Beyond clonal interference: Scrutinizing the complexity of the dynamics of three competing clones Sylvain Billiard
arXiv:1607.04656v1 [q-bio.PE] 15 Jul 2016
1
∗1
and Charline Smadi
†2,3
´ Eco-Pal´ ´ Unit´e Evoeo, UMR CNRS 8198, Universit´e des Sciences et Technologies Lille, 59655 Villeneuve d’Ascq Cedex, France 2
3
Irstea Lisc, 9 avenue Blaise Pascal, 63178 Aubi`ere, France
Institut des Syst`emes Complexes Paris Ile-de-France, 113 rue Nationale, 75013 Paris, France. July 19, 2016
Abstract In large adapting clonal species, several beneficial mutations can co-occur, affecting the process of adaptation. Several experimental and theoretical works showed that clonal interference can be an important factor limiting the rate of adaptation. However, models done so far do not embrace the diversity of observed dynamics in experiments, especially non-linear dynamics. We develop here a stochastic model with explicit competitive interactions between clones and describe the complexity of the emerging dynamics of the population, supposing that two mutants enter a resident population in a single copy at different times. These clones can either get fixed, be lost or be maintained in polymorphism, depending on their competitive abilities. We show that frequency-dependent selection can give rise to unexpected dynamics: competitive interactions between clones can foster adaptation by increasing or decreasing both the fixation probability and time of beneficial mutations. We finally estimate the likeliness of frequency-dependent selection and of the different potential final states of the population by assuming prior distributions of the ecological parameters. We show that under our assumptions, non-transitive fitness and non-linear dynamics are likely to play an important role into the adaptation of large clonal populations.
Key-words:clonal reinforcement, clonal interference, non-linear dynamics, fixation time, fixation probability, adaptation, cooperation, polymorphism. The authors wish to be identified to the reviewers.
∗ Corresponding
Author,
[email protected]
†
[email protected]
1
1
Introduction
Adaptation is often viewed as a cumulative succession of fixations of beneficial mutations, and the waiting time between two successive beneficial mutations is viewed as one of the main factor limiting adaptation rate (Fisher, 1930; Crow and Kimura, 1965). This is however not true in general since several beneficial mutations can compete in a single population, which generates different interference phenomena, such as the Muller’s ratchet (Muller, 1932, 1964) or more generally Hill-Robertson interfences (Hill and Robertson, 1966). Competition between several beneficial mutations is especially expected in large asexual populations with high mutation rates and adaptation is then expected to be slowed down, a phenomenon called “clonal interference” (Gerrish and Lenski, 1998). Clonal interference and its consequences on adaptation rates seems general as they have been observed in bacteria, viruses, yeasts or cancer tumors (e.g. Miralles et al., 1999; de Visser and Rozen, 2006; Greaves and Maley, 2012; Lang et al., 2013). Despite the ubiquity and importance of clonal interference, experiments showed that adaptation is not blocked and keep on even after thousands generations (e.g. Wiser et al., 2013). These observations led more recently to the development of an alternative view of clonal interference, considering that beneficial mutations can occur in a single lineage (Desai and Fisher, 2007), thus increasing fitness gradually. In large populations with high mutation rates, it is expected that adaptation follows a wave (Desai and Fisher, 2007; Good et al., 2012) bounded by the most and the least fit lineages. Adaptation rate is expected to be constant in this model and depends only on the width of the wave, in other words on the genetic variability present in the population. Models of clonal interference have been useful in understanding the dynamics of adaptation in asexual populations, and have known many empirical successes. However, many experiments have shown that the dynamics of beneficial mutations can be much more complex. Different experiments showed that dynamics can be nonlinear: lineages can show multiple frequency peaks during the course of adaptation (Lee and Marx, 2013) and different lineages can coexist for a long time in a single population (Lang et al., 2011; Maddamsetti et al., 2015). This suggests that frequency-dependent fitness can occur. Moreover, cooperation between lineages and niche construction have been observed multiple times in tumoral cancers and bacteria (Yang et al., 2014; Kinnersley et al., 2014), which suggests that taking into account different ecological interactions might be important for a global understanding of the evolution of asexual populations with large population size and high mutation rates. Clonal interference has been mostly theoretically studied using population genetics models with strong assumptions: beneficial mutations are assumed to have simple epistatic interactions, the effect of mutations on fitness is assumed to be transitive, selection is assumed not to depend on the mutants frequency, and the environment is assumed constant. Non-transitive fitness and non-linear dynamics are consequently considered as special cases, of less important interest. There is however a large literature dealing with Lotka-Volterra models with more than two species, especially with non-transitive competitive interactions. These studies try to define the conditions for the coexistence of multiple competitive species in a community (e.g. Huisman et al., 2001). Such studies are generally deterministic and investigate the conditions of stability of dynamical systems. When dealing with stochastic dynamics, simulations are generally performed in spatialized context (Laird, 2014) again in order 2
to determine the conditions for the coexistence of multiple competing species. Those studies do not however investigate the probability and time of fixation of mutations and are thus of limited interest regarding the effect of competitive interactions between several clones in the course of adaptation, especially on the adaptation rate. Here we propose a stochastic model with three different lineages under competition, where the competitive interactions are not necessarily transitive, thus relaxing one important assumption of the models studying clonal interference so far. Our model embraces a large variety of phenomena observed in the course of adaptation of asexual species. We recover classical results from clonal interference models and we also show that unexpected behaviours are expected. For instance, we show that in some cases competitive interactions between three clones can lead to a higher rate of adaptation. Our results generally show that non-linear dynamics are likely in large clonal populations, which challenges the interpretation of experimental results.
2 2.1
Model and methods Definitions and assumptions
We denote i ∈ [0, 1, 2] a type of individuals (types can be phenotypes, alleles, strains, clonal species, mutants, etc.). For the sake of simplicity, we will use in the rest of the paper the term mutant i when referring to type i individuals. N (t) = (Ni (t)) is a vector whose elements are the number of mutants i in the population at time t, with Ni (t) a random variable. We assume that the environment has a fixed quantity of available resources: the carrying capacity K modulates the intensity of competition between individuals. We investigate the population dynamics of three clonal types as a birth-death process with competition in continuous time. Each mutant i is characterized by its individual ecological parameters: βi and δi are respectively the individual birth and natural death rates, and Cij is the effect of competition of a single mutant j on a single mutant i, assuming Cij ≥ 0 and Cii > 0. For simplicity, we assume that competition between individuals affects mortality. The individual death rate of a mutant i thus depends on both an intrinsic component (δi ) and a component due to competition: di (N (t)) = δi + Ci0 N0 (t)/K + Ci1 N1 (t)/K + Ci2 N2 (t)/K. We suppose that the resident population is only composed of mutants 0, at its ecological steady-state equilibrium, say N0 (0). A single mutant 1 is introduced in the population (N1 (0) = 1). The population of mutants 1 follows a stochastic dynamics that depends on the ecological parameters and the competitive interactions between mutants 0 and mutants 1. The time taken for mutants 1 to invade the resident population, and eventually get fixed, is of order log K (Champagnat, 2006). Since the dynamics is stochastic, mutants 1 can either spread or be lost. Since we are interested in the dynamics of three competing clones, we will focus on cases where neither mutants 0 nor mutants 1 are lost when a single mutant 2 enters the population by mutation (or migration). We assume that the time at which this event occurs is α log K, log K being the time scale of the whole stochastic dynamics. There are two general cases. Either α is low enough that mutants 1 are still in too few numbers to affect the invasion of mutants 2, or α is large enough that mutants 1 have invaded the population and thus affect the invasion of mutants 2. We will investigate both situations and show that the time α log K at which mutant 2 enters the population is crucial and largely affect the final state of the population. 3
2.2
The stochastic dynamics is a succession of several phases
When the population is large, the dynamics followed by the population can be divided into a succession of two kinds of phases. First, when a mutation enters a population, say a mutant j enters in an single copy in a i resident population, the dynamics of mutants j is well approximated by a branching birth-death process without interactions until its population size N (j) is large enough, i.e. when it is of order K (Fournier and M´el´eard, 2004; Champagnat, 2006). In the case of the joint dynamics of three interacting clones, the dynamics of two mutants j and k in a resident population i is also well approximated by a branching birth-death process without interactions (see the proofs in the companion paper Billiard and Smadi (in press)). Second, when two or three mutants have a population size of order K, then the stochastic dynamics of these populations is well approximated by a competitive Lotka-Volterra deterministic system (Fournier and M´el´eard, 2004; Champagnat, 2006; Billiard and Smadi, in press). During this phase, if the population size of one of the mutants is of order lower than K, then its population size essentially does not change until the deterministic equilibrium is reached. Even though the whole dynamics is stochastic, we will respectively call these two kinds of phases “stochastic” and “deterministic”, for the sake of simplicity (see Fig. 1). The dynamics of the population can finally be described as a succession of “stochastic” and “deterministic” phases. Note that we only give in this paper the relevant biological results and a summarized version of the model (mathematical proofs and detailed computations are given in a companion paper Billiard and Smadi (in press)).
