEffects(genotFitness = gffd3,
frequencyDependentFitness = TRUE),
spPopSizes = c(WT = 100, A = 1, B = 11))
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.2
## 3 B 0.9
## 4 A, B 0.0
evalAllGenotypes(allFitnessEffects(genotFitness = gffd3,
frequencyDependentFitness = TRUE),
spPopSizes = c(WT = 100, A = 11, B = 1))
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.0
## 3 B 1.3
## 4 A, B 0.0In this simulation, the population collapses: genotype B is able to invade the population when there are some A’s around. But as soon as A disappears due to competition from B, B collapses as its birth rate becomes 0.9, less than the death rate; we are using the “McFLD” model (where death rate \(D(N) = \max(1, \log(1 + N/K))\)); see details in 3.2.1.1.
set.seed(1)
sfd3 <- oncoSimulIndiv(afd3,
model = "McFLD",
onlyCancer = FALSE,
finalTime = 200,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)plot(sfd3, show = "genotypes", type = "line")
plot(sfd3, show = "genotypes")
sfd3
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = afd3, model = "McFLD", mu = 1e-04, initSize = 5000,
## finalTime = 200, onlyCancer = FALSE, keepPhylog = FALSE,
## errorHitWallTime = FALSE, errorHitMaxTries = FALSE, seed = NULL)
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 3 0 0 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 173.825 7037
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.007210558 5000 5000
## OccurringDrivers
## 1
##
## Final population composition:
## Genotype N
## 1 0
## 2 A 0
## 3 B 0Had we used the “usual” death rate expression, that can lead to death rates below 1 (see 3.2.1.1), we would have obtained a population that stabilizes around a final value slightly below the initial one (the one that corresponds to a death rate equal to 0.9):
set.seed(1)
sfd4 <- oncoSimulIndiv(afd3,
model = "McFL",
onlyCancer = FALSE,
finalTime = 145,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)plot(sfd4, show = "genotypes", type = "line")
plot(sfd4, show = "genotypes")
sfd4
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = afd3, model = "McFL", mu = 1e-04, initSize = 5000,
## finalTime = 145, onlyCancer = FALSE, keepPhylog = FALSE,
## errorHitWallTime = FALSE, errorHitMaxTries = FALSE, seed = NULL)
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 3 7750 7716 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 145 5941
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.007544433 4858.963 5000
## OccurringDrivers
## 1
##
## Final population composition:
## Genotype N
## 1 0
## 2 A 34
## 3 B 7716## Check final pop size corresponds to birth = death
K <- 5000/(exp(1) - 1)
K
## [1] 2909.884
log1p(4290/K)
## [1] 0.905951810.4 Predator-prey, commensalism, and consumer-resource models
Since birth rates can be arbitrary functions of frequencies of other clones, we can easily model classical ecological models, such as predator-prey, competition, commensalism and, more generally, consumer-resource models (see, for example, section 3.4 in Otto and Day 2007).
OncoSimulR was originally designed as a forward-time genetic simulator; thus, we used to need to use a simple trick to get the system going. For example, suppose we want to model a predator-prey system; we could do this having a WT that can mutate into either preys or predators. (This is no longer necessary since we can specify starting the simulation from populations with arbitrary numbers of different clones; see sections 6.7 and 6.8. But we have not yet updated the examples).
Even with the There is, in fact, a small leakage in the system because both preys and predators are “leaking” a small number of children, via mutation to a non-viable predator-and-prey genotype, but this is negligible relative to their birth/death rates.
We will use below the usual consumer-resource model with Lotka-Volterra equations. To model them, we use the “Exp” model, which has a constant death rate of 1, and directly translate the usual expressions (e.g., expressions 3.15a and 3.15b in p. 73 of Otto and Day 2007) for rates of change into the birth rate. Of course, you can use any other function for how rates of change of each “species” depend on the numbers of the different species, including constant inflow and outflow, Type I, Type II, and Type III functional responses, etc. And you can model systems that involve multiple different types of preys, predators, commensals, etc (see the example in 10.2 for a four-type example).
What about sampleEvery? In “for real” work, and specially with complex
models, you might want to decrease it, or at least examine how much
results are affected by changes in sampleEvery. Decreasing sampleEvery
will result in birth rates being updated more frequently, and we use the
BNB algorithm, which updates all rates only when the whole population is
sampled. (Recall that by default, in the McFL and McFLD models the
setting for sampleEvery is smaller than in the Exp model). Further
discussion of these issues is provided in sections 10.7 and
15.6.
Since the WT genotype is just a trick used to get the system going, we will make sure it disappears from the system soon after we get it to have both preys and predators (and we use the max function to prevent birth rate from ever becoming negative or identically 0). Finally, note that we can make this much more sophisticated; for example, we could get the system going differently by having different mutations from WT to prey and predator. Remember that you can now start the simulation from arbitrary initial compositions (section 6.7). This is left here for historical purposes and to show additional use cases.
10.4.1 Competition
First, create a function that will generate the usual Lotka-Volterra expressions for competition models (see below for this example without using the WT, and directly starting from “S1” and “S2”).
G_fe_LV <- function(r1, r2, K1, K2, a_12, a_21, awt = 1e-4,
gt = c("WT", "S1", "S2")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("max(0.1, 1 - ", awt, " * (n_2 + n_1))"),
paste0("1 + ", r1,
" * ( 1 - (n_1 + ", a_12, " * n_2)/", K1,
")"),
paste0("1 + ", r2,
" * ( 1 - (n_2 + ", a_21, " * n_1)/", K2,
")")
))
}
## Show expressions for birth rates
G_fe_LV("r1", "r2", "K1", "K2", "a_12", "a_21", "awt")
## Genotype Fitness
## 1 WT max(0.1, 1 - awt * (n_2 + n_1))
## 2 S1 1 + r1 * ( 1 - (n_1 + a_12 * n_2)/K1)
## 3 S2 1 + r2 * ( 1 - (n_2 + a_21 * n_1)/K2)Note we use numbers, not letters, in the expressions above (n_1,
etc). This allows for reusing the function, but requires extra care
making sure that the numbers match the order of the genotypes (it is
not a problem with “S1” and “S2”, since there “S1” will always be 1
and “S2” 2; but what about “Predator” and “prey”?)
Remember the above are the birth rates for a model with death rate = 1 (the “Exp” model). That is why we added a 1 to the birth rate. If you subtract 1 from the expressions for the birth rates of predators and prey above you get the standard expressions for the differential equations for Lotka-Volterra model of competition: \(\frac{\mathrm{d}n_1}{\mathrm{d}t} = r_1 n_1 (1 - \frac{n_1 + \alpha_{12} n_2}{K_1})\). (Verbosely: we are simulating using a model where we have, in a rather general expression, \(\frac{\mathrm{d}n_1}{\mathrm{d}t} = (b\ - d)\ n_1\), where \(b\) and \(d\) are the birth and death rates (that could be arbitrary functions of other stuff); these are the birth and death rates used in the BNB algorithm. And when we simulate under the “Exp” model we have \(d = 1\). So just solve for \((b - d) n_1 = r_1 n_1 (1 - \frac{n_1 + \alpha_{12} n_2}{K_1})\), with \(b = 1\) and you get the \(b\) you need to use).
Now, run that model by setting appropriate parameters; see how we have \(a\_12 > 0\) and \(a\_21 > 0\).
fe_competition <-
allFitnessEffects(
genotFitness =
G_fe_LV(1.5, 1.4, 10000, 4000, 0.6, 0.2,
gt = c("WT","S1", "S2")),
frequencyDependentFitness = TRUE,
frequencyType = "abs")
competition <- oncoSimulIndiv(fe_competition,
model = "Exp",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
initSize = 40000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)If we plot the whole simulation, we of course see the WT:
plot(competition, show = "genotypes")
but we can avoid that by showing the plot after the WT are long gone:
plot(competition, show = "genotypes",
xlim = c(80, 100))
plot(competition, show = "genotypes", type = "line",
xlim = c(80, 100), ylim = c(1500, 12000))
10.4.2 Competition
We repeat the above, but starting directly from the two species, using the 6.8 logic and 6.7.
G_fe_LVm <- function(r1, r2, K1, K2, a_12, a_21, awt = 1e-4,
gt = c("S1", "S2")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ", r1,
" * ( 1 - (n_1 + ", a_12, " * n_2)/", K1,
")"),
paste0("1 + ", r2,
" * ( 1 - (n_2 + ", a_21, " * n_1)/", K2,
")")
))
}
## Show expressions for birth rates
G_fe_LVm("r1", "r2", "K1", "K2", "a_12", "a_21", "awt")
## Genotype Fitness
## 1 S1 1 + r1 * ( 1 - (n_1 + a_12 * n_2)/K1)
## 2 S2 1 + r2 * ( 1 - (n_2 + a_21 * n_1)/K2)fe_competitionm <-
allFitnessEffects(
genotFitness =
G_fe_LVm(1.5, 1.4, 10000, 4000, 0.6, 0.2,
gt = c("S1", "S2")),
frequencyDependentFitness = TRUE)
fe_competitionm$full_FDF_spec
## S1 S2 Genotype_as_numbers Genotype_as_letters Genotype_as_fvars
## 1 1 0 1 S1 n_1
## 2 0 1 2 S2 n_2
## Fitness_as_fvars
## 1 1 + 1.5 * ( 1 - (n_1 + 0.6 * n_2)/10000)
## 2 1 + 1.4 * ( 1 - (n_2 + 0.2 * n_1)/4000)
## Fitness_as_letters
## 1 1 + 1.5 * ( 1 - (n_1 + 0.6 * n_2)/10000)
## 2 1 + 1.4 * ( 1 - (n_2 + 0.2 * n_1)/4000)
competitionm <- oncoSimulIndiv(fe_competitionm,
model = "Exp",
initMutant = c("S1", "S2"),
initSize = c(5000, 2000),
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)plot(competitionm, show = "genotypes")
10.4.3 Predator-prey, first example
The simplest model would use the above Lotka-Volterra expressions
and set one of the a_ to be negative (see, for example, Otto and Day 2007, 73). Let’s turn former “S2” and thus a_21 < 0 (to make the
effects more salient, we also increase that value in magnitude).
We will also use a general function to generate fitness expressions. This is actually nicer than we did above, because it allows us to give the names of the species, not codes such as “n_1” that depend on how the names are ordered by R.
G_fe_LVm2 <- function(r1, r2, K1, K2, a_12, a_21, awt = 1e-4,
gt = c("S1", "S2")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ", r1,
" * ( 1 - (n_", gt[1], " + ", a_12, " * n_", gt[2], ")/", K1,
")"),
paste0("1 + ", r2,
" * ( 1 - (n_", gt[2], " + ", a_21, " * n_", gt[1], ")/", K2,
")")
))
}
## But notice that, because of ordering, "prey" ends up being n_2
## but that is not a problem.
fe_pred_preym2 <-
allFitnessEffects(
genotFitness =
G_fe_LVm2(1.5, 1.4, 10000, 4000, 1.1, -0.5, awt = 1,
gt = c("prey", "Predator")),
frequencyDependentFitness = TRUE)
## frequencyType set to 'auto'
## All single-gene genotypes as input to to_genotFitness_std
fe_pred_preym2$full_FDF_spec
## Predator prey Genotype_as_numbers Genotype_as_letters
## 1 0 1 2 prey
## 2 1 0 1 Predator
## Genotype_as_fvars Fitness_as_fvars
## 1 n_2 1 + 1.5 * ( 1 - (n_2 + 1.1 * n_1)/10000)
## 2 n_1 1 + 1.4 * ( 1 - (n_1 + -0.5 * n_2)/4000)
## Fitness_as_letters
## 1 1 + 1.5 * ( 1 - (n_prey + 1.1 * n_Predator)/10000)
## 2 1 + 1.4 * ( 1 - (n_Predator + -0.5 * n_prey)/4000)
## Change order and note how these are, of course, equivalent
fe_pred_preym3 <-
allFitnessEffects(
genotFitness =
G_fe_LVm2(1.4, 1.5, 4000, 10000, -0.5, 1.1, awt = 1,
gt = c("Predator", "prey")),
frequencyDependentFitness = TRUE)
## frequencyType set to 'auto'
## All single-gene genotypes as input to to_genotFitness_std
fe_pred_preym3$full_FDF_spec
## Predator prey Genotype_as_numbers Genotype_as_letters
## 1 1 0 1 Predator
## 2 0 1 2 prey
## Genotype_as_fvars Fitness_as_fvars
## 1 n_1 1 + 1.4 * ( 1 - (n_1 + -0.5 * n_2)/4000)
## 2 n_2 1 + 1.5 * ( 1 - (n_2 + 1.1 * n_1)/10000)
## Fitness_as_letters
## 1 1 + 1.4 * ( 1 - (n_Predator + -0.5 * n_prey)/4000)
## 2 1 + 1.5 * ( 1 - (n_prey + 1.1 * n_Predator)/10000)
evalAllGenotypes(fe_pred_preym2, spPopSizes = c(1000, 300))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.0000
## 2 Predator 2.4700
## 3 prey 2.3005
## 4 Predator, prey 0.0000
evalAllGenotypes(fe_pred_preym3, spPopSizes = c(300, 1000))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.0000
## 2 Predator 2.4700
## 3 prey 2.3005
## 4 Predator, prey 0.0000s_pred_preym2 <- oncoSimulIndiv(fe_pred_preym2,
model = "Exp",
initMutant = c("prey", "Predator"),
initSize = c(1000, 1000),
onlyCancer = FALSE,
finalTime = 200,
mu = 1e-3,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)plot(s_pred_preym2, show = "genotypes")
You can easily play with a range of parameters, say the carrying capacity of one of the species, to see how they affect the stochasticity of the system.
If you run the above model repeatedly, you will frequently find that
only one of the species is left; and, yes, that could be the
“Predator”, as in the Lotka-Volterra expressions above there can be
predators without prey. For example, you can check that the birth
rate of the predator is larger than 1 even if there are 0 prey
(identically 1 when n_1 = K_1:
evalAllGenotypes(fe_pred_preym2, spPopSizes = c(0, 300))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.0000
## 2 Predator 2.2950
## 3 prey 2.4505
## 4 Predator, prey 0.0000
evalAllGenotypes(allFitnessEffects(
genotFitness =
G_fe_LVm2(1.5, 1.4, 100, 40,
0.6, -0.5, awt = 0.1,
gt = c("prey", "Predator")),
frequencyDependentFitness = TRUE),
spPopSizes = c(0, 40))
## frequencyType set to 'auto'
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.00
## 2 Predator 1.00
## 3 prey 2.14
## 4 Predator, prey 0.0010.4.4 Predator-prey, second example
We use now the model in p. 76 of Otto and Day (2007), where prey grow exponentially in the absence of predators (and predators will eventually go extinct in the absence of prey):
\[\frac{\mathrm{d}n_1}{\mathrm{d}t} = r\ n_1 - a\ c\ n_1\ n_2\] \[\frac{\mathrm{d}n_2}{\mathrm{d}t} = \epsilon\ a\ c\ n_1\ n_2 - \delta\ n2\]
(Recall what we explained in section 10.4.1 for how we find the \(b\), birth rate, to use in our simulations when we are using and “Exp” model with death rate 1: basically, each of the birth rates, \(b_i\) is \(1 + expression\ above/n_i\)).
C_fe_pred_prey2 <- function(r, a, c, e, d,
gt = c("s1", "s2")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ", r, " - ", a,
" * ", c, " * n_2"),
paste0("1 + ", e, " * ", a,
" * ", c, " * n_1 - ", d)
))
}
C_fe_pred_prey2("r", "a", "c", "e", "d")
## Genotype Fitness
## 1 s1 1 + r - a * c * n_2
## 2 s2 1 + e * a * c * n_1 - dGiven how we wrote C_fe_pred_prey2, the prey is hardcoded as
n_1, so specify names of creatures so that the prey comes first,
in terms of order (note we avoided this problem in the example
above, 10.4.3, by always using the full name of the
genotype we refered to in the function to generate the fitness
effects, G_fe_LVm2). (Yes, we could have used a classic pair:
“Hare” and “Lynx”).
fe_pred_prey2 <-
allFitnessEffects(
genotFitness =
C_fe_pred_prey2(r = .7, a = 1, c = 0.005,
e = 0.02, d = 0.4,
gt = c("Fly", "Lizard")),
frequencyDependentFitness = TRUE)
## frequencyType set to 'auto'
## All single-gene genotypes as input to to_genotFitness_std
fe_pred_prey2$full_FDF_spec
## Fly Lizard Genotype_as_numbers Genotype_as_letters
## 1 1 0 1 Fly
## 2 0 1 2 Lizard
## Genotype_as_fvars Fitness_as_fvars
## 1 n_1 1 + 0.7 - 1 * 0.005 * n_2
## 2 n_2 1 + 0.02 * 1 * 0.005 * n_1 - 0.4
## Fitness_as_letters
## 1 1 + 0.7 - 1 * 0.005 * n_2
## 2 1 + 0.02 * 1 * 0.005 * n_1 - 0.4
## You want to make sure you start the simulation from
## a viable condition
evalAllGenotypes(fe_pred_prey2,
spPopSizes = c(5000, 100))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.0
## 2 Fly 1.2
## 3 Lizard 1.1
## 4 Fly, Lizard 0.0
set.seed(2)
pred_prey2 <- oncoSimulIndiv(fe_pred_prey2,
model = "Exp",
initMutant = c("Fly", "Lizard"),
initSize = c(500, 100),
sampleEvery = 0.1,
mu = 1e-3,
onlyCancer = FALSE,
finalTime = 100,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
op <- par(mfrow = c(1, 2))
## Nicer colors
plot(pred_prey2, show = "genotypes")
## But this shows better what is going on
plot(pred_prey2, show = "genotypes", type = "line")
par(op)If you run that model repeatedly, sometimes the system will go extinct quickly, or you will only get prey growing exponentially.
You could now (left as an exercise) build a more complex model to simulate arms-race scenarios between predators and prey (maybe by having mutations with possibly opposing effects on different coefficients above).
