\section{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 \citep{fisher1930, crowkimura1965}. 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  \citep{muller32, muller64} or more generally Hill-Robertson interfences \citep{hillrobertson66}. 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'' \citep{gerrishlenski98}. Clonal interference and its consequences on adaptation rates seems general as they have been observed in  bacteria, viruses, yeasts or cancer tumors \citep[e.g.][]{mirallesetal99, devisserrozen06,  greavesmaley12, langetal13}. \\ 
Despite the ubiquity and importance of clonal interference, experiments showed that adaptation is not blocked and keep on even after thousands generations \citep[e.g.][]{wiseretal13}. 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 \citep{desaifisher07}, thus increasing fitness gradually. In large populations with high mutation rates, it is expected that adaptation follows a wave \citep{desaifisher07, goodetal12} 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 non-linear: lineages can show multiple frequency peaks during the course of adaptation \citep{leemarx13} and different lineages can coexist for a long time in a single population \citep{langetal11, maddamsettietal15}. 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 \citep{yangetal14, kinnersleyetal14}, 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 \citep[e.g.][]{huismanetal01}. 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 \citep{laird14} again in order 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. \\ 




\section{Model and methods}

\subsection{Definitions and assumptions}
We denote $i \in [0,1,2]$ 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 \emph{mutant} $i$ when referring to type $i$ individuals. $N(t)=(N_i(t))$ is a vector whose elements are the number of mutants $i$ in the population at time $t$, with $N_i(t)$ 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: $\beta_i$ and $\delta_i$ are respectively the individual birth and natural death rates, and $C_{ij}$ is the effect of competition of a single mutant $j$ on a single mutant $i$, assuming $C_{ij} \geq 0$ and $C_{ii}>0$. 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 ($\delta_i$) and a component due to competition: $d_i(N(t))=\delta_i+C_{i0} N_0(t)/K +C_{i1}N_1(t)/K+C_{i2}N_2(t)/K$.\\
We suppose that the resident population is only composed of mutants $0$, at its ecological steady-state equilibrium, say $N_0(0)$. A single mutant $1$ is introduced in the population ($N_1(0)=1$). 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 $\log K$ \citep{champagnat06}. 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 $\alpha \log K$, $\log K$ being the time scale of the whole stochastic dynamics. There are two general cases. Either $\alpha$ is low enough that mutants $1$ are still in too few numbers to affect the invasion of mutants $2$, or $\alpha$ 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 $\alpha \log K$ at which mutant $2$ enters the population is crucial and largely affect the final state of the population.\\

\subsection{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 $K$ \citep{fourniermeleard04, champagnat06}. 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 \cite{billiardsmadi}). 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 \citep{fourniermeleard04, champagnat06, billiardsmadi}. 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. \ref{fig:phases}). 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 \cite{billiardsmadi}).\\

{\bf Deterministic phases.} We denote by $n_i$ the size of the population of mutants $i$ when dealing with the deterministic dynamics, while we will keep the notation $N_i$ when dealing with stochastic dynamics ($n_i$ is the population size rescaled by the carrying capacity $N_i / K$ when $N_i$ 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,
\begin{equation} \label{EDO}
\left\{\begin{array}{ll}
	\dot{n}_0=(\beta_0-\delta_0-C_{0,0}n_0-C_{0,1}n_1-C_{0,2}n_2)n_0,\\
        \dot{n}_1=(\beta_1-\delta_1-C_{1,0}n_0-C_{1,1}n_1-C_{1,2}n_2)n_1,\\
	\dot{n}_2=(\beta_2-\delta_2-C_{2,0}n_0-C_{2,1}n_1-C_{2,2}n_2)n_2.
       \end{array}
\right.
 \end{equation}