Deterministic phases. We denote by ni the size of the population of mutants i when dealing with the deterministic dynamics, while we will keep the notation Ni when dealing with stochastic dynamics (ni is the population size rescaled by the carrying capacity Ni /K when Ni is of order K). When the population size of the three mutants are of order K, the dynamics of the rescaled process can be well approximated by the following system of ordinary differential equations, n˙ = (β0 − δ0 − C0,0 n0 − C0,1 n1 − C0,2 n2 )n0 , 0 n˙ 1 = (β1 − δ1 − C1,0 n0 − C1,1 n1 − C1,2 n2 )n1 , n˙ = (β − δ − C n − C n − C n )n . 2
2
2
2,0 0
2,1 1
2,2 2
(1)
2
Under the assumptions that competitive parameters are Cij ≥ 0 and Cii > 0 for all {i, j}, this system of deterministic equations is a three-dimensions Lotka-Volterra competitive model. Such a three species population shows different possible dynamics: different equilibrium states (either monomorphic or polymorphic, with two or three coexistent mutants), or stable limit cycles (Zeeman, 1993; Zeeman and van den Driessche, 1998; Zeeman and Zeeman, 2003). The fate of a mutant i entering in a single copy a resident j population is associated to the so-called “invasion fitness”, denoted Sij = βi − δi − Cij n ¯ j , where n ¯j =
βj −δj Cjj
is the population size of mutant j at equilibrium
when there are only mutants j in the population. The invasion fitness corresponds to the initial growth rate of the mutant when it is rare. If the resident population is composed of both mutants i and j at equilibrium, 4
then the fate of a mutant k entering in a single copy is associated with the invasion fitness denoted Skij = βk − δk − Cki n ¯ iij − Ckj n ¯ jij , where
n ¯ iij =
Cjj (βi −δi )−Cij (βj −δj ) ,n ¯ jij Cii Cjj −Cij Cji
=
Cii (βj −δj )−Cji (βi −δi ) , Cii Cjj −Cij Cji
(2)
is the equilibrium of Eq. 1 when there are only mutants i and j in the population. If Skij > 0, mutation k is favorable when rare in the polymorphic resident population (i, j) and can invade.
The stochastic phase. When the population size of at least one mutant, say i, is low (i.e. its population size is of order lower than K), while the other mutants are at their deterministic steady state (their population size is of order K), the dynamics of mutants i is close to a pure birth-death process with birth and death rates P respectively βi and δi + j6=i Cij nj , thus neglicting the effect of the competition between individuals i. When a mutant i enters a resident j population in a single copy, the probability of invasion of the mutant i, defined as the probability that the population of mutants i reaches a size of order K, is Sij /βi when Sij > 0, and a probability 0 if Sij ≤ 0. The time taken by a mutant i which enters a j resident population to reach the threshold population size is of order log K/Sij . We can similarly define the probability of invasion of a mutant k in a resident population with both mutants i and j as Skij /βk when Skij > 0, and a probability 0 if Skij ≤ 0. The time taken by a mutant k which enters a i and j resident population to reach the threshold population size is of order log K/Skij . In both cases, the stochastic phase ends when the mutant is either lost or reaches a threshold population size of order K.
2.3
Stochastic dynamics and final states with three competitive clones
When a single favorable mutant enters a resident population, the stochastic dynamics can be decomposed into three successive phases (see Fig. 1a and 1b): First, a stochastic phase corresponding to the beginning of the mutant invasion and where the mutant has a population size of order lower than K; Second, a deterministic phase when the population size of the new mutant is large enough (of order K); Third, a new stochastic phase until the resident mutant is lost (if both mutants do not stably coexist at deterministic equilibrium). In the case of competition between three clones with two mutations entering a resident population, the succession of stochastic and deterministic phases must be decomposed into a higher and not limited number of phases.
What determines the different dynamics and final states when there is competition between three clones? The different dynamics and final states depend on the ecological parameters and also on the following conditions (see details of computation in Appendix A1 and illustrations in Fig. 1 and 2, simulation algorithm given in Appendix A3): i) Does mutant 2 enter the population during the first (Fig. 1a) or second (Fig. 1b) stochastic phase (depending on the time α log K)? If mutant 2 enters the population during the first stochastic phase then mutant 1 has a population size of order lower than K. Consequently, mutant 2 suffers the competitive effect of mutants 0 only. If mutant 2 enters the population during the second stochastic phase then, assuming mutant 1 is favorable in 5
the resident population, there are two possibilities: either mutants 0 and 1 stably coexist, in which case mutant 2 suffers the competitive effects of both mutants, either mutant 0 has a population size of order lower than K, in which case mutant 2 suffers the competitive effects of mutant 1 only; ii) When mutant 2 enters the population during the first stochastic phase (Fig. 1a), does mutant 1 or 2 reaches first a population size of order K? Since mutants 1 and 2 have a population size lower than order K, the speed at which they invade the resident population only depends on their competitive interactions with the resident mutants 0. Hence, which mutants reaches first a population size of order K depends on their invasion fitness S10 and S20 and on the time when mutant 2 enters the population α log K. The first mutant which reaches a population size of order K determines the initial state of the succeeding deterministic phase; iii) What is the equilibrium of the first deterministic phase: stable coexistence of two mutants, i.e. two mutants have a population size of order K, or a single mutant has a population size of order K? This only depends on the sign of the invasion fitnesses (Eq. 1). For instance, mutants 0 and 1 stably coexist if S01 > 0 and S10 > 0. iv) What is the population size of all mutants when the second stochastic phase begins? It depends on whether two mutants stably coexist or not at the end of the deterministic phase (step iii), and on the population size of the mutant which did not invade and still has a size lower than order K; v) Does a mutant go extinct before the start of the next deterministic phase? When a mutant has a population size of order lower than K and is deleterious in a given context, it is expected to go extinct. However, its time to extinction can take longer than the time for another rare mutation to reach a population size of order K. In this case, a new deterministic phase begins. The ecological context of the deleterious mutation can change before it goes extinct, which changes its fate. vi) Steps ii-v are again applied for the further succeeding phases (when applicable) as often as necessary until a final steady state is reached.
The different possible final states and their hitting times. Our goal is to investigate how clonal interference might affect the dynamics of mutant populations and thus adaptation. We will thus especially focus on cases where mutation 1 and 2 have a positive invasion fitness when 2 enters the population during the first stochastic phase ( S10 > 0and S20 > 0), and on cases where mutation 1 has a positive invasion fitness( S10 > 0) if mutation 2 enters the population during the second stochastic phase (in the latter case, the invasion fitness of mutant 2 depends on the currect state of the population). All the possible final states and their hitting times are compiled in Tables 1 and 2. We do not give all detailed calculations for all cases here, we only give one detailed example in Appendix A2 as an illustration (for complete and detailed computations see the companion paper Billiard and Smadi (in press)). Roughly, two classes of final states are possible: either one mutant goes to fixation (it can be either 0, 1 or 2), or two or three mutants coexist (in all possible combinations).
