The impact of random frequency-dependent mutations on the average population fitness

ABSTRACT: BACKGROUND: In addition to selection, the process of evolution is accompanied by stochastic effects, such as changing environmental conditions, genetic drift and mutations. Commonly it is believed that without genetic drift, advantageous mutations quickly fixate in a halpoid population due to strong selection and lead to a continuous increase of the average fitness. This conclusion is based on the assumption of constant fitness. However, for frequency dependent fitness, where the fitness of an individual depends on the interactions with other individuals in the population, this does not hold. RESULTS: We propose a mathematical model that allows to understand the consequences of random frequency dependent mutations on the dynamics of an infinite large population. The frequencies of different types change according to the replicator equations and the fitness of a mutant is random and frequency dependent. To capture the interactions of different types, we employ a payoff matrix of variable size and thus are able to accommodate an arbitrary number of mutations. We assume that at most one mutant type arises at a time. The payoff entries to describe the mutant type are random variables obeying a probability distribution which is related to the fitness of the parent type. CONCLUSIONS: We show that a random mutant can decrease the average fitness under frequency dependent selection, based on analytical results for two types, and on simulations for n types. Interestingly, in the case of at most two types the probabilities to increase or decrease the average fitness are independent of the concrete probability density function. Instead, they only depend on the probability that the payoff entries of the mutant are larger than the payoff entries of the parent type.     

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n×n payoff matrix. Players can use mixed strategies, which are given by the vector (p₁,…,p_n). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_n. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.     

SINGLE‐GENE SPECIATION WITH PLEIOTROPY: EFFECTS OF ALLELE DOMINANCE,POPULATION SIZE,AND DELAYED INHERITANCE

Single‐gene speciation is considered to be unlikely, but an excellent example is found in land snails, in which a gene for left‐right reversal has given rise to new species multiple times. This reversal might be facilitated by their small population sizes and maternal effect (i.e., “delayed inheritance,” in which an individual's phenotype is determined by the genotype of its mother). Recent evidence suggests that a pleiotropic effect of the speciation gene on antipredator survival may also promote speciation. Here we theoretically demonstrate that, without a pleiotropic effect, in small populations the fixation probability of a recessive mutant is higher than a dominant mutant, but they are identical for large populations and sufficiently weak selection. With a pleiotropic effect that increases mutant viability, a dominant mutant has a higher fixation probability if the strength of viability selection is sufficiently greater than that of reproductive incompatibility, whereas a recessive mutant has a higher fixation probability otherwise. Delayed inheritance increases the fixation probability of a mutant if viability selection is sufficiently weaker than reproductive incompatibility. Our results clarify the conflicting effects of viability selection and positive frequency‐dependent selection due to reproductive incompatibility and provide a new perspective to single‐gene speciation theory.     

Is Payoff Necessarily Weighted by Probability When Making a Risky Choice? Evidence from Functional Connectivity Analysis

How people make decisions under risk remains an as-yet-unresolved but fundamental question. Mainstream theories about risky decision making assume that the core processes involved in reaching a risky decision include weighting each payoff or reward magnitude by its probability and then summing the outcomes. However, recently developed theories question whether payoffs are necessarily weighted by probability when making a risky choice. Using functional connectivity analysis, we aimed to provide neural evidence to answer whether this key assumption of computing expectations holds when making a risky choice. We contrasted a trade-off instruction choice that required participants to integrate probability and payoff information with a preferential choice that did not. Based on the functional connectivity patterns between regions in which activity was detected during both of the decision-making tasks, we classified the regions into two networks. One network includes primarily the left and right lateral prefrontal cortices and posterior parietal cortices, which were found to be related to probability in previous reports, and the other network is composed of the bilateral basal ganglia, which have been implicated in payoff. We also found that connectivity between the payoff network and some regions in the probability network (including the left lateral prefrontal cortices and bilateral inferior parietal lobes) were stronger during the trade-off instruction choice task than during the preferential choice task. This indicates that the functional integration between the probability and payoff networks during preferential choice was not as strong as the integration during trade-off instruction choice. Our results provide neural evidence that the weighting process uniformly predicted by the mainstream theory is unnecessary during preferential choice. Thus, our functional integration findings can provide a new direction for the investigation of the principles of risky decision making.     

