The effect of deleterious alleles on adaptation in asexual populations

We calculate the fixation probability of a beneficial allele that arises as the result of a unique mutation in an asexual population that is subject to recurrent deleterious mutation at rate U. Our analysis is an extension of previous works, which make a biologically restrictive assumption that selection against deleterious alleles is stronger than that on the beneficial allele of interest. We show that when selection against deleterious alleles is weak, beneficial alleles that confer a selective advantage that is small relative to U have greatly reduced probabilities of fixation. We discuss the consequences of this effect for the distribution of effects of alleles fixed during adaptation. We show that a selective sweep will increase the fixation probabilities of other beneficial mutations arising during some short interval afterward. We use the calculated fixation probabilities to estimate the expected rate of fitness improvement in an asexual population when beneficial alleles arise continually at some low rate proportional to U. We estimate the rate of mutation that is optimal in the sense that it maximizes this rate of fitness improvement. Again, this analysis relaxes the assumption made previously that selection against deleterious alleles is stronger than on beneficial alleles.     

Soft sweeps: molecular population genetics of adaptation from standing genetic variation

A population can adapt to a rapid environmental change or habitat expansion in two ways. It may adapt either through new beneficial mutations that subsequently sweep through the population or by using alleles from the standing genetic variation. We use diffusion theory to calculate the probabilities for selective adaptations and find a large increase in the fixation probability for weak substitutions, if alleles originate from the standing genetic variation. We then determine the parameter regions where each scenario-standing variation vs. new mutations-is more likely. Adaptations from the standing genetic variation are favored if either the selective advantage is weak or the selection coefficient and the mutation rate are both high. Finally, we analyze the probability of "soft sweeps," where multiple copies of the selected allele contribute to a substitution, and discuss the consequences for the footprint of selection on linked neutral variation. We find that soft sweeps with weaker selective footprints are likely under both scenarios if the mutation rate and/or the selection coefficient is high.     

Fixation probability of a beneficial mutation conferring decreased generation time in changing environments

Background

One central building block of population genetics is the fixation probability. It is a probabilistic understanding of the eventual fate of new mutations. Moreover, the fixation probability of new beneficial mutations plays an important effect on the adaptation of populations to environmental challenges. Great progress has been made in the study of the beneficial mutations that increases offspring number. However, the fixation probability of beneficial mutations with a shorter generation time under various genetic and ecological conditions has not been explored.

Results

Here we extend the classical result of the fixation probability of beneficial mutations obtained by Haldane, and estimate the fixation probability of a beneficial mutation with a reduced generation time in a changing environment. Assuming that the selective advantage is very small, we concentrate all the changing factors of environment on a single quantity: effective selective advantage. Using a time-dependent branching process, we get the analytic approximation for the fixation probability of beneficial mutations that decrease the generation time. Then, we apply this approximation to four interesting biological cases.

Conclusions

In these instances, we show the comparison of the approximation with the accurate values. We find that they are consistent, demonstrating the effectiveness of our result for the fixation probability of beneficial mutations conferring a reduced replication time.

     

The Wright-Fisher model with temporally varying selection and population size

The Wright-Fisher model is considered in the case where the population size is random and the magnitude of the selective advantage of one of the alleles varies with time. The central question addressed is the possibility of ultimate genetic polymorphism. Partial results are obtained in the general case and complete results in the case where the population size and selective advantage are not density dependent. Bounds on the fixation probability are obtained when the selective advantage is constant.     