2.4
Likelihood of the final states
We want to estimate the likelihood of the different possible final states assuming ecological parameters are drawn in given prior distributions. The complexity of the model can be reduced to: ρi = βi − δi , the net individuals eij = reproductive rate of mutants i, and C
Cij Cjj
the ratio of the between and within competitive interactions. 6
We drew 106 different sets of parameters in prior distributions. For a given set of parameters and a given α, the final state is given by Tab. 1 and 2, which allows to estimate the posterior distribution of the final states among the 106 random sets of parameters. The time α log K at which the second mutation enters the population has a large impact on the final states. We can determine the final states for a given parameter set for any α using the results of our model (Tab. 1 and 2). Assuming mutation 2 enters the population during the first stochastic phase, we know that mutant 2 necessarily appears before mutant 1 spreads out, i.e. α < 1/S10 . If mutation 2 enters the population during the second stochastic phase, mutant 2 necessarily appears after mutant 1 spreads out and before mutant 0 goes extinct, i.e. 1/S10 < α < 1/S10 +1/|S01 |. We also know that there are two threshold values α < 1/S10 +1/S20 (S21 /|S01 |−1) and α < S02 S21 /(S10 |S12 ||S01 |) − 1/S01 (Tab. 1) which determine which dynamics is followed by the population when mutation 2 enters the population during the first stochastic phase. Similarly, there are two threshold values 1/S10 < α < 1/S10 + 1/|S01 | − 1/S21 and 1/S10 < α < 1/S10 + S02 /(|S12 ||S01 |) − 1/S21 (Tab. 2) which determine which dynamics is followed by the population when mutation 2 enters the population during the second stochastic phase (see above). We can thus finally compute the probability of any final states given an ecological parameter set and assuming α is uniformly drawn in the interval [0, 1/S10 ] or [1/S10 , 1/S10 + 1/|S01 |] respectively when mutation 2 enters the population during the first or second stochastic phase. Effect of mutations on the reproductive rates. In bacteria, yeasts or some eukaryotes, fitness is generally estimated as the initial growh rate (at low density) of mutants (see Table 2 in Martin and Lenormand (2006) and the Appendix in Manna et al. (2012)). We will thus assume that the effect of mutations on the growth rate of mutant i follows a Fisher’s geometric model. Given the net reproduction rate of mutants 0 is ρ0 , we assumed that the reproductive rate of mutant i is ρi = ρ0 + xi with xi , the effect of mutation i, being drawn in a shifted negative Gamma distribution which is an approximation of a Fisher’s geometric model for adaptation (Martin and Lenormand, 2006). Note that when mutation 2 enters the population during the second stochastic phase, mutation 2 is assumed to occur in the most frequent mutation at equilibrium: ρ2 = ρ1 + x2 when mutant 1 is more frequent than mutant 0, ρ2 = ρ0 + x2 otherwise. Effect of mutations on competition. There is, to our knowledge, no theoretical or empirical consensus on eij . Without any knowledge about the distributhe distribution of mutation effects on the competitive abilities C eij follows arbitrary tion of competitive abilities, we simply assumed that the ratio of competitive interaction C chosen distributions. First, we assumed that it follows a uniform distribution in the interval [1 − µ, 1 + µ], with 0 ≤ µ ≤ 1. Second we assumed it follows an exponential distribution with parameter µ > 0. When µ = 0, eij = 1, fitnesses are necessarily transitive, while if µ > 0, frequency-dependent fitnesses can occur. As µ all C eij also increases, i.e. the more different can the competitive increases, the variance of the competitive ratio C interactions be between mutants.
7
3
Results
Given that two favorable mutations successively enter in a single copy a resident population 0, Tables 1 and 2 show all possible dynamics and final states, depending on the sign of the invasion fitnesses Sij and Sijk , {i, j, k} ∈ {0, 1, 2}, and the time of appearance of the second mutation α log K. When the second mutation enters the population during the first stochastic phase, Tab. 1 shows that six final states are possible: fixation of either mutation 1 or 2, stable polymorphic equilibrium with two (mutants 0 and 1, 1 and 2 or 0 and 2) or three mutants (mutants 0, 1 and 2). Tab. 1 also shows that a given possible final state can be reached under different conditions, and consequently after different durations. For instance, mutant 2 can fix under three different cases (a subcase of B, and cases E and I), with potentially different fixation times. When mutation 2 enters the population during the second stochastic phase, Tab. 2 shows that a seventh final state is possible: the fixation of mutant 0, even if mutants 1 and 2 are advantageous when they enter the population. Interestingly, Rock-Paper-Scissors cyclical dynamics can only occur if the second mutation enters the population during the second stochastic phase. Assuming three co-occuring competing clones, our model can thus capture a large diversity of dynamics and allows to determine their duration, final states and likeliness. In the following, we first show that despite the complexity and variety of the possible stochastic dynamics with three competing clones, six possible general dynamics can be defined. Second, we focus on several special cases of particular interest. We especially argue that our model, despite its simple assumptions, captures a large range of dynamics diversity observed in experiments.
3.1
Beyond clonal interference: Six possible dynamics
Two categories of dynamics have been proposed in the empirical literature to explain observations in experiments with several interacting clones: clonal interference, when adaptation is slowed down because of the interaction between advantageous mutations (Gerrish and Lenski, 1998), or clonal reinforcement (Kinnersley et al., 2014), also called niche construction or frequency-dependent selection elsewhere (Yang et al., 2014), when several clones stably coexist. Using our model, focusing on the second mutation (mutant 2), we can embrace the two previous proposed categories, and propose an alternative categorization, with more accurate definitions. Competition between three clones can affect adaptation for three reasons: 1) It can promote or hinder polymorphism maintenance; 2) The invasion probability of mutation 2 can be increased or decreased (relatively to the case where mutant 2 enters alone the resident population, i.e. there are only two interacting clones); 3) In cases where mutation 2 goes to fixation, its fixation time can be be shorter or longer. Hence, dynamics with three competing clones can be classified in six general cases: When clonal interaction promotes polymorphism maintenance, we call the dynamics “clonal coexistence”. When the mutant 2 goes to fixation, we call “clonal assistance” when the duration of the sweep is shorter, and “clonal interference” (following Gerrish and Lenski (1998)) when it is larger than with only two competing clones. Finally, we call “soft” vs. “hard” the dynamics depending on wheter the invasion probability of the mutant 2 is lower vs. higher than with only two competing clones. This gives 6 possible general dynamics, summarized in Table 3. Rate of adaptation: fixation time vs. probability of invasion of beneficial mutants. Clonal inter8
ference is viewed in the literature as the phenomenon of the increase in fixation time of co-occuring beneficial mutants in a population, which consequently decreases the rate of adaptation of clonal species. However, the rate of adaptation can be affected also by the rate at which new mutants invade a population, i.e. by their probability of invasion. To our knowledge, in all models dealing with clonal interference so far, derived from population genetics models, the probability of invasion of a new mutant only depends on its own features: in general, a selection coefficient s is arbitrarily assigned to a mutant independently of the composition and state of the resident population when this mutant occurs. Its invasion probability and its time of fixation are approximately 2s and 1/s: increasing s necessarily both increases the probability of fixation and decreases the time of fixation. In our approach, we have a more general point of view: both the fixation time and probability of invasion depend on the state of the population when a mutant enters the population. We thus argue that fixation times and probabilities of invasion should be considered independently in order to evaluate to which extent interaction between various clones can affect rate of adaptation. Interestingly, depending on the state of the population when mutation occurs and on their own ecological specificities, the rate of adaptation can increase: the probability of invasion can be higher or the time to fixation can be shorter, what we propose to call “hard clonal assistance”. Clonal coexistence: cooperative interactions are not necessary. Several experiments of competition between clones have shown stable persistence of different strains in a single well-mixed population, which has been explained by frequency-dependent selection, niche construction or cooperative interactions. Especially, Kinnersley et al. (2014) introduced the concept of “clonal reinforcement” when “the emergence of one genotype favors the emergence and persistence of other genotypes via cooperative interactions”. Here we show that clones can favor either the emergence (“hard” vs. “slow” dynamics), or the persistence (“clonal coexistence” vs. “clonal assistance” or “clonal interference”) or both (“hard clonal coexistence”), without any cooperative interactions between clones, but only competition. We do not argue that persistence observed in Kinnersley et al. (2014) are not effectively due to cooperative interactions, rather we propose an alternative hypothesis: both facilitated emergence and stable persistence of clones can be due to non-transitive competitive interactions, or frequency-dependent selection. It would need specific experimental work to show whether or not clones effectively cooperate.