10.4.5 Commensalism
Modelling commensalism simply requires changing the values of the
\(\alpha\), the a_12 and a_21. Again, we can now avoid starting
from a WT and start the simulation directly from “A” and “Commensal”
(section 6.7).
For example (not run, as this is just repetitive):
fe_commens <-
allFitnessEffects(
genotFitness =
G_fe_LV(1.2, 1.3, 5000, 20000,
0, -0.2,
gt = c("WT","A", "Commensal")),
frequencyDependentFitness = TRUE,
frequencyType = "abs")
commens <- oncoSimulIndiv(fe_commens,
model = "Exp",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
initSize = 40000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(commens, show = "genotypes")
plot(commens, show = "genotypes",
xlim = c(80, 100))
plot(commens, show = "genotypes", type = "line",
xlim = c(80, 100), ylim = c(2000, 22000))
10.5 Frequency-dependent fitness: can I mix relative and absolute frequencies?
Yes, of course, since you can always use an absolute specification with the appropriate quotient. For example, the following two specifications are identical:
rar <- data.frame(Genotype = c("WT", "A", "B", "C"),
Fitness = c("1",
"1.1 + .3*f_2",
"1.2 + .4*f_1",
"1.0 + .5 * (f_1 + f_2)"))
afear <- allFitnessEffects(genotFitness = rar,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
evalAllGenotypes(afear, spPopSizes = c(100, 200, 300, 400))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.00
## 2 A 1.19
## 3 B 1.28
## 4 C 1.25
## 5 A, B 0.00
## 6 A, C 0.00
## 7 B, C 0.00
## 8 A, B, C 0.00
rar2 <- data.frame(Genotype = c("WT", "A", "B", "C"),
Fitness = c("1",
"1.1 + .3*(n_2/N)",
"1.2 + .4*(n_1/N)",
"1.0 + .5 * ((n_1 + n_2)/N)"))
afear2 <- allFitnessEffects(genotFitness = rar2,
frequencyDependentFitness = TRUE,
frequencyType = "abs")
evalAllGenotypes(afear2, spPopSizes = c(100, 200, 300, 400))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.00
## 2 A 1.19
## 3 B 1.28
## 4 C 1.25
## 5 A, B 0.00
## 6 A, C 0.00
## 7 B, C 0.00
## 8 A, B, C 0.00and simulating with them leads to identical results
set.seed(1)
tmp1 <- oncoSimulIndiv(afear,
model = "McFL",
onlyCancer = FALSE,
finalTime = 30,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
set.seed(1)
tmp2 <- oncoSimulIndiv(afear2,
model = "McFL",
onlyCancer = FALSE,
finalTime = 30,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
stopifnot(identical(print(tmp1), print(tmp2)))
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = afear, model = "McFL", mu = 1e-04, initSize = 5000,
## finalTime = 30, onlyCancer = FALSE, keepPhylog = FALSE, errorHitWallTime = FALSE,
## errorHitMaxTries = FALSE, seed = NULL)
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 4 5072 4748 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 30 1250
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.006620645 3219.745 3333.333
## OccurringDrivers
## 1
##
## Final population composition:
## Genotype N
## 1 4748
## 2 A 46
## 3 B 273
## 4 C 5
##
## Individual OncoSimul trajectory with call:
## oncoSimulIndiv(fp = afear2, model = "McFL", mu = 1e-04, initSize = 5000,
## finalTime = 30, onlyCancer = FALSE, keepPhylog = FALSE, errorHitWallTime = FALSE,
## errorHitMaxTries = FALSE, seed = NULL)
##
## NumClones TotalPopSize LargestClone MaxNumDrivers MaxDriversLast
## 1 4 5072 4748 0 0
## NumDriversLargestPop TotalPresentDrivers FinalTime NumIter
## 1 0 0 30 1250
## HittedWallTime HittedMaxTries errorMF minDMratio minBMratio
## 1 FALSE FALSE 0.006620645 3219.745 3333.333
## OccurringDrivers
## 1
##
## Final population composition:
## Genotype N
## 1 4748
## 2 A 46
## 3 B 273
## 4 C 5So you can always mix relative and absolute; here fitness of two genotypes depends on the relative frequencies of others, whereas fitness of the third on the absolute frequencies (number of cells):
rar3 <- data.frame(Genotype = c("WT", "A", "B", "C"),
Fitness = c("1",
"1.1 + .3*(n_2/N)",
"1.2 + .4*(n_1/N)",
"1.0 + .5 * ( n_1 > 20)"))
afear3 <- allFitnessEffects(genotFitness = rar3,
frequencyDependentFitness = TRUE,
frequencyType = "abs")
evalAllGenotypes(afear3, spPopSizes = c(100, 200, 300, 400))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.00
## 2 A 1.19
## 3 B 1.28
## 4 C 1.50
## 5 A, B 0.00
## 6 A, C 0.00
## 7 B, C 0.00
## 8 A, B, C 0.00
set.seed(1)
tmp3 <- oncoSimulIndiv(afear3,
model = "McFL",
onlyCancer = FALSE,
finalTime = 60,
mu = 1e-4,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(tmp3, show = "genotypes")
10.6 Frequency-dependent fitness: can I use genes with mutator effects?
Yes. The following examples show it:
## Relative
r1fd <- data.frame(Genotype = c("WT", "A", "B", "A, B"),
Fitness = c("1",
"1.4 + 1*(f_2)",
"1.4 + 1*(f_1)",
"1.6 + f_1 + f_2"))
afe4 <- allFitnessEffects(genotFitness = r1fd,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(1)
s1fd <- oncoSimulIndiv(afe4,
model = "McFL",
onlyCancer = FALSE,
finalTime = 40,
mu = 1e-4,
initSize = 5000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s1fd, show = "genotypes")
mtfd <- allMutatorEffects(epistasis = c("A" = 0.1,
"B" = 10))
set.seed(1)
s2fd <- oncoSimulIndiv(afe4,
muEF = mtfd,
model = "McFL",
onlyCancer = FALSE,
finalTime = 40,
mu = 1e-4,
initSize = 5000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s2fd, show = "genotypes")
plotClonePhylog(s1fd, keepEvents = TRUE)
plotClonePhylog(s2fd, keepEvents = TRUE)
Of course, it also works with absolute frequencies (code not executed for the sake of speed):
## Absolute
r5 <- data.frame(Genotype = c("WT", "A", "B", "A, B"),
Fitness = c("1",
"1.25 - .0025*(n_2)",
"1.25 - .0025*(n_1)",
"1.4"),
stringsAsFactors = FALSE)
afe5 <- allFitnessEffects(genotFitness = r5,
frequencyDependentFitness = TRUE,
frequencyType = "abs")
set.seed(8)
s5 <- oncoSimulIndiv(afe5,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
initSize = 5000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s5, show = "genotypes")
plot(s5, show = "genotypes", log = "y", type = "line")
mt <- allMutatorEffects(epistasis = c("A" = 0.1,
"B" = 10))
set.seed(8)
s6 <- oncoSimulIndiv(afe5,
muEF = mt,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
initSize = 5000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s6, show = "genotypes")
plot(s6, show = "genotypes", log = "y", type = "line")
plotClonePhylog(s5, keepEvents = TRUE)
plotClonePhylog(s6, keepEvents = TRUE)
Note that evalAllGenotypesFitAndMut currently works with frequency-dependent fitness:
evalAllGenotypes(allFitnessEffects(genotFitness = r1fd,
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(10, 20, 30, 40))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.7
## 3 B 1.6
## 4 A, B 2.1
evalAllGenotypesFitAndMut(allFitnessEffects(genotFitness = r1fd,
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
mtfd,
spPopSizes = c(10, 20, 30, 40))
## Warning in match_spPopSizes(spPopSizes, fitnessEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness MutatorFactor
## 1 WT 1.0 1.0
## 2 A 1.7 0.1
## 3 B 1.6 10.0
## 4 A, B 2.1 1.010.7 Can we use the BNB algorithm to model frequency-dependent fitness?
This question is similar to the one we address in 15.6. Briefly, the answer is yes. You can think of this as an approximation to an exact simulation of a stochastic system. You can also think of a delay in the system in the sense that the changes in rates due to changes in the frequencies of the different genotypes are updated at periodic intervals, not immediately.
11 Additional examples of frequency-dependent fitness
In this section, we provide additional examples that use frequency-dependent fitness. As mentioned also in 10,
Note also that in most of these examples we make rather arbitrary and simple assumptions about the genetic basis of the different phenotypes or strategies (most are one-mutation-away from WT); see 10.4 and 10.2 (where we change the ancestor-dependent relationships). In some examples mutation rates are also very high, to speed up processes and because a high mutation rate is used as a procedure (a hack?) to quickly obtain descendants from WT (i.e., to get the game started with some representatives of the non-WT types).
Examples 11.1, 11.2, 11.3, 11.4, 11.5 were originally prepared by Sara Dorado Alfaro, Miguel Hernández del Valle, Álvaro Huertas García, Diego Mañanes Cayero, Alejandro Martín Muñoz; example 11.6 was originally prepared by Marta Couce Iglesias, Silvia García Cobos, Carlos Madariaga Aramendi, Ana Rodríguez Ronchel, and Lucía Sánchez García; examples 11.7, 11.8, 11.9 were prepared by Yolanda Benítez Quesada, Asier Fernández Pato, Esperanza López López, Alberto Manuel Parra Pérez. All of these as an exercisse for the course Programming and Statistics with R (Master’s Degree in Bioinformatics and Computational Biology, Universidad Autónoma de Madrid), course 2019-20.
11.1 Rock-paper-scissors model in bacterial community
11.1.1 Introduction
This example is inspired by Kerr et al. (2002). It describes the relationship between three populations of Escherichia coli, that turns out to be very similar to a rock-paper-scissors game.
An E. coli community can have a specific strain of colicinogenic bacteria, that are capable of creating colicin, a toxin to which this special strain is resistant. The wild-type bacteria is killed by this toxin, but can mutate into a resistant strain.
So, there are three kinds of bacteria: wild-type (WT), colicinogenic (C) and resistant (R). The presence of C reduces the population of WT, but increases the population of R because R has an advantage over C, since R doesn’t have the cost of creating the toxin. At the same time, WT has an advantage over R, because by losing the toxin receptors, R loses also some important functions. Therefore, every strain “wins” against one strain and “loses” against the other, creating a rock-paper-scissors game.
crs <- function (a, b, c){
data.frame(Genotype = c("WT", "C", "R"),
Fitness = c(paste0("1 + ", a, " * f_R - ", b, " * f_C"),
paste0("1 + ", b, " * f_ - ", c, " * f_R"),
paste0("1 + ", c, " * f_C - ", a, " * f_")
))
}The equations are:
\[\begin{align} W\left(WT\right) = 1 + af_R - bf_C\\ W\left(C\right) = 1 + bf_{WT} - cf_R\\ W\left(R\right) = 1 + cf_C - af_{WT}\\ \end{align}\]
where \(f_{WT}\), \(f_C\) and \(f_R\) are the frequencies of WT, C and R, respectively.
crs("a", "b", "c")
## Genotype Fitness
## 1 WT 1 + a * f_R - b * f_C
## 2 C 1 + b * f_ - c * f_R
## 3 R 1 + c * f_C - a * f_11.1.1.1 Case 1
We are going to study the scenario in which all the relationships have the same relative weight.
afcrs1 <- allFitnessEffects(genotFitness = crs(1, 1, 1),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
resultscrs1 <- oncoSimulIndiv(afcrs1,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-2,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
op <- par(mfrow = c(1, 2))
plot(resultscrs1, show = "genotypes", type = "line", cex.lab=1.1,
las = 1)
plot(resultscrs1, show = "genotypes", type = "stacked")
par(op) An oscillatory equilibrium is reached, in which the same populations have a similar number of individuals but oscillates. This makes sense, because the rise on a particular strand will lead to a rise in the one that “wins” against it, and then to a rise in the one that “wins” against the second one, creating this cyclical behaviour. In the stacked plot we can see that the total population remains almost constant.
Note, though, that altering mutation rate (which is huge here) can change the results of the model.
11.1.1.2 Case 2
We are going to put a bigger weight in one of the coefficients, so a=10, b=1, c=1.
afcrs2 <- allFitnessEffects(genotFitness = crs(10, 1, 1),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
If we run multiple simulations, for example by doing
resultscrs2 <- oncoSimulPop(10,
afcrs2,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-2,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)we can verify there are two different scenarios.
The first one is the one in which all the strains coexist, with the colicinogenic bacteria having a much bigger population.
set.seed(1)
resultscrs2a <- oncoSimulIndiv(afcrs2,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-2,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(resultscrs2a, show = "genotypes", type = "line")
In the second one, the wild type and the colicinogenic bacteria dissapear, so the resistant strain is the only one that survives.
As above, though, decreasing the mutation rate can lead to a different solution and you will want to run the model for much longer to see the resistant strain appear and outcompete the others.
set.seed(3)
resultscrs2b <- oncoSimulIndiv(afcrs2,
model = "McFL",
onlyCancer = FALSE,
finalTime = 60,
mu = 1e-2,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(resultscrs2b, show = "genotypes", type = "line", cex.lab=1.1,
las = 1)
11.1.1.3 Case 3
Finally, we are going to put more weight in two coefficients, so a=1, b=5, c=5.
afcrs3 <- allFitnessEffects(genotFitness = crs(1, 5, 5),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
resultscrs3 <- oncoSimulIndiv(afcrs3,
model = "McFL",
onlyCancer = FALSE,
finalTime = 60,
mu = 1e-2,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(resultscrs3, show = "genotypes", type = "line", cex.lab=1.1,
las = 1)
In all the cases all three strains survive, with C having a much smaller population than the other two.
11.2 Hawk and Dove example
The example we are going to show is one of the first games that Maynard Smith analyzed, for example in his classic Maynard Smith (1982) (see also, e.g., https://en.wikipedia.org/wiki/Chicken_%28game%29 ).
In this game, the two competitors are subtypes of the same species but with different strategies. The Hawk first displays aggression, then escalates into a fight until it either wins or is injured (loses). The Dove first displays aggression, but if faced with major escalation runs for safety. If not faced with such escalation, the Dove attempts to share the resource (see the payoff matrix, for instance in https://en.wikipedia.org/wiki/Chicken_%28game%29#Hawk%E2%80%93dove ).
Given that the resource is given the value V, the damage from losing a fight is given cost C:
- If a Hawk meets a Dove he gets the full resource V to himself
- If a Hawk meets a Hawk – half the time he wins, half the time he loses… so his average outcome is then V/2 minus C/2
- If a Dove meets a Hawk he will back off and get nothing – 0
- If a Dove meets a Dove both share the resource and get V/2
The actual payoff however depends on the probability of meeting a Hawk or Dove, which in turn is a representation of the percentage of Hawks and Doves in the population when a particular contest takes place. That in turn is determined by the results of all of the previous contests. If the cost of losing C is greater than the value of winning V (the normal situation in the natural world) the mathematics ends in an stationary point (ESS), a mix of the two strategies where the population of Hawks is V/C.
In this case we assume a stable equilibrium in the population dynamics, that is, although there are external variations in the model, it recovers and returns to equilibrium.
We are going to simulate with OncoSimulR the situation in which the cost of losing C is greater than the value of gaining V (C = 10, V = 2). We assume that both Hawk and Dove are derived from WT by one mutation (see also 10.4) and we will use very high mutation rates to get some hawks and doves from WT quickly (see above).
Before performing the simulation, let’s look at the fitness of each competitor.
## Stablish Genotype-Fitnees mapping. D = Dove, H = Hawk
## With newer OncoSimulR functionality, using WT to start the simulation
## would no longer be needed.
H_D_fitness <- function(c, v,
gt = c("WT", "H", "D")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1"),
paste0("1 + f_H *", (v-c)/2, "+ f_D *", v),
paste0("1 + f_D *", v/2)))
}
## Fitness Effects specification
HD_competition <-allFitnessEffects(
genotFitness = H_D_fitness(10, 2,
gt = c("WT", "H", "D")),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## Plot fitness landscape of genotype "H, D" evaluation
data.frame("Doves_fitness" = evalGenotype(genotype = "D",
fitnessEffects = HD_competition,
spPopSizes = c(5000, 5000, 5000)),
"Hawks_fitness" = evalGenotype(genotype = "H",
fitnessEffects = HD_competition,
spPopSizes = c(5000, 5000, 5000))
)
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Doves_fitness Hawks_fitness
## 1 1.333333 0.3333333We observe that the penalty of fighting (C > V) benefits the dove in terms of fitness respect to the hawk.
## Simulated trajectories
## run only a few for the sake of speed
simulation <- oncoSimulPop(2,
mc.cores = 2,
HD_competition,
model = "McFL", # There is no collapse
onlyCancer = FALSE,
finalTime = 50,
mu = 1e-2, # Quick emergence of D and H
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
## Plot first trajectory as an example
plot(simulation[[1]], show = "genotypes", type = "line",
xlim = c(40, 50),
lwdClone = 2, ylab = "Number of individuals",
main = "Hawk and Dove trajectory",
col = c("#a37acc", "#f8776d", "#7daf00"),
font.main=2, font.lab=2,
cex.main=1.4, cex.lab=1.1,
las = 1)
As mentioned above, mathematically when a stationary point (ESS) is reached the relative frequency of hawks is V/C and doves 1-(V/C). Considering f_H as relative frecuency of hawks and f_D = 1-f_H as frequency of doves:
Hawk: \[1 + f_H*(v-c)/2 + (1-f_H)*v\]
Dove: \[1 + (1-f_H)*v/2\]
Hawk = Dove: \[1 + f_H*(v-c)/2 + (1-f_H)*v = 1 + (1-f_H)*v/2\]
Resolving for f_H: \[f_H = v/c\]
Therefore, the relative frequency of hawks in equilibrium is equal to V/C. In our case it would be 20% (C = 10, V = 2). Let’s check it:
## Recover the final result from first simulation
result <- tail(simulation[[1]][[1]], 1)
## Get the number of organisms from each species
n_WT <- result[2]
n_D <- result[3]
n_H <- result[4]
total <- n_WT + n_D + n_H
## Dove percentage
data.frame("Doves" = round(n_D/total, 2)*100,
"Hawks" = round(n_H/total, 2)*100 )
## Doves Hawks
## 1 79 21To sum up, this example shows that when the risks of contest injury or death (the Cost C) is significantly greater than the potential reward (the benefit value V), the stable population will be mixed between aggressors and doves, and the proportion of doves will exceed that of the aggressors. This explains behaviours observed in nature.