The evolution of payoff matrices: providing incentives to cooperate

Most of the work in evolutionary game theory starts with a model of a social situation that gives rise to a particular payoff matrix and analyses how behaviour evolves through natural selection. Here, we invert this approach and ask, given a model of how individuals behave, how the payoff matrix will evolve through natural selection. In particular, we ask whether a prisoner's dilemma game is stable against invasions by mutant genotypes that alter the payoffs. To answer this question, we develop a two-tiered framework with goal-oriented dynamics at the behavioural time scale and a diploid population genetic model at the evolutionary time scale. Our results are two-fold: first, we show that the prisoner's dilemma is subject to invasions by mutants that provide incentives for cooperation to their partners, and that the resulting game is a coordination game similar to the hawk-dove game. Second, we find that for a large class of mutants and symmetric games, a stable genetic polymorphism will exist in the locus determining the payoff matrix, resulting in a complex pattern of behavioural diversity in the population. Our results highlight the importance of considering the evolution of payoff matrices to understand the evolution of animal social systems.     

Finite populations choose an optimal language

This paper studies the evolution of a proto-language in a finite population under the frequency-dependent Moran process. A proto-language can be seen as a collection of concept-to-sign mappings. An efficient proto-language is a bijective mapping from objects of communication to used signs and vice versa. Based on the comparison of fixation probabilities, a method for deriving conditions of evolutionary stability in a finite population [Nowak et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 246-650], it is shown that efficient proto-languages are the only strategies that are protected by selection, which means that no mutant strategy can have a fixation probability that is greater than the inverse population size. In passing, the paper provides interesting results about the comparison of fixation probabilities as well as Maynard Smith's notion of evolutionary stability for finite populations [Maynard Smith, 1988. Can a mixed strategy be stable in a finite population? J. Theor. Biol. 130, 247-251] that are generally true for games with a symmetric payoff function.     

Decision-making in variable environments-- a case of group selection and inter-generational conflict?

Suppose organisms need to engage in a particular action exactly once during some fixed period of time. Further suppose they can time this action to optimise their fitness based on the expected current payoff and the probability distribution of later payoffs. For an example we consider the timing of the annual nuptial flight in eusocial insects. Using two population genetics models, we ask whether stochasticity leads to evolutionary conflict between the queen and her offspring. We find that the winning phenotype is independent of who controls the timing. The best response to any non-equilibrium population strategy is the same in both control scenarios, a result that carries over to the diploid case. Although inter-generational conflict is therefore ruled out, the models support a previous observation that at equilibrium some of the offspring have a lower expected payoff than others. By measuring fitness in terms of relative reproductive success, we show that all individuals are in fact equally well off making group-selectionist arguments unnecessary. As such, the models should improve our understanding of the difficult conceptual problem of the unit of natural selection in stochastic environments.     

Intracellular Selection, Conversion Bias, and the Expected Substitution Rate of Organelle Genes

A key step in the substitution of a new organelle mutant throughout a population is the generation of germ-line cells homoplasmic for that mutant. Given that each cell typically contains multiple copies of organelles, each of which in turn contains multiple copies of the organelle genome, processes akin to drift and selection in a population are responsible for producing homoplasmic cells. This paper examines the expected substitution rate of new mutants by obtaining the probability that a new mutant is fixed throughout a cell, allowing for arbitrary rates of genome turnover within an organelle and organelle turnover within the cell, as well as (possibly biased) gene conversion and genetic differences in genome and/or organelle replication rates. Analysis is based on a variation of Moran's model for drift in a haploid population. One interesting result is that if the rate of unbiased conversion is sufficiently strong, it creates enough intracellular drift to overcome even strong differences in the replication rates of wild-type and mutant genomes. Thus, organelles with very high conversion rates are more resistant to intracellular selection based on differences in genome replication and/or degradation rates. It is found that the amount of genetic exchange between organelles within the cell greatly influences the probability of fixation. In the absence of exchange, biased gene conversion and/or differences in genome replication rates do not influence the probability of fixation beyond the initial fixation within a single organelle. With exchange, both these processes influence the probability of fixation throughout the entire cell. Generally speaking, exchange between organelles accentuates the effects of directional intracellular forces.(ABSTRACT TRUNCATED AT 250 WORDS)     