3.2
Two specific dynamics when the second mutant lately enters the population: Rock-Paper-Scissors or annihilation of adaptation
Two related specific dynamics are encountered only when mutant 2 enters the resident population during the second stochastic phase (α > 1/S10 , Tab. 2): Rock-Paper-Scissor dynamics (final state J in Tab. 2), or a return to the initial state, i.e. a population fixed for mutant 0 (final state G in Tab. 2). These two different dynamics are illustrated in Fig. 1b and Fig. 2b. Both dynamics have the same parameters, except the time at which mutant 2 enters the population α log K. This illustrates the importance of considering stochastic dynamics: if mutant 2 enters the population early enough that mutant 1 is not extinct when mutant 2 invades, then RockPaper-Scissors cyclical dynamics take place, otherwise mutant 0 goes to fixation and adaptation is annihilated 9
despite the occurrence of two beneficial mutations. In a deterministic model, for the same parameters, a mutant can not go extinct and only Rock-Paper-Scissor dynamics is possible. Our results also show that Rock-Paper-Scissors dynamics can be obtained in a narrow set of parameters. In addition to specific parameters values regarding the competitive interactions between the three clones, the second mutant must occur in the population in a narrow time frame. First, it must occur after mutant 1 invaded, since mutant 2 is deleterious in a mutant 0 resident population. Second, if mutant 2 occurs too late during the second stochastic phase, then mutant 0 can be extinct before mutant 2 invades, in which case mutant 1 goes to fixation. These results have important consequences regarding our understanding of empirical RockPaper-Scissor dynamics observed in natural populations: either the three types of individuals involved in such stable cycles have effectively entered the population by mutation or migration in a single individual, in which case the third type of individuals has necessarily entered the population in a narrow time frame. Otherwise, the alternative explanation is that the three types of individuals went together in a single population with a sufficiently large enough population size such that the dynamics initially followed an almost-deterministic dynamics, which certainly occurred by a massive migration and mixing of three different and complementary types of individuals.
3.3
Likehood of the final states assuming prior distributions of the ecological parameters
Figures 3 and 4 show the posterior probability of the dynamics and final states when mutation 2 enters the population during the first and second stochastic phases, assuming that the competition abilities are respectively eij is low, all drawn in an uniform or exponential distribution. When the variance of the distribution of the C eij ' 1), i.e. invasion fitness are mostly transitive. We naturally clones have similar competitive abilities (C recover predictions from population genetics models: The likeliest scenari are the fixation either of mutant 1 or 2 (Fig. 3c, 3d, 4c, 4d). Rapidly, when the variance of the uniform and exponential distributions increases, polymorphic final states become the likeliest. When the effect of mutation on competitive abilities become eij ) are drawn in large, the likeliness of all dynamics rapidly reaches a plateau when competitive abilities (C an uniform distribution (Fig. 3). When competitive abilities are drawn in an exponential distribution, the likeliest state is the one with three coexisting clones. Our results suggest that non-transitive fitness are mostly expected to occur when several clones are interacting as soon as mutations affect their competitive abilities. This further supports that clonal coexistence is likely to occur even when considering only competitive interactions: cooperative interactions are not necessary to explain the stable coexistence of several clones. Finally, our results show that Rock-Paper-Scissors dynamics and annihilation of adaptation are weakly probable. Comparing left and right columns in Fig. 3 and 4 shows that the time at which mutation 2 enters the population only marginally affects the dynamics and the final states. Interestingly, comparing the final states between cases with two or three interacting clones (Fig.5) shows that more polymorphic final states are expected when three clones are interacting, even though the difference is small. Whether a further increase in the number of interacting clones could even more promote the maintenance of polymorphism because of non-transitive fitness. 10
Fig. 3, 4 and 5 also show that the prior distribution of the competitive abilities has important effect: polymorhic final states are more expected when the distribution is exponential. Finally, Fig. 3e-f and 4e-f show the likeliness of clonal interference vs. clonal assistance. When competitive abilities follow an uniform distribution, clonal interference, i.e. the slowing down of fixation of beneficial mutation, is the most probable, especially when the competitive abilities are similar between clones (small µ). However, rapidly when the difference between competitive abilities increases (large µ) the likeliness of clonal assistance increases, and reaches a plateau. When mutation 2 enters the population during the second stochastic phase, clonal assistance is even likelier than clonal interference. The situation is different and more complex when competitive abilities follow an exponential distribution, especially when the second mutation enters the population during the second stochastic phase. Contrarily to the uniform distribution, the exponential distribution is asymetrical around the mean, which generates asymetrical invasion fitness distributions and explains the difference observed in our results. Indeed, when the mean µ is low, the distribution of the invasion fitness is skewed towards negative values, while it is skewed towards positive values when µ is large (not shown), which explains why polymorphic states are the likeliest for large µ. Globally, our results thus suggest that clonal interference might indeed be an important factor affecting adaptation rate, but clonal assistance can be as important given non-transitive fitnesses are possible.
4
Discussion
Both theory and experimental observations tend to agree regarding the rate of adaptation in evolving large asexual populations: the speed at which new beneficial mutations go to fixation should decrease during the course of adaptation and reach a plateau (e.g. Gerrish and Lenski, 1998; Desai and Fisher, 2007), what is effectively found in experimental evolution of bacteria (de Visser and Rozen, 2006). Clonal interference is believed to be the most important factor causing this limit to adaptation: different beneficial mutations occurring simultaneously in competing lineages tend to mutually decrease their probability of fixation and increase their time to fixation, what has been effectively observed in viruses (e.g. Pandit and de Boer, 2014), bacteria (de Visser and Rozen, 2006) or cancer (Greaves and Maley, 2012). This congruence between models and data can be challenged regarding our results. Indeed, by explicitly taking into account competitive interactions between different clones, we showed that the evolutionary dynamics can be much more complex: competitive interactions between clones can either impede or foster adaptation by modifying fixation times and probabilities, and can promote coexistence of clones in the long-term. Our results are different from previous works because in our model nontransitive fitnesses are allowed, giving rise to frequency-dependent selection, non-linear and cyclical dynamics. Our model is in fact more general than previous works since they are a subcase where all competitive interaction eij = 1). Most importantly, we showed that frequency-dependent selection is very likely when abilities are equal (C mutations have an effect on competitive interaction abilities (Fig. 3 and 4), giving rise to almost equiprobable dynamics with either clonal interference or clonal assistance, i.e. respectively a decrease or an increase in the rate of adaptation. We might thus wonder why a decreasing rate of adaptation is generally observed in experimental evolution. 11