Under the assumptions that competitive parameters are  $C_{ij} \geq 0$ and $C_{ii}>0$ 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 \citep{zeeman93,zeemanvandendriessche98,zeemanzeeman03}.\\
The fate of a mutant $i$ entering in a single copy a resident $j$ population is associated to the so-called ``invasion fitness'', denoted $S_{ij}=\beta_i-\delta_i-C_{ij} \bar{n}^j$, where $\bar{n}^j=\frac{\beta_j-\delta_j}{C_{jj}}$ 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, then the fate of a mutant $k$ entering in a single copy is associated with the invasion fitness denoted $S_{kij}=\beta_k-\delta_k-C_{ki} \bar{n}^{i}_{ij}-C_{kj} \bar{n}^{j}_{ij}$, where
\begin{equation} \label{eqpop}
\begin{array}{ll}\\
        \bar{n}^i_{ij}=\frac{C_{jj}(\beta_i-\delta_i)-C_{ij}(\beta_j-\delta_j)} {C_{ii}C_{jj}- C_{ij}C_{ji}}, \bar{n}^j_{ij}=\frac{C_{ii}(\beta_j-\delta_j)-C_{ji}(\beta_i-\delta_i)}{C_{ii}C_{jj}- C_{ij}C_{ji}},
       \end{array}
 \end{equation}
is the equilibrium of Eq. \ref{EDO} when there are only mutants $i$ and $j$ in the population. If $S_{kij}>0$, mutation $k$ is favorable when rare in the polymorphic resident population $(i,j)$ and can invade. \\

{\bf The stochastic phase.} When the population size of at least one mutant, say $i$, is low (\emph{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 respectively $\beta_i$ and $\delta_i+\sum_{j \neq i}C_{ij}n_j$, 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 $S_{ij}/\beta_i$ when $S_{ij}>0$, and a probability 0 if  $S_{ij}\leq 0$.  The time taken by a mutant $i$ which enters a $j$ resident population to reach the threshold population size is of order $\log K / S_{ij}$. We can similarly define the probability of invasion of a mutant $k$ in a resident population with both mutants $i$ and $j$ as $S_{kij}/\beta_k$ when $S_{kij}>0$, and a probability 0 if $S_{kij}\leq 0$. The time taken by a mutant $k$ which enters a $i$ and $j$ resident population to reach the threshold population size is of order $\log K / S_{kij}$. In both cases, the stochastic phase ends when the mutant is either lost or reaches a threshold population size of order $K$.
 
\subsection{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. \ref{fig:phases}a and \ref{fig:phases}b): 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.\\

{\bf 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. \ref{fig:phases} and \ref{fig:clonalreinforcement}, simulation algorithm given in Appendix A3):\\
i) Does mutant $2$ enter the population during the first (Fig. \ref{fig:phases}a) or second (Fig. \ref{fig:phases}b) stochastic phase (depending on the time $\alpha \log K$)? 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 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. \ref{fig:phases}a), 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 $S_{10}$ and $S_{20}$ and on the time when mutant 2 enters the population $\alpha \log K$. 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. \ref{EDO}). For instance, mutants 0 and 1 stably coexist if $S_{01}>0$ and $S_{10}>0$. \\
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.\\

{\bf 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 ( $S_{10}>0$and  $S_{20}>0$), and on cases where mutation $1$ has a positive invasion fitness( $S_{10}>0$) 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 \ref{tableass1} and \ref{tableass2}. 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 \citet{billiardsmadi}). 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).
 
