The distribution of fitness effects among beneficial mutations

We know little about the distribution of fitness effects among new beneficial mutations, a problem that partly reflects the rarity of these changes. Surprisingly, though, population genetic theory allows us to predict what this distribution should look like under fairly general assumptions. Using extreme value theory, I derive this distribution and show that it has two unexpected properties. First, the distribution of beneficial fitness effects at a gene is exponential. Second, the distribution of beneficial effects at a gene has the same mean regardless of the fitness of the present wild-type allele. Adaptation from new mutations is thus characterized by a kind of invariance: natural selection chooses from the same spectrum of beneficial effects at a locus independent of the fitness rank of the present wild type. I show that these findings are reasonably robust to deviations from several assumptions. I further show that one can back calculate the mean size of new beneficial mutations from the observed mean size of fixed beneficial mutations.     

The distribution of beneficial and fixed mutation fitness effects close to an optimum

The distribution of the selection coefficients of beneficial mutations is pivotal to the study of the adaptive process, both at the organismal level (theories of adaptation) and at the gene level (molecular evolution). A now famous result of extreme value theory states that this distribution is an exponential, at least when considering a well-adapted wild type. However, this prediction could be inaccurate under selection for an optimum (because fitness effect distributions have a finite right tail in this case). In this article, we derive the distribution of beneficial mutation effects under a general model of stabilizing selection, with arbitrary selective and mutational covariance between a finite set of traits. We assume a well-adapted wild type, thus taking advantage of the robustness of tail behaviors, as in extreme value theory. We show that, under these general conditions, both beneficial mutation effects and fixed effects (mutations escaping drift loss) are beta distributed. In both cases, the parameters have explicit biological meaning and are empirically measurable; their variation through time can also be predicted. We retrieve the classic exponential distribution as a subcase of the beta when there are a moderate to large number of weakly correlated traits under selection. In this case too, we provide an explicit biological interpretation of the parameters of the distribution. We show by simulations that these conclusions are fairly robust to a lower adaptation of the wild type and discuss the relevance of our findings in the context of adaptation theories and experimental evolution.     

Mutational neighbourhood and mutation supply rate constrain adaptation in Pseudomonas aeruginosa

Understanding adaptation by natural selection requires understanding the genetic factors that determine which beneficial mutations are available for selection. Here, using experimental evolution of rifampicin-resistant Pseudomonas aeruginosa, we show that different genotypes vary in their capacity for adaptation to the cost of antibiotic resistance. We then use sequence data to show that the beneficial mutations associated with fitness recovery were specific to particular genetic backgrounds, suggesting that genotypes had access to different sets of beneficial mutations. When we manipulated the supply rate of beneficial mutations, by altering effective population size during evolution, we found that it constrained adaptation in some selection lines by restricting access to rare beneficial mutations, but that the effect varied among the genotypes in our experiment. These results suggest that mutational neighbourhood varies even among genotypes that differ by a single amino acid change, and this determines their capacity for adaptation as well as the influence of population biology processes that alter mutation supply rate.     

The Properties of Adaptive Walks in Evolving Populations of Fungus

The rarity of beneficial mutations has frustrated efforts to develop a quantitative theory of adaptation. Recent models of adaptive walks, the sequential substitution of beneficial mutations by selection, make two compelling predictions: adaptive walks should be short, and fitness increases should become exponentially smaller as successive mutations fix. We estimated the number and fitness effects of beneficial mutations in each of 118 replicate lineages of Aspergillus nidulans evolving for approximately 800 generations at two population sizes using a novel maximum likelihood framework, the results of which were confirmed experimentally using sexual crosses. We find that adaptive walks do indeed tend to be short, and fitness increases become smaller as successive mutations fix. Moreover, we show that these patterns are associated with a decreasing supply of beneficial mutations as the population adapts. We also provide empirical distributions of fitness effects among mutations fixed at each step. Our results provide a first glimpse into the properties of multiple steps in an adaptive walk in asexual populations and lend empirical support to models of adaptation involving selection towards a single optimum phenotype. In practical terms, our results suggest that the bulk of adaptation is likely to be accomplished within the first few steps.     