In other words, why does adaptation in clonal species mostly follow a single subcase of our model where all eij ' 1). Several hypotheses can be made. First, as Lenski competitive interaction abilities are almost equal (C and colleagues claimed regarding their long-term experimental evolution, the experiments were designed to avoid frequency-dependent selection: no horizontal gene transfer, a well-mixed environment with no spatial structure and a low concentration of the density-limiting resource (Maddamsetti et al., 2015). One can thus imagine that these experimental conditions were effectively fulfilled and their goal is achieved. However, several experiments showed that even in ideal conditions, frequency-dependent selection occurs (Maddamsetti et al., 2015; Kinnersley et al., 2014; Lang et al., 2011). Hence, even if the experimental design limits the occurrence of mutations with non-transitive effects on fitness, it does not obliterate it. A question thus remains: why is frequency-dependent selection so rare in experimental evolution? Second, we only considered three interacting clones in our model, while much more can be competing in a large clonal population. To our knowledge, it is not known whether an increasing number of competing clones decreases the plausibility of non-transitive interactions. However, we showed that non-transitive interactions are more probable with three than two interacting clones (Fig. 5). Deterministic models with four interacting clones have also shown that even more complex dynamics can be expected, especially chaotic ones (Arneodo et al., 1982; Vano et al., 2006). This suggests that a large number of interacting clones could not be the main cause for the rarity of non-transitive interactions. Third, our estimation of the likeliness of frequency-dependent selection are based on two strong hypotheses: the effect of mutations of competitive interaction abilities follows either an uniform or exponential distribution, and a mutation shows no trade-offs between its effect on the growth rate and the competitive abilities. To our knowledge there is neither empirical nor theoretical treatments about the distribution of the competitive interactions and the trade-offs between species or clones. Maharjan et al. (2006) estimated strength of frequency dependent selection in a limited range of frequencies in a well-mixed environment. They especially showed that non-transitive interactions can have similar extent than transitive interactions. Gallet et al. (2012) detected no intransitive interactions between clones three pairs of bacterial clones in a simple experimental sets, suggesting that competitive interactions are weak. Yet, a general treatment is lacking in order to evaluate the potential importance of non-transitive interactions in adaptive populations. Whether non-transitive interactions can be expected to be frequent or not is an important question since it can challenge the generality of the evolutionary dynamics observed in controlled experimental evolution of clonal species. Non-linear dynamics have been observed in various experiments and different explanations and interpretations have been proposed. Lang et al. (2011) observed different dynamics in different replicates. They especially observed cases where a lineage showed two successive frequency peaks. They explained this observation by the occurrence of a third cryptic mutation affecting a preexisting lineage, what they called “multiple mutations” dynamics. We have shown in our model that such a dynamics can be explained with three interacting clones only: such cyclical dynamics can be observed in cases of Rock-Paper-Scissor. Importantly, we show that such cyclical dynamics can only be observed if the second mutation occurs during the second stochastic phase, i.e. when the first mutation is near fixation. It would be interesting to look thoroughly into the data to challenge this prediction. Maharjan et al. (2006), Lang et al. (2011) and Maddamsetti et al. (2015) also showed the possible coexistence of clones in the long-term, which can effectively be explained by negative 12
frequency-dependent selection, as proposed here. Rosenzweig et al. (1994) and Kinnersley et al. (2014) showed the long-term coexistence of three lineages derived by mutation from a single initial Escherichia coli clone, in a long-run experimental evolution in a chemostat. They interpreted this coexistence as due to cooperative rather than competitive interactions, yet we showed that non-transitive competitive interactions can explain the emergence by mutation and stable coexistence of three clones. Rosenzweig et al. (1994) showed that the three clones can stably coexist by pairs. Under our framework, it means that Sij > 0 for all {i, j} ∈ {0, 1, 2}, and we predict that the three clones should indeed coexist because of non-transitive competitive interactions. The three clones consume the same resources (glucose, acetate, glycerol, Helling et al. (1987), Rosenzweig et al. (1994)) supporting the existence of competitive interactions. However, looking thoroughly at the data, it is not obvious that one of the clones is maintained in coexistence when in competition with another: its frequency is less than 1% and sometimes even not detected in the experiments (see Fig. 1d in Rosenzweig et al. (1994)). Taking an opposite point of view than the authors of the experiments, we can conclude that mutant 2 is not maintained when in competition with mutant 1 (i.e. S12 < 0). In this case, our model predicts that the three clones can not stably coexist, and consequently cooperation between the three clones can be an exclusive explanation for the observed coexistence.
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‘rock–paper–scissors’. Oikos 123:472–480. Lang, G. I., Botstein, D., and Desai, M. M. 2011. Genetic variation and the fate of beneficial mutations in asexual populations. Genetics 188:647–661. Lang, G. I., D. P. Rice, M. J. H., Sodergren, E., Weinstock, G. M., Botstein, D., and Desai., M. M. 2013. Pervasive genetic hitchhiking and clonal interference in forty evolving yeast populations. Nature 500:571–574. Lee, M.-C. and Marx, C. J. 2013. Synchronous waves of failed soft sweeps in the laboratory: Remarkably rampant clonal interference of alleles at a single locus. Genetics 193:943–952. Maddamsetti, R., Lenski, R. E., and E., J. 2015. Adaptation, clonal interference, and frequency-dependent interactions in a long-term evolution experiment with Escherichia coli. Genetics 200:619–631. Maharjan, R., Seeto, S., Notley-McRobb, L., and Ferenci, T. 2006. Clonal adaptive radiation in a constant environment. Science 313:514–517. Manna, F., Gallet, R., Martin, G., and Lenormand, T. 2012. The high-throughput yeast deletion fitness data and the theories of dominance. Journal of Evolutionary Biology 25:892–903. Martin, G. and Lenormand, T. 2006. A general multivariate extension of fisher’s geometrical model and the distribution of mutation fitness effects across species. Evolution 60:893–907. Miralles, R., Gerrish, P., Moya, A., and Elena, S. 1999. Clonal interference and the evolution of RNA viruses. Science 285:1745–1747. 14
Muller, H. 1932. Some genetic aspects of sex. The American Naturalist 8:118–138. Muller, H. 1964. The relation of recombination to mutational advance. Mutation Research 1:2–9. Pandit, A. and de Boer, R. J. 2014. Reliable reconstruction of hiv-1 whole genome haplotypes reveals clonal interference and genetic hitchhiking among immune escape variants. Retrovirology 11:56. Rosenzweig, R. F., Sharp, R. R., Treves, D. S., and Adams, J. 1994. Microbial evolution in a simple unstructured environment: Genetic differentiation in Escherichia coli. Genetics 137:903–917. Vano, J. A., Wildenberg, J. C., Anderson, M. B., Noel, J. K., and Sprott., J. C. 2006. Chaos in low-dimensional Lotka-Volterra models of competition. Nonlinearity 19:2391. Wiser, M. J., Ribeck, N., and Lenski., R. E. 2013. Long-term dynamics of adaptation in asexual populations. Science 342:1364–1367. Yang, K. R., Mooney, S. M., Zarif, J. C., Coffey, D. S., Taichman, R. S., and Pienta, K. J. 2014. Niche inheritance: A cooperative pathway to enhance cancer cell fitness through ecosystem engineering. Journal of Cellular Biochemistry 115:1478–1485. Zeeman, E. and Zeeman, M. 2003. From local to global behavior in competitive Lotka-Volterra systems. Transactions of the American Mathematical Society 355:713–734. Zeeman, M. L. 1993. Hopf bifurcations in competitive three-dimensional lotka-volterra systems. Dynamics and Stability of Systems 8:189–216. Zeeman, M. L. and van den Driessche, P. 1998. Three-dimensional competitive lotka-volterra systems with no periodic orbits. SIAM Journal on Applied Mathematics 58:227–234.
Appendix A1. Conditions for the different final states In the case of competition between three clones, with two mutations entering a resident population in a single copy, the stochastic dynamics of the population can be decomposed into a varying number of successive “stochastic” and “deterministic” phases, depending on the ecological parameters and 1) The state of the population when mutant 2 enters the population, i.e. whether mutant 2 enters the population during the first or second stochastic phase (depending on the time α log K); 2) Does mutant 1 or mutant 2 reach first a population size of order K. 3) the deterministic equilibrium, i.e. either stable coexistence of two or three mutants (two or three mutants have a population size of order K), or a single mutant (a single mutant has a population size of order K); 4) what is the population size of all mutants when the second stochastic phase begins and 5) whether a mutant goes extinct before a succeeding deterministic phase begins. In the following, we detail what determines he different final states. 15
1. When does mutant 2 enters the population? When two mutants 1 and 2 enter a 0 population in a single copy two different situations must be considered: either the second mutation 2 enters the population during the first stochastic phase (Fig. 1(a)), i.e. α < 1/S10 , or during the second stochastic phase (Fig. 1(a)), i.e. α > 1/S10 . When K is large, the duration time of the stochastic phases are of order log K whereas the deterministic phase has a duration of order 1, hence we can consider that mutation 2 has a negligible probability to enter the population during the deterministic phase.
