Testing the extreme value domain of attraction for distributions of beneficial fitness effects

In modeling evolutionary genetics, it is often assumed that mutational effects are assigned according to a continuous probability distribution, and multiple distributions have been used with varying degrees of justification. For mutations with beneficial effects, the distribution currently favored is the exponential distribution, in part because it can be justified in terms of extreme value theory, since beneficial mutations should have fitnesses in the extreme right tail of the fitness distribution. While the appeal to extreme value theory seems justified, the exponential distribution is but one of three possible limiting forms for tail distributions, with the other two loosely corresponding to distributions with right-truncated tails and those with heavy tails. We describe a likelihood-ratio framework for analyzing the fitness effects of beneficial mutations, focusing on testing the null hypothesis that the distribution is exponential. We also describe how to account for missing the smallest-effect mutations, which are often difficult to identify experimentally. This technique makes it possible to apply the test to gain-of-function mutations, where the ancestral genotype is unable to grow under the selective conditions. We also describe how to pool data across experiments, since we expect few possible beneficial mutations in any particular experiment.     

The distribution of fitness effects among beneficial mutations in Fisher's geometric model of adaptation

Recent models of adaptation at the DNA sequence level assume that the fitness effects of new mutations show certain statistical properties. In particular, these models assume that the distribution of fitness effects among new mutations is in the domain of attraction of the so-called Gumbel-type extreme value distribution. This assumption has not, however, been justified on any biological or theoretical grounds. In this note, I study random mutation in one of the simplest models of mutation and adaptation-Fisher's geometric model. I show that random mutation in this model yields a distribution of mutational effects that belongs to the Gumbel type. I also show that the distribution of fitness effects among rare beneficial mutations in Fisher's model is asymptotically exponential. I confirm these analytic findings with exact computer simulations. These results provide some support for the use of Gumbel-type extreme value theory in studies of adaptation and point to a surprising connection between recent phenotypic- and sequence-based models of adaptation: in both, the distribution of fitness effects among rare beneficial mutations is approximately exponential.     

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.     

The Distribution of Fitness Effects of Beneficial Mutations in Pseudomonas aeruginosa

Understanding how beneficial mutations affect fitness is crucial to our understanding of adaptation by natural selection. Here, using adaptation to the antibiotic rifampicin in the opportunistic pathogen Pseudomonas aeruginosa as a model system, we investigate the underlying distribution of fitness effects of beneficial mutations on which natural selection acts. Consistent with theory, the effects of beneficial mutations are exponentially distributed where the fitness of the wild type is moderate to high. However, when the fitness of the wild type is low, the data no longer follow an exponential distribution, because many beneficial mutations have large effects on fitness. There is no existing population genetic theory to explain this bias towards mutations of large effects, but it can be readily explained by the underlying biochemistry of rifampicin–RNA polymerase interactions. These results demonstrate the limitations of current population genetic theory for predicting adaptation to severe sources of stress, such as antibiotics, and they highlight the utility of integrating statistical and biophysical approaches to adaptation.     

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 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.     

Quantifying the adaptive potential of an antibiotic resistance enzyme

For a quantitative understanding of the process of adaptation, we need to understand its "raw material," that is, the frequency and fitness effects of beneficial mutations. At present, most empirical evidence suggests an exponential distribution of fitness effects of beneficial mutations, as predicted for Gumbel-domain distributions by extreme value theory. Here, we study the distribution of mutation effects on cefotaxime (Ctx) resistance and fitness of 48 unique beneficial mutations in the bacterial enzyme TEM-1 β-lactamase, which were obtained by screening the products of random mutagenesis for increased Ctx resistance. Our contributions are threefold. First, based on the frequency of unique mutations among more than 300 sequenced isolates and correcting for mutation bias, we conservatively estimate that the total number of first-step mutations that increase Ctx resistance in this enzyme is 87 [95% CI 75-189], or 3.4% of all 2,583 possible base-pair substitutions. Of the 48 mutations, 10 are synonymous and the majority of the 38 non-synonymous mutations occur in the pocket surrounding the catalytic site. Second, we estimate the effects of the mutations on Ctx resistance by determining survival at various Ctx concentrations, and we derive their fitness effects by modeling reproduction and survival as a branching process. Third, we find that the distribution of both measures follows a Fréchet-type distribution characterized by a broad tail of a few exceptionally fit mutants. Such distributions have fundamental evolutionary implications, including an increased predictability of evolution, and may provide a partial explanation for recent observations of striking parallel evolution of antibiotic resistance.     