11.3 Game Theory with social dilemmas of tumour acidity and vasculature
This example is based on Kaznatcheev et al. (2017). In this work, it is explained that the progression of cancer is marked by the acquisition of a number of hallmarks, including self-sufficiency of growth factor production for angiogenesis and reprogramming energy metabolism for aerobic glycolysis. Moreover, there is evidence of intra-tumour heterogeneity. Given that some cancer cells can not invest in something that benefits the whole tumor while others can free-ride on the benefits created by them (evolutionary social dilemmas), how do these population level traits evolve, and how are they maintained? The authors answer this question with a mathematical model that treats acid production through glycolysis as a tumour-wide public good that is coupled to the club good of oxygen from better vascularisation.
The cell types of the model are:
- VOP: VEGF (over)-producers.
- GLY: glycolytic cells.
- DEF: aerobic cells that do not call for more vasculature.
On the other hand, the micro-environmental parameters of the model are:
- a: the benefit per unit of acidification.
- v: the benefit from oxygen per unit of vascularisation.
- c: the cost of (over)-producing VEGF.
The fitness equations derived from those populations and parameters are:
\[\begin{equation}
W\left(GLY\right) = 1 + a * \left(f_1 + 1\right)
\end{equation}\]
\[\begin{equation}
W\left(VOP\right) = 1 + a * f_1 + v * \left(f_2 + 1\right) - c
\end{equation}\]
\[\begin{equation}
W\left(DEF\right) = 1 + a * f_1 + v * f_2
\end{equation}\]
Where \(f_1\) is the GLY cells’ frequency and \(f_2\) is the VOF cells’ frequency at a given time. All fitness equations start from balance by the sum of 1.
Finally, depending of the parameter’s values, the model can lead to three different situations (as in other examples, the different types are one mutation away from WT):
11.3.1 Fully glycolytic tumours:
If the fitness benefit of a single unit of acidification is higher than the maximum benefit from the club good for aerobic cells, then GLY cells will always have a strictly higher fitness than aerobic cells, and be selected for. In this scenario, the population will converge towards all GLY, regardless of the initial proportions (as long as there is at least some GLY in the population).
# Definition of the function for creating the corresponding dataframe.
avc <- function (a, v, c) {
data.frame(Genotype = c("WT", "GLY", "VOP", "DEF"),
Fitness = c("1",
paste0("1 + ",a," * (f_GLY + 1)"),
paste0("1 + ",a," * f_GLY + ",v," * (f_VOP + 1) - ",c),
paste0("1 + ",a," * f_GLY + ",v," * f_VOP")
))
}
# Specification of the different effects on fitness.
afavc <- allFitnessEffects(genotFitness = avc(2.5, 2, 1),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## For real, you would probably want to run
## this multiple times with oncoSimulPop
simulation <- oncoSimulIndiv(afavc,
model = "McFL",
onlyCancer = FALSE,
finalTime = 15,
mu = 1e-3,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
# Representation of the plot of one simulation as an example (the others are
# highly similar).
plot(simulation, show = "genotypes", type = "line",
ylab = "Number of individuals", main = "Fully glycolytic tumours",
font.main=2, font.lab=2, cex.main=1.4, cex.lab=1.1, las = 1)
11.3.2 Fully angiogenic tumours:
If the benefit to VOP from their extra unit of vascularisation is higher than the cost c to produce that unit, then VOP will always have a strictly higher fitness than DEF, selecting the proportion of VOP cells towards 1. In addition, if the maximum possible benefit of the club good to aerobic cells is higher than the benefit of an extra unit of acidification, then for sufficiently high number of VOP, GLY will have lower fitness than aerobic cells. When both conditions are satisfied, the population will converge towards all VOP.
# Definition of the function for creating the corresponding dataframe.
avc <- function (a, v, c) {
data.frame(Genotype = c("WT", "GLY", "VOP", "DEF"),
Fitness = c("1",
paste0("1 + ",a," * (f_GLY + 1)"),
paste0("1 + ",a," * f_GLY + ",v, " * (f_VOP + 1) - ",c),
paste0("1 + ",a," * f_GLY + ",v, " * f_VOP")
))
}
# Specification of the different effects on fitness.
afavc <- allFitnessEffects(genotFitness = avc(2.5, 7, 1),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
simulation <- oncoSimulIndiv(afavc,
model = "McFL",
onlyCancer = FALSE,
finalTime = 15,
mu = 1e-4,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)## We get a huge number of VOP very quickly
## (too quickly?)
plot(simulation, show = "genotypes", type = "line",
ylab = "Number of individuals", main = "Fully angiogenic tumours",
font.main=2, font.lab=2, cex.main=1.4, cex.lab=1.1, las = 1)
11.3.3 Heterogeneous tumours:
If the benefit from an extra unit of vascularisation in a fully aerobic group is lower than the cost c to produce that unit, then for a sufficiently low proportion of GLY and thus sufficiently large number of aerobic cells sharing the club good, DEF will have higher fitness than VOP. This will lead to a decrease in the proportion of VOP among aerobic cells and thus a decrease in the average fitness of aerobic cells. A lower fitness in aerobic cells will lead to an increase in the proportion of GLY until the aerobic groups (among which the club good is split) get sufficiently small and fitness starts to favour VOP over DEF, swinging the dynamics back.
# Definition of the function for creating the corresponding dataframe.
avc <- function (a, v, c) {
data.frame(Genotype = c("WT", "GLY", "VOP", "DEF"),
Fitness = c("1",
paste0("1 + ",a," * (f_GLY + 1)"),
paste0("1 + ",a," * f_GLY + ",v," * (f_VOP + 1) - ",c),
paste0("1 + ",a," * f_GLY + ",v," * f_VOP")
))
}
# Specification of the different effects on fitness.
afavc <- allFitnessEffects(genotFitness = avc(7.5, 2, 1),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
# Launching of the simulation (20 times).
simulation <- oncoSimulIndiv(afavc,
model = "McFL",
onlyCancer = FALSE,
finalTime = 25,
mu = 1e-4,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
# Representation of the plot of one simulation as an example (the others are
# highly similar).
plot(simulation, show = "genotypes", type = "line",
ylab = "Number of individuals", main = "Heterogeneous tumours",
font.main=2, font.lab=2, cex.main=1.4, cex.lab=1.1, las = 1)
11.4 Prostate cancer tumour–stroma interactions
This example is based on Basanta et al. (2012). The authors apply evolutionary game theory to model the behavior and progression of a prostate tumour formed by three different cell populations: stromal cells, a dependant tumour phenotype capable of co-opting stromal cells to support its growth and an independent tumour phenotype that does not require microenvironmental support, be it stromal associated or not. To enable this, the model has four variables, which is the minimun necessary to describe the relationships in terms of costs and benefits between the different types of cells and, of course, to describe the progression of the cancer.
The different cell types, hence, are as follows:
- S: stromal cells.
- D: microenvironmental-dependent tumour cells.
- I: microenvironmental-independent tumour cells.
And the parameters that describe the relationships are as follows:
- \(\alpha\): benefit derived from the cooperation between a S cell and a D cell.
- \(\beta\): cost of extracting resources from the microenvironment.
- \(\gamma\): cost of being microenvironmentally independent.
- \(\rho\): benefit derived by D from paracrine growth factors produced by I cells.
Table 11.1 shows the payoffs for each cell type when interacting with others. We consider no other phenotypes are relevant in the context of the game and disregard spatial considerations.
| . | S | D | I |
|---|---|---|---|
| S | \(0\) | \(\alpha\) | \(0\) |
| D | \(1+\alpha-\beta\) | \(1-2\beta\) | \(1-\beta+\rho\) |
| I | \(1-\gamma\) | \(1-\gamma\) | \(1-\gamma\) |
As in Basanta et al. (2012), the I cells are relatively independent from the microenvironment and produce their own growth factors (e.g. testosterone) and thus are considered to have a comparatively constant fitness \((1-\gamma)\), where \(\gamma\) represents the fitness cost for I cells to be independent. The D cells rely more on their microenvironment for survival and growth at a fitness cost \((\beta)\) that represents the scarcity of resources or space that I cells can procure themselves. A resource-poor microenvironment would then be characterised by a higher value of \(\beta\). As I cells produce space and shareable growth factors, this model assumes that D cells derive a fitness advantage from their interactions with I cells represented by the variable \(\rho\). On the other hand, D cells interacting with other D cells will have a harder time sharing existing microenvironmental resources with other equally dependant cells and thus are assumed to have double the cost \(2\beta\) for relying on the microenvironment for survival and growth and thus have a fitness of \(1–2\beta\). The S cells can interact with tumour cells. In a normal situation, this population are relatively growth quiescent with low rates of proliferation and death. For this reason the fitness benefit derived by stromal cells from the interactions with tumour cells is assumed to be zero. However, they are able to undergo rapid proliferation and produce growth factors if they are stimulated by factors produced by I cells, giving rise to a mutualistic relationship. This relationship is represented by the parameter \(\alpha\). A low \(\alpha\) represents tumours in which the stroma cannot be co-opted and vice versa.
From these variables, the fitness of each cell population \((W\left(S\right), W\left(I\right), W\left(D\right))\) is as follows:
\[\begin{equation}
W\left(S\right) = 1 + f_3\alpha
\tag{11.1}
\end{equation}\]
\[\begin{equation}
W\left(I\right) = 2 - \gamma
\tag{11.2}
\end{equation}\]
\[\begin{equation}
W\left(D\right) = 1 + \left(1 - f_2 - f_3\right)\left(1 - \beta +
\alpha\right) +\\
f_2\left(1 - \beta + \rho\right) + f_3(1 - 2\beta) + \\
1 - \beta + \alpha + f_2\left(\rho - \alpha\right) - f_3\left(\beta +
\alpha\right)
\tag{11.3}
\end{equation}\]
where \(f_2\) is the frequency of I cells and \(f_3\) is the frequency of D cells at a given time. All fitness equations start from balance by the sum of 1.
11.4.1 Simulations
First, we define the fitness of the different genotypes (see Equations (11.1), (11.2) and (11.3)) through the function fitness_rel that builds a data frame.
It is important to note that this program models a situation where, from a WT cell population, the rest of the cell population types are formed. However, this model has also stromal cells that are not formed from a WT, since they are not tumour cells although interacting with it. Hence, for this model, we can not represent scenarios with total biological accuracy, something that we must consider when interpreting the results.
fitness_rel <- function(a, b, r, g, gt = c("WT", "S", "I", "D")) {
data.frame(
Genotype = gt,
Fitness = c("1",
paste0("1 + ", a, " * f_D"),
paste0("1 + 1 - ", g),
paste0("1 + (1 - f_I - f_D) * (1 - ", b, " + ",
a, ") + f_I * (1 - ", b, " + ", r,
") + f_D * (1 - 2 * ", b, ") + 1 - ", b,
" + ", a, " + f_I * (", r, " - ",
a, ") - f_D * (", b, " + ", a, ")"))
)
}Then, we are going to model different scenarios that represent different biological situations. In this case, we are going to explain four possible situations.
Note: for these simulations the values of paratemers are normalised in the range (0 : 1) so 1 represents the maximum value for any parameter being positive of negative to fitness depending on the parameter.
11.4.1.1 First scenario
In this simulation, we are modelling a situation where the environment is relatively resource-poor. In addition, we set a intermediate cooperation between D-D and D-I and a very low benefit from coexistence of D with I.
- \(\alpha\) (a) = 0.5: intermediate cooperation between D and D cells.
- \(\beta\) (b) = 0.7: relatively resource-poor microenvironment.
- \(\rho\) (p) = 0.1: low benefit of D cells.
- \(\gamma\) (g) = 0.8: high cost of independence of I cells.
We can observe that high values of \(\alpha\) and low values of \(\rho\) are translated in a larger profit of D cells from his interaction with S cells than from his interaction with I cells. Also, because of the high cost of independence of I cells (\(\gamma\)), it is not surprise that this population ends up becoming extinct. Finally, the tumour is composed by two cellular types: D and S cells.
scen1 <- allFitnessEffects(genotFitness = fitness_rel(a = 0.5, b = 0.7,
r = 0.1, g = 0.8),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(1)
simulScen1 <- oncoSimulIndiv(scen1,
model = "McFL",
onlyCancer = FALSE,
finalTime = 70,
mu = 1e-4,
initSize = 5000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
op <- par(mfrow = c(1, 2))
plot(simulScen1, show = "genotypes", type = "line",
main = "First scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
plot(simulScen1, show = "genotypes",
main = "First scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
par(op)To understand the stability of the results, we should run multiple simulations. We will not pursue that here. Note that the results can be sensitive to the initial population size and the mutation rate.
11.4.1.2 Second scenario
In this case, we set \(\alpha\) lower than in the first scenario and we enable the indepenence of I cells through a lower \(\gamma\).
- \(\alpha\) (a) = 0.3: low cooperation between D cells.
- \(\beta\) (b) = 0.7: relatively resource-poor microenvironment.
- \(\rho\) (r) = 0.1: low benefit of D cells from coexisting with I cells.
- \(\gamma\) (g) = 0.7: lower cost of independence of I cells than in the first scenario.
Because of we are easing the possibility of independence of I cells, instead of extinguishing as in the first scenario, they compose the bulk of the tumour along with D cells in spite of the low benefit of cooperation between them (low \(\rho\)). Besides, we can observe that the population of I cells is bigger than the population of D cells, being at the end of the simulation in balance. On the other hand, stromal cells drop at the beginning of the simulation.
scen2 <- allFitnessEffects(genotFitness = fitness_rel(a = 0.3, b = 0.7,
r = 0.1, g = 0.7),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(1)
simulScen2 <- oncoSimulIndiv(scen2,
model = "McFL",
onlyCancer = FALSE,
finalTime = 70,
mu = 1e-4,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
op <- par(mfrow = c(1, 2))
plot(simulScen2, show = "genotypes", type = "line",
main = "Second scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
plot(simulScen2, show = "genotypes",
main = "Second scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
par(op)11.4.1.3 Third scenario
In this case, we have a extreme situation where the microenvironment is rich (high \(\beta\)) and the independence costs are very low (\(\gamma\)) in relation with the previous scenarios.
- \(\alpha\) (a) = 0.2: low cooperation between D cells.
- \(\beta\) (b) = 0.3: rich microenvironment, being beneficial for D cells.
- \(\rho\) (r) = 0.1: low benefit of D cells from coexisting with I cells.
- \(\gamma\) (g) = 0.3: independence costs very low, being beneficial for I cells.
Although \(\gamma\) and \(\rho\) are very low, which could make us think that I cells will control the tumour, we can observe that the fact that the microenvironment is very rich (with a low value of \(\beta\)) allows to D cells lead the progression of the tumour over the rest of cell populations, including I cells.
scen3 <- allFitnessEffects(genotFitness = fitness_rel(a = 0.2, b = 0.3,
r = 0.1, g = 0.3),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(1)
simulScen3 <- oncoSimulIndiv(scen3,
model = "McFL",
onlyCancer = FALSE,
finalTime = 50,
mu = 1e-4,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
op <- par(mfrow = c(1, 2))
plot(simulScen3, show = "genotypes", type = "line",
main = "Third scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
plot(simulScen3, show = "genotypes",
main = "Third scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
par(op)11.4.1.4 Fourth scenario
This is a variation of the third scenario to illustrate that, if we set a microenvironment more rich than in the previous scenario, we get a cooperation between D and I cells, although we can still observe the superiority of D cells over I cells.
- \(\alpha\) (a) = 0.2: low cooperation between D cells.
- \(\beta\) (b) = 0.4: lower richness than in the previous scenario.
- \(\rho\) (r) = 0.1: low benefit of D cells from coexisting with I cells.
- \(\gamma\) (g) = 0.3: independence costs very low, being beneficial for I cells.
scen4 <- allFitnessEffects(genotFitness = fitness_rel(a = 0.2, b = 0.4,
r = 0.1, g = 0.3),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## Set a different seed to show the results better since
## with set.seed(1) the progression of I cells was not shown
set.seed(2)
simulScen4 <- oncoSimulIndiv(scen4,
model = "McFL",
onlyCancer = FALSE,
finalTime = 40,
mu = 1e-4,
initSize = 4000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
op <- par(mfrow = c(1, 2))
plot(simulScen4, show = "genotypes", type = "line",
main = "Fourth scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
plot(simulScen4, show = "genotypes",
main = "Fourth scenario",
cex.main = 1.4, cex.lab = 1.1,
las = 1)
par(op)In this case, we can see that there is more variation in the size of population of I cells. There are cases where the I cell population cooperates with D cells and, in others, there is not cooperation. You can examine this running multiple simulations (or manually rerun the example above changing the seed).
11.5 Evolutionary Dynamics of Tumor-Stroma Interactions in Multiple Myeloma
This example is based on Sartakhti et al. (2016). The authors provide a frequency-dependent model to study the growth of malignant plasma cells in multiple myeloma. Assuming that cancer cells and stromal cells cooperate by exchanging diffusible factors, the study is carried out in the framework of evolutionary game theory.
We first need to define a payoff strategy for this kind of scenario. The following definitions are needed:
- There are \(n\) phenotypes in a population denoted by \(\{ P_1, \ldots, P_n \}\).
- Each phenotype can produce one diffusible factor \(\{ G_1, \ldots, G_n \}\).
- Each diffusible factor \(j\) has a different effect \(r_{i,j}\) on the other phenotypes \(i\).
- The cost for \(P_i\) for growth factor \(G_i\) is denoted as \(c_i\).
- \(M\) is the number of cells within the diffusion range.
- There are \(M_j\) individuals of type \(P_j\) among the other group members.