The Evolution of Recombination: Removing the Limits to Natural Selection

One of the oldest hypotheses for the advantage of recombination is that recombination allows beneficial mutations that arise in different individuals to be placed together on the same chromosome. Unless recombination occurs, one of the beneficial alleles is doomed to extinction, slowing the rate at which adaptive mutations are incorporated within a population. We model the effects of a modifier of recombination on the fixation probability of beneficial mutations when beneficial alleles are segregating at other loci. We find that modifier alleles that increase recombination do increase the fixation probability of beneficial mutants and subsequently hitchhike along as the mutants rise in frequency. The strength of selection favoring a modifier that increases recombination is proportional to λ(2)Sδr/r when linkage is tight and λ(2)S(3)δ r/N when linkage is loose, where λ is the beneficial mutation rate per genome per generation throughout a population of size N, S is the average mutant effect, r is the average recombination rate, and δr is the amount that recombination is modified. We conclude that selection for recombination will be substantial only if there is tight linkage within the genome or if many loci are subject to directional selection as during periods of rapid evolutionary change.     

Recombination Speeds Adaptation by Reducing Competition between Beneficial Mutations in Populations of Escherichia coli

Identification of the selective forces contributing to the origin and maintenance of sex is a fundamental problem in biology. The Fisher–Muller model proposes that sex is advantageous because it allows beneficial mutations that arise in different lineages to recombine, thereby reducing clonal interference and speeding adaptation. I used the F plasmid to mediate recombination in the bacterium Escherichia coli and measured its effect on adaptation at high and low mutation rates. Recombination increased the rate of adaptation ∼3-fold more in the high mutation rate treatment, where beneficial mutations had to compete for fixation. Sequencing of candidate loci revealed the presence of a beneficial mutation in six high mutation rate lines. In the absence of recombination, this mutation took longer to fix and, over the course of its substitution, conferred a reduced competitive advantage, indicating interference between competing beneficial mutations. Together, these results provide experimental support for the Fisher–Muller model and demonstrate that plasmid-mediated gene transfer can accelerate bacterial adaptation.     

Recombination speeds adaptation by reducing competition between beneficial mutations in populations of Escherichia coli

Identification of the selective forces contributing to the origin and maintenance of sex is a fundamental problem in biology. The Fisher–Muller model proposes that sex is advantageous because it allows beneficial mutations that arise in different lineages to recombine, thereby reducing clonal interference and speeding adaptation. I used the F plasmid to mediate recombination in the bacterium Escherichia coli and measured its effect on adaptation at high and low mutation rates. Recombination increased the rate of adaptation ~3-fold more in the high mutation rate treatment, where beneficial mutations had to compete for fixation. Sequencing of candidate loci revealed the presence of a beneficial mutation in six high mutation rate lines. In the absence of recombination, this mutation took longer to fix and, over the course of its substitution, conferred a reduced competitive advantage, indicating interference between competing beneficial mutations. Together, these results provide experimental support for the Fisher–Muller model and demonstrate that plasmid-mediated gene transfer can accelerate bacterial adaptation.     

Ploidy controls the success of mutators and nature of mutations during budding yeast evolution

BACKGROUND: We used the budding yeast Saccharomyces cerevisiae to ask how elevated mutation rates affect the evolution of asexual eukaryotic populations. Mismatch repair defective and nonmutator strains were competed during adaptation to four laboratory environments (rich medium, low glucose, high salt, and a nonfermentable carbon source). RESULTS: In diploids, mutators have an advantage over nonmutators in all conditions, and mutators that win competitions are on average fitter than nonmutator winners. In contrast, haploid mutators have no advantage when competed against haploid nonmutators, and haploid mutator winners are less fit than nonmutator winners. The diploid mutator winners were all superior to their ancestors both in the condition they had adapted to, and in two of the other conditions. This phenotype was due to a mutation or class of mutations that confers a large growth advantage during the respiratory phase of yeast cultures that precedes stationary phase. This generalist mutation(s) was not selected in diploid nonmutator strains or in haploid strains, which adapt primarily by fixing specialist (condition-specific) mutations. In diploid mutators, such mutations also occur, and the majority accumulates after the fixation of the generalist mutation. CONCLUSIONS: We conclude that the advantage of mutators depends on ploidy and that diploid mutators can give rise to beneficial mutations that are inaccessible to nonmutators and haploid mutators.     