FISHER'S MODEL AND THE GENOMICS OF ADAPTATION: RESTRICTED PLEIOTROPY,HETEROGENOUS MUTATION,AND PARALLEL EVOLUTION

Genetic theories of adaptation generally overlook the genes in which beneficial substitutions occur, and the likely variation in their mutational effects. We investigate the consequences of heterogeneous mutational effects among loci on the genetics of adaptation. We use a generalization of Fisher's geometrical model, which assumes multivariate Gaussian stabilizing selection on multiple characters. In our model, mutation has a distinct variance–covariance matrix of phenotypic effects for each locus. Consequently, the distribution of selection coefficients s varies across loci. We assume each locus can only affect a limited number of independent linear combinations of phenotypic traits (restricted pleiotropy), which differ among loci, an effect we term “orientation heterogeneity.” Restricted pleiotropy can sharply reduce the overall proportion of beneficial mutations. Orientation heterogeneity has little impact on the shape of the genomic distribution, but can substantially increase the probability of parallel evolution (the repeated fixation of beneficial mutations at the same gene in independent populations), which is highest with low pleiotropy. We also consider variation in the degree of pleiotropy and in the mean s across loci. The latter impacts the genomic distribution of s, but has a much milder effect on parallel evolution. We discuss these results in the light of evolution experiments.     

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.     

The influence of deleterious mutations on adaptation in asexual populations

We study the dynamics of adaptation in asexual populations that undergo both beneficial and deleterious mutations. In particular, how the deleterious mutations affect the fixation of beneficial mutations was investigated. Using extensive Monte Carlo simulations, we find that in the "strong-selection weak mutation (SSWM)" regime or in the "clonal interference (CI)" regime, deleterious mutations rarely influence the distribution of "selection coefficients of the fixed mutations (SCFM)"; while in the "multiple mutations" regime, the accumulation of deleterious mutations would lead to a decrease in fitness significantly. We conclude that the effects of deleterious mutations on adaptation depend largely on the supply of beneficial mutations. And interestingly, the lowest adaptation rate occurs for a moderate value of selection coefficient of deleterious mutations.     

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.     

Complexity, pleiotropy, and the fitness effect of mutations

One of the assumptions underlying many theoretical predictions in evolutionary biology concerns the distribution of the fitness effect of mutations. Approximations to this distribution have been derived using various theoretical approaches, of which Fisher's geometrical model is among the most popular ones. Two key concepts in this model are complexity and pleiotropy. Recent studies have proposed different methods for estimating how complexity varies across species, but their results have been contradictory. Here, we show that contradictory results are to be expected when the assumption of universal pleiotropy is violated. We develop a model in which the two key parameters are the total number of traits and the mean number of traits affected by a single mutation. We derive approximations for the distribution of the fitness effect of mutations when populations are either well-adapted or away from the optimum. We also consider drift load in a well-adapted population and show that it is independent of the distribution of the fitness effect of mutations. We show that mutation accumulation experiments can only measure the effect of the mean number of traits affected by mutations, whereas drift load only provides information about the total number of traits. We discuss the plausibility of the model.     

The evolution of mutation rates: separating causes from consequences

Natural selection can adjust the rate of mutation in a population by acting on allelic variation affecting processes of DNA replication and repair. Because mutation is the ultimate source of the genetic variation required for adaptation, it can be appealing to suppose that the genomic mutation rate is adjusted to a level that best promotes adaptation. Most mutations with phenotypic effects are harmful, however, and thus there is relentless selection within populations for lower genomic mutation rates. Selection on beneficial mutations can counter this effect by favoring alleles that raise the mutation rate, but the effect of beneficial mutations on the genomic mutation rate is extremely sensitive to recombination and is unlikely to be important in sexual populations. In contrast, high genomic mutation rates can evolve in asexual populations under the influence of beneficial mutations, but this phenomenon is probably of limited adaptive significance and represents, at best, a temporary reprieve from the continual selection pressure to reduce mutation. The physiological cost of reducing mutation below the low level observed in most populations may be the most important factor in setting the genomic mutation rate in sexual and asexual systems, regardless of the benefits of mutation in producing new adaptive variation. Maintenance of mutation rates higher than the minimum set by this "cost of fidelity" is likely only under special circumstances.     

Fitness landscapes: an alternative theory for the dominance of mutation

Deleterious mutations tend to be recessive. Several theories, notably those of Fisher (based on selection) and Wright (based on metabolism), have been put forward to explain this pattern. Despite a long-lasting debate, the matter remains unresolved. This debate has focused on the average dominance of mutations. However, we also know very little about the distribution of dominance coefficients among mutations, and about its variation across environments. In this article we present a new approach to predicting this distribution. Our approach is based on a phenotypic fitness landscape model. First, we show that under a very broad range of conditions (and environments), the average dominance of mutation of small effects should be approximately one-quarter as long as adaptation of organisms to their environment can be well described by stabilizing selection on an arbitrary set of phenotypic traits. Second, the theory allows predicting the whole distribution of dominance coefficients among mutants. Because it provides quantitative rather than qualitative predictions, this theory can be directly compared to data. We found that its prediction on mean dominance (average dominance close to 0.25) agreed well with the data, based on a meta-analysis of dominance data for mildly deleterious mutations. However, a simple landscape model does not account for the dominance of mutations of large effects and we provide possible extension of the theory for this class of mutations. Because dominance is a central parameter for evolutionary theory, and because these predictions are quantitative, they set the stage for a wide range of applications and further empirical tests.     

