Mutation-selection Equilibrium Research Articles

Evolution with recombination as Gibbs sampling

This work presents a population genetic model of evolution, which includes haploid selection, mutation, recombination, and drift. The mutation-selection equilibrium can be expressed exactly in closed form for arbitrary fitness functions without resorting to diffusion approximations. Tractability is achieved by generating new offspring using n-parent rather than 2-parent recombination. While this enforces linkage equilibrium among offspring, it allows analysis of the whole population under linkage disequilibrium. We derive a general and exact relationship between fitness fluctuations and response to selection. Our assumptions allow analytical calculation of the stationary distribution of the model for a variety of non-trivial fitness functions. These results allow us to speak to genetic architecture, i.e., what stationary distributions result from different fitness functions. This paper presents methods for exactly deriving stationary states for finite and infinite populations. This method can be applied to many fitness functions, and we give exact calculations for four of these. These results allow us to investigate metastability, tradeoffs between fitness functions, and even consider error-correcting codes.

An Improved Codon Modeling Approach for Accurate Estimation of the Mutation Bias.

Phylogenetic codon models are routinely used to characterize selective regimes in coding sequences. Their parametric design, however, is still a matter of debate, in particular concerning the question of how to account for differing nucleotide frequencies and substitution rates. This problem relates to the fact that nucleotide composition in protein-coding sequences is the result of the interactions between mutation and selection. In particular, because of the structure of the genetic code, the nucleotide composition differs between the three coding positions, with the third position showing a more extreme composition. Yet, phylogenetic codon models do not correctly capture this phenomenon and instead predict that the nucleotide composition should be the same for all three positions. Alternatively, some models allow for different nucleotide rates at the three positions, an approach conflating the effects of mutation and selection on nucleotide composition. In practice, it results in inaccurate estimation of the strength of selection. Conceptually, the problem comes from the fact that phylogenetic codon models do not correctly capture the fixation bias acting against the mutational pressure at the mutation–selection equilibrium. To address this problem and to more accurately identify mutation rates and selection strength, we present an improved codon modeling approach where the fixation rate is not seen as a scalar, but as a tensor. This approach gives an accurate representation of how mutation and selection oppose each other at equilibrium and yields a reliable estimate of the mutational process, while disentangling the mean fixation probabilities prevailing in different mutational directions.

Deep Sequencing of MHC-Adapted Viral Lines Reveals Complex Recombinational Exchanges With Endogenous Retroviruses Leading to High-Frequency Variants.

Experimental evolution (serial passage) of Friend virus complex (FVC) in mice demonstrates phenotypic adaptation to specific host major histocompatibility complex (MHC) genotypes. These evolved viral lines show increased fitness and virulence in their host-genotype-of-passage, but display fitness and virulence tradeoffs when infecting unfamiliar host MHC genotypes. Here, we deep sequence these viral lines in an attempt to discover the genetic basis of FVC adaptation. The principal prediction for genotype-specific adaptation is that unique mutations would rise to high frequency in viral lines adapted to each host MHC genotype. This prediction was not supported by our sequencing data as most observed high-frequency variants were present in each of our independently evolved viral lines. However, using a multi-variate approach to measure divergence between viral populations, we show that populations of replicate evolved viral lines from the same MHC congenic mouse strain were more similar to one another than to lines derived from different MHC congenic mouse strains, suggesting that MHC genotype does predictably act on viral evolution in our model. Sequence analysis also revealed rampant recombination with endogenous murine leukemia virus sequences (EnMuLVs) that are encoded within the BALB/c mouse genome. The highest frequency variants in all six lines contained a 12 bp insertion from a recombinant EnMuLV source, suggesting such recombinants were either being favored by selection or were contained in a recombinational hotspot. Interestingly, they did not reach fixation, as if they are low fitness. The amount of background mutations linked to FVC/EnMuLV variable sites indicated that FVC/EnMuLV recombinants had not reached mutation selection equilibrium and thus, that EnMuLV sequences are likely continuously introgressing into the replicating viral population. These discoveries raise the question: is the expression of EnMuLV sequences in mouse splenocytes that permit recombination with exogenous FVC a pathogen or host adaptation?