The application of a linear algebra to the analysis of mutation rates.

Cells and bacteria growing in culture are subject to mutation, and as this mutation is the ultimate substrate for selection and evolution, the factors controlling the mutation rate are of some interest. The mutational event is not observed directly, but is inferred from the phenotype of the original mutant or of its descendants; the rate of mutation is inferred from the number of such mutant phenotypes. Such inference presumes a knowledge of the probability distribution for the size of a clone arising from a single mutation. We develop a mathematical formulation that assists in the design and analysis of experiments which investigate mutation rates and mutant clone size distribution, and we use it to analyse data for which the classical Luria-Delbrück clone-size distribution must be rejected.     

The different limits of weak selection and the evolutionary dynamics of finite populations

Evolutionary theory often resorts to weak selection, where different individuals have very similar fitness. Here, we relate two ways to introduce weak selection. The first considers evolutionary games described by payoff matrices with similar entries. This approach has recently attracted a lot of interest in the context of evolutionary game dynamics in finite populations. The second way to introduce weak selection is based on small distances in phenotype space and is a standard approach in kin-selection theory. Whereas both frameworks are interchangeable for constant fitness, frequency-dependent selection shows significant differences between them. We point out the difference between both limits of weak selection and discuss the condition under which the differences vanish. It turns out that this condition is fulfilled by the popular parametrization of the prisoner's dilemma in benefits and costs. However, for general payoff matrices differences between the two frameworks prevail.     

On the survival probabilities of mutant genes and the evolution of dominance.

Intrinsic biological resemblance between two types is measured in terms of a correlation between their fitnesses under various possible environmental conditions. A tendency toward dominance is defined as the intrinsic biological resemblance between homozygote and heterozygote. The effect of a tendency toward dominance on the à priori survival probability of a mutant gene is studied when the fitnesses of the mutated forms are given only by their distributions. Close intrinsic resemblance between homozygous and heterozygous forms of a new mutant gene is shown to substantially increase the à priori survival probability of this gene. A probabilistic effect of selection is, thus, shown to statistically favor dominant or near dominant mutant genes to start with.This probabilistic effect is suggested as complementary to the Fisherian process of selection on the heterozygote modifiers, taking place at further stages of the progress of a mutant gene.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

Influence of Opinion Dynamics on the Evolution of Games

Under certain circumstances such as lack of information or bounded rationality, human players can take decisions on which strategy to choose in a game on the basis of simple opinions. These opinions can be modified after each round by observing own or others payoff results but can be also modified after interchanging impressions with other players. In this way, the update of the strategies can become a question that goes beyond simple evolutionary rules based on fitness and become a social issue. In this work, we explore this scenario by coupling a game with an opinion dynamics model. The opinion is represented by a continuous variable that corresponds to the certainty of the agents respect to which strategy is best. The opinions transform into actions by making the selection of an strategy a stochastic event with a probability regulated by the opinion. A certain regard for the previous round payoff is included but the main update rules of the opinion are given by a model inspired in social interchanges. We find that the fixed points of the dynamics of the coupled model are different from those of the evolutionary game or the opinion models alone. Furthermore, new features emerge such as the independence of the fraction of cooperators with respect to the topology of the social interaction network or the presence of a small fraction of extremist players.     

Evolution of Cooperation in a Heterogeneous Graph: Fixation Probabilities under Weak Selection

It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.     