The probability of parallel evolution

Abstract How often will natural selection drive parallel evolution at the DNA sequence level? More precisely, what is the probability that selection will cause two populations that live in identical environments to substitute the same beneficial mutation? Here I show that, under fairly general conditions, the answer is simple: if a wild‐type sequence can mutate to n different beneficial mutations, replicate populations will on average fix the same mutation with probability P= 2/(n + 1). This probability, which is derived using extreme value theory, is independent of most biological details, including the length of the gene in question and the precise distribution of fitness effects among alleles. I conclude that the probability of parallel evolution under natural selection is nearly twice as large as that under neutrality.     

Quantitative Genetic Analysis Indicates Natural Selection on Leaf Phenotypes Across Wild Tomato Species (Solanum sect. Lycopersicon; Solanaceae)

Adaptive evolution requires both raw genetic material and an accessible path of high fitness from one fitness peak to another. In this study, we used an introgression line (IL) population to map quantitative trait loci (QTL) for leaf traits thought to be associated with adaptation to precipitation in wild tomatoes (Solanum sect. Lycopersicon; Solanaceae). A QTL sign test showed that several traits likely evolved under directional natural selection. Leaf traits correlated across species do not share a common genetic basis, consistent with a scenario in which selection maintains trait covariation unconstrained by pleiotropy or linkage disequilibrium. Two large effect QTL for stomatal distribution colocalized with key genes in the stomatal development pathway, suggesting promising candidates for the molecular bases of adaptation in these species. Furthermore, macroevolutionary transitions between vastly different stomatal distributions may not be constrained when such large-effect mutations are available. Finally, genetic correlations between stomatal traits measured in this study and data on carbon isotope discrimination from the same ILs support a functional hypothesis that the distribution of stomata affects the resistance to CO₂ diffusion inside the leaf, a trait implicated in climatic adaptation in wild tomatoes. Along with evidence from previous comparative and experimental studies, this analysis indicates that leaf traits are an important component of climatic niche adaptation in wild tomatoes and demonstrates that some trait transitions between species could have involved few, large-effect genetic changes, allowing rapid responses to new environmental conditions.     

The population genetics of adaptation: the adaptation of DNA sequences

I describe several patterns characterizing the genetics of adaptation at the DNA level. Following Gillespie (1983, 1984, 1991), I consider a population presently fixed for the ith best allele at a locus and study the sequential substitution of favorable mutations that results in fixation of the fittest DNA sequence locally available. Given a wild type sequence that is less than optimal, I derive the fitness rank of the next allele typically fixed by natural selection as well as the mean and variance of the jump in fitness that results when natural selection drives a substitution. Looking over the whole series of substitutions required to reach the best allele, I show that the mean fitness jumps occurring throughout an adaptive walk are constrained to a twofold window of values, assuming only that adaptation begins from a reasonably fit allele. I also show that the first substitution and the substitution of largest effect account for a large share of the total fitness increase during adaptation. I further show that the distribution of selection coefficients fixed throughout such an adaptive walk is exponential (ignoring mutations of small effect), a finding reminiscent of that seen in Fisher's geometric model of adaptation. Last, I show that adaptation by natural selection behaves in several respects as the average of two idealized forms of adaptation, perfect and random.     

Distributions of beneficial fitness effects in RNA

Beneficial mutations are the driving force of evolution by natural selection. Yet, relatively little is known about the distribution of the fitness effects of beneficial mutations in populations. Recent work of Gillespie and Orr suggested some of the first generalizations for the distributions of beneficial fitness effects and, surprisingly, they depend only weakly on biological details. In particular, the theory suggests that beneficial mutations obey an exponential distribution of fitness effects, with the same exponential parameter across different regions of genotype space, provided only that few possible beneficial mutations are available to that genotype. Here we tested this hypothesis with a quasi-empirical model of RNA evolution in which fitness is based on the secondary structures of molecules and their thermodynamic stabilities. The fitnesses of randomly selected genotypes appeared to follow a Gumbel-type distribution and thus conform to a basic assumption of adaptation theory. However, the observed distributions of beneficial fitness effects conflict with specific predictions of the theory. In particular, the distributions of beneficial fitness effects appeared exponential only when the vast majority of small-effect beneficial mutations were ignored. Additionally, the distribution of beneficial fitness effects varied with the fitness of the parent genotype. We believe that correlation of the fitness values among similar genotypes is likely the cause of the departure from the predictions of recent adaptation theory. Although in conflict with the current theory, these results suggest that more complex statistical generalizations about beneficial mutations may be possible.     

