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.     

Soft selection reduces loss of heterozygosity in asexual reproduction

The adaptive value of sexual reproduction is still debated in evolutionary theory. It has been proposed that the advantage of sexual reproduction over asexual reproduction is to promote genetic diversity, to prevent the accumulation of harmful mutations or to preserve heterozygosity. Since these hypothetical advantages depend on the type of asexual reproduction, understanding how selection affects the taxonomic distribution of each type could help us discriminate between existing hypotheses. Here, I argue that soft selection, competition among embryos or offspring in selection arenas prior to the hard selection of the adult phase, reduces loss of heterozygosity in certain types of asexual reproduction. Since loss of heterozygosity leads to the unmasking of recessive deleterious mutations in the progeny of asexual individuals, soft selection facilitates the evolution of these types of asexual reproduction. Using a population genetics model, I calculate how loss of heterozygosity affects fitness for different types of apomixis and automixis, and I show that soft selection significantly reduces loss of heterozygosity, hence increases fitness, in apomixis with suppression of the first meiotic division and in automixis with central fusion, the most common types of asexual reproduction. Therefore, if sexual reproduction evolved to preserve heterozygosity, soft selection should be associated with these types of asexual reproduction. I discuss the evidence for this prediction and how this and other observations on the distribution of different types of asexual reproduction in nature is consistent with the heterozygosity hypothesis.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

Stochastic sampling of interaction partners versus deterministic payoff assignment

Evolutionary game dynamics describes how successful strategies spread in a population. In well-mixed populations, the usual assumption, e.g. underlying the replicator dynamics, is that individuals obtain a payoff from interactions with a representative sample of the population. This determines their fitness. Here, we analyze a situation in which payoffs are obtained through a single interaction, so that individuals of the same type can have different payoffs. We show analytically that for weak selection, this scenario is identical to the usual approach in which an individual interacts with the whole population. For strong selection, however, differences arise that are reflected in the fixation probabilities and lead to deviating evolutionary dynamics.     

The Evolution and Consequences of Sex-Specific Reproductive Variance

Natural selection favors alleles that increase the number of offspring produced by their carriers. But in a world that is inherently uncertain within generations, selection also favors alleles that reduce the variance in the number of offspring produced. If previous studies have established this principle, they have largely ignored fundamental aspects of sexual reproduction and therefore how selection on sex-specific reproductive variance operates. To study the evolution and consequences of sex-specific reproductive variance, we present a population-genetic model of phenotypic evolution in a dioecious population that incorporates previously neglected components of reproductive variance. First, we derive the probability of fixation for mutations that affect male and/or female reproductive phenotypes under sex-specific selection. We find that even in the simplest scenarios, the direction of selection is altered when reproductive variance is taken into account. In particular, previously unaccounted for covariances between the reproductive outputs of different individuals are expected to play a significant role in determining the direction of selection. Then, the probability of fixation is used to develop a stochastic model of joint male and female phenotypic evolution. We find that sex-specific reproductive variance can be responsible for changes in the course of long-term evolution. Finally, the model is applied to an example of parental-care evolution. Overall, our model allows for the evolutionary analysis of social traits in finite and dioecious populations, where interactions can occur within and between sexes under a realistic scenario of reproduction.     

Cell lineage dynamics in stratified shoot apical meristems

We devise a stochastic and spatially explicit model for the dynamics of the initials cells in a stratified shoot apical meristem (SAM). The meristem is composed of three layers with seven initials per layer. We investigate the probability and number of divisions for a mutant lineage to either reach fixation or becoming purged through selection or drift. In contrast to previous studies our results show that the functional organization of the initials in stratified SAMs acts as an efficient purging mechanism particularly of deleterious mutations. All mutants are rapidly purged when deleterious. The probability of fixation for mutants with a higher fitness than the wild type increases linearly up to 70%. The median number of divisions to fixation of both genotypes is insensitive to the mutant's fitness. The median number of divisions to wild-type fixation is less than 100, with the upper quartile below 200. The largest number of divisions to wild-type fixation are in the order of 100 000 divisions. Our results indicate that the spatial organization of SAM enables the efficient purging of mutant lineages, particularly if they are deleterious. On the other hand, long-lived chimeric stages are common when mutant lineages succeed to overcome the initial numerical disadvantage.     

