The genetics of adaptation: the roles of pleiotropy, stabilizing selection and drift in shaping the distribution of bidirectional fixed mutational effects

Pleiotropy allows for the deterministic fixation of bidirectional mutations: mutations with effects both in the direction of selection and opposite to selection for the same character. Mutations with deleterious effects on some characters can fix because of beneficial effects on other characters. This study analytically quantifies the expected frequency of mutations that fix with negative and positive effects on a character and the average size of a fixed effect on a character when a mutation pleiotropically affects from very few to many characters. The analysis allows for mutational distributions that vary in shape and provides a framework that would allow for varying the frequency at which mutations arise with deleterious and positive effects on characters. The results show that a large fraction of fixed mutations will have deleterious pleiotropic effects even when mutation affects as little as two characters and only directional selection is occurring, and, not surprisingly, as the degree of pleiotropy increases the frequency of fixed deleterious effects increases. As a point of comparison, we show how stabilizing selection and random genetic drift affect the bidirectional distribution of fixed mutational effects. The results are then applied to QTL studies that seek to find loci that contribute to phenotypic differences between populations or species. It is shown that QTL studies are biased against detecting chromosome regions that have deleterious pleiotropic effects on characters.     

Lack of Evidence for Sign Epistasis Between Beneficial Mutations in an RNA Bacteriophage

In previous work (Betancourt, Genetics 181:1535, 2009), I propagated three large laboratory populations of an RNA phage (MS2) as they adapted to a controlled laboratory environment. These populations were large enough so that evolution might be expected to be mostly repeatable, but they nevertheless fixed different suites of mutations over the course of the experiment. Here, I investigate one possible explanation for these results: epistasis, in which the effect of a mutation depends on its genetic background, may have prevented populations with different initial substitutions from fixing the same set of subsequent mutations. I show that two mutations that previously occurred in different genetic backgrounds are beneficial on either background. This result suggests that sign epistasis-in which a mutation is beneficial on one background, but deleterious on another-is not the cause of different evolutionary trajectories observed in the Betancourt (2009) experiment. However, they can be explained by either magnitude epistasis-in which mutations have stronger or weaker beneficial effects depending on the background-or by the simultaneous fixation of multiple beneficial mutations. Surprisingly, the large populations of the previous experiment showed less parallel evolution than the small populations of this experiment, which lends support to the fixation of multiple beneficial mutations contributing to the patterns seen in both experiments.     

Effects of Pleiotropy on Predictions concerning Mutation-Selection Balance for Polygenic Traits

Previous mathematical analyses of mutation-selection balance for metric traits assume that selection acts on the relevant loci only through the character(s) under study. Thus, they implicitly assume that all of the relevant mutation and selection parameters are estimable. A more realistic analysis must recognize that many of the pleiotropic effects of loci contributing variation to a given character are not known. To explore the consequences of these hidden effects, I analyze models of two pleiotropically connected polygenic traits, denoted P1 and P2. The actual equilibrium genetic variance for P1, based on complete knowledge of all mutation and selection parameters for both P1 and P2, can be compared to a prediction based solely on observations of P1. This extrapolation mimics empirically obtainable predictions because of the inevitability of unknown pleiotropic effects. The mutation parameters relevant to P1 are assumed to be known, but selection intensity is estimated from the within-generation reduction of phenotypic variance for P1. The extrapolated prediction is obtained by substituting these parameters into formulas based on single-character analyses. Approximate analytical and numerical results show that the level of agreement between these univariate extrapolations and the actual equilibrium variance depends critically on both the genetic model assumed and the relative magnitudes of the mutation and selection parameters. Unless per locus mutation rates are extremely high, i.e., generally greater than 10(-4), the widely used gaussian approximation for genetic effects at individual loci is not applicable. Nevertheless, the gaussian approximations predict that the true and extrapolated equilibria are in reasonable agreement, i.e., within a factor of two, over a wide range of parameter values. In contrast, an alternative approximation that applies for moderate and low per locus mutation rates predicts that the extrapolation will generally overestimate the true equilibrium variance unless there is little selection associated with hidden effects. The tendency to overestimate is understandable because selection acts on all of the pleiotropic manifestations of a new mutation, but equilibrium covariances among the characters affected may not reveal all of this selection. This casts doubt on the proposal that much of the additive polygenic variance observed in natural populations can be explained by mutation-selection balance. It also indicates the difficulty of critically evaluating this hypothesis.     