Mutations of intermediate effect are responsible for adaptation in evolving Pseudomonas fluorescens populations

The fixation of a beneficial mutation represents the first step in adaptation, and the average effect of such mutations is therefore a fundamental property of evolving populations. It is nevertheless poorly characterized because the rarity of beneficial mutations makes it difficult to obtain reliable estimates of fitness. We obtained 68 genotypes each containing a single fixed beneficial mutation from experimental populations of Pseudomonas fluorescens, evolving in medium with serine as the sole carbon source and estimated the selective advantage of each by competition with the ancestor. The distribution of selection coefficients is modal and closely resembles the Weibull distribution. The average selection coefficient (2.1) and beneficial mutation rate (3.8x10(-8)) are high relative to previous studies, possibly because the ancestral population grows poorly in serine-limited medium. Our experiment suggests that the initial stages of adaptation to stressful environments will involve the substitution of mutations with large effect on fitness.     

Fixation probabilities when generation times are variable: the burst death model

Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a "burst-death" life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst-death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst-death model predicts that fixation probability depends only on burst rate.     

Fixation probability for lytic viruses: the attachment-lysis model

The fixation probability of a beneficial mutation is extremely sensitive to assumptions regarding the organism's life history. In this article we compute the fixation probability using a life-history model for lytic viruses, a key model organism in experimental studies of adaptation. The model assumes that attachment times are exponentially distributed, but that the lysis time, the time between attachment and host cell lysis, is constant. We assume that the growth of the wild-type viral population is controlled by periodic sampling (population bottlenecks) and also include the possibility that clearance may occur at a constant rate, for example, through washout in a chemostat. We then compute the fixation probability for mutations that increase the attachment rate, decrease the lysis time, increase the burst size, or reduce the probability of clearance. The fixation probability of these four types of beneficial mutations can be vastly different and depends critically on the time between population bottlenecks. We also explore mutations that affect lysis time, assuming that the burst size is constrained by the lysis time, for experimental protocols that sample either free phage or free phage and artificially lysed infected cells. In all cases we predict that the fixation probability of beneficial alleles is remarkably sensitive to the time between population bottlenecks.     

Soft sweeps II--molecular population genetics of adaptation from recurrent mutation or migration

In the classical model of molecular adaptation, a favored allele derives from a single mutational origin. This ignores that beneficial alleles can enter a population recurrently, either by mutation or migration, during the selective phase. In this case, descendants of several of these independent origins may contribute to the fixation. As a consequence, all ancestral haplotypes that are linked to any of these copies will be retained in the population, affecting the pattern of a selective sweep on linked neutral variation. In this study, we use analytical calculations based on coalescent theory and computer simulations to analyze molecular adaptation from recurrent mutation or migration. Under the assumption of complete linkage, we derive a robust analytical approximation for the number of ancestral haplotypes and their distribution in a sample from the population. We find that so-called "soft sweeps," where multiple ancestral haplotypes appear in a sample, are likely for biologically realistic values of mutation or migration rates.     

The influence of local extinctions on the probability of fixation of chromosomal rearrangements

We investigated the influence of local extinctions in a subdivided population on the probability of fixation of an initially rare allele, for different migration rates. The selective regimes considered were strict underdominance, meiotic drive, and underdominance associated with meiotic drive. We show that local extinctions can increase the probability of fixation of initially rare alleles in underdominant loci for relatively high migration rates, even when both homozygotes have the same fitness. This increase is due to drift during founder events. On the contrary, local extinctions decrease the probability of fixation of alleles favoured by meiotic drive. For a locus where both meiotic drive and underdominance act, the effect of local extinctions depends on the relative strength of the two selective regimes and the initial frequency of the rare allele. For parameter values such that the rare allele is initially selected against, local extinctions decrease the probability of fixation for low migration rates while they cause an increase for moderate migration rates. When the parameter values are such that the rare allele should always be favoured by selection, local extinctions always decrease the probability of fixation. In this latter case, we show the existence of an optimal migration rate which maximizes the probability of fixation.     