The Evolutionarily Stable Distribution of Fitness Effects

The distribution of fitness effects (DFE) of new mutations is a key parameter in determining the course of evolution. This fact has motivated extensive efforts to measure the DFE or to predict it from first principles. However, just as the DFE determines the course of evolution, the evolutionary process itself constrains the DFE. Here, we analyze a simple model of genome evolution in a constant environment in which natural selection drives the population toward a dynamic steady state where beneficial and deleterious substitutions balance. The distribution of fitness effects at this steady state is stable under further evolution and provides a natural null expectation for the DFE in a population that has evolved in a constant environment for a long time. We calculate how the shape of the evolutionarily stable DFE depends on the underlying population genetic parameters. We show that, in the absence of epistasis, the ratio of beneficial to deleterious mutations of a given fitness effect obeys a simple relationship independent of population genetic details. Finally, we analyze how the stable DFE changes in the presence of a simple form of diminishing-returns epistasis.     

ADAPTIVE WALKS AND DISTRIBUTION OF BENEFICIAL FITNESS EFFECTS

We study the adaptation dynamics of a maladapted asexual population on rugged fitness landscapes with many local fitness peaks. The distribution of beneficial fitness effects is assumed to belong to one of the three extreme value domains, viz. Weibull, Gumbel, and Fréchet. We work in the strong selection‐weak mutation regime in which beneficial mutations fix sequentially, and the population performs an uphill walk on the fitness landscape until a local fitness peak is reached. A striking prediction of our analysis is that the fitness difference between successive steps follows a pattern of diminishing returns in the Weibull domain and accelerating returns in the Fréchet domain, as the initial fitness of the population is increased. These trends are found to be robust with respect to fitness correlations. We believe that this result can be exploited in experiments to determine the extreme value domain of the distribution of beneficial fitness effects. Our work here differs significantly from the previous ones that assume the selection coefficient to be small. On taking large effect mutations into account, we find that the length of the walk shows different qualitative trends from those derived using small selection coefficient approximation.     

Mutational bias for body size in rhabditid nematodes

Mutational bias is a potentially important agent of evolution, but it is difficult to disentangle the effects of mutation from those of natural selection. Mutation-accumulation experiments, in which mutations are allowed to accumulate at very small population size, thus minimizing the efficiency of natural selection, are the best way to separate the effects of mutation from those of selection. Body size varies greatly among species of nematode in the family rhabditidae; mutational biases are both a potential cause and a consequence of that variation. We report data on the cumulative effects of mutations that affect body size in three species of rhabditid nematode that vary fivefold in adult size. Results are very consistent with previous studies of mutations underlying fitness in the same strains: two strains of Caenorhabditis briggsae decline in body size about twice as fast as two strains of C. elegans, with a concomitant higher point estimate of the genomic mutation rate; the confamilial Oscheius myriophila is intermediate. There is an overall mutational bias, such that mutations reduce size on average, but the bias appears consistent between species. The genetic correlation between mutations that affect size and those underlying fitness is large and positive, on average.     

Beneficial fitness effects are not exponential for two viruses

The distribution of fitness effects for beneficial mutations is of paramount importance in determining the outcome of adaptation. It is generally assumed that fitness effects of beneficial mutations follow an exponential distribution, for example, in theoretical treatments of quantitative genetics, clonal interference, experimental evolution, and the adaptation of DNA sequences. This assumption has been justified by the statistical theory of extreme values, because the fitnesses conferred by beneficial mutations should represent samples from the extreme right tail of the fitness distribution. Yet in extreme value theory, there are three different limiting forms for right tails of distributions, and the exponential describes only those of distributions in the Gumbel domain of attraction. Using beneficial mutations from two viruses, we show for the first time that the Gumbel domain can be rejected in favor of a distribution with a right-truncated tail, thus providing evidence for an upper bound on fitness effects. Our data also violate the common assumption that small-effect beneficial mutations greatly outnumber those of large effect, as they are consistent with a uniform distribution of beneficial effects.     