Pleiotropic mutation, modularity and evolvability

The relationship between pleiotropy and the rate of evolution of a phenotypic character (evolvability) in a population is explored using computer simulations. I present results that suggest the rate of evolution of a phenotypic character may not decline when that character is pleiotropically associated to an increasing number of other characters, provided that the characters are under pure directional selection such that they are far from their optima relative to the average magnitude of a mutation. These conditions may be relevant during adaptive radiations. Adding pleiotropic associations to a set of characters in which one is under directional selection and the other is under stabilizing selection increases the rate of adaptation of the character under directional selection provided that the new characters that come to be pleiotropically associated are under directional selection. Thus, increasing the number of pleiotropic associations under these conditions increases the rate of adaptation of a character.     

Two steps forward, one step back: the pleiotropic effects of favoured alleles

Pleiotropy is one of the most commonly observed attributes of genes. Yet the extent and influence of pleiotropy have been underexplored in population genetics models. In this paper, I quantify the extent to which pleiotropy inhibits the spread of alleles in response to directional selection on a focal trait. Under the assumption that pleiotropic effects are extensive and deleterious, the fraction of alleles that are beneficial overall is severely limited by pleiotropy and rises nearly linearly with the strength of directional selection on the focal trait. Over a broad class of distribution of pleiotropic effects, the mean selective effect of those alleles that are beneficial overall is halved, on average, by pleiotropy. The fraction of new mutant alleles that are beneficial overall and that succeed in fixing within a population is even more severely limited when directional selection is weak, but it rises quadratically with the strength of directional selection. Finally, the mean selective effect of mutant alleles that are beneficial and succeed in fixing is reduced by one-third, on average, by pleiotropy. These results help to shape our understanding of the evolutionary inertia caused by pleiotropy.     

Pleiotropy or linkage? Their relative contributions to the genetic correlation of quantitative traits and detection by multitrait GWA studies

Genetic correlations between traits may cause correlated responses to selection. Previous models described the conditions under which genetic correlations are expected to be maintained. Selection, mutation, and migration are all proposed to affect genetic correlations, regardless of whether the underlying genetic architecture consists of pleiotropic or tightly linked loci affecting the traits. Here, we investigate the conditions under which pleiotropy and linkage have different effects on the genetic correlations between traits by explicitly modeling multiple genetic architectures to look at the effects of selection strength, degree of correlational selection, mutation rate, mutational variance, recombination rate, and migration rate. We show that at mutation-selection(-migration) balance, mutation rates differentially affect the equilibrium levels of genetic correlation when architectures are composed of pairs of physically linked loci compared to architectures of pleiotropic loci. Even when there is perfect linkage (no recombination within pairs of linked loci), a lower genetic correlation is maintained than with pleiotropy, with a lower mutation rate leading to a larger decrease. These results imply that the detection of causal loci in multitrait association studies will be affected by the type of underlying architectures, whereby pleiotropic variants are more likely to be underlying multiple detected associations. We also confirm that tighter linkage between nonpleiotropic causal loci maintains higher genetic correlations at the traits and leads to a greater proportion of false positives in association analyses.     

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.     