Most Undirected Random Graphs Are Amplifiers of Selection for Birth-Death Dynamics,but Suppressors of Selection for Death-Birth Dynamics

We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-death updating, where an individual is selected to reproduce with probability proportional to its fitness and one of its neighbors is replaced by that offspring at random. If we instead focus on death-Birth updating, in which a random individual is removed and its neighbors compete for the empty spot, then the same ensemble of graphs consists of almost only suppressors of selection for which the fixation probability is decreased for advantageous mutants and increased for disadvantageous mutants. Thus, the impact of population structure on evolutionary dynamics is a subtle issue that will depend on seemingly minor details of the underlying evolutionary process.     

Probability of fixation of an advantageous mutant in a viral quasispecies

The probability that an advantageous mutant rises to fixation in a viral quasispecies is investigated in the framework of multitype branching processes. Whether fixation is possible depends on the overall growth rate of the quasispecies that will form if invasion is successful rather than on the individual fitness of the invading mutant. The exact fixation probability can be calculated only if the fitnesses of all potential members of the invading quasispecies are known. Quasispecies fixation has two important characteristics: First, a sequence with negative selection coefficient has a positive fixation probability as long as it has the potential to grow into a quasispecies with an overall growth rate that exceeds that of the established quasispecies. Second, the fixation probabilities of sequences with identical fitnesses can nevertheless vary over many orders of magnitudes. Two approximations for the probability of fixation are introduced. Both approximations require only partial knowledge about the potential members of the invading quasispecies. The performance of these two approximations is compared to the exact fixation probability on a network of RNA sequences with identical secondary structure.     

Sexual selection and its effect on the fixation of an asexual clone

Sexual selection is a powerful and ubiquitous force in sexual populations. It has recently been argued that sexual selection can eliminate the twofold cost of sex even with low genomic mutation rates. By means of differential male mating success, deleterious mutations in males become more deleterious than in females, and it has been shown that sexual selection can drastically reduce the mutational load in a sexual population, with or without any form of epistasis. However, any mechanism that claims to maintain sexual reproduction must be able to prevent the fixation of an asexual mutant clone with a twofold fitness advantage. Here, I show that despite very strong sexual selection, the fixation of an asexual mutant cannot be prevented under reasonable genomic mutation rates. Sexual selection can have a strong effect on the average mutational load in a sexual population, but as it cannot prevent the fixation of an asexual mutant, it is unlikely to play a key role on the maintenance of sexual reproduction.     

Systems approach to studying animal sociality: individual position versus group organization in dynamic social network models

Social networks can be used to represent group structure as a network of interacting components, and also to quantify both the position of each individual and the global properties of a group. In a series of simulation experiments based on dynamic social networks, we test the prediction that social behaviors that help individuals reach prominence within their social group may conflict with their potential to benefit from their social environment. In addition to cases where individuals were able to benefit from improving both their personal relative importance and group organization, using only simple rules of social affiliation we were able to obtain results in which individuals would face a trade-off between these factors. While selection would favor (or work against) social behaviors that concordantly increase (or decrease, respectively) fitness at both individual and group level, when these factors conflict with each other the eventual selective pressure would depend on the relative returns individuals get from their social environment and their position within it. The presented results highlight the importance of a systems approach to studying animal sociality, in which the effects of social behaviors should be viewed not only through the benefits that those provide to individuals, but also in terms of how they affect broader social environment and how in turn this is reflected back on an individual's fitness.     