Recombination and hitchhiking of deleterious alleles

When new advantageous alleles arise and spread within a population, deleterious alleles at neighboring loci can hitchhike alongside them and spread to fixation in areas of low recombination, introducing a fixed mutation load. We use branching processes and diffusion equations to calculate the probability that a deleterious allele hitchhikes and fixes alongside an advantageous mutant. As expected, the probability of fixation of a deleterious hitchhiker rises with the selective advantage of the sweeping allele and declines with the selective disadvantage of the deleterious hitchhiker. We then use computer simulations of a genome with an infinite number of loci to investigate the increase in load after an advantageous mutant is introduced. We show that the appearance of advantageous alleles on genetic backgrounds loaded with deleterious alleles has two potential effects: it can fix deleterious alleles, and it can facilitate the persistence of recombinant lineages that happen to occur. The latter is expected to reduce the signals of selection in the surrounding region. We consider these results in light of human genetic data to infer how likely it is that such deleterious hitchhikers have occurred in our recent evolutionary past.     

Estimation of the rate and effect of new beneficial mutations in asexual populations

The rate and effect of available beneficial mutations are key parameters in determining how a population adapts to a new environment. However, these parameters are poorly known, in large part because of the difficulty of designing and interpreting experiments to examine the rare and intrinsically stochastic process of mutation occurrence. We present a new approach to estimate the rate and selective advantage of beneficial mutations that underlie the adaptation of asexual populations. We base our approach on the analysis of experiments that track the effect of newly arising beneficial mutations on the dynamics of a neutral marker in evolving bacterial populations and develop efficient estimators of mutation rate and selective advantage. Using extensive simulations, we evaluate the accuracy of our estimators and conclude that they are quite robust to the use of relatively low experimental replication. To validate the predictions of our model, we compare theoretical and experimentally determined estimates of the selective advantage of the first beneficial mutation to fix in a series of ten replicate populations. We find that our theoretical predictions are not significantly different from experimentally determined selection coefficients. Application of our method to suitably designed experiments will allow estimation of how population evolvability depends on demographic and initial fitness parameters.     

Linkage and the Limits to Natural Selection

The probability of fixation of a favorable mutation is reduced if selection at other loci causes inherited variation in fitness. A general method for calculating the fixation probability of an allele that can find itself in a variety of genetic backgrounds is applied to find the effect of substitutions, fluctuating polymorphisms, and deleterious mutations in a large population. With loose linkage, r, the effects depend on the additive genetic variance in relative fitness, var (W), and act by reducing effective population size by (N/N(e)) = 1 + var (W)/2r(2). However, tightly linked loci can have a substantial effect not predictable from N(e). Linked deleterious mutations reduce the fixation probability of weakly favored alleles by exp(-2U/R), where U is the total mutation rate and R is the map length in Morgans. Substitutions can cause a greater reduction: an allele with advantage s < s(crit) = (π(2)/6) log(e) (S/s)[var(W)/R] is very unlikely to be fixed. (S is the advantage of the substitution impeding fixation.) Fluctuating polymorphisms at many (n) linked loci can also have a substantial effect, reducing fixation probability by exp [ &2Kn var(W)/R] [K = -1/E((u - u)(2)/uv) depending on the frequencies (u,v) at the selected polymorphisms]. Hitchhiking due to all three kinds of selection may substantially impede adaptation that depends on weakly favored alleles.     