Inferences of Demography and Selection in an African Population of Drosophila melanogaster

It remains a central problem in population genetics to infer the past action of natural selection, and these inferences pose a challenge because demographic events will also substantially affect patterns of polymorphism and divergence. Thus it is imperative to explicitly model the underlying demographic history of the population whenever making inferences about natural selection. In light of the considerable interest in adaptation in African populations of Drosophila melanogaster, which are considered ancestral to the species, we generated a large polymorphism data set representing 2.1 Mb from each of 20 individuals from a Ugandan population of D. melanogaster. In contrast to previous inferences of a simple population expansion in eastern Africa, our demographic modeling of this ancestral population reveals a strong signature of a population bottleneck followed by population expansion, which has significant implications for future demographic modeling of derived populations of this species. Taking this more complex underlying demographic history into account, we also estimate a mean X-linked region-wide rate of adaptation of 6 × 10⁻¹¹/site/generation and a mean selection coefficient of beneficial mutations of 0.0009. These inferences regarding the rate and strength of selection are largely consistent with most other estimates from D. melanogaster and indicate a relatively high rate of adaptation driven by weakly beneficial mutations.     

Fitness effects of fixed beneficial mutations in microbial populations

Beneficial mutations are intuitively relevant to understanding adaptation, yet not all beneficial mutations are of consequence to the long-term evolutionary outcome of adaptation. Many beneficial mutations-mostly those of small effect-are lost due either to (1) genetic drift or to (2) competition among clones carrying different beneficial mutations, a phenomenon called the "Hill-Robertson effect" for sexual populations and "clonal interference" for asexual populations. Competition among clones becomes more prevalent with increasing genetic linkage and increasing population size, and it is thus generally characteristic of microbial populations. Together, these two phenomena suggest that only those beneficial mutations of large fitness effect should achieve fixation, despite the fact that most beneficial mutations produced are predicted to have very small fitness effects. Here, we confirm this prediction-both empirically and theoretically-by showing that fitness effects of fixed beneficial mutations follow a distribution whose mode is positive.     

Adaptive protein evolution of X-linked and autosomal genes in Drosophila: implications for faster-X hypotheses

Patterns of sex chromosome and autosome evolution can be used to elucidate the underlying genetic basis of adaptative change. Evolutionary theory predicts that X-linked genes will adapt more rapidly than autosomes if adaptation is limited by the availability of beneficial mutations and if such mutations are recessive. In Drosophila, rates of molecular divergence between species appear to be equivalent between autosomes and the X chromosome. However, molecular divergence contrasts are difficult to interpret because they reflect a composite of adaptive and nonadaptive substitutions between species. Predictions based on faster-X theory also assume that selection is equally effective on the X and autosomes; this might not be true because the effective population sizes of X-linked and autosomal genes systematically differ. Here, population genetic and divergence data from Drosophila melanogaster, Drosophila simulans, and Drosophila yakuba are used to estimate the proportion of adaptive amino acid substitutions occurring in the D. melanogaster lineage. After gene composition and effective population size differences between chromosomes are controlled, X-linked and autosomal genes are shown to have equivalent rates of adaptive divergence with approximately 30% of amino acid substitutions driven by positive selection. The results suggest that adaptation is either unconstrained by a lack of beneficial genetic variation or that beneficial mutations are not recessive and are thus highly visible to natural selection whether on sex chromosomes or on autosomes.     

Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves

The dynamics of adaptation are difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from random genetic drift arising in a vanguard of particularly fit individuals of the population. Several approaches have been developed to analyze the crucial role of genetic drift on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, little is known about how genetic drift causes fluctuations to emerge on the population level, where it becomes palpable as variations in the adaptation speed and the fitness distribution. Yet these phenomena control the decay of genetic diversity and variability in evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we show that correlations induced by these emergent fluctuations can be computed at any arbitrary order by a suitable choice of a dynamical constraint. The resulting linear equations exhibit fluctuation-induced terms that amplify short-distance correlations and suppress long-distance ones. These terms, which are in general not small, control the decay of genetic diversity and, for wave-tip dominated (“pulled”) waves, lead to anticorrelations between the tip of the wave and the lagging bulk of the population. While it is natural to consider the process of adaptation as a branching random walk in fitness space subject to a constraint (due to finite resources), we show that other traveling wave phenomena in ecology and evolution likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach toward analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions, or epidemics.     