Mutability of demographic noise in microbial range expansions

Demographic noise, the change in the composition of a population due to random birth and death events, is an important driving force in evolution because it reduces the efficacy of natural selection. Demographic noise is typically thought to be set by the population size and the environment, but recent experiments with microbial range expansions have revealed substantial strain-level differences in demographic noise under the same growth conditions. Many genetic and phenotypic differences exist between strains; to what extent do single mutations change the strength of demographic noise? To investigate this question, we developed a high-throughput method for measuring demographic noise in colonies without the need for genetic manipulation. By applying this method to 191 randomly-selected single gene deletion strains from the E. coli Keio collection, we find that a typical single gene deletion mutation decreases demographic noise by 8% (maximal decrease: 81%). We find that the strength of demographic noise is an emergent trait at the population level that can be predicted by colony-level traits but not cell-level traits. The observed differences in demographic noise from single gene deletions can increase the establishment probability of beneficial mutations by almost an order of magnitude (compared to in the wild type). Our results show that single mutations can substantially alter adaptation through their effects on demographic noise and suggest that demographic noise can be an evolvable trait of a population.Subject terms: Bacterial evolution, Population genetics, Population dynamics, Population genetics, Biofilms     

Exploiting the Adaptation Dynamics to Predict the Distribution of Beneficial Fitness Effects

Adaptation of asexual populations is driven by beneficial mutations and therefore the dynamics of this process, besides other factors, depends on the distribution of beneficial fitness effects. It is known that on uncorrelated fitness landscapes, this distribution can only be of three types: truncated, exponential and power law. We performed extensive stochastic simulations to study the adaptation dynamics on rugged fitness landscapes, and identified two quantities that can be used to distinguish the underlying distribution of beneficial fitness effects. The first quantity studied here is the fitness difference between successive mutations that spread in the population, which is found to decrease in the case of truncated distributions, remains nearly a constant for exponentially decaying distributions and increases when the fitness distribution decays as a power law. The second quantity of interest, namely, the rate of change of fitness with time also shows quantitatively different behaviour for different beneficial fitness distributions. The patterns displayed by the two aforementioned quantities are found to hold good for both low and high mutation rates. We discuss how these patterns can be exploited to determine the distribution of beneficial fitness effects in microbial experiments.     

An evolutionary quantitative genetics model for phenotypic (co)variances under limited dispersal,with an application to socially synergistic traits

Darwinian evolution consists of the gradual transformation of heritable traits due to natural selection and the input of random variation by mutation. Here, we use a quantitative genetics approach to investigate the coevolution of multiple quantitative traits under selection, mutation, and limited dispersal. We track the dynamics of trait means and of variance–covariances between traits that experience frequency‐dependent selection. Assuming a multivariate‐normal trait distribution, we recover classical dynamics of quantitative genetics, as well as stability and evolutionary branching conditions of invasion analyses, except that due to limited dispersal, selection depends on indirect fitness effects and relatedness. In particular, correlational selection that associates different traits within‐individuals depends on the fitness effects of such associations between‐individuals. We find that these kin selection effects can be as relevant as pleiotropy for the evolution of correlation between traits. We illustrate this with an example of the coevolution of two social traits whose association within‐individuals is costly but synergistically beneficial between‐individuals. As dispersal becomes limited and relatedness increases, associations between‐traits between‐individuals become increasingly targeted by correlational selection. Consequently, the trait distribution goes from being bimodal with a negative correlation under panmixia to unimodal with a positive correlation under limited dispersal.     

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.     

The relationship between the pleiotropic phenotypic effects of a mutation fixed by selection

A pleiotropic model of mutation is presented that allows for correlations between the effects of a new mutation and for the distribution of mutational effects to vary from being leptokurtic to normally distributed. Using this model I quantify how selection transforms the correlation between the effects of a new (random) mutation into the correlation between the effects of a mutation that is fixed by selection and contributes to an adaptation. Results suggest that under most conditions the correlation between the effects of a fixed mutation is less than the correlation between the effects of a new mutation. I also generalize previous results that quantified the expected size of a fixed mutation's effect on a character given an observed effect of that mutation on another character. In agreement with previous results, work here suggests that as the observed effect becomes large and beneficial the expected effect on another character approaches the expected effect of a new (random) mutation given the observed effect. Lastly, these theoretical results are related to recent empirical work that found beneficial mutations had a positive correlation in their pleiotropic effects.     

A general extreme value theory model for the adaptation of DNA sequences under strong selection and weak mutation

Recent theoretical studies of the adaptation of DNA sequences assume that the distribution of fitness effects among new beneficial mutations is exponential. This has been justified by using extreme value theory and, in particular, by assuming that the distribution of fitnesses belongs to the Gumbel domain of attraction. However, extreme value theory shows that two other domains of attraction are also possible: the Fréchet and Weibull domains. Distributions in the Fréchet domain have right tails that are heavier than exponential, while distributions in the Weibull domain have right tails that are truncated. To explore the consequences of relaxing the Gumbel assumption, we generalize previous adaptation theory to allow all three domains. We find that many of the previously derived Gumbel-based predictions about the first step of adaptation are fairly robust for some moderate forms of right tails in the Weibull and Fréchet domains, but significant departures are possible, especially for predictions concerning multiple steps in adaptation.