Pleiotropic effects of beneficial mutations in Escherichia coli

Micromutational models of adaptation have placed considerable weight on antagonistic pleiotropy as a mechanism that prevents mutations of large effect from achieving fixation. However, there are few empirical studies of the distribution of pleiotropic effects, and no studies that have examined this distribution for a large number of adaptive mutations. Here we examine the form and extent of pleiotropy associated with beneficial mutations in Escherichia coli. To do so, we used a collection of independently evolved genotypes, each of which contains a beneficial mutation that confers increased fitness in a glucose-limited environment. To determine the pleiotropic effects of these mutations, we examined the fitnesses of the mutants in five novel resource environments. Our results show that the majority of mutations had significant fitness effects in alternative resources, such that pleiotropy was common. The predominant form of this pleiotropy was positive--that is, most mutations that conferred increased fitness in glucose also conferred increased fitness in novel resources. We did detect some deleterious pleiotropic effects, but they were primarily limited to one of the five resources, and within this resource, to only a subset of mutants. Although pleiotropic effects were generally positive, fitness levels were lower and more variable on resources that differed most in their mechanisms of uptake and catabolism from that of glucose. Positive pleiotropic effects were strongly correlated in magnitude with their direct effects, but no such correlation was found among mutants with deleterious pleiotropic effects. Whereas previous studies of populations evolved on glucose for longer periods of time showed consistent declines on some of the resources used here, our results suggest that deleterious pleiotropic effects were limited to only a subset of the beneficial mutations available.     

Payoffs,Not Tradeoffs,in the Adaptation of a Virus to Ostensibly Conflicting Selective Pressures

The genetic architecture of many phenotypic traits is such that genes often contribute to multiple traits, and mutations in these genes can therefore affect multiple phenotypes. These pleiotropic interactions often manifest as tradeoffs between traits where improvement in one property entails a cost in another. The life cycles of many pathogens include periods of growth within a host punctuated with transmission events, such as passage through a digestive tract or a passive stage of exposure in the environment. Populations exposed to such fluctuating selective pressures are expected to acquire mutations showing tradeoffs between reproduction within and survival outside of a host. We selected for individual mutations under fluctuating selective pressures for a ssDNA microvirid bacteriophage by alternating selection for increased growth rate with selection on biophysical properties of the phage capsid in high-temperature or low-pH conditions. Surprisingly, none of the seven unique mutations identified showed a pleiotropic cost; they all improved both growth rate and pH or temperature stability, suggesting that single mutations even in a simple genetic system can simultaneously improve two distinct traits. Selection on growth rate alone revealed tradeoffs, but some mutations still benefited both traits. Tradeoffs were therefore prevalent when selection acted on a single trait, but payoffs resulted when multiple traits were selected for simultaneously. We employed a molecular-dynamics simulation method to determine the mechanisms underlying beneficial effects for three heat-shock mutations. All three mutations significantly enhanced the affinities of protein-protein interfacial bindings, thereby improving capsid stability. The ancestral residues at the mutation sites did not contribute to protein-protein interfacial binding, indicating that these sites acquired a new function. Computational models, such as those used here, may be used in future work not only as predictive tools for mutational effects on protein stability but, ultimately, for evolution.     

The speed of adaptation in large asexual populations

In large asexual populations, beneficial mutations have to compete with each other for fixation. Here, I derive explicit analytic expressions for the rate of substitution and the mean beneficial effect of fixed mutations, under the assumptions that the population size N is large, that the mean effect of new beneficial mutations is smaller than the mean effect of new deleterious mutations, and that new beneficial mutations are exponentially distributed. As N increases, the rate of substitution approaches a constant, which is equal to the mean effect of new beneficial mutations. The mean effect of fixed mutations continues to grow logarithmically with N. The speed of adaptation, measured as the change of log fitness over time, also grows logarithmically with N for moderately large N, and it grows double-logarithmically for extremely large N. Moreover, I derive a simple formula that determines whether at given N beneficial mutations are expected to compete with each other or go to fixation independently. Finally, I verify all results with numerical simulations.     