Empirical Complexities in the Genetic Foundations of Lethal Mutagenesis

From population genetics theory, elevating the mutation rate of a large population should progressively reduce average fitness. If the fitness decline is large enough, the population will go extinct in a process known as lethal mutagenesis. Lethal mutagenesis has been endorsed in the virology literature as a promising approach to viral treatment, and several in vitro studies have forced viral extinction with high doses of mutagenic drugs. Yet only one empirical study has tested the genetic models underlying lethal mutagenesis, and the theory failed on even a qualitative level. Here we provide a new level of analysis of lethal mutagenesis by developing and evaluating models specifically tailored to empirical systems that may be used to test the theory. We first quantify a bias in the estimation of a critical parameter and consider whether that bias underlies the previously observed lack of concordance between theory and experiment. We then consider a seemingly ideal protocol that avoids this bias—mutagenesis of virions—but find that it is hampered by other problems. Finally, results that reveal difficulties in the mere interpretation of mutations assayed from double-strand genomes are derived. Our analyses expose unanticipated complexities in testing the theory. Nevertheless, the previous failure of the theory to predict experimental outcomes appears to reside in evolutionary mechanisms neglected by the theory (e.g., beneficial mutations) rather than from a mismatch between the empirical setup and model assumptions. This interpretation raises the specter that naive attempts at lethal mutagenesis may augment adaptation rather than retard it.     

Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection

With a small effective population size, random genetic drift is more important than selection in determining the fate of new alleles. Small populations therefore accumulate deleterious mutations. Left unchecked, the effect of these fixed alleles is to reduce the reproductive capacity of a species, eventually to the point of extinction. New beneficial mutations, if fixed by selection, can restore some of this lost fitness. This paper derives the overall change in fitness due to fixation of new deleterious and beneficial alleles, as a function of the distribution of effects of new mutations and the effective population size. There is a critical effective size below which a population will on average decline in fitness, but above which beneficial mutations allow the population to persist. With reasonable estimates of the relevant parameters, this critical effective size is likely to be a few hundred. Furthermore, sexual selection can act to reduce the fixation probability of deleterious new mutations and increase the probability of fixing new beneficial mutations. Sexual selection can therefore reduce the risk of extinction of small populations.     

How does adaptation sweep through the genome? Insights from long-term selection experiments

A major goal in evolutionary biology is to understand the origins and fates of adaptive mutations. Natural selection may act to increase the frequency of de novo beneficial mutations, or those already present in the population as standing genetic variation. These beneficial mutations may ultimately reach fixation in a population, or they may stop increasing in frequency once a particular phenotypic state has been achieved. It is not yet well understood how different features of population biology, and/or different environmental circumstances affect these adaptive processes. Experimental evolution is a promising technique for studying the dynamics of beneficial alleles, as populations evolving in the laboratory experience natural selection in a replicated, controlled manner. Whole-genome sequencing, regularly obtained over the course of sustained laboratory selection, could potentially reveal insights into the mutational dynamics that most likely occur in natural populations under similar circumstances. To date, only a few evolution experiments for which whole-genome data are available exist. This review describes results from these resequenced laboratory-selected populations, in systems with and without sexual recombination. In asexual systems, adaptation from new mutations can be studied, and results to date suggest that the complete, unimpeded fixation of these mutations is not always observed. In sexual systems, adaptation from standing genetic variation can be studied, and in the admittedly few examples we have, the complete fixation of standing variants is not always observed. To date, the relative frequency of adaptation from new mutations versus standing variation has not been tested using a single experimental system, but recent studies using Caenorhabditis elegans and Saccharomyces cerevisiae suggest that this a realistic future goal.