Then, the payoff for strategy \(P_j\) is: \[\begin{align} \pi_{P_j}(M_1,\ldots,M_n)=\frac{(M_j+1)\times c_j}{M}r_{j,j} + \sum_{i=1, i \neq j}^n \frac{M_i \times c_i}{M}r_{j,i} - c_j \;. \end{align}\]
In multiple Myeloma we have three different types of cells that have autocrine and paracrine effects on the cells within their diffusion range: Malignant plasma cells (MM), Osteoblasts (OB) and Osteoclasts (OC). The specification of fitness is the following (see [10]):
\[\begin{align} W_{OC} &= \frac{(f_{1}(M-1)+1)c_{1}}{M}r11+ \frac{((1-f_{3})(M-1)-f_{1}(M-1)-1)c_{2}}{M}r12 \\ &+\frac{(M-(1-f_3)(M-1))c_{3}}{M}r13-c_{1} \\ W_{OB} &= \frac{(f_{2}(M-1)+1)c_{2}}{M}r22+ \frac{((1-f_{1})(M-1)-f_{2}(M-1)-1)c_{3}}{M}r23 \\ &+\frac{(M-(1-f_{1})(M-1))c_{1}}{M}r21-c_{2} \\ W_{MM} &= \frac{(f_{3}(M-1)+1)c_{3}}{M}r33+ \frac{((1-f_{2})(M-1)-f_{3}(M-1)-1)c_{1}}{M}r31 \\ &+\frac{(M-(1-f_{2})(M-1))c_{2}}{M}r32-c_{3} \;, \end{align}\]
where \(f_1\), \(f_2\) and \(f_3\) denote the frequency of the phenotype OC, OB and MM in the population. The multiplication factors for diffusible factors produced by the cells are shown in the following table (taken from Sartakhti et al. (2016)):
Figure 11.1: Multiplication factor in myeloma interaction. Table 1 in Sartakhti et al., 2016, ‘Evolutionary Dynamics of Tumor-Stroma Interactions in Multiple Myeloma’ https://doi.org/10.1371/journal.pone.0168856 .
Several scenarios varying the values of the parameters are shown in Sartakhti et al. (2016). Here we reproduce some of them.
11.5.1 Simulations
First, we define the fitness of the different genotypes through the function fitness_rel.
f_cells <- function(c1, c2, c3, r11, r12, r13,
r21, r22, r23, r31, r32, r33, M, awt = 1e-4,
gt = c("WT", "OC", "OB", "MM")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("max(0.1, 1 - ", awt, " * (f_OC + f_OB+f_MM)*N)"),
paste0("1", "+(((f_OC * (", M, "-1)+1)*", c1, ")/", M, ")*",r11,
"+((((1-f_MM) * (", M, "-1)-f_OC*(", M, "-1)-1)*", c2, ")/", M, ")*", r12,
"+(((", M, "-(1-f_MM)*(", M, "-1))*", c3, ")/", M, ")*", r13,
"-", c1
),
paste0("1", "+(((f_OB*(", M, "-1)+1)*", c2, ")/", M, ")*", r22,
"+((((1-f_OC)*(", M, "-1)-f_OB*(", M, "-1)-1)*", c3, ")/", M, ")*", r23,
"+(((", M, "-(1-f_OC)*(", M, "-1))*", c1, ")/", M, ")*", r21,
"-", c2
),
paste0("1", "+(((f_MM*(", M, "-1)+1)*", c3, ")/", M, ")*", r33,
"+((((1-f_OB)*(", M, "-1)-f_MM*(", M, "-1)-1)*", c1, ")/", M, ")*", r31,
"+(((", M, "-(1-f_OB)*(", M, "-1))*", c2, ")/", M, ")*", r32,
"-", c3
)
)
,stringsAsFactors = FALSE
)
}It is important to note that, in order to exactly reproduce the experiments of the paper, we need to create an initial population with three different types of cell, but we do not need the presence of a wild type. For this reason, we will increase the probability of mutation of the wild type, which will disappear in early stages of the simulation; this is a procedure we have used before in several cases too (e.g., 11.2). This is something that we must consider when interpreting the results.
11.5.2 Scenario 1
Here we model a common situation in multiple myeloma in which \(c_1<c_2<c_3\). In the presence of a small number of MM cells, the stable point on the OB-OC border becomes a saddle point and clonal selection leads to a stable coexistence of OC and MM cells. Parameters for the simulation can be seen in the R code.
N <- 40000
M <- 10
c1 <- 1
c2 <- 1.2
c3 <- 1.4
r11 <- 0
r12 <- 1
r13 <- 2.5
r21 <- 1
r22 <- 0
r23 <- -0.3
r31 <- 2.5
r32 <- 0
r33 <- 0
fe_cells <-
allFitnessEffects(
genotFitness =
f_cells(c1, c2, c3, r11, r12, r13,
r21, r22, r23, r31, r32, r33, M,
gt = c("WT", "OC", "OB", "MM")),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
## Simulated trajectories
set.seed(2)
simulation <- oncoSimulIndiv(fe_cells,
model = "McFL",
onlyCancer = FALSE,
finalTime = 20, ## 30
mu = c("OC"=1e-1, "OB"=1e-1, "MM"=1e-4),
initSize = N,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
## Plot trajectories
## op <- par(mfrow = c(1, 2))
## plot(simulation, show = "genotypes", type = "line")
plot(simulation, show = "genotypes", thinData = TRUE)
## par(op)Clearly, the appearance of MM cells quickly brings the system to an equilibrium point, which is stable. OB cells are extinct and cancer has propelled.
11.5.3 Scenario 2
As in the second scenario of [10] (\(c1=c2=c3\)) configuration A (upper row in the grid of images). We should find one stable point on the OC-OB edge under certain conditions, which are met in the example. Further information about the parameters can be found in the R code shown below.
N <- 40000
M <- 10
c1 <- 1
c2 <- 1
c3 <- 1
r11 <- 0
r12 <- 1
r13 <- 0.5
r21 <- 1
r22 <- 0
r23 <- -0.3
r31 <- 0.5
r32 <- 0
r33 <- 0
fe_cells <-
allFitnessEffects(
genotFitness =
f_cells(c1, c2, c3, r11, r12, r13,
r21, r22, r23, r31, r32, r33, M,
gt = c("WT", "OC", "OB", "MM")),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
## Simulated trajectories
set.seed(1)
simulation <- oncoSimulIndiv(fe_cells,
model = "McFL",
onlyCancer = FALSE,
finalTime = 15, ## 25
mu = c("OC"=1e-1, "OB"=1e-1, "MM"=1e-4),
initSize = N,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
#Plot trajectorie
plot(simulation, show = "genotypes", thinData = TRUE)
As expected, under these condtitions, MM cells are not able to propagate. The equilibrium point in the OB-OC edge is stable, resisting small variations in the number of MM cells.
11.6 An example of modellization in Parkinson disease related cell community
The following example is based on Masliah (2007), and it discusses the coexistance between cells that produce \(\alpha\)-synuclein and \(\beta\)-synuclein, related with pore-like oligomer development and Parkinson disease.
park1<- data.frame(Genotype = c("WT", "A", "B", "A,B"),
Fitness = c("1",
"1 + 3*(f_1 + f_2 + f_1_2)",
"1 + 2*(f_1 + f_2 + f_1_2)", ## We establish
## the fitness of B smaller than the one of A because
## it is an indirect cause of the disease and not a direct one.
"1.5 + 4.5*(f_1 + f_2 + f_1_2)")) ## The baseline
## of the fitness is higher in the
## AB population (their growth is favored).
parkgen1<- allFitnessEffects(genotFitness = park1,
frequencyDependentFitness = TRUE,
frequencyType = "rel") - \(\alpha\)- – normal a-synuclein production
- \(\beta\)- – normal b-synuclein production
- \(\alpha\)+ – deleterious a-synuclein production, causing aggregation and the rest of mentioned effects
- \(\beta\)+ – deleterious b-synuclein production, preventing it from slowing down the aggregation of a-synuclein
- WT - \(\alpha\)- and \(\beta\)- (balance)
- A - \(\alpha\)+ and \(\beta\)- (increases [slight] the probability of Parkinson’s disease)
- B – \(\alpha\)- and \(\beta\)+ (increases [slight] the probability of Parkinson’s disease)
- AB – \(\alpha\)+ and \(\beta\)+ (massive increase of probability of Parkinson’s disease)
In this simulation, at the very end, the only cells that remain alive are those from AB population (which means that both \(\alpha\) and \(\beta\) are mutated, \(\alpha\)+ and \(\beta\)+, which means that the individual has a high risk of developing the disease, and all the cells keep this mutation). Genotype AB is able to invade the population when there are some A’s and B’s around. Cooperation increases the fitness at 1.5 level respecting the fitness for just A’s or just B’s, so the rest of population (appart from AB) collapses. We can observe this cell behaviour in the following code and graphic:
set.seed(1)
fpark1 <- oncoSimulIndiv(parkgen1,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
initSize = 5000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(fpark1, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue"))
Analyzing the graphic obtain from OncoSimulR we can see that A, B and the WT population dissapear at (more or less) the same point that AB population increases drastically, which makes sense because the fitness of AB population is greater than the other three population fitnesses. As the birth rate of the population depends directly of the fitness, at the end, just the AB population survives. We have a model of frequency-dependent fitness here, but the results are not really surprising given the fitness of each type, and no coexistence is possible.
The AB population comes mainly from the A population, as would be expected, because of its fitness, and abundance, relative to that of the B population:
plotClonePhylog(fpark1, N = 0, keepEvents=TRUE, timeEvents=TRUE)
11.7 Evolutionary Game between Commensal and Pathogenic Microbes in Intestinal Microbiota
The following adapted example is based on Wu and Ross (2016). As explained in p. 2 of the aforementioned paper, the commensal microbiota has been simplified into two phenotypic groups: antibiotic-sensitive bacteria (CS) [WT in the code below] and antibiotic-tolerant bacteria (CT). In addition, a third phenotypic group of pathogenic bacteria (PA) is considered which are kept in low numbers in absence of intestinal microbiota disturbances. We assume CS and CT bacteria cooperate and depend on each other for optimal proliferation, leading to a benefit (bG) for both of them as well as to a cost for factor growth (cG) production which permits a stable coexistence between CS and CT cells if the fraction of PA cells is negligible. Meanwhile, PA possess a reproductive or metabolic advantage relative to CS and CT. Without antibiotic administration CS population inhibits PA population via the release of a chemical compound which harms PA (iPA) when their relative frequency is equal to or greater than 0.2, carrying a production cost (cI). However, PA population takes over CS population in the presence of antibiotic (ab). In this situation PA and CT compete for resources. Finally, a cohabit cost (cS) is considered for all three cell types.
The adapted payoff matrix obtained is shown in the next table.
| CS | CT | PA | |
|---|---|---|---|
| CS | bG – cG – cS - ab | bG – cG – cS - ab | bG – cG – cS -cI - ab |
| CT | bG – cG – cS | bG – cG – cS | bG – cG – cS |
| PA | bPA – iPA - cS | bPA – cS | bPA – cS |
On the basis of the interactions between the different microbe populations and the payoff matrix, we establish fitness as:
- CS (Fcs = bG([CS] + [CT]) - cS*([CS] + [CT] + [PA]) - cI - cG - ab)
- CT (Fct = bG([CS] + [CT]) - cS*([CS] + [CT] + [PA]) - cG)
- PA (Fpa = bPA([PA]) - cS([CS] + [CT] + [PA]) - iPA[CS])
A function to create the data frame of frequency-dependent fitnesses is defined in order to be able to model situations with different values for the parameters, and it is assumed that WT derive to the other cell types by a single mutation.
create_fe <- function(bG, cG, iPA, cI, cS, bPA, ab,
gt = c("WT", "CT", "PA")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ", bG, " * (f_ + f_CT) - ", cS,
" * (f_ + f_CT + f_PA) - ", cI, "(f_PA > 0.2) - ", cG,
" - ", ab),
paste0("1 + ", bG, " * (f_ + f_CT) - ", cS,
" * (f_ + f_CT + f_PA) - ", cG),
paste0("1 +", bPA, " - ", cS, " * (f_ + f_CT + f_PA) - ", iPA,
" *(f_(f_PA > 0.2))")),
stringsAsFactors = FALSE)
}We can check we recover the Table 11.2 :
create_fe("bG", "cG", "iPA", "cI", "cS", "bPA", "ab")
## Genotype
## 1 WT
## 2 CT
## 3 PA
## Fitness
## 1 1 + bG * (f_ + f_CT) - cS * (f_ + f_CT + f_PA) - cI(f_PA > 0.2) - cG - ab
## 2 1 + bG * (f_ + f_CT) - cS * (f_ + f_CT + f_PA) - cG
## 3 1 +bPA - cS * (f_ + f_CT + f_PA) - iPA *(f_(f_PA > 0.2))We verify what we have specified executing evalAllGenotypes. We specify
populations sizes for evaluating fitness at different moments in the
total population evolution.
In the absence of antibiotic we observe how, even though CS (WT) and CT population size is equal, CT fitness is greater than CS fitness due to the PA inhibitory factor releasing cost. PA fitness is decreased by means of this inhibitor produced by CS.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(7, 1, 9, 0.5, 2, 5, 0),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(1000, 1000, 1000))
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 2.166667
## 2 CT 2.666667
## 3 PA 1.000000
## 4 CT, PA 0.000000CS (WT) fitness decrease in the presence of antibiotic, while PA and CT fitness are not affected, and CT fitness is the largest.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(7, 1, 9, 0.5, 2, 5, 2),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(1000, 1000, 1000))
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.1666667
## 2 CT 2.6666667
## 3 PA 1.0000000
## 4 CT, PA 0.0000000In the extreme situation in in which antibiotic is administrated and only PA is present, neither WT (CS) nor CT would be able to grow:
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(7, 1, 9, 0.5, 2, 5, 2),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(0, 0, 1000))
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0
## 2 CT 0
## 3 PA 4
## 4 CT, PA 0From a PA population size much greater than WT (CS) population size, the presence of WT decreases PA fitness via inhibitor production while WT fitness does not increase due to antibiotic administration.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(7, 1, 9, 0.5, 2, 5, 2),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(100, 0, 1000))
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.000000
## 2 CT 0.000000
## 3 PA 3.181818
## 4 CT, PA 0.000000Starting from a population of WT and with high antibiotic doses, WT does not have the capacity of growing while CT shows the largest fitness (and PA fitness is decreased by the inhibitory compound produced by WT):
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(7, 1, 9, 0.5, 2, 5, 5),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(1000, 0, 0))
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0
## 2 CT 5
## 3 PA 4
## 4 CT, PA 0At the same WT and PA population size and with high antibiotic dose administration, the fitness of both of them is decreased to the same extent since they inhibit each other while CT grow a little.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(7, 1, 9, 0.5, 2, 5, 5),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(1000, 0, 1000))
## All single-gene genotypes as input to to_genotFitness_std
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 0.0
## 2 CT 1.5
## 3 PA 0.0
## 4 CT, PA 0.0We have verified that fitness specification gives the expected results:
Now we create allFitnessEffects object and simulate two different
situations: microbiota evolution in the absence and presence of
antibiotic. The model used is McFL so density dependence in the
death rate is considered.
11.7.1 Antibiotic absence situation
We observe how CT and PA cells emerge at time 0 from WT mutations. PA population starts to grow but when its frequency is greater than the established threshold, 0.2, WT population produces inhibitory compounds which harm PA and affect its fitness; when its frequency is under that threshold, WT stop releasing inhibitory PA growth factor and PA starts to grow again. This loop remains over time. Meanwhile, WT population decreases slowly due to the cost of producing the inhibitor when PA frequency exceed the established 0.2 and is reached by CT population, which grow until WT and CT are stabilized and cohabit taking in account the cost for sharing space.
woAntib <- allFitnessEffects(
genotFitness = create_fe(7, 1, 9, 0.5, 2, 5, 0),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## We do not run this for speed but load it below
set.seed(2)
woAntibS <- oncoSimulIndiv(woAntib,
model = "McFL",
onlyCancer = FALSE,
finalTime = 2000,
mu = 1e-4,
initSize = 1000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE,
keepEvery = 2 ## store a smaller object
)
## Load stored results
data(woAntibS)
plot(woAntibS, show = "genotypes", type = "line",
col = c("black", "green", "red"))
11.7.2 Antibiotic presence situation
We observe how CT and PA cells emerge from WT mutations. WT cells decrease in number due to antibiotic administration and inhibitory PA growth factor when the frequency of this latest surpass the threshold imposed, so PA population grow with difficulty in comparison to CT. WT population finally disappear and CT and PA compete for resources, but CT takes over PA given its larger population size and CT population remain stable over time.
wiAntib <- allFitnessEffects(
genotFitness = create_fe(7, 1, 9, 0.5, 2, 5, 2),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(2)
wiAntibS <- oncoSimulIndiv(wiAntib,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 1e-4,
initSize = 1000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(wiAntibS, show = "genotypes", type = "line",
col = c("black", "green", "red"))
11.8 Modeling of breast cancer through evolutionary game theory.
The following example is based on Barton and Sendova (2018). This model assumes that there are four different types of cells in the body: (a) the native cells (NC), which are the healthy stromal cells; (b) the macrophages (Mph), which are part of the immune system; (c) the benign tumor cells (BTC), lump-forming cancer cells that lack the ability to metastasize; (d) the motile tumor cells (MTC), metastatic cancer cells that can invade neighboring tissues.
Both the native cells and macrophages produce growth factor, which benefits all types of cells.The cost of producing the growth factor, cG, and the benefits of the growth factor, bG, will be assumed to be the same for all types of the cells. The macrophages and motile tumor cells can move and we will assume that the ability comes at the costs cM,Mph, and cM,MTC respectively. The native cells and benign tumor cells stay in place and thus have to share the resources with other native and benign tumor cells, which comes at the cost cS. The cancer cells can reproduce faster than native cells or macrophages, which we model by additional benefit bR to the cancer cells, but the cancer cells can be destroyed by macrophages, which we model by additional cost cD to the cancer cells.