Establishment of new mutations in changing environments

Understanding adaptation in changing environments is an important topic in evolutionary genetics, especially in the light of climatic and environmental change. In this work, we study one of the most fundamental aspects of the genetics of adaptation in changing environments: the establishment of new beneficial mutations. We use the framework of time-dependent branching processes to derive simple approximations for the establishment probability of new mutations assuming that temporal changes in the offspring distribution are small. This approach allows us to generalize Haldane's classic result for the fixation probability in a constant environment to arbitrary patterns of temporal change in selection coefficients. Under weak selection, the only aspect of temporal variation that enters the probability of establishment is a weighted average of selection coefficients. These weights quantify how much earlier generations contribute to determining the establishment probability compared to later generations. We apply our results to several biologically interesting cases such as selection coefficients that change in consistent, periodic, and random ways and to changing population sizes. Comparison with exact results shows that the approximation is very accurate.     

Analytical Results for Individual and Group Selection of Any Intensity

The idea of evolutionary game theory is to relate the payoff of a game to reproductive success (= fitness). An underlying assumption in most models is that fitness is a linear function of the payoff. For stochastic evolutionary dynamics in finite populations, this leads to analytical results in the limit of weak selection, where the game has a small effect on overall fitness. But this linear function makes the analysis of strong selection difficult. Here, we show that analytical results can be obtained for any intensity of selection, if fitness is defined as an exponential function of payoff. This approach also works for group selection (= multi-level selection). We discuss the difference between our approach and that of inclusive fitness theory.     

Directionality of contact networks suppresses selection pressure in evolutionary dynamics

Individuals of different types, may it be genetic, cultural, or else, with different levels of fitness often compete for reproduction and survival. A fitter type generally has higher chances of disseminating their copies to other individuals. The fixation probability of a single mutant type introduced in a population of wild-type individuals quantifies how likely the mutant type spreads. How much the excess fitness of the mutant type increases its fixation probability, namely, the selection pressure, is important in assessing the impact of the introduced mutant. Previous studies mostly based on undirected and unweighted contact networks of individuals showed that the selection pressure depends on the structure of networks and the rule of reproduction. Real networks underlying ecological and social interactions are usually directed or weighted. Here we examine how the selection pressure is modulated by directionality of interactions under several update rules. Our conclusions are twofold. First, directionality discounts the selection pressure for different networks and update rules. Second, given a network, the update rules in which death events precede reproduction events significantly decrease the selection pressure than the other rules.     

Stochastic evolutionary dynamics of direct reciprocity

Evolutionary game theory is the study of frequency-dependent selection. The success of an individual depends on the frequencies of strategies that are used in the population. We propose a new model for studying evolutionary dynamics in games with a continuous strategy space. The population size is finite. All members of the population use the same strategy. A mutant strategy is chosen from some distribution over the strategy space. The fixation probability of the mutant strategy in the resident population is calculated. The new mutant takes over the population with this probability. In this case, the mutant becomes the new resident. Otherwise, the existing resident remains. Then, another mutant is generated. These dynamics lead to a stationary distribution over the entire strategy space. Our new approach generalizes classical adaptive dynamics in three ways: (i) the population size is finite; (ii) mutants can be drawn non-locally and (iii) the dynamics are stochastic. We explore reactive strategies in the repeated Prisoner''s Dilemma. We perform ‘knock-out experiments’ to study how various strategies affect the evolution of cooperation. We find that ‘tit-for-tat’ is a weak catalyst for the emergence of cooperation, while ‘always cooperate’ is a strong catalyst for the emergence of defection. Our analysis leads to a new understanding of the optimal level of forgiveness that is needed for the evolution of cooperation under direct reciprocity.     

The replicator equation on graphs

We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called 'birth-death', 'death-birth' and 'imitation'. A fourth update rule, 'pairwise comparison', is shown to be equivalent to birth-death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner's Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors-Paper game.