Fixation of beneficial mutations in the presence of epistatic interactions

We investigate the effect of deleterious mutations on the process of fixation of new advantageous mutants in an asexual population. In particular we wish to study the dependence of the process on the strength of the deleterious mutations. We suppose the existence of epistatic interaction between the genes. We study the model by means of branching process theory and also by numerical simulations. Our results show the occurrence of two distinct regimes of behavior for the probability of fixation of these variants. The occurrence of either regime depends on the ratio between the selective advantage of the beneficial mutation s _b and on the selective parameter for deleterious mutations s _b . In the former, which takes place for s _b /s _d ≲ 1, the probability of fixation increases with the epistasis parameter α, whereas for s _b /s _d ≫ 1 the probability of fixation is a complex function of α and the mutation rate U. Surprisingly, we find that for the multiplicative landscape (α = 1) the probability of fixation P _fix is given by where π (s _b ) is the probability of fixation for the two-allele model in the absence of mutations as calculated by Haldane (1927, Proc. Camb. Phil. Soc., 26, 220–230) and Kimura (1962, Genetics, 47, 713–719).     

The age and evolution of an antiviral resistance mutation in Drosophila melanogaster

What selective processes underlie the evolution of parasites and their hosts? Arms-race models propose that new host-resistance mutations or parasite counter-adaptations arise and sweep to fixation. Frequency-dependent models propose that selection favours pathogens adapted to the most common host genotypes, conferring an advantage to rare host genotypes. Distinguishing between these models is empirically difficult. The maintenance of disease-resistance polymorphisms has been studied in detail in plants, but less so in animals, and rarely in natural populations. We have made a detailed study of genetic variation in host resistance in a natural animal population, Drosophila melanogaster, and its natural pathogen, the sigma virus. We confirm previous findings that a single (albeit complex) mutation in the gene ref(2)P confers resistance against sigma and show that this mutation has increased in frequency under positive selection. Previous studies suggested that ref(2)P polymorphism reflects the progress of a very recent selective sweep, and that in Europe during the 1980s, this was followed by a sweep of a sigma virus strain able to infect flies carrying this mutation. We find that the ref(2)P resistance mutation is considerably older than the recent spread of this viral strain and suggest that—possibly because it is recessive—the initial spread of the resistance mutation was very slow.     

Effects of dominance on the probability of fixation of a mutant allele

We consider whether the fixation probability of an allele in a two-allele diploid system is always a monotonic function of the selective advantage of the allele. We show that while this conjecture is correct for intermediate dominance, it is not correct in general for either overdominant or underdominant alleles, and that for some parameter ranges the fixation probability can initially decrease and then increase as a function of the amount of selection. We have partial results that characterize the ranges of parameters for which this happens.      

Adaptive evolution of asexual populations under Muller's ratchet

We study the population genetics of adaptation in nonequilibrium haploid asexual populations. We find that the accumulation of deleterious mutations, due to the operation of Muller's ratchet, can considerably reduce the rate of fixation of advantageous alleles. Such reduction can be approximated reasonably well by a reduction in the effective population size. In the absence of Muller's ratchet, a beneficial mutation can only become fixed if it creates the best possible genotype; if Muller's ratchet operates, however, mutations initially arising in a nonoptimal genotype can also become fixed in the population, since the loss of the least-loaded class implies that an initially nonoptimal background can become optimal. We show that, while the rate at which adaptive mutations become fixed is reduced, the rate of fixation of deleterious mutations due to the ratchet is not changed by the presence of beneficial mutations as long as the rate of their occurrence is low and the deleterious effects of mutations (s(d)) are higher than the beneficial effects (s(a)). When s(a) > s(d), the advantage of a beneficial mutation can outweigh the deleterious effects of associated mutations. Under these conditions, a beneficial allele can drag to fixation deleterious mutations initially associated with it at a higher rate than in the absence of advantageous alleles. We propose analytical approximations for the rates of accumulation of deleterious and beneficial mutations. Furthermore, when allowing for the possible occurrence of interference between beneficial alleles, we find that the presence of deleterious mutations of either very weak or very strong effect can marginally increase the rate of accumulation of beneficial mutations over that observed in the absence of such deleterious mutations.