2. Which mutant invades first? If mutant 2 enters the population during the first stochastic phase, then the population size of mutants 1 is lower than order K, thus the invasion fitness of mutation 2 in a 0 resident population with a low number of 1 individuals is simply S20 = β2 − δ2 − C20 n ¯ 0 . If α + 1/S20 < 1/S10 the population of mutants 2 reaches a size of order K first, hence phase 1 has a duration of log K(α + 1/S20 ). Otherwise, if α + 1/S20 > 1/S10 the population of mutants 1 reaches a size of order K first, hence phase 1 has a duration of log K/S10 . At the end of phase 1, three scenarios are thus possible regarding the start of the deterministic phase - Scenario 1a: If 1/S10 − 1/S20 < α < 1/S10 , populations of individuals 0 and 1 both have a size of order K, engage in an approximatively deterministic dynamics, while the population size of mutants 2 does not change; - Scenario 1b: If 0 < α < 1/S10 − 1/S20 , populations of individuals 0 and 2 both have a size of order K, engage in an approximatively deterministic dynamics, while the population size of mutants 1 does not change; - Scenario 1c: If α > 1/S10 , populations of individuals 0 and 1 both have a size of order K, engage in an approximatively deterministic dynamics, while mutants 2 are not yet present in the population. Under scenarios 1a and 1b, in order to know the starting conditions of the second stochastic phase, it is necessary to know the population size of the mutant which did not invade when phase 1 is over. We denote Xij a given number of mutants i in a resident population j, and T (Xij ) the time taken for a population of mutants i to reach size Xij after the introduction of a single individual i, given it did not go to extinction. According to the branching process theory (Athreya and Ney), since phase 1 can be approximated by a birth and death process without interactions, we have T (Xij )/ log Xij ' 1/Sij .
(3)
Hence, the time taken for a population of mutants i to reach the threshold size of order K in a resident population j is log K/Sij . Under scenario 1a, mutants 1 reach the threshold population size of order K first. Suppose that the size of the population of mutants 2 at the end of phase 1 is X20 . Since the mutants 2 occurs after a time α, we have T (X20 ) = (1/S10 − α) log K = log X20 /S20 , which gives the size of the population of mutants 2 at the end of phase 1, X20 = K S20 (1/S10 −α) . Under scenario 1b, similar computations give the size of the population of mutants 1 at the end of phase 1, X10 = K S10 (1/S20 +α) .
16
3. Does the invading mutant stably coexist with the resident population? As shown in the previous section, given the ecological parameters and the time of appearance of the second mutation, three scenarios are possible at the beginning of the deterministic phase. Yet, we have to consider only two possible cases at the end of the deterministic phase: the resident population is being invaded either by a mutant 1 or 2. The dynamics during the deterministic phase thus corresponds to a competitive Lotka-Volterra systems with two species, and its final state only depends on the relative invasion fitnesses. Under scenarios 1a and 1c, if S01 > 0 then individuals 0 and 1 coexist at the end of the deterministic phase, i.e. both have a size of order K, while if S01 < 0, then the number of mutants 0 is lower than order K. Under scenario 1b, the final states of phase 2 are analog: coexistence of 0 and 2 if S02 > 0, invasion of mutant 2 individuals if S02 < 0. At the end of the deterministic phase, two situations are thus possible: - Scenario 2a: If S0j < 0, only the size of the population of mutants j is of order K; - Scenario 2b: If S0j > 0, both mutants 0 and j populations are of order K, they thus stably coexist. 4. What is the population size of each mutant at the beginning of the second stochastic phase? At the start of the second stochastic phase, three situations are possible: i) A single mutant (either 1 or 2, depending on which mutant invades first during the first stochastic phase) has a population size of order K and the two others have a population size lower than K (mutants 0 and either 1 or 2). Under this situation, the dynamics of the second stochastic phase is similar to the dynamics of the first stochastic phase, except that the starting population sizes are of order K for mutants 0 and X10 = K S10 (1/S20 +α) or X20 = K S20 (1/S10 −α) as shown before. ii) Two mutants have a population size of order K (mutants 0 and either 1 or 2) and the other has a population size lower than K (either 1 or 2 with respectively size X10 or X20 ). iii) Mutant 2 has not yet entered the population (if α > 1/S10 ). When it does at time α, the starting conditions are identical to situation i) (given mutants 0 did not go extinct, i.e. assuming 1/S10 < α < 1/S10 + 1/ |S01 |, since the time it takes for mutants 0 to be lost in a 1 resident population is 1/ |S01 |), or to situation ii) except that the initial mutant 2 population size is equal to 1. When the size of the population of mutants 2 reaches a threshold of order K, then the second stochastic phase ends and a new deterministic phase begins. The succession of stochastic and deterministic phases can thus continue until a steady state is reached (because of the loss of at least one mutant, or because of a stable equilibrium or cycle limits are reached).
5. Does a mutant go extinct before the successive deterministic phase begins? Finally, the last conditions to determine the dynamics and the final states is whether or not one of the mutant goes extinct before a deterministic phase begins. These conditions are shown in Tab. 1 and 2 as the inequalities involving α, the time at which mutation 2 enters the population. In both Tab. 1 and 2, the first inequalitiy involving α refers to whether or not mutation 0 goes extinct before the second deterministic phase begins, i.e. when mutation 2 spreads out the population. The second inequalitiy involving α refers to whether or not mutation 1 goes extinct when mutation 0 begins to spread out again in a later deterministic phase. These conditions depend on the time at which mutation 2 enters the population because the time taken for mutation 0 or 1 to go extinct before a deterministic phase begins depends on population size of all mutants, and therefore on the 17
time when mutation 2 appears. The conditions for the extinction of mutant 0 before the second deterministic phase begins are different when mutation 2 enters the population in the first or the second stochastic phase. Following similar calculations than previously, we have α < 1/S10 + 1/S20 (S21 /|S01 | − 1) and α < 1/S10 + 1/|S01 | − 1/S21 the conditions under which mutation 0 does not go extinct before the successive deterministic phase begins (where mutation 2 spreads out) respectively when mutation 2 enters the population during the first or second stochastic phase. Similarly, we have α < S02 S21 /(S10 |S12 ||S01 |) − 1/S10 and α < 1/S10 + S02 /(|S12 ||S01 |) − 1/S21 the conditions under which mutation 1 does not go extinct before the successive deterministic phase begins (where mutation 0 spreads out) respectively when mutation 2 enters the population during the first or second stochastic phase.
A2. An illustration of the calculation of the hitting times of the final states All the possible final states and their hitting times are compiled in Tab. 1 and 2. We do not detail all calculations for all cases here, we only give one example in the following as an illustration (for complete and detailed computations see Billiard and Smadi (in press)). We focus on the following case: when mutation 2 goes to fixation. Consider scenario 1a and 2a (the mutant 2 enters the population during the first stochastic phase and, at the beginning of the second stochastic phase, only the population of mutants 1 has a size of order K). The mutants 1 enters first the resident 0 population in a single copy. According to the branching process theory, the time it would take for the population size of mutants 1 to reach a threshold size of order K is close to log K/S10 . If the mutation 2 appears in the population after a time α and before the population enters the deterministic phase, i.e. if α < 1/S10 , then the dynamics is changed and there are two possibilities, depending on the ecological parameters: either mutants 1 or 2 reach first the threshold population size of order K. Without loss of generality, we assume that mutants 1 first reach the threshold population size (computations are similar in the other case). We need to compute the size of the population of mutants 2 when the first stochastic phase ends in order to know the state of the population at the beginning of the second stochastic phase. Let us denote X the population size of mutants 2 at the end of the first stochastic phase, and denote TX the time taken for mutant 2 population to reach size X. Since we know that the time for the population of mutants 1 to reach the threshold size K is log K/S10 and that mutation 2 occurs after a time α log K then we know that (0)
TX = log K(1/S10 −α). According to the branching process theory, during phase 1, the populations of mutants 1 and 2 are negligible relatively to mutants 0, hence the dynamics of mutants 2 only depends on mutants 0. Hence, the time taken for the population of mutants 2 to reach size X if it was in a resident population 0 would (0)
be TX = log X/S20 . Resolving log K(1/S10 − α) = log X/S20 gives the size of the population of mutants 2 at the beginning of phase 2: X = K S20 (1/S10 −α) . During the deterministic phase, there is a probability near 1 that the size of the population of mutants 2 remains unchanged. We now compute the duration of the second stochastic phase, i.e. the time it takes for the mutants 2 population to reach a size of order K, where the population will once again enter a deterministic phase. At the beginning of the second stochastic phase, the size of the population of mutant 0 is negligible and the dynamics of mutants 2 only depends on mutants 1. The time it would take for the population of mutants 2 to reach a 18
(1)
size of order K in a resident population 1 is TK = log K/S21 . The time to reach size X = K S20 (1/S10 −α) in a (1)
mutants 1 population is TX = log K S20 (1/S10 −α) /S21 . We thus can conclude that the duration of the second stochastic phase is (1)
(1)
TK − TX = 1/S21 (log K 1−S20 (1/S10 −α) ) =
1 − S20 (1/S10 − α) log K. S21
Finally, since the duration of the first stochastic in this case is 1/S10 and since mutant 2 enters the population after a time α log K, the total duration for mutant 2 phase to reach a population size of order K is 1 1 S20 log K + −α 1− . S21 S10 S21
(4)
We will consider this duration as the “fixation time” of mutant 2 since at the beginning of the successive stochastic phase, other mutants have a negligible population size. Mutants 0 and 1 thus do not have an ecological effect of mutants 2 anymore. Moreover, one can think that this duration time is the one observed experimentally: mutants 0 and 1 are negligible and can thus be detected with difficulty: most probably, this state will be considered as the fixation of mutant 2. One can however computes the total duration including the extinction time of mutant 0 and 1 with similar computations (see Tab. 4). In order to determine the conditions under which competitive interactions can slow down the invasion and fixation of mutant 2, i.e. clonal interference, we can solve the following inequality from Eq.4 1 1 S20 1 + −α 1− . > S21 S10 S21 S20
(5)
A3. Simulation algorithm To illustrate our results, we performed individuals based stochastic simulations of our model. At time T , the total rate of possible events is given by E(T ) =
2 X
βi + δi (T )Ni (T ) +
i=0
X
Cij Nj (T ) Ni (T ).