Deleterious Mutations, Apparent Stabilizing Selection and the Maintenance of Quantitative Variation

Apparent stabilizing selection on a quantitative trait that is not causally connected to fitness can result from the pleiotropic effects of unconditionally deleterious mutations, because as N. Barton noted, "...individuals with extreme values of the trait will tend to carry more deleterious alleles...." We use a simple model to investigate the dependence of this apparent selection on the genomic deleterious mutation rate, U; the equilibrium distribution of K, the number of deleterious mutations per genome; and the parameters describing directional selection against deleterious mutations. Unlike previous analyses, we allow for epistatic selection against deleterious alleles. For various selection functions and realistic parameter values, the distribution of K, the distribution of breeding values for a pleiotropically affected trait, and the apparent stabilizing selection function are all nearly Gaussian. The additive genetic variance for the quantitative trait is kQa2, where k is the average number of deleterious mutations per genome, Q is the proportion of deleterious mutations that affect the trait, and a2 is the variance of pleiotropic effects for individual mutations that do affect the trait. In contrast, when the trait is measured in units of its additive standard deviation, the apparent fitness function is essentially independent of Q and a2; and beta, the intensity of selection, measured as the ratio of additive genetic variance to the "variance" of the fitness curve, is very close to s = U/k, the selection coefficient against individual deleterious mutations at equilibrium. Therefore, this model predicts appreciable apparent stabilizing selection if s exceeds about 0.03, which is consistent with various data. However, the model also predicts that beta must equal Vm/VG, the ratio of new additive variance for the trait introduced each generation by mutation to the standing additive variance. Most, although not all, estimates of this ratio imply apparent stabilizing selection weaker than generally observed. A qualitative argument suggests that even when direct selection is responsible for most of the selection observed on a character, it may be essentially irrelevant to the maintenance of variation for the character by mutation-selection balance. Simple experiments can indicate the fraction of observed stabilizing selection attributable to the pleiotropic effects of deleterious mutations.     

The estimation of selection coefficients in Afrikaners: Huntington disease, porphyria variegata, and lipoid proteinosis.

The effects of mutation, migration, random drift, and selection on the change in frequency of the alleles associated with Huntington disease, porphyria variegata, and lipoid proteinosis have been assessed in the Afrikaner population of South Africa. Although admixture cannot be completely discounted, it was possible to exclude migration and new mutation as major sources of changes in the frequency of these alleles by limiting analyses to pedigrees descendant from founding families. Calculations which overestimated the possible effect of random drift demonstrated that drift did not account for the observed changes in gene frequencies. Therefore these changes must have been caused by natural selection, and a coefficient of selection was estimated for each trait. For the rare, dominant, deleterious allele associated with Huntington disease, the coefficient of selection was estimated to be .34, indicating that this allele has a selective disadvantage, contrary to some recent studies. For the presumed dominant and probably deleterious allele associated with porphyria variegata, the coefficient of selection lies between .07 and .02. The coefficient of selection for the rare, clinically recessive allele associated with lipoid proteinosis was estimated to be .07. Calculations based on a model system indicate that the observed decrease in allele frequency cannot be explained solely on the basis of selection against the homozygote. Thus, this may be an example of a pleiotropic gene which has a dominant effect in terms of selection even though its known clinical effect is recessive.     

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.     

Increasing sequence correlation limits the efficiency of recombination in a multisite evolution model

The accumulation of preexisting beneficial alleles in a haploid population, under selection and infrequent recombination, and in the absence of new mutation events is studied numerically by means of detailed Monte Carlo simulations. On the one hand, we confirm our previous work, in that the accumulation rate follows modified single-site kinetics, with a timescale set by an effective selection coefficient s(eff) as shown in a previous work, and we confirm the qualitative features of the dependence of s(eff) on the population size and the recombination rate reported therein. In particular, we confirm the existence of a threshold population size below which evolution stops before the emergence of best-fit individuals. On the other hand, our simulations reveal that the population dynamics is essentially shaped by the steady accumulation of pairwise sequence correlation, causing sequence congruence in excess of what one would expect from a uniformly random distribution of alleles. By sequence congruence, we understand here the opposite of genetic distance, that is, the fraction of monomorphic sites of specified allele type in a pair of genomes (individual sequences). The effective selection coefficient changes more rapidly with the recombination rate and has a higher threshold in this parameter than found in the previous work, which neglected correlation effects. We examine this phenomenon by monitoring the time dependence of sequence correlation based on a set of sequence congruence measures and verify that it is not associated with the development of linkage disequilibrium. We also discuss applications to HIV evolution in infected individuals and potential implications for drug therapy.     