They provide also the payoff matrix reproduced in next table :
| MTC | Mph | NC | BTC | |
|---|---|---|---|---|
| MTC | bR - cMMTC | bR - cMMTC - cD + bG | bR - cMMTC + bG | bR - cMMTC |
| Mph | - cG - cMMph | bG - cG - cMMph | bG - cG - cMMph | -cG - cMMph |
| NC | - cG | bG - cG | bG - cG - cS | - cG - cS |
| BTC | bR | bR + bG - cD | bR + bG - cS | bR - cS |
Overall, when the concentrations of the cells are [NC], [Mph], [BTC] and [MTC], the net benefits (benefits minus the costs) to each type of the cells are:
- W(NC) = bG([NC] + [Mph]) - cG - cS([NC] + [BTC])
- W(Mph) = bG([NC] + [Mph]) - cG - cMMph
- W(BTC) = bR + bG([NC] + [Mph]) - cS([NC] + [BTC]) - cD[Mph]
- W(MTC) = bR + bG([NC] + [Mph]) - cMMTC - cD[Mph]
To allow modelling scenarios with different values for the parameters above we will define a function to create the data frame of frequency-dependent fitnesses. First, we will consider the NC cell type as the WT. Moreover, we will assume that each one of the other cell types types (Mph , BTC and MTC) are all derived from WT by a single mutation in one of three genes, say, O, B, M, respectively.
create_fe <- function(cG, bG, cS, cMMph, cMMTC, bR, cD,
gt = c("WT", "Mph", "BTC", "MTC")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ", bG, "*(f_ + f_Mph) - ", cG, " - ", cS, "*(f_ + f_BTC)"),
paste0("1 + ", bG, "*(f_ + f_Mph) - ", cG, " - ", cMMph),
paste0("1 + ", bR, " + ", bG, "*(f_ + f_Mph) - ", cS, "* (f_ + f_BTC) -",
cD , " * f_Mph"),
paste0("1 + ", bR, " + ", bG, " *(f_ + f_Mph) -", cMMTC, " - ",
cD , " * f_Mph")
),
stringsAsFactors = FALSE)
}We can check we recover the Table 11.3 :
create_fe("cG", "bG","cS", "cMMph", "cMMTC", "bR", "cD")
## Genotype Fitness
## 1 WT 1 + bG*(f_ + f_Mph) - cG - cS*(f_ + f_BTC)
## 2 Mph 1 + bG*(f_ + f_Mph) - cG - cMMph
## 3 BTC 1 + bR + bG*(f_ + f_Mph) - cS* (f_ + f_BTC) -cD * f_Mph
## 4 MTC 1 + bR + bG *(f_ + f_Mph) -cMMTC - cD * f_MphWe check that we have correctly specified the different parameters
executing evalAllGenotypes. For this, we specify populations
sizes for evaluating fitness at different moments in the total
population evolution.
When there are only wild-type cells, the fitness of the cancer cells is higher than the other type of cells’ fitness. This makes sense, since there are no macrophages that can affect them.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(2, 5, 1, 0.8, 1, 1, 9),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(WT = 1000, Mph = 0, BTC = 0, MTC = 0))
## All single-gene genotypes as input to to_genotFitness_std
## Genotype Fitness
## 1 WT 3.0
## 2 BTC 6.0
## 3 MTC 6.0
## 4 Mph 3.2
## 5 BTC, MTC 0.0
## 6 BTC, Mph 0.0
## 7 MTC, Mph 0.0
## 8 BTC, MTC, Mph 0.0In case that there are wild-type cells and macrophages, fitness of cancer cells is lower than before.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(2, 5, 1, 0.8, 1, 1, 9),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(WT = 1000, Mph = 1000, BTC = 0, MTC = 0))
## All single-gene genotypes as input to to_genotFitness_std
## Genotype Fitness
## 1 WT 3.5
## 2 BTC 2.0
## 3 MTC 1.5
## 4 Mph 3.2
## 5 BTC, MTC 0.0
## 6 BTC, Mph 0.0
## 7 MTC, Mph 0.0
## 8 BTC, MTC, Mph 0.0When cancer cells start to grow, the fitness of wild type cells and macrophages decrease. This makes sense for wild-type cells, since they will share space with BTC cells, reducing their resources; and for macrophages, because the decrease of wild-type cells will affect the benefit of growth factor produced by them.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(2, 5, 1, 0.8, 1, 1, 9),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(WT = 1000, Mph = 1000, BTC = 100, MTC = 100))
## All single-gene genotypes as input to to_genotFitness_std
## Genotype Fitness
## 1 WT 3.045455
## 2 BTC 1.954545
## 3 MTC 1.454545
## 4 Mph 2.745455
## 5 BTC, MTC 0.000000
## 6 BTC, Mph 0.000000
## 7 MTC, Mph 0.000000
## 8 BTC, MTC, Mph 0.000000So, since the model seems correct, we create now the allFitnessEffects object and do the simulation. Here, we use the McFL model, with death density dependence in addition to the frequency dependence in the birth rates. In this case, we model three different situations: cancer being controlled, development of a non-metastatic cancer, and development of metastatic cancer.
11.8.1 Cancer kept under control
In this example, cD value is increased in order to represent a highly functioning immune system that helps fighting against cancer cells, while the cost of producing growth factor (cG) is low to allow a better fitness of non-cancer cells. This results in the fitness of wild-type cells and macrophages being kept in high levels, helping to control the proliferation of cancer cells, which is maintained under acceptable levels.
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(0.5, 4, 1, 0.2, 1, 0.5, 4),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 50,
mu = 1e-4,
initSize = 10000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue", "yellow"))
11.8.2 Development of a non-metastatic cancer
In this second scenario, bR and cMMTC are increased, promoting a higher proliferative capacity of cancer cells but increasing the cost of the ability to move by metastatic cells (MTC), while Cs is slightly decreased, allowing a better fitness of non-motile cells that have to share resources (WT and BTC). This situation leads to the appearance of a non-metastatic cancer due to the higher fitness of BTC cells, thanks to their absence of mobility cost and their increased proliferative capacity, specially in the absence of other cell types that compete for resources.
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(1, 4, 0.5, 1, 1.5, 1, 4),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 50,
mu = 1e-4,
initSize = 10000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue", "yellow"))
11.8.3 Development of a metastatic cancer
In this last example, cS value is increased, hindering the growth and fitness of non-motile cells that compete for space and resources, whereas the cost of mobility of metastatic cancer cells (cMMTC) is considerably reduced, thus favoring their proliferation. This scenario leads to the development of metastatic cancer, due to a rapid increase in the proliferation of metastatic cancer cells, thanks to their low mobility cost, and a slightly slower increase in the fitness of benign tumor cells, that have to compete with resources with the wild-type cells until their disappearance.
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(0.5, 4, 2, 0.5, 0.5, 1, 4),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 50,
mu = 1e-4,
initSize = 10000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue", "yellow"))
11.9 Improving the previous example. Modeling of breast cancer with the presence chemotherapy and resistance.
With the aim of obtaining a more reliable and representative picture of what happens in a situation of cancer development and treatment, we model a more complex scenario inproving the previous example, 11.8.
This alternative model considers the presence of seven cell types, four of which are the same that have been described in the previous model: the native cells (NC), macrophages (Mph), benign tumor cells (BTC) and metastatic cancer cells (MTC). Moreover, we include here the chemotherapy-resistant normal cells (R), chemotherapy-resistant benign tumor cells (BTC,R) and chemotherapy-resistant metastatic cancer cells (MTC,R).
Most of the costs and benefits of this new model are similar to those described in the previous situation. Native cells and macrophages, but also the chemotherapy-resistant normal cells, produce growth factor bG, which benefits all cell types. These three cells types are considered to have the same cost of producing the growth factor, cG. The macrophages and motile tumor cells (both MTC and MTC,R) have the ability to move, assuming that the cost of this ability is cM,Mph, cM,MTC and cM,MTC,R respectively. The native cells and benign tumor cells (both BTC and BTC,R) have no motile capacity and have to stay in place, competing for the available resources with other native and benign tumor cells, which comes at the cost cS for all these cell types. As occurred in the previous model, considering that cancer cells have a higher reproductive rate than native cells or macrophages, we add an additional benefit bR to the cancer cells. However, as they can also be destroyed and attacked by macrophages, we add an additional cost cD to the cancer cells. Moreover, in this model we add an additional factor Q that represents the effect of chemotherapeutic treatment, which will not have any effect on cells that have been able to develop resistance to it, but it will have a negative effect on the rest of the cell types, being especially important on cancer cells, as they have a higher rate of reproduction.
The payoff matrix will also introduce slight changes, which reproduced in next table:
| MTC | Mph | NC | BTC | R | BTC,R | MTC,R | |
|---|---|---|---|---|---|---|---|
| MTC | bR - cMMTC - Q | bR - cMMTC - cD + bG -Q | bR - cMMTC + bG -Q | bR - cMMTC - Q | bR - cMMTC + bG - Q | bR - cMMTC - Q | bR - cMMTC -Q |
| Mph | - cG - cMMph - 0.01 * Q | bG - cG - cMMph - 0.01 * Q | bG - cG - cMMph - 0.01 * Q | -cG - cMMph - 0.01 * Q | bG - cG - cMMph- 0.01 * Q | -cG - cMMph - 0.01 * Q | - cG - cMMph - 0.01 * Q |
| NC | - cG - 0.01 * Q | bG - cG - 0.01 * Q | bG - cG - cS - 0.01 * Q | - cG - cS - 0.01 * Q | bG - cG - cS - 0.01 * Q | - cG - cS - 0.01 * Q | - cG - 0.01 * Q |
| BTC | bR - Q | bR + bG - cD - Q | bR + bG - cS - Q | bR - cS - Q | bR + bG - cS -Q | bR - cS - Q | bR - Q |
| R | - cG | bG - cG | bG - cG - cS | - cG - cS | bG - cG - cS | - cG - cS | - cG |
| BTC,R | bR | bR + bG - cD | bR + bG - cS | bR - cS | bR + bG - cS | bR - cS | bR |
| MTC,R | bR - cMMTC | bR - cMMTC - cD + bG | bR - cMMTC + bG | bR - cMMTC | bR - cMMTC + bG | bR - cMMTC | bR - cMMTC |
Overall, when the concentrations of the cells are ǪNC, ǪMph, ǪBTC, ǪMTC, ǪR, ǪBTC,R and ǪMTC,R, the net benefits (benefits minus the costs) to each cell type are:
- W(NC) = bG([NC] + [Mph] + [R]) - cG - cS([NC] + [BTC] + [R] + [BTC,R]) - 0.01*Q
- W(Mph) = bG([NC] + [Mph] + [R]) - cG - cMMph - 0.001*Q
- W(BTC) = bR + bG([NC] + [Mph] + [R]) - cS([NC] + [Mph] + [R] + [BTC,R]) - cD[Mph] - Q
- W(MTC) = bR + bG([NC] + [Mph] + [R]) - cMMTC - cD[Mph] - Q
- W(R) = bG ([NC] + [Mph] + [R]) - cG - cS ([NC] + [BTC] + [R] + [BTC,R])
- W(BTC,R) = bR + bG([NC] + [Mph] + [R]) - cS([NC] + [Mph] + [R] + [BTC,R]) - cD[Mph]
- W(MTC,R) = bR + bG([NC] + [Mph] + [R]) - cMMTC - cD[Mph]
Now, we will define a function to create the data frame of frequency-dependent fitnesses to allow modelling several situations. As in the previous model, we will consider the NC cell type as the WT and will assume that Mph, BTC and MTC cells are all derived from WT by a single mutation, as previously described. Likewise, we consider that the chemotherapy-resistant normal cells (R) are derived from a single mutation in a gene, say, R. Furthermore, we will assume that the two chemotherapy-resistant cancer cells (BTC,R and MTC,R) are all derived from WT by two different mutations.
create_fe <- function(cG, bG, cS, cMMph, cMMTC, bR, cD, Q,
gt = c("WT", "BTC", "R", "MTC", "Mph", "BTC,R", "MTC,R")) {
data.frame(Genotype = gt,
Fitness = c(
paste0("1 + ", bG, "(f_ + f_R + f_Mph) - ", cG, " - ", cS, "(f_ + f_BTC + f_R + f_BTC_R) -",
"0.01*", Q),
paste0("1 + ", bR, " + ", bG, "(f_ + f_R + f_Mph) - ", cS, " (f_ + f_BTC + f_R + f_BTC_R) -",
cD , " * f_Mph -", Q),
paste0("1 + ", bG, "(f_ + f_R + f_Mph) - ", cG, " - ", cS, "(f_ + f_BTC + f_R + f_BTC_R)"),
paste0("1 + ", bR, " + ", bG, " *(f_ + f_R + f_Mph) -", cMMTC, " - ",
cD , " * f_Mph -", Q),
paste0("1 + ", bG, "(f_ + f_R + f_Mph) - ", cG, " - ", cMMph, "- 0.01*",Q),
paste0("1 + ", bR, " + ", bG, "(f_ + f_R + f_Mph) - ", cS, " (f_ + f_BTC + f_R + f_BTC_R) -",
cD , " * f_Mph"),
paste0("1 + ", bR, " + ", bG, " *(f_ + f_R + f_Mph) -", cMMTC, " - ",
cD , " * f_Mph")
),
stringsAsFactors = FALSE)
}We verify that we recover Table 11.4 :
create_fe("cG", "bG","cS", "cMMph", "cMMTC", "bR", "cD", "Q")
## Genotype
## 1 WT
## 2 BTC
## 3 R
## 4 MTC
## 5 Mph
## 6 BTC,R
## 7 MTC,R
## Fitness
## 1 1 + bG(f_ + f_R + f_Mph) - cG - cS(f_ + f_BTC + f_R + f_BTC_R) -0.01*Q
## 2 1 + bR + bG(f_ + f_R + f_Mph) - cS (f_ + f_BTC + f_R + f_BTC_R) -cD * f_Mph -Q
## 3 1 + bG(f_ + f_R + f_Mph) - cG - cS(f_ + f_BTC + f_R + f_BTC_R)
## 4 1 + bR + bG *(f_ + f_R + f_Mph) -cMMTC - cD * f_Mph -Q
## 5 1 + bG(f_ + f_R + f_Mph) - cG - cMMph- 0.01*Q
## 6 1 + bR + bG(f_ + f_R + f_Mph) - cS (f_ + f_BTC + f_R + f_BTC_R) -cD * f_Mph
## 7 1 + bR + bG *(f_ + f_R + f_Mph) -cMMTC - cD * f_MphWe check once again that we have correctly specified the different parameters executing evalAllGenotypes. We are trying the same case than in the previous example, having wild-type cells, macrophages and both cancer cells. But in this case, we are introducing the parameter “Q” (chemotherapy), with a value of 5.
evalAllGenotypes(allFitnessEffects(genotFitness =
create_fe(2,5,1,0.8,1,1,9,5),
frequencyDependentFitness = TRUE,
frequencyType = "rel"),
spPopSizes = c(WT = 1000, BTC = 100, R = 0,
MTC = 100, Mph = 1000,
"BTC, R" = 0, "MTC, R" = 0))
## Genotype Fitness
## 1 WT 2.995455
## 2 BTC 0.000000
## 3 MTC 0.000000
## 4 Mph 2.695455
## 5 R 3.045455
## 6 BTC, MTC 0.000000
## 7 BTC, Mph 0.000000
## 8 BTC, R 1.954545
## 9 MTC, Mph 0.000000
## 10 MTC, R 1.454545
## 11 Mph, R 0.000000
## 12 BTC, MTC, Mph 0.000000
## 13 BTC, MTC, R 0.000000
## 14 BTC, Mph, R 0.000000
## 15 MTC, Mph, R 0.000000
## 16 BTC, MTC, Mph, R 0.000000As we can see, BTC and MTC have a fitness of 0, so the will not proliferate. But cancer cells mutants with a double mutation (BTC, R and MTC, R) are resistant to chemotherapy, so they can still grow. This is what we were expecting to happen, so we started with the simulation.
Now, we create the allFitnessEffects object and do the simulation using the McFL model. In this case, we specify gene-specific mutations rates, so we can model a more realistic scenario where the appearence of chemotherapy-resistant cells is hindered. Once again, we simulate different situations by changing the values of the different parameters.
11.9.1 Absence of chemotherapy
In the first scenario, we simulate a situation where no chemotherapy is applied. In this condition, there is a very low R mutation rate, which hampers the proliferation of chemotherapy-resistant cells. A fast appearance of macrophages is observed, as well as non-resistant tumor cells. As there is no chemoterapy treatment, the fitness of non-resistant tumor cells is favoured against macrophages, leading to the appearance of cancer, where BTC cells show the greatest fitness due to the mobility cost of MTC cells.
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(2, 5, 1, 0.8, 1, 1, 9, 0),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
#Set mutation rates
muvar2 <- c("Mph" = 1e-2, "BTC" = 1e-3, "MTC"=1e-3, "R" = 1e-7)
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 20,
mu = muvar2,
initSize = 10000,
keepPhylog = TRUE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue", "pink", "orange", "brown"))
11.9.2 Chemotherapy with low R mutation rate
In the second scenario, we perform a simulation including chemotherapy as a treatment. However, here we maintain a low R mutation rate, hampering once again the appearance of chemotherapy-resistant cells. This situation can be reflecting the application of a combination chemotherapy that reduces or limits the appearance of resistance. We observe that the fitness of wild-type cells and macrophages increases rapidly and remains elevated, while tumor cells undergo some proliferation at first, but it remains under control over time thanks to the negative effect on them of chemotherapy.