j
The probability that at time T + ∆T , the next event is a birth (resp. a death) of an individual i is given by βi Ni (T )/E(T ) (resp. di (N (T ))Ni (T )/E(t)). The time ∆T is drawn in an exponential distribution with parameter E(T ). If an individual i is born (resp. is dead) then the size of the population of mutants i becomes Ni (T + ∆T ) = Ni (T ) + 1 (resp. Ni (T + ∆T ) = Ni (T ) − 1). The simulations finally just consist in randomly determining the succession of events and the time taken for each event. Note that we did not use this stochastic algorithm to determine the distribution of the different final states since given the ecological parameters summarized in Sij , and α the time of appearance of mutation 2, we can determine the final states using Tab. 1 and 2.
19
Tables and Figures
20
21 1 S10
21
+ S1
1 S10
1+
1+
|S01 | S02
1−S20
1 S10
1 S10
−α
1 S10
−α
−α
1 S10
1 S10
−α
−α
1 S10
|S01 | |S01 |S|12 | S02 + S02 S102
21
+ S1
|
012
|S
1 S10
1+ S 01
1−S20
+ S1 21
+ S1 21
21
+ S1
1 S10
201
+S1
1 S10
1 S10
1 S10
1 S10
−α
Duration of the sweep: log K.
S02 S21 S12
S21 , S12 , S012 S02 , S12 , S102 S102 , S21 , S02
S102
S102
S21 , S02
S12
S21 , S12 , S02 , S012 , S102
S102
S02 , S12 , S21
S21 , S102
S02 , S12
S21 , S12
S12 , S102
S02 , S21
S02
S012
S12 , S02
0
(1)
(2)
(0, 0, n ¯2) (2)
(2)
(0)
(2)
(2)
(2)
mutants i are lost. The notation “012” means that the three mutants coexist in the final state of the population, but we do not know whether population
equilibrium population size of mutant u individuals in a population where mutants i, j and k coexist at equilibrium. When n ¯ uijk = 0, then in the final state,
dynamics are denoted by capital letters from A to K. The different possible final states are denoted by the vector (¯ n0ijk , n ¯ 1ijk , n ¯ 2ijk ), where n ¯ uijk is the
(2)
(0, n ¯ 1 , 0)
012
(0)
(¯ n02 , 0, n ¯ 02 )
(0, 0, n ¯2)
012
(0)
(¯ n02 , 0, n ¯ 02 )
(1) (2) (0, n ¯ 12 , n ¯ 12 )
012
012
012
012
012
¯ 02 ) (¯ n02 , 0, n
(0)
(0)
¯ 02 ) (¯ n02 , 0, n
¯ 02 ) (¯ n02 , 0, n
¯ 02 ) (¯ n02 , 0, n
(0)
¯ 12 ) (0, n ¯ 12 , n
and their hitting times in function of the invasion fitnesses Sij , Sijk and the time α at which mutation 2 enters the population. The different possible
sizes reach a stable fixed point or stable limit cycles.
(1)
(¯ n01 , n ¯ 01 , 0)
(0)
Final state
Table 1: When mutation 2 enters the population during the first stochastic phase: the different possible dynamics, the possible final states of the population
E I F H G K
C J D
B
Case A
22 21
|S
|
|S
|
|S | α+ S1 + S01 21 02
α+ S1 + S01 21 02
|
1 S10
|S | 1+ S12 10
102
21
α− S1 + S1 10 21
10
α− S1 + S1
α− S1 + S1 10 21
|S
|
21
α− S1 + S1 10 21
1+ S 12
α+ S1 + S01 21 02
|S
|S | α+ S1 + S01 21 02
10
α− S1 + S1
α+ S1 21
012
α+ S1 + S 01
201
α+ S 1
210
α+ S 1
1 S10
Duration of the sweep: log K.
S12 , S02 , S20 S02 S12 S21 , S102 S20 , S012 S21 , S12 S20 , S02 S102 S012 S21 , S02 S20 , S12 S02 S12 S21 S20
S21 S20 , S12 , S21 S20 , S02 , S21 S20 , S02 , S12 S21 , S12 , S02 S20 , S02 S21 , S12 S20 , S02 , S12 , S21 S20 , S02 , S12 , S21 S20 , S12 S21 , S02 S20 , S21 , S12 , S012 S20 , S21 , S02 , S102 S20 , S02 , S12 , S102 S21 , S02 , S12 , S012
S20 , S102
S20
S20
S102
S12 , S02 , S21
S20
Sij , Sijk , i, j, k ∈ {0, 1, 2}
S12 , S02
0
(2)
(2)
(2)
(0)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(0, n ¯ 1 , 0)
cycles RPS
012
(¯ n0 , 0, 0)
(0)
(¯ n02 , 0, n ¯ 02 )
(0, 0, n ¯2)
012
012
(¯ n02 , 0, n ¯ 02 )
(0)
(1) (2) (0, n ¯ 12 , n ¯ 12 )
012
012
012
012
012
012
012
(0, n ¯ 12 , n ¯ 12 )
(1)
(¯ n02 , 0, n ¯ 02 )
(0)
(0, n ¯ 12 , n ¯ 12 )
(1)
(¯ n02 , 0, n ¯ 02 )
(0)
(0, n ¯ 12 , n ¯ 12 )
(1)
(¯ n02 , 0, n ¯ 02 )
¯ 02 ) (¯ n02 , 0, n
(0)
¯ 12 ) (0, n ¯ 12 , n
(1)
(0, 0, n ¯2)
(0, 0, n ¯2)
(0, 0, n ¯2)
“Rock-Paper-Scissors” refers to the case where the final state follow stable limit cycles. Other notations and definitions are the same than in the legend of
population and their hitting times in function of the invasion fitnesses Sij , Sijk and the time α at which mutation 2 enters the population. The notation
Tab. 2.