The G matrix under fluctuating correlational mutation and selection

Theoretical quantitative genetics provides a framework for reconstructing past selection and predicting future patterns of phenotypic differentiation. However, the usefulness of the equations of quantitative genetics for evolutionary inference relies on the evolutionary stability of the additive genetic variance-covariance matrix (G matrix). A fruitful new approach for exploring the evolutionary dynamics of G involves the use of individual-based computer simulations. Previous studies have focused on the evolution of the eigenstructure of G. An alternative approach employed in this paper uses the multivariate response-to-selection equation to evaluate the stability of G. In this approach, I measure similarity by the correlation between response-to-selection vectors due to random selection gradients. I analyze the dynamics of G under several conditions of correlational mutation and selection. As found in a previous study, the eigenstructure of G is stabilized by correlational mutation and selection. However, over broad conditions, instability of G did not result in a decreased consistency of the response to selection. I also analyze the stability of G when the correlation coefficients of correlational mutation and selection and the effective population size change through time. To my knowledge, no prior study has used computer simulations to investigate the stability of G when correlational mutation and selection fluctuate. Under these conditions, the eigenstructure of G is unstable under some simulation conditions. Different results are obtained if G matrix stability is assessed by eigenanalysis or by the response to random selection gradients. In this case, the response to selection is most consistent when certain aspects of the eigenstructure of G are least stable and vice versa.     

Molecular evolution, mutation size and gene pleiotropy: a geometric reexamination

The influence of phenotypic effects of genetic mutations on molecular evolution is not well understood. Neutral and nearly neutral theories of molecular evolution predict a negative relationship between the evolutionary rate of proteins and their functional importance; nevertheless empirical studies seeking relationships between evolutionary rate and the phenotypic role of proteins have not produced conclusive results. In particular, previous studies have not found the expected negative correlation between evolutionary rate and gene pleiotropy. Here, we studied the effect of gene pleiotropy and the phenotypic size of mutations on the evolutionary rate of genes in a geometrical model, in which gene pleiotropy was characterized by n molecular phenotypes that affect organismal fitness. For a nearly neutral process, we found a negative relationship between evolutionary rate and mutation size but pleiotropy did not affect the evolutionary rate. Further, for a selection model, where most of the substitutions were fixed by natural selection in a randomly fluctuating environment, we also found a negative relationship between evolutionary rate and mutation size, but interestingly, gene pleiotropy increased the evolutionary rate as √n. These findings may explain part of the disagreement between empirical data and traditional expectations.     

UNPREDICTABILITY OF CORRELATED RESPONSE TO SELECTION: PLEIOTROPY AND SAMPLING INTERACT

Given a set of loci that contribute additive genetic variation for a trait being selected, the pleiotropic effects of these loci on a second trait may vary. I simulated selection on genetic systems having different combinations of pleiotropic effects to investigate the variability of correlated responses to selection. The simulation shows that there are many possible combinations of pleiotropic effects that are characterized by the same value of the genetic correlation; the genetic correlation does not uniquely determine a set of pleiotropic effects. Furthermore, for a given value of the genetic correlation, differences in pleiotropic effects have a substantial impact on the variation in correlated responses. Some combinations of pleiotropic effects constrain correlated response to a narrow range of possible values; others allow a wide range, including some correlated responses in a direction opposite the sign of the genetic correlation. The genetic correlation is not a reliable predictor of pleiotropic constraint. Whereas it has been previously established that genetic correlations are not necessarily constraints, the alternative is also true: correlated response can be strictly constrained despite a genetic correlation of zero. Given the frequency of correlated responses in a direction opposite to the one predicted by the genetic correlation, it follows that correlated response is not a reliable predictor of genetic correlation in the base population.     