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(2, 5, 1, 0.8, 1, 1, 9, 2),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
muvar2 <- c("Mph" = 1e-2, "BTC" = 1e-3, "MTC"=1e-3, "R" = 1e-7)
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 20,
mu = muvar2,
initSize = 10000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue", "pink", "orange", "brown"))
11.9.3 Chemotherapy with considerable R mutation rate
Finally, we simulate a scenario in the presence of chemotherapy as a treatment and also a considerable R mutation rate. This would allow the appearance of chemotherapy-resistant cells. This simulation reflects the situation of using chemotherapy treatments against which tumor cells develop resistance. Here, we observe a similar fitness evolution of wild-type cells and macrophages to the previous example, rapidly increasing their population and remaining elevated for a period of time. The proliferation of non-resistant tumor cells is also maintained under acceptable levels thanks to the effect of chemotherapy. However, due to the increased R mutation rate, chemotherapy-resistant cells begin to appear in low levels, until the fitness of chemotherapy-resistant benign tumor cells stats to increase considerably, leading to the disappearance of the other cell types and allowing the development of a non-metastatic cancer.
afe_3_a <- allFitnessEffects(
genotFitness =
create_fe(2, 5, 1, 0.8, 1, 1, 9, 2),
frequencyDependentFitness = TRUE,
frequencyType = "rel")
muvar2 <- c("Mph" = 1e-2, "BTC" = 1e-3, "MTC"=1e-3, "R" = 1e-5)
set.seed(2)
s_3_a <- oncoSimulIndiv(afe_3_a,
model = "McFL",
onlyCancer = FALSE,
finalTime = 20, ## short for speed; increase for "real"
mu = muvar2,
initSize = 10000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
plot(s_3_a, show = "genotypes", type = "line",
col = c("black", "green", "red", "blue", "pink", "orange", "brown"))
12 Simulating therapeutic interventions that depend on time
In this section we show some examples using the time dependent functionality; with it, fitness can be made to depend on T, the current time defined in the simulation. These examples were originally prepared by Niklas Endres, Rafael Barrero Rodríguez, Rosalía Palomino Cabrera and Silvia Talavera Marcos, as an exercisse for the course Programming and Statistics with R (Master’s Degree in Bioinformatics and Computational Biology, Universidad Autónoma de Madrid), course 2019-20; Niklas Endres had the idea of accessing T from exprTk.
This first example is an artificial simulation, but it shows how the fitness of a genotype can suddenly increase at a certain given timepoint.
## Fitness definition
fl <- data.frame(
Genotype = c("WT", "A", "B"),
Fitness = c("1", #WT
"if (T>50) 1.5; else 0;", #A
"0*f_") , #B
stringsAsFactors = FALSE
)
fe <- allFitnessEffects(genotFitness = fl,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
## Evaluate the fitness before and after the specified currentTime
evalAllGenotypes(fe, spPopSizes = c(100, 100, 100))
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1
## 2 A 0
## 3 B 0
## 4 A, B 0
evalAllGenotypes(fe, spPopSizes = c(100, 100, 100), currentTime = 80)
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 1 WT 1.0
## 2 A 1.5
## 3 B 0.0
## 4 A, B 0.0
## Simulation
sim <- oncoSimulIndiv(fe,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 0.01,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL,
errorHitMaxTries = FALSE,
errorHitWallTime = FALSE)
## Plot the results
plot(sim, show = "genotypes") 
12.1 Adaptive control of competitive release and chemotherapeutic resistance
The code structure of the previous example can be used, for instance, to simulate the effect of different chemotherapy treatment protocols.
An example of using these game theory concepts is the adaptive theory. The primary goal is to maximize the time of tumor control by using the tumor cells that are sensitive to treatment as agents that can supress the proliferation of the resistant cells. Thus, a significant residual populations of tumor cells can be under control to inhibit the growth of cells that otherwise cannot control: killed vs resistant.
Newton and Ma Newton and Ma (2019) built simulations for a tumor consisting in two types of cells: resistant to chemotherapy and sensitive to chemotherapy. The idea is to promote competition among tumor cells in order to prevent tumor growth.
To do this, they developed a three-component Prisoner’s Dilemma scenario including healthy (H) cells as well as chemoresistant (R) and chemosensitive (S) cancer cells. Healthy cells are cooperators, while cancer cells are defectors.
This is the system’s payoff matrix:
a <- 1; b <- 0.5; c <- 0.5 ## a b c
d <- 1; e <- 1.25; f <- 0.7 ## d e f
g <- 0.975; h <- -0.5; i <- 0.75 ## g h i
payoff_m <- matrix(c(a,b,c,d,e,f,g,h,i), ncol=3, byrow=TRUE)
colnames(payoff_m) <- c("Healthy", "Chemo-sensitive", "Chemo-resistant")
rownames(payoff_m) <- c("Healthy", "Chemo-sensitive", "Chemo-resistant")
print(payoff_m <- as.table(payoff_m))
## Healthy Chemo-sensitive Chemo-resistant
## Healthy 1.000 0.500 0.500
## Chemo-sensitive 1.000 1.250 0.700
## Chemo-resistant 0.975 -0.500 0.750The numerical values in the matrix are selected to satisfy the following theoretical constraints:
g > a > i > c; d > a > e > b; f > i > e > h and d > g (cost to resistance).
Thus, the fitness definitions for the three types of cells could be written as in the following data frame:
print( df <- data.frame(
CellType = c("H", "S", "R"),
Fitness = c("F(H) = ax(H) + bx(S) + cx(R)", #Healthy
"F(S) = dx(H) + ex(S) + fx(R)", #Sensitive
"F(R) = gx(H) + hx(S) + ix(R)")), #Resistant
row.names = FALSE )
## CellType Fitness
## H F(H) = ax(H) + bx(S) + cx(R)
## S F(S) = dx(H) + ex(S) + fx(R)
## R F(R) = gx(H) + hx(S) + ix(R)In summary:
- The fitness of H cells is lower than the chemosensitive cells.
- Without any therapy, S cells have a higher fitness value than R cells, because of the cost to resistance
- In the presence of therapy, R cells take over the S cells
One of the challenges is to optimize the drug dosage intervals. In practice, this would mean to infer the growth rates of the different cell types from a frequent monitoring of the tumor environment. With OncoSimulR is possible to try out different intervals on simulations of the collected data.
12.1.1 Scenario without chemotherapy
First of all, we can simulate the growth of a tumor from H cells without any treatment. We can consider that R tumor cells are generated from S and WT cells. Thus, as we expected, we can observe in the simulation results that S cells grow (tumor) and R cells cannot subsist because of their fitness disadvantage: the cost of being resistant.
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
## Coefficients
# Healthy Sensitive Resistant
a=3; b=1.5; c=1.5 # Healthy
d=4; e=5; f=2.8 # Sensitive
g=3.9; h=-2; i=2.2 # Resistant
# Here we divide coefficients to reduce the amount of cells obtained in the simulation.
# We have divided a, b and c by 3, and d, e and i by 4.
# Healthy Sensitive Resistant
a <- 1; b <- 0.5; c <- 0.5 # Healthy
d <- 1; e <- 1.25; f <- 0.7 # Sensitive
g <- 0.975; h <- -0.5; i <- 0.75 # Resistant
## Fitness definition
players <- data.frame(Genotype = c("WT","S","R","S,R"),
Fitness = c(paste0(a, "*f_+", b, "*f_S+", c, "*f_S_R"), #WT
paste0(d,"*f_+",e,"*f_S+",f,"*f_S_R"), #S
"0", #R
paste0(g,"*f_+",h,"*f_S+",i,"*f_S_R")), #S,R
stringsAsFactors = FALSE)
game <- allFitnessEffects(genotFitness = players,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## Plot the first scenario
eag <- evalAllGenotypes(game, spPopSizes = c(10,1,0,10))[c(1, 3, 4),]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
plot(eag)
## Simulation
gamesimul <- oncoSimulIndiv(game,
model = "McFL",
onlyCancer = FALSE,
finalTime = 40,
mu = 0.01,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL)
## Plot 2
plot(gamesimul, show = "genotypes", type = "line",
col = c("black", "green", "red"), ylim = c(20, 50000))
plot(gamesimul, show = "genotypes")
12.1.2 Scenario with continuous chemotherapy: fixed dose
For this simulation, we add the effect of chemotherapy as a fixed coefficient (drug_eff), representing a fixed dose. The dose is delivered only when a tumor has grown up, so we perform the drug effect after starting the simulation. For this, we apply the time funcionality used in the first section of this chapter.
# Effect of drug on fitness sensible tumor cells
drug_eff <- 0.01
wt_fitness <- paste0(a, "*f_+", b, "*f_S+", c, "*f_S_R")
sens_fitness <- paste0(d, "*f_+", e, "*f_S+", f, "*f_S_R")
res_fitness <- paste0(g, "*f_+", h, "*f_S+", i, "*f_S_R")
players_1 <- data.frame(Genotype = c("WT", "S", "R", "S, R"),
Fitness = c(wt_fitness, #WT
paste0("if (T>50) ", drug_eff, "*(",sens_fitness, ")",";
else ", sens_fitness, ";"), #S
"0", #R
res_fitness), #S,R
stringsAsFactors = FALSE)
period_1 <- allFitnessEffects(genotFitness = players_1,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(2)
final_time <- 170 ## for speed
simul_period_1 <- oncoSimulIndiv(period_1,
model = "McFL",
onlyCancer = FALSE,
finalTime = final_time,
mu = 0.01,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL)
# ylim has been adapted to number of cells
plot(simul_period_1, show = "genotypes", type = "line",
col = c("black", "green", "red"), ylim = c(20, 300000),
thinData = TRUE)
## plot(simul_period_1, show = "genotypes", ylim = c(20, 12000))As expected, the simulation results show how sensitive cells suddenly decrease when the current time of the simulation reach a value of 50, which is the consequence of the chemotherapy. In this regard, the code illustrates how sensitive cells fitness is multiplied by drug_eff variable after 50 units of time, and then resistant cells start to grow exponentially until reaching an equilibrium where chemoterapy does not affect them anymore.
With this example, we can show how chemotherapy usage could be counterproductive under certain situations, especially in those cases in which resistant tumor cells are more aggressive than sensitive cells.
12.1.3 Scenario with switching doses of chemotherapy
The original model by Newton and Ma Newton and Ma (2019) includes chemotherapeutic dosage as a time-dependent controller. The main idea is to increase or decrease the dose according to a periodic cancer growth in order to avoid the fixation of both sensitive and resistant cells, and keep the tumor trajectory enclosed in a loop.
The model developed by this group is a is a cubic nonlinear system based on Hamiltonian orbits. They use time-dependent chemotherapeutic parameters w, whose values are different in each one of the several and carefully chosen intervals that depends on C(t), which is the chemo-concentration parameter. For simplicity’s sake, we will just define the fitness of sensitive cells as dependent on a sine time function.
During this simulation, we will see in our results small oscilations doses that keep sensitive cells population at a minimum value and the R cells progress is prevented.
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
# Healthy Sensitive Resistant
a <- 1; b <- 0.5; c <- 0.5 # Healthy
d <- 1; e <- 1.25; f <- 0.7 # Sensitive
g <- 0.975; h <- -0.5; i <- 0.75 # Resistant
wt_fitness <- paste0(a, "*f_+", b, "*f_S+", c, "*f_S_R")
sens_fitness <- paste0(d, "*f_+", e, "*f_S+", f, "*f_S_R")
res_fitness <- paste0(g, "*f_+", h, "*f_S+", i, "*f_S_R")
fitness_df <-data.frame(Genotype = c("WT", "S", "R", "S, R"),
Fitness = c(wt_fitness, #WT
paste0("if (T>50) (sin(T+2)/10) * (", sens_fitness,")",
"; else ", sens_fitness, ";"), #S
"0", #R
res_fitness), #S,R
stringsAsFactors = FALSE)
afe <- allFitnessEffects(genotFitness = fitness_df,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
switching_sim <- oncoSimulIndiv(afe,
model = "McFL",
onlyCancer = FALSE,
finalTime = 100,
mu = 0.01,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL)
plot(switching_sim, show = "genotypes", type = "line",
col = c("black", "green", "red"), ylim = c(20, 200000))
## plot(switching_sim, show = "genotypes", ylim = c(20, 1000))12.2 Growth factors as chemotherapy target
It has been reported in the literature and verified in the above simulations, that resistance acquisition is almost unavoidable. A new approach to avoid this evolutionary adaptation, proposes to change the chemotherapy target, from the cell subclones to the growing factors (GF) they produce. These molecules, which are secreted to the medium by cooperators, help to grow their own subclone types besides the GF defectors. Thus, the fitness of the whole population increases and the tumor grows.
An attended side effect of this type of treatment is the emergence of cooperators that overproduce GF. This will increase their fitness and reduce the impact of GF sequestering agents. However, it would also increase the cost of its production, which will decrease this benefitial impact.
Archetti (2013) develops a series of formulas to relate the impact of cooperation in tumor composition and fitness of subclones, which they divide as cooperators and defectors. Although GFs are not always distributed homogeneously, we will assume they do in the following simulations for the sake of simplicity.
Attending to the theoretical foundations mentioned, we will create a cooperation scenario, including wild-type healthy cells (WT) and tumour cells, which are cooperators (C), defectors (D) and overproducers (P).
The system payoff matrix is the following:
# WT Cooperators Defectors Overproducers
a <- 1; b <- 0.5; c <- 0.5; m <- 0.75 ## a b c m
d <- 1; e <- 1.25; f <- 0.7; o <- 0.185 ## d e f o
g <- 1; h <- 1.5; i <- 0.5; p <- 2.5 ## g h i p
j <- 0.8; k <- 1; l <- 0.5; q <- 1.5 ## j k l q
payoff_m <- matrix(c(a,b,c,m,d,e,f,o,g,h,i,p,j,k,l,q), ncol=4, byrow=TRUE)
colnames(payoff_m) <- c("WT", "Cooperators", "Defectors", "Overproducers")
rownames(payoff_m) <- c("WT", "Cooperators", "Defectors", "Overproducers")
print(payoff_m <- as.table(payoff_m))
## WT Cooperators Defectors Overproducers
## WT 1.000 0.500 0.500 0.750
## Cooperators 1.000 1.250 0.700 0.185
## Defectors 1.000 1.500 0.500 2.500
## Overproducers 0.800 1.000 0.500 1.500The fitness definitions for the four types of cells would be the following:
print( df <- data.frame(
CellType = c("WT", "C", "D", "P"),
Fitness = c("F(WT) = ax(WT) + bx(C) + cx(D) + mx(P)",
"F(C) = dx(WT) + ex(C) + fx(D) + ox(P)",
"F(D) = gx(WT) + hx(C) + ix(D) + px(P)",
"F(P) = jx(WT) + kc(C) + lx(D) + qx(P)")),
row.names = FALSE )
## CellType Fitness
## WT F(WT) = ax(WT) + bx(C) + cx(D) + mx(P)
## C F(C) = dx(WT) + ex(C) + fx(D) + ox(P)
## D F(D) = gx(WT) + hx(C) + ix(D) + px(P)
## P F(P) = jx(WT) + kc(C) + lx(D) + qx(P)Ordered by decreasing fitness:
In summary:
- The fitness of WT cells is lower than the tumour cells.
- Without any therapy, C cells fitness is higher than P cells, so the overproducers will not overtake them.
- D subclones benefit from the high concentration of cooperators (C or P).
12.2.1 Scenario without chemotherapy
First, we will study the fitness of the subclones types based on hypothetical frequencies. Then, we will simulate the growth of the tumor without any treatment. For this, we are considering that C cells are the original tumor cells and they can mutate and lose by deletion the GF gene (D cells grow) or duplicate it (P cells grow).
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
## Coefficients
## New coefficients for the interaction with overproducing sensitive:
# WT COOPERATOR DEFECTOR OVERPRODUCER
a <- 1; b <- 0.5; c <- 0.5; m <- 0.75 # WT
wt_fitness <- paste0(a, "*f_+", b, "*f_C+", c, "*f_C_D+", m, "*f_C_P")
d <- 1; e <- 1.25; f <- 0.7; o <- 1.875 # Cooperator
coop_fitness <- paste0(d, "*f_+", e, "*f_C+", f, "*f_C_D+", o, "*f_C_P")
g <- 1; h <- 1.5; i <- 0.5; p <- 2.5 # Defector
def_fitness <- paste0(g, "*f_+", h, "*f_C+", i, "*f_C_D+", p, "*f_C_P")
j <- 0.8; k <- 1; l <- 0.5; q <- 1.5 # Cooperator overproducing
over_fitness <- paste0(j, "*f_+", k, "*f_C+", l, "*f_C_D+", q, "*f_C_P")
## No-chemotherapy
## Fitness definition
coop_no <- data.frame(Genotype = c("WT", "C", "D", "P", "C,D", "C,P", "D,P", "C,D,P"),
Fitness = c(
wt_fitness, #WT
coop_fitness, #S
"0", #D
"0", #P
def_fitness, #S,D
over_fitness, #S,P
"0", #D,P
"0" #C,D,P
),
stringsAsFactors = FALSE)
game_no <- allFitnessEffects(genotFitness = coop_no,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## First plot
eag <- evalAllGenotypes(game_no,
spPopSizes = c(WT = 10, C = 10, D = 0, P = 0,
"C, D" = 10, "C, P" = 1,
"D, P" = 0, "C, D, P" = 0))[c(1, 2, 5, 6),]
plot(eag)
## Simulation
gamesimul_no <- oncoSimulIndiv(game_no,
model = "McFL",
onlyCancer = FALSE,
finalTime = 35,
mu = 0.01,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL)
## Second plot
plot(gamesimul_no, show = "genotypes", type = "line",
col = c("blue", "red", "green", "purple"), ylim = c(20, 50000),
thinData = TRUE)
# Third plot
## plot(gamesimul_no, show = "genotypes")The resulting plots show that D cells are highly benefited by the second most common subclone activity (C cells), which is a cooperator. On the other hand, subclone P has a lower fitness value because of the GF cost. However, as we can see in the first plot, a small number of clones remain and survive.
12.2.2 Scenario with GF as target for chemotherapy
In this simulation, we will add the drug effect (at time 50) to the fitness of each subclone. We assume that this will cause a reduction of the fitness by the level of GF dependency for each subclone. We now introduce some constants to make it more accurate. As cooperators will keep producing GF, they will be less affected by the sequestration of this molecule, so their coefficients will be greater than 1. In addition, P also will be less affected since its production of GF is greater, so its coefficient will be greater than the C one (1.5 > 1.1). On the other hand, as D is a defector, it will be more affected by the treatment, so its coefficient will set to 0.9, smaller than 1.