An Evolutionary Model that Satisfies Detailed Balance

We propose a class of evolution models that involves an arbitrary exchangeable process as the breeding process and different selection schemes. In those models, a new genome is born according to the breeding process, and after that a genome is removed according to the selection scheme that involves fitness. Thus, the population size remains constant. The process evolves according to a Markov chain, and, unlike in many other existing models, the stationary distribution – so called mutation-selection equilibrium – can easily found and studied. As a special case our model contains a (sub) class of Moran models. The behaviour of the stationary distribution when the population size increases is our main object of interest. Several phase-transition theorems are proved.

Mutational Load and the Functional Fraction of the Human Genome.

The fraction of the human genome that is functional is a question of both evolutionary and practical importance. Studies of sequence divergence have suggested that the functional fraction of the human genome is likely to be no more than ∼15%. In contrast, the ENCODE project, a systematic effort to map regions of transcription, transcription factor association, chromatin structure, and histone modification, assigned function to 80% of the human genome. In this article, we examine whether and how an analysis based on mutational load might set a limit on the functional fraction. In order to do so, we characterize the distribution of fitness of a large, finite, diploid population at mutation-selection equilibrium. In particular, if mean fitness is ∼1, the fitness of the fittest individual likely to occur cannot be unreasonably high. We find that at equilibrium, the distribution of log fitness has variance nus, where u is the per-base deleterious mutation rate, n is the number of functional sites (and hence incorporates the functional fraction f), and s is the selection coefficient of deleterious mutations. In a large ( reproducing population, the fitness of the fittest individual likely to exist is . These results apply to both additive and recessive fitness schemes. Our approach is different from previous work that compared mean fitness at mutation-selection equilibrium with the fitness of an individual who has no deleterious mutations; we show that such an individual is exceedingly unlikely to exist. We find that the functional fraction is not very likely to be limited substantially by mutational load, and that any such limit, if it exists, depends strongly on the selection coefficients of new deleterious mutations.

Differences in adaptive dynamics determine the success of virus variants that propagate together.

Virus fitness is a complex parameter that results from the interaction of virus-specific characters (e.g. intracellular growth rate, adsorption rate, virion extracellular stability, and tolerance to mutations) with others that depend on the underlying fitness landscape and the internal structure of the whole population. Individual mutants usually have lower fitness values than the complex population from which they come from. When they are propagated and allowed to attain large population sizes for a sufficiently long time, they approach mutation-selection equilibrium with the concomitant fitness gains. The optimization process follows dynamics that vary among viruses, likely due to differences in any of the parameters that determine fitness values. As a consequence, when different mutants spread together, the number of generations experienced by each of them prior to co-propagation may determine its particular fate. In this work we attempt a clarification of the effect of different levels of population diversity in the outcome of competition dynamics. To this end, we analyze the behavior of two mutants of the RNA bacteriophage Qβ that co-propagate with the wild-type virus. When both competitor viruses are clonal, the mutants rapidly outcompete the wild type. However, the outcome in competitions performed with partially optimized virus populations depends on the distance of the competitors to their clonal origin. We also implement a theoretical population dynamics model that describes the evolution of a heterogeneous population of individuals, each characterized by a fitness value, subjected to subsequent cycles of replication and mutation. The experimental results are explained in the framework of our theoretical model under two non-excluding, likely complementary assumptions: (1) The relative advantage of both competitors changes as populations approach mutation-selection equilibrium, as a consequence of differences in their growth rates and (2) one of the competitors is more robust to mutations than the other. The main conclusion is that the nearness of an RNA virus population to mutation-selection equilibrium is a key factor determining the fate of particular mutants arising during replication.

Fisher–Wright model with deterministic seed bank and selection

Seed banks are common characteristics to many plant species, which allow storage of genetic diversity in the soil as dormant seeds for various periods of time. We investigate an above-ground population following a Fisher–Wright model with selection coupled with a deterministic seed bank assuming the length of the seed bank is kept constant and the number of seeds is large. To assess the combined impact of seed banks and selection on genetic diversity, we derive a general diffusion model. The applied techniques outline a path of approximating a stochastic delay differential equation by an appropriately rescaled stochastic differential equation. We compute the equilibrium solution of the site-frequency spectrum and derive the times to fixation of an allele with and without selection. Finally, it is demonstrated that seed banks enhance the effect of selection onto the site-frequency spectrum while slowing down the time until the mutation–selection equilibrium is reached.