Hitchhiking effect of a beneficial mutation spreading in a subdivided population

A central problem in population genetics is to detect and analyze positive natural selection by which beneficial mutations are driven to fixation. The hitchhiking effect of a rapidly spreading beneficial mutation, which results in local removal of standing genetic variation, allows such an analysis using DNA sequence polymorphism. However, the current mathematical theory that predicts the pattern of genetic hitchhiking relies on the assumption that a beneficial mutation increases to a high frequency in a single random-mating population, which is certainly violated in reality. Individuals in natural populations are distributed over a geographic space. The spread of a beneficial allele can be delayed by limited migration of individuals over the space and its hitchhiking effect can also be affected. To study this effect of geographic structure on genetic hitchhiking, we analyze a simple model of directional selection in a subdivided population. In contrast to previous studies on hitchhiking in subdivided populations, we mainly investigate the range of sufficiently high migration rates that would homogenize genetic variation at neutral loci. We provide a heuristic mathematical analysis that describes how the genealogical structure at a neutral locus linked to the locus under selection is expected to change in a population divided into two demes. Our results indicate that the overall strength of genetic hitchhiking--the degree to which expected heterozygosity decreases--is diminished by population subdivision, mainly because opportunity for the breakdown of hitchhiking by recombination increases as the spread of the beneficial mutation across demes is delayed when migration rate is much smaller than the strength of selection. Furthermore, the amount of genetic variation after a selective sweep is expected to be unequal over demes: a greater reduction in expected heterozygosity occurs in the subpopulation from which the beneficial mutation originates than in its neighboring subpopulations. This raises a possibility of detecting a "hidden" geographic structure of population by carefully analyzing the pattern of a selective sweep.     

Pleiotropic Models of Quantitative Variation

It is widely held that each gene typically affects many characters, and that each character is affected by many genes. Moreover, strong stabilizing selection cannot act on an indefinitely large number of independent traits. This makes it likely that heritable variation in any one trait is maintained as a side effect of polymorphisms which have nothing to do with selection on that trait. This paper examines the idea that variation is maintained as the pleiotropic side effect of either deleterious mutation, or balancing selection. If mutation is responsible, it must produce alleles which are only mildly deleterious (s approximately 10(-3)), but nevertheless have significant effects on the trait. Balancing selection can readily maintain high heritabilities; however, selection must be spread over many weakly selected polymorphisms if large responses to artificial selection are to be possible. In both classes of pleiotropic model, extreme phenotypes are less fit, giving the appearance of stabilizing selection on the trait. However, it is shown that this effect is weak (of the same order as the selection on each gene): the strong stabilizing selection which is often observed is likely to be caused by correlations with a limited number of directly selected traits. Possible experiments for distinguishing the alternatives are discussed.     

An explicit model for the inbreeding load in the evolutionary analysis of selfing

I present analytical predictions for the equilibrium inbreeding load expected in a population under mutation, selection, and a regular mating system for any population size and for any magnitude and recessivity of the deleterious effects. Using this prediction, I deduce the relative fitness of mutant alleles with small effect on selfing to explore the situations where selfing or outcrossing are expected to evolve. The results obtained are in agreement with previous literature, showing that natural selection is expected to lead to stable equilibria where populations show either complete outcrossing or complete selfing, and that selfing is promoted by large deleterious mutation rates. I find that the evolution of selfing is favored by a large recessivity of deleterious effects, while the magnitude of homozygous deleterious effects only becomes relevant in relatively small populations. This result contradicts the standard assumption that purging in large populations will only promote selfing when homozygous deleterious effects are large, and implies that previously published results obtained assuming lethal mutations in large populations can be extrapolated to nonlethal alleles of similar recessivity. This conclusion and the general approach used in this analysis can be useful in the study of the evolution of mating systems.