## Chemotherapy - GF Impairing
# Effect of drug on GF availability
# This term is multiplied by the fitness, and reduces the GF available
drug_eff <- 0.25
coop_fix <- data.frame(Genotype = c("WT", "C", "D", "P", "C,D", "C,P", "D,P", "C,D,P"),
Fitness = c(
wt_fitness, #WT
paste0("if (T>50) ", drug_eff, "* 1.2 *(", coop_fitness, ")",
"; else ", coop_fitness, ";"), #C
"0", #D
"0", #P
paste0("if (T>50) ", drug_eff, "*(", def_fitness, ")",
"; else ", def_fitness, ";"), #C,D
paste0("if (T>50) ", drug_eff, "* 1.5 * (", over_fitness, ")",
"; else ", over_fitness, ";"), #C,P **
"0", #D,P
"0" #C,D,P
),
stringsAsFactors = FALSE)
## ** The drug effect is 1.5 times the original because of the overproduction of GF and the
## full availability of this molecule inside the producing subclone.
period_fix <- allFitnessEffects(genotFitness = coop_fix,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
set.seed(2)
final_time <- 30 ## you'd want this longer; short for speed of vignette
simul_period_fix <- oncoSimulIndiv(period_fix,
model = "McFL",
onlyCancer = FALSE,
finalTime = final_time,
mu = 0.01,
initSize = 5000,
keepPhylog = FALSE,
seed = NULL)
# First plot
plot(simul_period_fix, show = "genotypes", type = "line",
col = c("blue", "red", "green", "purple"), ylim = c(20, 50000),
thinData = TRUE)
# Second plot
## plot(simul_period_fix, show = "genotypes", ylim = c(20, 8000))As expected by Archetti and Pienta Archetti and Pienta (2019), the tumour will be biased towards P mutant. Even if its fitness is lower than C because of its overproduction of GF, it will assure its survival and proliferation when chemotherapy is applied. As a result of this, D does not completely disappear, but reduces dramatically its number.
12.3 Examples using time dependent frequency definition
Here in this chapter we will comment some others approaches that can have this funcionality: increasing or decreasing the fitness as therapeutic interventions, or slow down the collapse of a subpopulation of cells.
12.3.1 Increasing fitness at a certain timepoint
It is possible to increase the fitness value by using T functionality. In the following example we can see how fitness value in genotypes A and B increases when the simulation time reaches a specific value. Since we have used the exponential model, that is the reason why we observe some delay between the specified time and when A or B populations start to grow.
dfT1 <- data.frame(Genotype = c("WT", "A", "B"),
Fitness = c("1",
"if (T>50) 1 + 2.35*f_; else 0.50;",
"if (T>200) 1 + 0.45*(f_ + f_1); else 0.50;"),
stringsAsFactors = FALSE)
afeT1 <- allFitnessEffects(genotFitness = dfT1,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(1)
simT1 <- oncoSimulIndiv(afeT1,
model = "Exp",
mu = 1e-5,
initSize = 1000,
finalTime = 500,
onlyCancer = FALSE,
seed = NULL)
plot(simT1, show = "genotypes", type = "line")
We can check if the fitness values have increased by evaluating the genotypes in the simulation time intervals.
evalAllGenotypes(afeT1, spPopSizes = c(10,10,10), currentTime = 49)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 0.5
## 3 B 0.5
evalAllGenotypes(afeT1, spPopSizes = c(10,10,10), currentTime = 51)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.783333
## 3 B 0.500000
evalAllGenotypes(afeT1, spPopSizes = c(10,10,10), currentTime = 201)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.783333
## 3 B 1.30000012.3.2 Intervention at a certain point to stop subpopulation growth
On the other hand, besides using T functionality to create time intervals, we can also use it as an intervention. In the following example we have used the functionality to increase the fitness value of genotype A, and suddenly decreases it as an intervention, where B population takes advantage to grow, since its fitness is greater now. When the intervention elapses, we can see how A population starts to grow again and outcompetes with wild-type population whose fitness does not change during the simulation.
dfT2 <- data.frame(Genotype = c("WT", "A", "B"),
Fitness = c(
"1",
"if (T>0 and T<50) 0; else if (T>100 and T<150) 0.05; else 1.2 + 0.35*f_;",
"0.8 + 0.45*(f_)"
),
stringsAsFactors = FALSE)
afeT2 <- allFitnessEffects(genotFitness = dfT2,
frequencyDependentFitness = TRUE,
frequencyType = "rel")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(1)
simT2 <- oncoSimulIndiv(afeT2,
model = "McFL",
mu = 1e-5,
initSize = 10000,
finalTime = 225,
onlyCancer = FALSE,
seed = NULL)
plot(simT2, show = "genotypes", thinData = TRUE)
In this case, just like above, we can evaluate the fitness in each time interval and observe how fitness is differente in each one of them.
evalAllGenotypes(afeT2, spPopSizes = c(100,10,10), currentTime = 49)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 0.000
## 3 B 1.175
evalAllGenotypes(afeT2, spPopSizes = c(100,10,10), currentTime = 51)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.491667
## 3 B 1.175000
evalAllGenotypes(afeT2, spPopSizes = c(100,10,10), currentTime = 101)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 0.050
## 3 B 1.175
evalAllGenotypes(afeT2, spPopSizes = c(100,10,10), currentTime = 201)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.491667
## 3 B 1.17500012.3.3 Intervention to slow down collapsing populations
We can use time dependent frequency functionality to slow down a collapsing population by doing an intervention at a certain time interval. In this case, we observe that genotype B has previously a higher fitness than genotype A (as long as the number of cells is greater than 10), but at certain point we can reduce its fitness in order to make A grow and reach B. However, after the time interval, B recovers its initial fitness again and overtakes A, but eventually collapses due to there are no more A’s around. This could be a way to slow down the collapse of a population when there is a size dependency between them.
dfT3 <- data.frame(Genotype = c("WT", "A", "B"),
Fitness = c(
"1",
"1 + 0.2 * (n_2 > 10)",
"if (T>50 and T<80) 0.80; else 0.9 + 0.4 * (n_1 > 10)"),
stringsAsFactors = FALSE)
afeT3 <- allFitnessEffects(genotFitness = dfT3,
frequencyDependentFitness = TRUE,
frequencyType = "abs")
## All single-gene genotypes as input to to_genotFitness_std
set.seed(2)
simT3 <- oncoSimulIndiv(afeT3,
model = "McFLD",
mu = 1e-4,
initSize = 5000,
finalTime = 500,
onlyCancer = FALSE,
seed = NULL,
errorHitWallTime = FALSE,
errorHitMaxTries = FALSE)
plot(simT3, show = "genotypes", type = "line")
We evaluate the fitness before and after genotype B has a higher and lower fitness respectively, and finally ends up collapsing because the condition of “n_1 > 10” is no longer accomplished.
evalAllGenotypes(afeT3, spPopSizes = c(10,10,10), currentTime = 30)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.0
## 3 B 0.9
evalAllGenotypes(afeT3, spPopSizes = c(11,11,11), currentTime = 79)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.2
## 3 B 0.8
evalAllGenotypes(afeT3, spPopSizes = c(11,11,11), currentTime = 81)[c(2,3), ]
## Warning in match_spPopSizes(spPopSizes, fmEffects): spPopSizes
## unnamed: cannot check genotype names.
## Genotype Fitness
## 2 A 1.2
## 3 B 1.313 Measures of evolutionary predictability and genotype diversity
Several measures of evolutionary predictability have been proposed in the literature (see, e.g., Szendro, Franke, et al. (2013) and references therein). We provide two, Lines of Descent (LOD) and Path of the Maximum (POM), following Szendro, Franke, et al. (2013); we also provide a simple measure of diversity of the actual genotypes sampled.
In Szendro, Franke, et al. (2013) “(…) paths defined as the time
ordered sets of genotypes that at some time contain the largest
subpopulation” are called “Path of the Maximum” (POM) (see their
p. 572). In our case, POM are obtained by finding the clone with
largest population size whenever we sample and, thus, the POMs will be
affected by how often we sample (argument sampleEvery), since we
are running a continuous time process.
Szendro, Franke, et al. (2013) also define Lines of Descent (LODs) which “(…) represent the lineages that arrive at the most populated genotype at the final time”. In that same page (572) they provide the details on how the LODs are obtained. Starting with version 2.9.2 of OncoSimulR I only provide an implementation where a single LOD per simulation is returned, with the same meaning as in Szendro, Franke, et al. (2013).
To briefly show some output, we will use again the 5.5 example.
pancr <- allFitnessEffects(
data.frame(parent = c("Root", rep("KRAS", 4), "SMAD4", "CDNK2A",
"TP53", "TP53", "MLL3"),
child = c("KRAS","SMAD4", "CDNK2A",
"TP53", "MLL3",
rep("PXDN", 3), rep("TGFBR2", 2)),
s = 0.05, sh = -0.3, typeDep = "MN"))
pancr16 <- oncoSimulPop(16, pancr, model = "Exp",
mc.cores = 2)
## Look a the first POM
str(POM(pancr16)[1:3])
## List of 3
## $ : chr [1:2] "" "KRAS"
## $ : chr [1:2] "" "KRAS"
## $ : chr [1:3] "" "KRAS" "KRAS, MLL3"
LOD(pancr16)[1:2]
## [[1]]
## [1] "" "KRAS"
##
## [[2]]
## [1] "" "KRAS"
## The diversity of LOD (lod_single) and POM might or might not
## be identical
diversityPOM(POM(pancr16))
## [1] 0.8222653
diversityLOD(LOD(pancr16))
## [1] 0.4634136
## Show the genotypes and their diversity (which might, or might
## not, differ from the diversity of LOD and POM)
sampledGenotypes(samplePop(pancr16))
##
## Subjects by Genes matrix of 16 subjects and 7 genes.
## Genotype Freq
## 1 KRAS 14
## 2 KRAS, MLL3 1
## 3 KRAS, SMAD4 1
##
## Shannon's diversity (entropy) of sampled genotypes: 0.4634136Beware, however, that if you use multiple initial mutants (section 6.7) the LOD function will probably not do what you want. It is not even clear that the LOD is well defined in this case. We are working on this.
14 Generating random DAGs for restrictions
You might want to randomly generate DAGs like those often found in the
literature on Oncogenetic trees et al. Function simOGraph
might help here.
## No seed fixed, so reruns will give different DAGs.
(a1 <- simOGraph(10))
## Root 1 2 3 4 5 6 7 8 9 10
## Root 0 1 1 1 0 0 0 0 0 0 0
## 1 0 0 0 0 1 1 1 0 0 0 0
## 2 0 0 0 0 0 1 1 1 0 0 0
## 3 0 0 0 0 1 1 1 0 0 0 0
## 4 0 0 0 0 0 0 0 1 0 0 0
## 5 0 0 0 0 0 0 0 0 1 1 1
## 6 0 0 0 0 0 0 0 0 1 1 0
## 7 0 0 0 0 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0 0 0 0 0
## 9 0 0 0 0 0 0 0 0 0 0 0
## 10 0 0 0 0 0 0 0 0 0 0 0
library(graph) ## for simple plotting
plot(as(a1, "graphNEL"))
Once you obtain the adjacency matrices, it is for now up to you to convert them into appropriate posets or fitnessEffects objects.
Why this function? I searched for, and could not find any that did what I
wanted, in particular bounding the number of parents, being able to
specify the approximate depth15 of the graph, and optionally
being able to have DAGs where no node is connected to another both
directly (an edge between the two) and indirectly (there is a path between
the two through other nodes). So I wrote my own code. The code is fairly
simple to understand (all in file generate-random-trees.R). I
would not be surprised if this way of generating random graphs has been
proposed and named before; please let me know, best if with a reference.
Should we remove direct connections if there are indirect? Or,
should we set removeDirectIndirect = TRUE? Setting
removeDirectIndirect = TRUE is basically asking for
the
transitive reduction of
the generated DAG. Except for Farahani and Lagergren (2013) and
Ramazzotti et al. (2015), none of the DAGs I’ve seen in the context of
CBNs, Oncogenetic trees, etc, include both direct and indirect
connections between nodes. If these exist, reasoning about the model
can be harder. For example, with CBN (AND or CMPN or monotone
relationships) adding a direct connection makes no difference iff we
assume that the relationships encoded in the DAG are fully respected
(e.g., all \(s_h = -\infty\)). But it can make a difference if we
allow for deviations from the monotonicity, specially if we only
check for the satisfaction of the presence of the immediate
ancestors. And things get even trickier if we combine XOR with
AND. Thus, I strongly suggest you leave the
default removeDirectIndirect = TRUE. If you change it, you should
double check that the fitnesses of the possible genotypes are what
you expect. In fact, I would suggest that, to be sure you get what
you think you should get, you convert the fitness from the DAG to a
fitness table, and pass that to the simulations, and this requires
using non-exposed user functions; to give you an idea, this could
work (but you’ve been warned: this is dangerous!)
g2 <- simOGraph(4, out = "rT", removeDirectIndirect = FALSE)
fe_from_d <- allFitnessEffects(g2)
fitness_d <- evalAllGenotypes(fe_from_d)
fe_from_t <- allFitnessEffects(genotFitness =
OncoSimulR:::allGenotypes_to_matrix(fitness_d))
## Compare
fitness_d
(fitness_t <- evalAllGenotypes(fe_from_t))
identical(fitness_d, fitness_t)
## ... but to be safe use fe_from_t as the fitnessEffects object for simulations
15 FAQ, odds and ends
15.1 What we mean by “clone”; and “I want clones disregarding passengers”
In this vignette we often use “clone” or “genotype” interchangeably. A clone denotes a set of cells that have identical genotypes. So if you are using a fitness specification with four genes (i.e., your genome has only four loci), there can be up to \(16 = 2^4\) different genotypes or clones. Any two entities that differ in the genotype are different clones. And this applies regardless of whether or not you declare that some genes (loci) are drivers or not. So if you have four genes, it does not matter whether only the first or all four are regarded as drivers; you will always have at most 16 different clones or 16 different genotypes. Of course you can arrive at the same clone/genotype by different routes. Just think about loci A and B in our four-loci genome, and how you can end up with a cell with both A and B mutated.
Analogously, if you have 100 genes, 10 drivers and 90 passengers, you can have up to \(2^{100}\) different clones or genotypes. Sure, one cell might have driver A mutated and passenger B mutated, and another cell might have driver A mutated and passenger C mutated. So if you only look at drivers you might be tempted to say that they are “the same clone for all practical purposes”; but they really are not the same clone as they differ in their genotype and this makes a lot of difference computationally.
If you want summaries of simulations that collapse over some genes
(say, some “passengers”, the 90 passengers we just mentioned) look
at the help for samplePop, argument geneNames. This would allow
you, for instance, to look at the diversity of clones/genotypes,
considering as identical those genotypes that only differ in genes
you deem relevant; something similar to defining a “drivers’ clone”
as the set formed from the union of all sets of cells that have
identical genotype with respect to only the drivers (so that in the
example of “A, B” and “A, C” just mentioned both cells would be
considered “the same clone” as they only differ with respect to
passengers). However, this “disregard some genes” only applies to
summaries of simulations once we are done simulating
data. OncoSimulR will always track clones, as defined above,
regardless of whether many of those clones have the same genotype if
you were to only focus on driver genes; see also section
15.2.
Labeling something as a “driver”, therefore, does not affect what we mean
by clone. Yes, labeling something as a driver can affect when you stop
simulations if you use detectionDrivers as a stopping mechanism (see
section 6.3). But, again, this has nothing to do with the
definition of “clone”.
If this is all obvious to you, ignore it. I am adding it here because I’ve seen strange misunderstandings that eventually could be traced to the apparently multiple meanings of clone. (And to make the story complete, Mather, Hasty, and Tsimring (2012) use the expression “class” —e.g., Algorithm 4 in the paper, Algorithm 5 in the supplementary material).
15.2 Does OncoSimulR keep track of individuals or of clones? And how can it keep track of such large populations?
OncoSimulR keeps track of clones, where a clone is a set of cells that are genetically identical (note that this means completely identical over the whole set of genes/markers you are using; see section 15.1). We do not need to keep track of individual cells because, for all purposes, and since we do not consider spatial structure, two or more cells that are genetically identical are interchangeable. This means, for instance, that the computational cost of keeping a population of a single clone with 1 individual or with \(10^9\) individuals is exactly the same: we just keep track of the genotype and the number of cells. (Sure, it is much more likely we will see a mutation soon in a clone with \(10^9\) cells than in a clone with 1, but that is a different issue.)
Of course, the entities that die, reproduce, and mutate are individual cells. This is of course dealt with by tracking clones (as is clearly shown by Algorithms 4 and 5 in Mather, Hasty, and Tsimring (2012)). Tracking individuals, as individuals, would provide no advantage, but would increase the computational burden by many orders of magnitude.
15.2.1 sampleEvery, keepPhylog, and pruning
At each sampling time (where sampleEvery determines the time units
between sampling times) the abundance of all the clones with number of
cells \(>0\) is recorded. This is the structure that at the end of the run
is converted into the pops.by.time matrix.
Now, some clones might arise from mutation between successive
population samples but these clones might be extinct by the time we
take a population sample. These clones do not appear in the
pops.by.time matrix because, as we just said, they have 0 cells at
the time of sampling. Of course, some of these clones might appear
again later and reach a size larger than 0 at some posterior
sampling time; it is at this time when this/these clone(s) will
appear in the pops.by.time matrix. This pruning of clones with 0
cells can allow considerable savings in computing time (OncoSimulR
needs to track the genotype of clones, their population sizes,
their birth, death, and mutation rates, their next mutation time and
the last time they were updated and thus it is important that we
only loop over structures with information that is really needed).