(1)
(¯ n01 , n ¯ 01 , 0)
(0)
Final state
Table 2: When mutation 2 enters the population during the second stochastic phase: the different possible dynamics, the possible final states of the
J M
G I
F H
E K
D
C L
B
A
Case
Lower
Higher
invasion probability
invasion probability
Final state
(S21 < S20 or S201 < S20 )
(S21 > S20 or S201 > S20 )
Polymorphism
Soft clonal coexistence
Hard clonal coexistence
Shorter fixation time
Soft clonal assistance
Hard clonal assistance
Longer fixation time
Soft clonal interference
Hard clonal interference
Fixation
Table 3
Extinction Time when mutation 2 occurs in Case
First Stochastic Phase
B1
I
max( |S102 | , |S112 | ) | 1 max( |S112 | , |S102 | 1 − |SS01 (1 − S ( − α) 20 S 21 10 max( |S102 | , |S112 | S10 α + S120 )
Case
Second Stochastic Phase
E
B1 E K
max( |S102 | , |S112 | ) max( |S112 | , |S102 | 1 − |S01 |(α −
1 S10
+
1 S21 )
1 |S12 |
Table 4
23
24
all i, C0,0 = 1.8,C1,0 = C2,1 = 1, C1,1 = 2.3,C1,2 = C2,0 = 2, C0,1 = 5, C0,2 = 3, C2,2 = 2.1, K = 1000, α = 1.9.
into the population. Parameters of the simulations: (a) β0 = 1, β1 = 2, β2 = 3, δi = 0 and Cij = 1 for all (i, j), K=1000, α = 0.5; (b)βi = 2 and δi = 0 for
eventually invades the population and get fixed, while in figure (b) mutation 0 finally get fixed, even though mutation 1 and 2 are favorable when they enter
well approximated by a succession of “stochastic” phases (birth-death processes without competition) and “deterministic” phases. In figure (a), mutation 2
single mutant 2 is introduced in the population. The red dot shows when the mutant 1 is lost. Both figures show that the stochastic dynamics can be
The population sizes of the different mutants are respectively black, red and green for mutant 0, 1 and 2 individuals. The green crosses show when a
Figure 1: Simulations of the stochastic process when mutation 2 enters the population during (a) the first stochastic phase, (b) the second stochastic phase.
25
α = 1.1.
C1,1 = 2.3,C1,2 = 3, C2,0 = 1.5, C0,1 = 4, C0,2 = 3, C2,2 = 2.1, α = 0.5; (c): C0,0 = C1,1 = C2,2 = 2, C0,1 = 2.5, C0,2 = C1,0 = C2,1 = 1, C1,2 = C2,0 = 3,
stochatic phase giving a Rock-Paper-Scissor dynamics. Parameters: βi = 2 and δi = 0 for all i and K = 1000; (a) and (b): C0,0 = 1.8,C1,0 = C2,1 = 1,
without (compare the green curves between (a) and (b) ; (c) Simulations of the stochastic process when mutation 2 enters the population during the second
Figure 2: Simulations of the stochastic process with (a) one mutation only, (b) clonal assistance, fixation of mutation 2 is faster with mutation 1 than
1.0
1.0
(a)
(b) 0.8
0.6
A
I
B
J
C
K
D E
0.4
Density
Density
0.8
0.6
0.4
F
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
M
1.0
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
(c)
(d) 0.8
Fixation 1
0.6
Fixation 2 Polym 0-1 Polym 0-2
0.4
Fixation 0
Density
Density
L
E
1.0
0.8
0.6
Fixation 1 Fixation 2 Polym 0-1
0.4
Polym 0-2
Polym 1-2
Polym 1-2
Polym 0-1-2 0.2
Polym 0-1-2
0.2
0.0 0.0
0.2
0.4
0.6
0.8
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
0.0 0.0
1.0
1.0
0.2
0.4
0.6
0.8
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
1.0
1.0
(e)
(f) 0.8
0.6
Clonal Interference Clonal
0.4
Density
0.8
Density
K
D
H
0.0 0.0
1.0
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
J
C
G 0.2
H
0.0 0.0
I
B
F
G 0.2
A
0.6
Clonal Interference Clonal
0.4
Assistance 0.2
0.0 0.0
Assistance 0.2
0.2
0.4
0.6
0.8
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
0.0 0.0
1.0
0.2
0.4
0.6
0.8
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
1.0
Figure 3: Posterior probability when competition coefficient is drawn in a uniform distribution. Left and right columns: the second mutation enters in the first (α < 1/S10 ) or second stochastic phase (1/S10 < α < 1/S10 + 1/|S01 |), respectively. (a)-(b) Dynamics (see Tab. 1 2 for the meaning of the letters); (c)-(d) Final states; (e)-(f) Clonal interference vs. clonal assistance. The growth rate of mutant 0 ρ0 = 2, supposed to be at 50% from the optimum in a Fisher’s Geometric adaptive landscape. The effect of mutations 1 and 2 on growth rate ρ1 and ρ2 are randomly drawn in an approximation of the adaptive landscape (Martin and Lenormand, eij = Cij /Cjj between mutant i and j are drawn in an uniform 2006). The ratio of competitive abilities C distribution with range [1 − µ, 1 + µ].
26
1.0
1.0
(a)
(b) 0.8
0.6
A
I
B
J
C
K
D E
0.4
Density
Density
0.8
0.6
0.4
F
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
M
1.0
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
(c)
(d) 0.8
Fixation 1
0.6
Fixation 2 Polym 0-1 Polym 0-2
0.4
Fixation 0
Density
Density
L
E
1.0
0.8
0.6
Fixation 1 Fixation 2 Polym 0-1
0.4
Polym 0-2
Polym 1-2
Polym 1-2
Polym 0-1-2 0.2
Polym 0-1-2
0.2
0.0 0.0
0.2
0.4
0.6
0.8
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
0.0 0.0
1.0
1.0
0.2
0.4
0.6
0.8
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
1.0
1.0
(e)
(f) 0.8
0.6
Clonal Interference Clonal
0.4
Density
0.8
Density
K
D
H
0.0 0.0
1.0
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
J
C
G 0.2
H
0.0 0.0
I
B
F
G 0.2
A
0.6
Clonal Interference Clonal
0.4
Assistance 0.2
0.0 0.0
Assistance 0.2
0.2
0.4
0.6
0.8
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
0.0 0.0
1.0
0.2
0.4
0.6
0.8
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
1.0
Figure 4: Posterior probability when competition coefficient is drawn in an exponential distribution. The ratio eij = Cij /Cjj between mutant i and j are drawn in an exponential distribution with of competitive abilities C mean µ. See legend of Fig. 3 for further details.
27
1.0
1.0
Two clones
Two clones
(a)
0.6
Beneficial mutations Fixation
0.4
Density
Density
(b)
0.8
0.8
0.6
Beneficial mutations Fixation
0.4
Polymorphism
Polymorphism 0.2
0.2
0.0 0.0
0.0 0.2
0.4
0.6
0.8
1.0
0.0
μ, maximum effect on competition ratio Cij/Cjj
0.5
1.0
1.5
1.0
1.0
(c)
(d) 0.8
0.6
Beneficial mutations Fixation
0.4
Density
0.8
Density
2.0
μ, mean effect on competition ratio Cij/Cjj
0.6
Beneficial mutations Fixation
0.4
Polymorphism
Polymorphism
0.2
0.0 0.0
0.2
0.2
0.4
0.6
0.8
0.0 0.0
1.0
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
1.0
0.5
1.0
1.5
1.0
(e)
(f) 0.8
0.6
Beneficial mutations Fixation
0.4
Density
0.8
Density
2.0
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
0.6
Beneficial mutations Fixation
0.4
Polymorphism 0.2
0.0 0.0
Polymorphism 0.2
0.2
0.4
0.6
0.8
˜ μ, maximum effect on competition ratio C ij =Cij /Cjj
0.0 0.0
1.0
0.5
1.0
1.5
˜ μ, mean effect on competition ratio C ij =Cij /Cjj
2.0
Figure 5: Polymorphism vs. fixation when (a)-(b) there are only two interacting clones (mutants 0 and 1, first row) or (c)-(f) three interacting clones (mutants 0, 1 and 2, second and third rows). The proportion of beneficial mutations among all simulations is also shown. Left and right columns: when competition coefficient is drawn in an uniform or exponential distribution, respectively; (c) and (e): the second mutant enters in the first stochastic phase (α < 1/S10 ), (d) and (f) the second mutation enters in the second stochastic phase (1/S10 < α < 1/S10 + 1/|S01 |).
28