Deterministic evolution of an asexual population under the action of beneficial and deleterious mutations on additive fitness landscapes

We study a continuous time model for the frequency distribution of an infinitely large asexual population in which both beneficial and deleterious mutations occur and fitness is additive. When beneficial mutations are ignored, the exact solution for the frequency distribution is known to be a Poisson distribution. Here we include beneficial mutations and obtain exact expressions for the frequency distribution at all times using an eigenfunction expansion method. We find that the stationary distribution is non-Poissonian and related to the Bessel function of the first kind. We also provide suitable approximations for the stationary distribution and the time to relax to the steady state. Our exact results, especially at mutation–selection equilibrium, can be useful in developing semi-deterministic approaches to understand stochastic evolution.

The evolution of sperm competition genes: The effect of mating system on levels of genetic variation within and between species.

It is widely established that proteins involved in reproduction diverge between species more quickly than other proteins. For male sperm proteins, rapid divergence is believed to be caused by postcopulatory sexual selection and/or sexual conflict. Here, we derive the expected levels of gene diversity within populations and divergence between them for male sperm protein genes evolving by postcopulatory, prezygotic fertility competition, i.e. the function imputed for some sperm and seminal fluid genes. We find that, at the mutation-selection equilibrium, both gene diversity within species and divergence between them are elevated relative to genes with similar selection coefficients expressed by both sexes. We show that their expected level of diversity is a function of the harmonic mean number of mates per female, which affects the strength of fertility selection stemming from male-male sperm competition. Our predictions provide a null hypothesis for distinguishing between other selective hypotheses accounting for the rapid evolution of male reproductive genes.

Fixation probability of rare nonmutator and evolution of mutation rates.

Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation-selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators and find that a nonmutator is more likely to fix when the deleterious mutation rate of the mutator population is high. Compensatory mutations in the mutator population are found to decrease the fixation probability of a nonmutator when the selection coefficient is large. But, surprisingly, the fixation probability changes nonmonotonically with increasing compensatory mutation rate when the selection is mild. Using these results for the fixation probability and a drift-barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments.

Tipping points and early warning signals in the genomic composition of populations induced by environmental changes

We live in an ever changing biosphere that faces continuous and often stressing environmental challenges. From this perspective, much effort is currently devoted to understanding how natural populations succeed or fail in adapting to evolving conditions. In a different context, many complex dynamical systems experience critical transitions where their dynamical behaviour or internal structure changes suddenly. Here we connect both approaches and show that in rough and correlated fitness landscapes, population dynamics shows flickering under small stochastic environmental changes, alerting of the existence of tipping points. Our analytical and numerical results demonstrate that transitions at the genomic level preceded by early-warning signals are a generic phenomenon in constant and slowly driven landscapes affected by even slight stochasticity. As these genomic shifts are approached, the time to reach mutation-selection equilibrium dramatically increases, leading to the appearance of hysteresis in the composition of the population. Eventually, environmental changes significantly faster than the typical adaptation time may result in population extinction. Our work points out several indicators that are at reach with current technologies to anticipate these sudden and largely unavoidable transitions.

Evolution of cooperation in a multidimensional phenotype space

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much theoretical work. An important model to study cooperation is the phenotypic model, where individuals are characterized by phenotypic properties that are visible to others. The phenotype of an individual can be represented for instance by a vector x=(x1,…,xn), where x1,…,xn are integers. The population can be well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies of the individuals can depend on their distance in the phenotype space. A cooperator can choose to help other individuals exhibiting the same phenotype and defects otherwise. Cooperation is said to be favored by selection if it is more abundant than defection in the stationary state. This means that the average frequency of cooperators in the stationary state strictly exceeds 1/2. Antal et al. (2009c) found conditions that ensure that cooperation is more abundant than defection in a one-dimensional (i.e. n=1) and an infinite-dimensional (i.e. n=∞) phenotype space in the case of the Prisoner’s Dilemma under weak selection. However, reality lies between these two limit cases. In this paper, we derive the corresponding condition in the case of a phenotype space of any finite dimension. This is done by applying a perturbation method to study a mutation-selection equilibrium under weak selection. This condition is obtained in the limit of a large population size by using the ancestral process. The best scenario for cooperation to be more likely to evolve is found to be a high population-scaled phenotype mutation rate, a low population-scaled strategy mutation rate and a high phenotype space dimension. The biological intuition is that a high population-scaled phenotype mutation rate reduces the quantity of interactions between cooperators and defectors, while a high population-scaled strategy mutation rate introduces newly mutated defectors that invade groups of cooperators. Finally it is easier for cooperation to evolve in a phenotype space of higher dimension because it becomes more difficult for a defector to migrate to a group of cooperators. The difference is significant from n=1 to n=2 and from n=2 to n=3, but becomes small as soon as n≥3.