However, we still need to track clones as clones, not simply as classes such as “number of mutated genes”. Therefore, very large genomes can represent a problem if they lead to the creation and tracking of many different clones (even if they have the same number of mutated genes), as we have seen, for instance, in section 2.3. In this case, programs that only keep track of numbers of mutated genes or of drivers, not individual clones, can of course achieve better speed.
What about the genealogy? If you ask OncoSimulR to keep track of the
complete parent-child relationships (keepPhylog = TRUE), you might
see in the genealogy clones that are not present in pops.by.time
if these are clones that never had a population size larger than 0
at any sampling time. To give an example, suppose that we will take
population samples at times 0, 1, and 2. Clone A, with a population
size larger than 0 at time 1, gives rise at time 1.5 to clone B;
clone B then gives rise to clone C at time 1.8. Finally, suppose
that at time 2 only clone C is alive. In other words, when we carry
out the update of the population with Algorithm 5 from Mather, Hasty, and Tsimring (2012),
clones A and B have size 0. Now, at time 1 clones B and C did not
yet exist, and clone B is never alive at times 1 or 2. Thus, clone B
is not present in pops.by.time. But we cannot remove clone B from
our genealogy if we want to reflect the complete genealogy of C. Thus,
pops.by.time will show only clones A and C (not B) but the
complete genealogy will show clones A, B, C (and will show that B
appeared from A at time 1.5 and C appeared from B at time
1.8). Since function plotClonePhylog offers a lot of flexibility
with respect to what clones to show depending on their population
sizes at different times, you can prevent being shown B, but its
existence is there should you need it (see also
2.3.5).
15.3 Dealing with errors in “oncoSimulPop”
When running OncoSimulR under Windows mclapply does not use
multiple cores, and errors from oncoSimulPop are reported
directly. For example:
## This code will only be evaluated under Windows
if(.Platform$OS.type == "windows")
try(pancrError <- oncoSimulPop(10, pancr,
initSize = 1e-5,
detectionSize = 1e7,
keepEvery = 10,
mc.cores = 2))Under POSIX operating systems (e.g., GNU/Linux or Mac OSX)
oncoSimulPop can ran parallelized by calling
mclapply. Now, suppose you did something like
## Do not run under Windows
if(.Platform$OS.type != "windows")
pancrError <- oncoSimulPop(10, pancr,
initSize = 1e-5,
detectionSize = 1e7,
keepEvery = 10,
mc.cores = 2)
## Warning in mclapply(seq.int(Nindiv), function(x) oncoSimulIndiv(fp =
## fp, : all scheduled cores encountered errors in user codeThe warning you are seeing tells you there was an error in the functions
called by mclapply. If you check the help for
mclpapply you’ll see that it returns a try-error object, so we
can inspect it. For instance, we could do:
pancrError[[1]]But the output of this call might be easier to read:
pancrError[[1]][1]And from here you could see the error that was returned by
oncoSimulIndiv: initSize < 1 (which is indeed true:
we pass initSize = 1e-5).
15.4 Whole tumor sampling, genotypes, and allele counts: what gives? And what about order?
You are obtaining genotypes, regardless of order. When we use “whole tumor sampling”, it is the frequency of the mutations in each gene that counts, not the order. So, for instance, “c, d” and “c, d” both contribute to the counts of “c” and “d”. Similarly, when we use single cell sampling, we obtain a genotype defined in terms of mutations, but there might be multiple orders that give this genotype. For example, \(d > c\) and \(c > d\) both give you a genotype with “c” and “d” mutated, and thus in the output you can have two columns with both genes mutated.
15.5 Doesn’t the BNB algorithm require small mutation rates for it to be advantageous?
As discussed in the original paper by Mather, Hasty, and Tsimring (2012) (see also their supplementary material), the BNB algorithm can achieve considerable speed advantages relative to other algorithms especially when mutation events are rare relative to birth and death events; the larger the mutation rate, the smaller the gains compared to other algorithms. As mentioned in their supplementary material (see p.5) “Note that the ‘cost’ of each step in BNB is somewhat higher than in SSA [SSA is the original Gillespie’s Stochastic Simulation Algorithm] since it requires generation of several random numbers as compared to only two uniform random numbers for SSA. However this cost increase is small compared with significant benefits of jumping over birth and death reactions for the case of rare mutations.”
Since the earliest versions, OncoSimulR has provided information to
assess these issues. The output of function oncoSimulIndiv
includes a list called “other” that itself includes two lists
named “minDMratio” and “minBMratio”, the smallest ratio, over
all simulations, of death rate to mutation rate or birth rate to
mutation rate, respectively. As explained above, the BNB algorithm
thrives when those are large. Note, though, we say “it thrives”:
these ratios being large is not required for the BNB algorithm to be
an exact simulation algorithm; these ratios being large make BNB
comparatively much faster than other algorithms.
15.6 Can we use the BNB algorithm with state-dependent birth or death rates?
As discussed in the original paper by Mather, Hasty, and Tsimring (2012) (see sections 2.6
and 3.2 of the paper and section E of the supplementary material),
the BNB algorithm can be used as an approximate stochastic
simulation algorithm “(…) with non-constant birth, death, and
mutation rates by evolving the system with a BNB step restricted to
a short duration t.” (p. 9 in supplementary material). The
justification is that “(…) the propensities for reactions can be
considered approximately constant during some short interval.”
(p. 1234). This is the reason why, when we use McFarland’s model, we
set a very short sampleEvery. In addition, the output of the
simulation functions contains the simple summary statistic errorMF
that can be used to assess the quality of the
approximation16.
Note that, as the authors point out, approximations are common with stochastic simulation algorithms when there is density dependence, but the advantage of the BNB algorithm compared to, say, most tau-leap methods is that clones of different population sizes are treated uniformly. Mather, Hasty, and Tsimring (2012) further present results from simulations comparing the BNB algorithm with the original direct SSA method and the tau-leaps (see their Fig. 5), which shows that the approximation is very accurate as soon as the interval between samples becomes reasonably short.
15.7 Sometimes I get exceptions when running with mutator genes
Yes, sure, the following will cause an exception; this is similar to the example used in 1.3.4 but there is one crucial difference:
sd <- 0.1 ## fitness effect of drivers
sm <- 0 ## fitness effect of mutator
nd <- 20 ## number of drivers
nm <- 5 ## number of mutators
mut <- 50 ## mutator effect THIS IS THE DIFFERENCE
fitnessGenesVector <- c(rep(sd, nd), rep(sm, nm))
names(fitnessGenesVector) <- 1:(nd + nm)
mutatorGenesVector <- rep(mut, nm)
names(mutatorGenesVector) <- (nd + 1):(nd + nm)
ft <- allFitnessEffects(noIntGenes = fitnessGenesVector,
drvNames = 1:nd)
mt <- allMutatorEffects(noIntGenes = mutatorGenesVector)Now, simulate using the fitness and mutator specification. We fix
the number of drivers to cancer, and we stop when those numbers of
drivers are reached. Since we only care about the time it takes to
reach cancer, not the actual trajectories, we set keepEvery = NA:
ddr <- 4
set.seed(2)
RNGkind("L'Ecuyer-CMRG")
st <- oncoSimulPop(4, ft, muEF = mt,
detectionDrivers = ddr,
finalTime = NA,
detectionSize = NA,
detectionProb = NA,
onlyCancer = TRUE,
keepEvery = NA,
mc.cores = 2, ## adapt to your hardware
seed = NULL) ## for reproducibility
## set.seed(NULL) ## return things to their "usual state"What happened? That you are using five mutator genes, each with an effect of multiplying by 50 the mutation rate. So the genotype with all those five genes mutated will have an increased mutation rate of \(50^5 = 312500000\). If you set the mutation rate to the default of \(1e-6\) you have a mutation rate of 312 which makes no sense (and leads to all sorts of numerical issues down the road and an early warning).
Oh, but you want to accumulate mutator effects and have some, or the early ones, have a large effects and the rest progressively smaller effects? You can do that using epistatic effects for mutator effects.
15.8 What are good values of sampleEvery?
First, we need to differentiate between the McFarland and the exponential models. If you use the McFarland model, you should read section 15.6 but, briefly, the small default is probably a good choice.
With the exponential model, however, simulations can often be much
faster if sampleEvery is large. How large? As large as you can
make it. sampleEvery should not be larger than your desired
keepEvery, where keepEvery determines the resolution or
granularity of your samples (i.e., how often you take a snapshot of
the population). If you only care about the final state, then set
keepEvery = NA.
The other factors that affects choosing a reasonable sampleEvery
are mutation rate and population size. If population growth is very
fast or mutation rate very large, you need to sample frequently to
avoid the “Recoverable exception ti set to DBL_MIN. Rerunning.”
issue (see discussion in section 2.4).
15.9 mutationPropGrowth and is mutation associated to division?
With BNB mutation is actually “mutate after division”: p. 1232 of
Mather et al., 2012 explains: “(…) mutation is simply defined as
the creation and subsequent departure of a single individual from
the class”. Thus, if we want individuals of
clones/genotypes/populations that divide faster to also produce
more mutants per unit time (per individual) we have to set
mutationPropGrowth = TRUE.
When mutationPropGrowth = FALSE, two individuals, one from a
fast growing genotype, and the other from a slow growing genotype,
would be “emiting” (giving rise to) different numbers of identical
(non-mutated) descendants per unit time, but they would be giving
rise to the same number of mutated descendants per unit time.
There is an example in Mather et al, p. 1234, section 3.1.1 where “Mutation rate is proportional to growth rate (faster growing species also mutate faster)”.
Of course, this only makes sense in models where birth rate changes.
16 Session info and packages used
This is the information about the version of R and packages used:
sessionInfo()
## R version 4.1.1 (2021-08-10)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.3 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.14-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.14-bioc/R/lib/libRlapack.so
##
## Random number generation:
## RNG: L'Ecuyer-CMRG
## Normal: Inversion
## Sample: Rejection
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_GB LC_COLLATE=C
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] parallel stats graphics grDevices utils datasets
## [7] methods base
##
## other attached packages:
## [1] igraph_1.2.7 graph_1.72.0 BiocGenerics_0.40.0
## [4] OncoSimulR_3.2.0 pander_0.6.4 BiocStyle_2.22.0
##
## loaded via a namespace (and not attached):
## [1] ggrepel_0.9.1 Rcpp_1.0.7 lattice_0.20-45
## [4] png_0.1-7 gtools_3.9.2 assertthat_0.2.1
## [7] digest_0.6.28 utf8_1.2.2 R6_2.5.1
## [10] cellranger_1.1.0 stats4_4.1.1 evaluate_0.14
## [13] ggplot2_3.3.5 highr_0.9 pillar_1.6.4
## [16] rlang_0.4.12 curl_4.3.2 readxl_1.3.1
## [19] data.table_1.14.2 car_3.0-11 Rgraphviz_2.38.0
## [22] jquerylib_0.1.4 Matrix_1.3-4 rmarkdown_2.11
## [25] stringr_1.4.0 foreign_0.8-81 munsell_0.5.0
## [28] compiler_4.1.1 xfun_0.27 pkgconfig_2.0.3
## [31] htmltools_0.5.2 tidyselect_1.1.1 tibble_3.1.5
## [34] bookdown_0.24 rio_0.5.27 fansi_0.5.0
## [37] crayon_1.4.1 dplyr_1.0.7 grid_4.1.1
## [40] jsonlite_1.7.2 gtable_0.3.0 lifecycle_1.0.1
## [43] DBI_1.1.1 magrittr_2.0.1 scales_1.1.1
## [46] zip_2.2.0 carData_3.0-4 stringi_1.7.5
## [49] farver_2.1.0 smatr_3.4-8 bslib_0.3.1
## [52] ellipsis_0.3.2 generics_0.1.1 vctrs_0.3.8
## [55] openxlsx_4.2.4 RColorBrewer_1.1-2 tools_4.1.1
## [58] forcats_0.5.1 glue_1.4.2 purrr_0.3.4
## [61] hms_1.1.1 abind_1.4-5 fastmap_1.1.0
## [64] yaml_2.2.1 colorspace_2.0-2 BiocManager_1.30.16
## [67] knitr_1.36 haven_2.4.3 sass_0.4.016.1 Time it takes to build the vignette and most time consuming chunks
Time to build the vignette:
## [1] "2.41963748931885 minutes"## The 15 most time consuming chunks
sort(unlist(all_times), decreasing = TRUE)[1:15]
## exTimeDec hurl3a examplesTCol fixedChemo1 hurl3b
## 7.480113 5.934512 5.409620 4.561698 3.941259
## scenGF1 scenGFChemo1 fdf2c switchChemo1 prbau003bb
## 3.577316 3.565021 3.265533 3.156772 3.078669
## fdf2d hurl3b2 smyelo3v57 mcflsx2 mcflsx3
## 3.061954 3.014428 2.771629 2.609288 2.594515paste("Sum times of chunks = ", sum(unlist(all_times))/60, " minutes")
## [1] "Sum times of chunks = 2.30820204814275 minutes"17 Funding
Supported by BFU2015-67302-R (MINECO/FEDER, EU) and PID2019-111256RB-I00/AEI/10.13039/501100011033 to RDU. S. Sanchez-Carrillo partially supported by a “Beca de Colaboración” from Universidad Autónoma de Madrid (2017-2018); J. Antonio Miguel supported by PEJ-2019-AI/BMD-13961 from Comunidad de Madrid.
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It is of course possible to do this with the carrying capacity (or gompertz-like) models, but there probably is little reason to do it. McFarland et al. (2013) discuss this has little effect on their results, for example. In addition, decreasing the death rate will more easily lead to numerical problems as shown in section 3.11.2.↩
Again, these are not necessarily reasonable or common settings. We are using them to understand what and how affects running time and space consumption.↩
By easily accessible I mean that there are many, preferably short, paths of non-decreasing fitness from the wildtype to this genotype. See definitions and discussion in, e.g., Franke et al. (2011).↩
These matrices do not exist during most of the execution of the C++ code; they are generated right before returning from the C++ code.↩
Given the dependence of death rates on population size in McFarland’s model (section 3.2.1 and 3.2.1.1), if all mutations have the same fitness effects we can calculate the equilibrium population size (where birth and death rates are equal) for a given number of mutated genes as: \(K * (e^{(1 + s)^p} - 1)\), where \(K\) is the initial equilibrium size, \(s\) the fitness effect of each mutation, and \(p\) the number of mutated genes.↩
Note for curious readers: it used to be the case that we converted the table of fitness of genotypes to a fitness specification with all possible epistatic interactions; you can take a look at the test file
test.genot_fitness_to_epistasis.Rthat uses thefem6object. We no longer do that but instead pass directly the fitness landscape.↩You can change this if you really want to.↩
This is a shortcut that we take because we think that it is what you mean. Note, however, that technically a clone with birth rate of 0 might have a non-zero probability of mutating before becoming extinct because in the continuous time model we use mutation is not linked to reproduction. In the present code, we are not allowing for any mutation when birth rate is 0. There are other options, but none which I find really better. An alternative implementation makes a clone immediately extinct if and only if any of the \(s_i = -\infty\). However, we still need to handle the case with \(s_i < -1\) as a special case. We either make it identical to the case with any \(s_i = -\infty\) or for any \(s_i > -\infty\) we set \((1 + s_i) = \max(0, 1 + s_i)\) (i.e., if \(s_i < -1\) then \((1 + s_i) = 0\)), to avoid obtaining negative birth rates (that make no sense) and the problem of multiplying an even number of negative numbers. I think only the second would make sense as an alternative.↩
We said “a few times”. For a clone of population size 1 —which is the size at which all clones start from mutation—, if death rate is, say, 90 but birth rate is 1, the probability of mutating before becoming extinct is very, very close to zero for all reasonable values of mutation rate}. How do we signal immediate extinction or no viability in this case? You can set the value of \(s = -\infty\).↩
OTs and CBNs have some other technical differences about the underlying model they assume, such as the exponential waiting time in CBNs. We will not discuss them here.↩
Of course, the “reach cancer” idea and the
onlyCancerargument are generic names; this could have been labeled “reach whatever interests me”.↩Setting
detectionDriversanddetectionSizeto “NA” is in fact equivalent to setting them to the largest possible numbers for these variables: \(2^{32} -1\) and \(\infty\), respectively.↩We assess probability of exiting at every sampling time, as given by
sampleEvery, that is the smallest possible sampling time that is separated from the previous time of assessment by at leastcheckSizePEvery. In other words, the interval between successive assessments will be the smallest multiple integer ofsampleEverythat is larger thancheckSizePEvery. For example, supposesampleEvery = 2andcheckSizePEvery = 3: we will assess exiting at times \(4, 8, 12, 16, \ldots\). IfsampleEvery = 3andcheckSizePEvery = 3: we will assess exiting at times \(6, 12, 18, \ldots\).↩There are several packages in R devoted to phylogenetic inference and related issues. For instance, ape. I have not used that infrastructure because of our very specific needs and circumstances; for instance, internal nodes are observed, we can have networks instead of trees, and we have no uncertainty about when events occurred.↩
Where depth is defined in the usual way to mean smallest number of nodes —or edges— to traverse to get from the bottom to the top of the DAG.↩
Death rates are affected by density dependence and, thus, it is on the death rates where the approximation that they are constant over a short interval plays a role. Thus, we examine how large the difference between successive death rates is. More precisely, let \(A\) and \(C\) denote two successive sampling periods, with \(D_A = log(1 + N_A/K)\) and \(D_C= log(1 + N_C/K)\) their death rates.
errorMF_sizestores the largest \(abs(D_C - D_A)\) between any two sampling periods ever seen during a simulation.errorMFstores the largest \(abs(D_C - D_A)/D_A\). Additionally, a simple procedure to use is to run the simulations with different values ofsampleEvery, say the default value of 0.025 and values that are 10, 20, and 50 times larger or smaller, and assess their effects on the output of the simulations and theerrorMFstatistic itself. You can check that using asampleEverymuch smaller than 0.025 rarely makes any difference inerrorMFor in the simulation output (though it increases computing time significantly). And, just for the fun of it, you can also check that using huge values forsampleEverycan lead to trouble and will be manifested too in the simulation output with large and unreasonable jumps in total population sizes and sudden extinctions.↩