\subsection{ 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: $\rho_i=\beta_i-\delta_i$, the net individuals reproductive rate of mutants $i$, and $\widetilde{C}_{ij}=\frac{C_{ij}}{C_{jj}}$ the ratio of the between and within competitive interactions. We drew $10^6$ different sets of parameters in prior distributions. For a given set of parameters and a given $\alpha$, the final state is given by Tab. \ref{tableass1} and  \ref{tableass2}, which allows to estimate the posterior distribution of the final states among the $10^6$ random sets of parameters. \\
The time $\alpha \log K$ 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 $\alpha$ using the results of our model (Tab. \ref{tableass1} and \ref{tableass2}). 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. $\alpha<1/S_{10}$. 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. $1/S_{10}<\alpha<1/S_{10}+1/|S_{01}|$. We also know that there are two threshold values $\alpha < 1/S_{10}+1/S_{20}(S_{21}/|S_{01}|-1)$ and $\alpha < S_{02} S_{21}/(S_{10} |S_{12}| |S_{01}|)-1/S_{01}$ (Tab. \ref{tableass1}) 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 $1/S_{10}<\alpha < 1/S_{10}+1/|S_{01}|-1/S_{21}$ and $1/S_{10}<\alpha <1/S_{10}+S_{02}/(|S_{12}||S_{01}|)-1/S_{21}$ (Tab. \ref{tableass2}) 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 $\alpha$ is uniformly drawn in the interval $\left[ 0, 1/S_{10}\right]$ or $\left[ 1/S_{10}, 1/S_{10}+1/|S_{01}|\right]$ respectively when mutation 2 enters the population during the first or second stochastic phase.     \\
{\bf 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 \cite{martinlenormand06} and the Appendix in \cite{mannaetal12}). 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 $\rho_0$, we assumed that the reproductive rate of mutant $i$ is $\rho_i=\rho_0 + x_i$ with $x_i$, 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 \citep{martinlenormand06}.  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: $\rho_2=\rho_1 + x_2$ when mutant 1 is more frequent than mutant 0, $\rho_2=\rho_0 + x_2$ otherwise.\\ 
{\bf Effect of mutations on competition.} There is, to our knowledge, no theoretical or empirical consensus on the distribution of mutation effects on the competitive abilities $\widetilde{C}_{ij}$. Without any knowledge about the distribution of competitive abilities, we simply assumed that the ratio of competitive interaction $\widetilde{C}_{ij}$ follows arbitrary chosen distributions. First, we assumed that it follows a uniform distribution in the interval $\left[1-\mu, 1+\mu \right]$, with $0 \leq \mu \leq 1$. Second we assumed it follows an exponential distribution with parameter $\mu>0$. When $\mu=0$, all $\widetilde{C}_{ij}=1$, fitnesses are necessarily transitive, while if $\mu>0$, frequency-dependent fitnesses can occur. As $\mu$ increases, the variance of the competitive ratio  $\widetilde{C}_{ij}$ also increases, i.e. the more different can the competitive interactions be between mutants. \\


\section{Results }
Given that two favorable mutations successively enter in a single copy a resident population 0, Tables \ref{tableass1} and \ref{tableass2} show all possible dynamics and final states, depending on the sign of the invasion fitnesses $S_{ij} \text{ and } S_{ijk}, \{i,j,k\} \in \{0,1,2 \}$, and the time of appearance of the second mutation $\alpha \log K$. When the second mutation enters the population during the first stochastic phase, Tab. \ref{tableass1} 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. \ref{tableass1} 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. \ref{tableass2} 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.

\subsection{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 \citep{gerrishlenski98}, or clonal reinforcement \citep{kinnersleyetal14}, also called niche construction or frequency-dependent selection elsewhere \citep{yangetal14}, 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 \cite{gerrishlenski98}) 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 \ref{tab:cat}.\\
{\bf Rate of adaptation: fixation time vs. probability of invasion of beneficial mutants.} Clonal interference 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''. \\
{\bf 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, \citet{kinnersleyetal14} 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 \citet{kinnersleyetal14} 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.