Virus genome dynamics under different propagation pressures: reconstruction of whole genome haplotypes of West Nile viruses from NGS data.

BackgroundExtensive focus is placed on the comparative analyses of consensus genotypes in the study of West Nile virus (WNV) emergence. Few studies account for genetic change in the underlying WNV quasispecies population variants. These variants are not discernable in the consensus genome at the time of emergence, and the maintenance of mutation-selection equilibria of population variants is greatly underestimated. The emergence of lineage 1 WNV strains has been studied extensively, but recent epidemics caused by lineage 2 WNV strains in Hungary, Austria, Greece and Italy emphasizes the increasing importance of this lineage to public health. In this study we explored the quasispecies dynamics of minority variants that contribute to cell-tropism and host determination, i.e. the ability to infect different cell types or cells from different species from Next Generation Sequencing (NGS) data of a historic lineage 2 WNV strain.ResultsMinority variants contributing to host cell membrane association persist in the viral population without contributing to the genetic change in the consensus genome. Minority variants are shown to maintain a stable mutation-selection equilibrium under positive selection, particularly in the capsid gene region.ConclusionsThis study is the first to infer positive selection and the persistence of WNV haplotype variants that contribute to viral fitness without accompanying genetic change in the consensus genotype, documented solely from NGS sequence data. The approach used in this study streamlines the experimental design seeking viral minority variants accurately from NGS data whilst minimizing the influence of associated sequence error.

Mutation-selection equilibrium in finite populations playing a Hawk–Dove game

We study the evolution of a finite population playing a Hawk–Dove game with mixed strategies. Players have a fixed strategy and their offspring inherit the parental strategy, with a probability u of mutating to another strategy. Payoff in the game is the only variation in fitness among individuals, and a selection coefficient δ measures the importance of the game in the overall fitness. Population evolution is carried out through a Moran process. We compare our numerical simulations with theoretical predictions in earlier work by Tarnita et al. (2009). Our results show that the effect of selection on the abundances of favored strategies is nonlinear, being less intense as δ increases. The mutation rate u has an opposite and stronger effect to that of selection. Heuristic theoretical arguments are given in order to explain this nonlinear relationship.

Mutation Rate Evolution in Replicator Dynamics

The mutation rate of an organism is itself evolvable. In stable environments, if faithful replication is costless, theory predicts that mutation rates will evolve to zero. However, positive mutation rates can evolve in novel or fluctuating environments, as analytical and empirical studies have shown. Previous work on this question has focused on environments that fluctuate independently of the evolving population. Here we consider fluctuations that arise from frequency-dependent selection in the evolving population itself. We investigate how the dynamics of competing traits can induce selective pressure on the rates of mutation between these traits. To address this question, we introduce a theoretical framework combining replicator dynamics and adaptive dynamics. We suppose that changes in mutation rates are rare, compared to changes in the traits under direct selection, so that the expected evolutionary trajectories of mutation rates can be obtained from analysis of pairwise competition between strains of different rates. Depending on the nature of frequency-dependent trait dynamics, we demonstrate three possible outcomes of this competition. First, if trait frequencies are at a mutation-selection equilibrium, lower mutation rates can displace higher ones. Second, if trait dynamics converge to a heteroclinic cycle-arising, for example, from "rock-paper-scissors" interactions-mutator strains succeed against non-mutators. Third, in cases where selection alone maintains all traits at positive frequencies, zero and nonzero mutation rates can coexist indefinitely. Our second result suggests that relatively high mutation rates may be observed for traits subject to cyclical frequency-dependent dynamics.

Mixing Times in Evolutionary Game Dynamics

Without mutation and migration, evolutionary dynamics ultimately leads to the extinction of all but one species. Such fixation processes are well understood and can be characterized analytically with methods from statistical physics. However, many biological arguments focus on stationary distributions in a mutation-selection equilibrium. Here, we address the mixing time required to reach stationarity in the presence of mutation. We show that mixing times in evolutionary games have the opposite behavior from fixation times when the intensity of selection increases: in coordination games with bistabilities, the fixation time decreases, but the mixing time increases. In coexistence games with metastable states, the fixation time increases, but the mixing time decreases. Our results are based on simulations and the Wentzel-Kramers-Brillouin approximation of the master equation.