Measurement of concurrent selection episodes

Current methods for measuring selection with longitudinal data have been developed with the assumption that episodes of selection are sequential. However, a number of empirical examinations have demonstrated that natural and sexual selection may act concurrently and in opposing directions. Other recent work has highlighted the difficulty of assigning fitness values for survival when reproduction and mortality within a population temporally overlap. I treat these as facets of a single problem; how to analyze selection where mortality and reproduction are concurrent. To address this problem, I formalize a method to estimate total fitness of individuals over a period of time utilizing longitudinal data. I then show how the fitness may be partitioned to provide two separate estimates of fitness for reproductive opportunity and reproductive success. In addition, another total fitness estimate for the period can be obtained from the two partitioned estimates. This procedure will allow calculation of total fitness where there are some missing datapoints for reproductive success of an individual. A simulation indicates that bias is generally low for the various fitness estimates. These methods should expand our ability to understand the interaction of different selection episodes.     

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.     

The impact of male-killing bacteria on host evolutionary processes

Male-killing bacteria are maternally inherited endosymbionts that selectively kill male offspring of their arthropod hosts. Using both analytical techniques and computer simulations, we studied the impact of these bacteria on the population genetics of their hosts. In particular, we derived and corroborated formulas for the fixation probability of mutant alleles, mean times to fixation and fixation or extinction, and heterozygosity for varying male-killer prevalence. Our results demonstrate that infections with male-killing bacteria impede the spread of beneficial alleles, facilitate the spread of deleterious alleles, and reduce genetic variation. The reason for this lies in the strongly reduced fitness of infected females combined with no or very limited gene flow from infected females to uninfected individuals. These two properties of male-killer-infected populations reduce the population size relevant for the initial emergence and spread of mutations. In contrast, use of Wright's equation relating sex ratio to effective population size produces misleading predictions. We discuss the relationship to the similar effect of background selection, the impact of other sex-ratio-distorting endosymbionts, and how our results affect the interpretation of empirical data on genetic variation in male-killer-infected populations.     

Selection for altruism through random drift in variable size populations

ABSTRACT: BACKGROUND: Altruistic behavior is defined as helping others at a cost to oneself and a lowered fitness. The lower fitness implies that altruists should be selected against, which is in contradiction with their widespread presence is nature. Present models of selection for altruism (kin or multilevel) show that altruistic behaviors can have 'hidden' advantages if the 'common good' produced by altruists is restricted to some related or unrelated groups. These models are mostly deterministic, or assume a frequency dependent fitness. RESULTS: Evolutionary dynamics is a competition between deterministic selection pressure and stochastic events due to random sampling from one generation to the next. We show here that an altruistic allele extending the carrying capacity of the habitat can win by increasing the random drift of "selfish" alleles. In other terms, the fixation probability of altruistic genes can be higher than those of a selfish ones, even though altruists have a smaller fitness. Moreover when populations are geographically structured, the altruists advantage can be highly amplified and the fixation probability of selfish genes can tend toward zero. The above results are obtained both by numerical and analytical calculations. Analytical results are obtained in the limit of large populations. CONCLUSIONS: The theory we present does not involve kin or multilevel selection, but is based on the existence of random drift in variable size populations. The model is a generalization of the original Fisher-Wright and Moran models where the carrying capacity depends on the number of altruists.     

The fixation of locally beneficial alleles in a metapopulation

Extinction, recolonization, and local adaptation are common in natural spatially structured populations. Understanding their effect upon genetic variation is important for systems such as genetically modified organism management or avoidance of drug resistance. Theoretical studies on the effect of extinction and recolonization upon genetic variance started appearing in the 1970s, but the role of local adaptation still has no good theoretical basis. Here we develop a model of a haploid species in a metapopulation in which a locally adapted beneficial allele is introduced. We study the effect of different spatial patterns of local adaptation, and different metapopulation dynamics, upon the fixation probability of the beneficial allele. Controlling for the average selection pressure, we find that a small area of positive selection can significantly increase the global probability of fixation. However, local adaptation becomes less important as extinction rate increases. Deme extinction and recolonization have a spatial smoothing effect that effectively reduces spatial variation in fitness.     