\subsection{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 ($\alpha>1/S_{10}$, Tab. \ref{tableass2}): Rock-Paper-Scissor dynamics (final state J in Tab. \ref{tableass2}), or a return to the initial state, i.e. a population fixed for mutant 0 (final state G in Tab. \ref{tableass2}). These two different dynamics are illustrated in Fig. \ref{fig:phases}b and Fig. \ref{fig:clonalreinforcement}b. Both dynamics have the same parameters, except the time at which mutant 2 enters the population $\alpha \log K$. 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 Rock-Paper-Scissors cyclical dynamics take place, otherwise mutant 0 goes to fixation and adaptation is annihilated 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 Rock-Paper-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

\subsection{Likehood of the final states assuming prior distributions of the ecological parameters}
Figures \ref{fig:Distrib1} and  \ref{fig:Distrib2} 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 drawn in an uniform or exponential distribution. When the variance of the distribution of the $\widetilde{C}_{ij}$ is low, all clones have similar competitive abilities ($\widetilde{C}_{ij} \simeq 1$), \emph{i.e.} invasion fitness are mostly transitive. We naturally recover predictions from population genetics models: The likeliest scenari are the fixation either of mutant 1 or 2 (Fig. \ref{fig:Distrib1}c, \ref{fig:Distrib1}d, \ref{fig:Distrib2}c, \ref{fig:Distrib2}d). 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 large, the likeliness of all dynamics rapidly reaches a plateau when competitive abilities ($\widetilde{C}_{ij}$) are drawn in an uniform distribution (Fig. \ref{fig:Distrib1}). 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. \ref{fig:Distrib1} and  \ref{fig:Distrib2} 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.\ref{fig:Distrib2clones}) 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. Fig. \ref{fig:Distrib1},  \ref{fig:Distrib2} and \ref{fig:Distrib2clones} 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. \ref{fig:Distrib1}e-f and \ref{fig:Distrib2}e-f show the likeliness of clonal interference \emph{vs.} clonal assistance. When competitive abilities follow an uniform distribution, clonal interference, \emph{i.e.} the slowing down of fixation of beneficial mutation, is the most probable,  especially when the competitive abilities are similar between clones (small $\mu$). However, rapidly when the difference between competitive abilities increases (large $\mu$) 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 $\mu$ is low, the distribution of the invasion fitness is skewed towards negative values, while it is skewed towards positive values when $\mu$ is large (not shown), which explains why polymorphic states are the likeliest for large $\mu$. 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.

\section{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 \cite[e.g.][]{gerrishlenski98, desaifisher07}, what is effectively found in experimental evolution of bacteria \citep{devisserrozen06}. 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 \citep[e.g.][]{panditdeboer14}, bacteria \citep{devisserrozen06} or cancer \citep{greavesmaley12}. 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 non-transitive 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 abilities are equal ($\widetilde{C}_{ij}=1$). Most importantly, we showed that frequency-dependent selection is very likely when mutations have an effect on competitive interaction abilities (Fig. \ref{fig:Distrib1} and \ref{fig:Distrib2}), 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. In other words, why does adaptation in clonal species mostly follow a single subcase of our model where all competitive interaction abilities are almost equal ($\widetilde{C}_{ij} \simeq 1$). Several hypotheses can be made. First, as Lenski 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 \citep{maddamsettietal15}. 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 \citep{maddamsettietal15, kinnersleyetal14, langetal11}. 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. \ref{fig:Distrib2clones}). Deterministic models with four interacting clones have also shown that even more complex dynamics can be expected, especially chaotic ones \citep{arneodoetal82, vanoetal06}. 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. \citet{maharjanetal06} 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. \citet{galletetal12} 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.  \citet{langetal11} 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. \citet{maharjanetal06}, \citet{langetal11} and \citet{maddamsettietal15} also showed the possible coexistence of clones in the long-term, which can effectively be explained by negative frequency-dependent selection, as proposed here.  \citet{rosenzweigetal1994} and \citet{kinnersleyetal14} showed the long-term coexistence of three lineages derived by mutation from a single initial \emph{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. \citet{rosenzweigetal1994} showed that the three clones can stably coexist by pairs. Under our framework, it means that $S_{ij}>0$ for all $\{i,j\} \in \{ 0,1,2\}$, 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, \cite{hellingetal87}, \cite{rosenzweigetal1994}) 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 \citet{rosenzweigetal1994}). 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 (\emph{i.e.} $S_{12}<0$). 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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