Mutation accumulation may be a minor force in shaping life history traits.

Is senescence the adaptive result of tradeoffs between younger and older ages or the nonadaptive burden of deleterious mutations that act at older ages? To shed new light on this unresolved question we combine adaptive and nonadaptive processes in a single model. Our model uses Penna's bit-strings to capture different age-specific mutational patterns. Each pattern represents a genotype and for each genotype we find the life history strategy that maximizes fitness. Genotypes compete with each other and are subject to selection and to new mutations over generations until equilibrium in gene-frequencies is reached. The mutation-selection equilibrium provides information about mutational load and the differential effects of mutations on a life history trait - the optimal age at maturity. We find that mutations accumulate only at ages with negligible impact on fitness and that mutation accumulation has very little effect on the optimal age at maturity. These results suggest that life histories are largely determined by adaptive processes. The non-adaptive process of mutation accumulation seems to be unimportant at evolutionarily relevant ages.

Genotype-Phenotype Maps Maximizing Evolvability: Modularity Revisited

The mechanisms translating genetic to phenotypic variation determine the distribution of heritable phenotypic variance available to selection. Pleiotropy is an aspect of this structure that limits independent variation of characters. Modularization of pleiotropy has been suggested to promote evolvability by restricting genetic covariance among unrelated characters and reducing constraints due to correlated response. However, modularity may also reduce total genetic variation of characters. We study the properties of genotype-phenotype maps that maximize average conditional evolvability, measured as the amount of unconstrained genetic variation in random directions of phenotypic space. In general, maximal evolvability occurs by maximizing genetic variance and minimizing genetic covariance. This does not necessarily require modularity, only patterns of pleiotropy that cancel on average. The detailed structure of the most evolvable genotype-phenotype maps depends on the distribution of molecular variance. When molecular variance is determined by mutation-selection equilibrium either highly pleiotropic or highly modular genotype-phenotype maps can be optimal, depending on the mutation rate and the relative strengths of stabilizing selection on the characters.

Strategy abundance in evolutionary many-player games with multiple strategies

Evolutionary game theory is an abstract and simple, but very powerful way to model evolutionary dynamics. Even complex biological phenomena can sometimes be abstracted to simple two-player games. But often, the interaction between several parties determines evolutionary success. Rather than pair-wise interactions, in this case we must take into account the interactions between many players, which are inherently more complicated than the usual two-player games, but can still yield simple results. In this manuscript we derive the composition of a many-player multiple strategy system in the mutation–selection equilibrium. This results in a simple expression which can be obtained by recursions using coalescence theory. This approach can be modified to suit a variety of contexts, e.g. to find the equilibrium frequencies of a finite number of alleles in a polymorphism or that of different strategies in a social dilemma in a cultural context.

Variable Mutation Rates as an Adaptive Strategy in Replicator Populations

For evolving populations of replicators, there is much evidence that the effect of mutations on fitness depends on the degree of adaptation to the selective pressures at play. In optimized populations, most mutations have deleterious effects, such that low mutation rates are favoured. In contrast to this, in populations thriving in changing environments a larger fraction of mutations have beneficial effects, providing the diversity necessary to adapt to new conditions. What is more, non-adapted populations occasionally benefit from an increase in the mutation rate. Therefore, there is no optimal universal value of the mutation rate and species attempt to adjust it to their momentary adaptive needs. In this work we have used stationary populations of RNA molecules evolving in silico to investigate the relationship between the degree of adaptation of an optimized population and the value of the mutation rate promoting maximal adaptation in a short time to a new selective pressure. Our results show that this value can significantly differ from the optimal value at mutation-selection equilibrium, being strongly influenced by the structure of the population when the adaptive process begins. In the short-term, highly optimized populations containing little variability respond better to environmental changes upon an increase of the mutation rate, whereas populations with a lower degree of optimization but higher variability benefit from reducing the mutation rate to adapt rapidly. These findings show a good agreement with the behaviour exhibited by actual organisms that replicate their genomes under broadly different mutation rates.