Age-specific fitness components and their temporal variation in the barn owl

Theory predicts that temporal variability plays an important role in the evolution of life histories, but empirical studies evaluating this prediction are rare. In constant environments, fitness can be measured by the population growth rate lambda, and the sensitivity of lambda to changes in fitness components estimates selection on these traits. In variable environments, fitness is measured by the stochastic growth rate lambda(S), and stochastic sensitivities estimate selection pressure. Here we examine age-specific schedules for reproduction and survival in a barn owl population (Tyto alba). We estimated how temporal variability affected fitness and selection, accounting for sampling variance. Despite large sample sizes of old individuals, we found no strong evidence for senescence. The most variable fitness components were associated with reproduction. Survival was less variable. Stochastic simulations showed that the observed variation decreased fitness by about 30%, but the sensitivities of lambda and lambda(S) to changes in all fitness components were almost equal, suggesting that temporal variation had negligible effects on selection. We obtained these results despite high observed variability in the fitness components and relatively short generation time of the study organism, a situation in which temporal variability should be particularly important for natural selection and early senescence is expected.     

Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods

In recent years, three related methods have been used to model the phenotypic dynamics of traits under the influence of natural selection. The first is based on an approximation to quantitative genetic recursion equations for sexual populations. The second is based on evolution in asexual lineages with mutation-generated variation. The third method finds an evolutionarily stable set of phenotypes for species characterized by a given set of fitness functions, assuming that the mode of reproduction places no constraints on the number of distinct types that can be maintained in the population. The three methods share the property that the rate of change of a trait within a homogeneous population is approximately proportional to the individual fitness gradient. The methods differ in assumptions about the potential magnitude of phenotypic differences in mutant forms, and in their assumptions about the probability that invasion or speciation occurs when a species has a stable, yet invadable phenotype. Determining the range of applicability of the different methods is important for assessing the validity of optimization methods in predicting the evolutionary outcome of ecological interactions. Methods based on quantitative genetic models predict that fitness minimizing traits will often be evolutionarily stable over significant time periods, while other approaches suggest this is likely to be rare. A more detailed study of cases of disruptive selection might reveal whether fitness-minimizing traits occur frequently in natural communities.     

Offspring social network structure predicts fitness in families

Social structures such as families emerge as outcomes of behavioural interactions among individuals, and can evolve over time if families with particular types of social structures tend to leave more individuals in subsequent generations. The social behaviour of interacting individuals is typically analysed as a series of multiple dyadic (pair-wise) interactions, rather than a network of interactions among multiple individuals. However, in species where parents feed dependant young, interactions within families nearly always involve more than two individuals simultaneously. Such social networks of interactions at least partly reflect conflicts of interest over the provision of costly parental investment. Consequently, variation in family network structure reflects variation in how conflicts of interest are resolved among family members. Despite its importance in understanding the evolution of emergent properties of social organization such as family life and cooperation, nothing is currently known about how selection acts on the structure of social networks. Here, we show that the social network structure of broods of begging nestling great tits Parus major predicts fitness in families. Although selection at the level of the individual favours large nestlings, selection at the level of the kin-group primarily favours families that resolve conflicts most effectively.     

Stochastic evolutionary dynamics on two levels

We consider a population that is subdivided into groups. Individuals reproduce proportional to their fitness. When a group reaches a certain size it has a probability to split into two groups while another group is eliminated. In this stochastic process, the number of groups is constant, while the total population size fluctuates between well-defined bounds. We calculate the fixation probability of newly introduced mutants under constant selection. We show that the described population structure acts as a suppressor of selection compared to an unstructured population of the same size. The maximum suppression of selection is obtained, when the number of groups equals the number of individuals per group. We also study opposing selection on two or more levels by analysing the evolutionary dynamics of hierarchically embedded Moran processes.     

Interaction rates,vital rates,background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk–Dove example.