Epistasis in polygenic traits and the evolution of genetic architecture under stabilizing selection

We consider the effects of epistasis in a polygenic trait in the balance of mutation and stabilizing selection. The main issues are the genetic variation maintained in equilibrium and the evolution of the mutational effect distribution. The model assumes symmetric mutation and a continuum of alleles at all loci. Epistasis is modeled proportional to pairwise products of the single-locus effects. A general analytical formalism is developed. Assuming linkage equilibrium, we derive results for the equilibrium mutation load and the genetic and mutational variance in the house of cards and the Gaussian approximation. The additive genetic variation maintained in mutation-selection balance is reduced by any pattern of the epistatic interactions. The mutational variance, in contrast, is often increased. Large differences in mutational effects among loci emerge, and a negative correlation among (standard mean) locus mutation effects and mutation rates is predicted. Contrary to the common view since Waddington, we find that stabilizing selection in general does not lead to canalization of the trait. We propose that canalization as a target of selection instead occurs at the genic level. Here, primarily genes with a high mutation rate are buffered, often at the cost of decanalization of other genes. An intuitive interpretation of this view is given in the discussion.     

SPONTANEOUS MUTATION ACCUMULATION IN MULTIPLE STRAINS OF THE GREEN ALGA,CHLAMYDOMONAS REINHARDTII

Estimates of mutational parameters, such as the average fitness effect of a new mutation and the rate at which new genetic variation for fitness is created by mutation, are important for the understanding of many biological processes. However, the causes of interspecific variation in mutational parameters and the extent to which they vary within species remain largely unknown. We maintained multiple strains of the unicellular eukaryote Chlamydomonas reinhardtii, for approximately 1000 generations under relaxed selection by transferring a single cell every ～10 generations. Mean fitness of the lines tended to decline with generations of mutation accumulation whereas mutational variance increased. We did not find any evidence for differences among strains in any of the mutational parameters estimated. The overall change in mean fitness per cell division and rate of input of mutational variance per cell division were more similar to values observed in multicellular organisms than to those in other single‐celled microbes. However, after taking into account differences in genome size among species, estimates from multicellular organisms and microbes, including our new estimates from C. reinhardtii, become substantially more similar. Thus, we suggest that variation in genome size is an important determinant of interspecific variation in mutational parameters.     

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.     

SPONTANEOUS MUTATION PARAMETERS FOR ARABIDOPSIS THALIANA MEASURED IN THE WILD

Mutations are the ultimate source of genetic diversity and their contributions to evolutionary process depend critically on their rate and their effects on traits, notably fitness. Mutation rate and mutation effect can be measured simultaneously through the use of mutation accumulation lines, and previous mutation accumulation studies measuring these parameters have been performed in laboratory conditions. However, estimation of mutation parameters for fitness in wild populations requires assays in environments where mutations are exposed to natural selection and natural environmental variation. Here we quantify mutation parameters in both the wild and greenhouse environments using 100 25th generation Arabidopsis thaliana mutation accumulation lines. We found significantly greater mutational variance and a higher mutation rate for fitness under field conditions relative to greenhouse conditions. However, our field estimates were low when scaled to natural environmental variation. Many of the mutation accumulation lines have increased fitness, counter to the expectation that nearly all mutations decrease fitness. A high mutation rate and a low mutational contribution to phenotypic variation may explain observed levels of natural genetic variation. Our findings indicate that mutation parameters are not fixed, but are variables whose values may reflect the specific environment in which mutations are tested.     

Transcriptome‐wide effects of sexual selection on the fate of new mutations

Sexual selection on males is predicted to have widespread effects on genetic variation as a consequence of the pleiotropic allelic effects on sexual and nonsexual traits. We manipulated the opportunity for sexual selection on males during 27 generations of mutation accumulation in inbred lines of Drosophila serrata, and used a microarray platform to investigate the effect of sexual selection on the expression of 2689 genes. While gene expression signal was, on average, higher in the absence of sexual selection, this difference was small (0.1%). In contrast, sexual selection impacted substantially on the mutational variance in gene expression. Over all genes, mutational variance in gene expression was, on average, 42% higher when sexual selection operated than when it was absent. Our results indicate that sexual selection on males can generate widespread effects across the genome. An increase in mutational variance without a corresponding change in mean suggested that most expression traits were unlikely to be under direct sexual selection. Instead, the mutational variance in gene expression traits is consistent with divergence generated by widespread pleiotropic associations with traits affecting male mating success.     

Quantifying the decanalizing effects of spontaneous mutations in rhabditid nematodes

The evolution of canalization, the robustness of the phenotype to environmental or genetic perturbation, has attracted considerable recent interest. A key step toward understanding the evolution of any phenotype is characterizing the rate at which mutation introduces genetic variation for the trait (the mutational variance, V(M)) and the average directional effects of mutations on the trait mean (DeltaM). In this study, the mutational parameters for canalization of productivity and body volume are quantified in two sets of mutation accumulation lines of nematodes in the genus Caenorhabditis and are compared with the mutational parameters for the traits themselves. Four results emerge: (1) spontaneous mutations consistently decanalize the phenotype; (2) the mutational parameters for decanalization, V(M) (quantified as mutational heritability) and DeltaM, are of the same order of magnitude as the same parameters for the traits themselves; (3) the mutational parameters for canalization are roughly correlated with the parameters for the traits themselves across taxa; and (4) there is no evidence that residual segregating overdominant loci contribute to the decay of canalization. These results suggest that canalization is readily evolvable and that any evolutionary factor that causes mutations to accumulate will, on average, decanalize the phenotype.     

Quantitative Genetic Variation in Body Size of Mice from New Mutations

To measure the amount of new genetic variation in 6-week weight of mice arising each generation from mutation, selection lines derived from an initially inbred strain were maintained for 25 generations. An analysis using an animal model with restricted maximum likelihood was applied to estimate a mutational genetic component of variance for the infinitesimal model of many genes of small effect. Assuming that the inbred base population was at a mutation-drift equilibrium, it is estimated that the heritability for body size has increased by 1.0% per generation, with lower and upper confidence limits of 0.6% and 1.6%, respectively. A model which includes a mutational genetic component of variance fits the data much better than one involving only base population genetic variance. A model with no genetic component fits the data very poorly. An environmental covariance of body size of mother and offspring was included in the model and accounts for 10% of the variance. By using information only from the observed response to selection, the estimated increase in heritability from mutation is 0.3% per generation. These values are higher than published estimates for the increase in variance from spontaneous mutations in bristle traits of Drosophila, for which there are extensive data, but similar to estimates for various skeletal traits in mice.     

Quantifying natural seasonal variation in mutation parameters with mutation accumulation lines

Mutations create novel genetic variants, but their contribution to variation in fitness and other phenotypes may depend on environmental conditions. Furthermore, natural environments may be highly heterogeneous. We assessed phenotypes associated with survival and reproductive success in over 30,000 plants representing 100 mutation accumulation lines of Arabidopsis thaliana across four temporal environments at a single field site. In each of the four assays, environmental variance was substantially larger than mutational variance. For some traits, whether mutational variance was significantly varied between seasons. The founder genotype had mean trait values near the mean of the distribution of the mutation accumulation lines in all field experiments. New mutations also contributed more phenotypic variation than would be predicted, given phenotypic and sequence‐level divergence among natural populations of A. thaliana. The combination of large environmental variance with a mean effect of mutation near zero suggests that mutations could contribute substantially to standing genetic variation.     

Stabilizing selection,mutational bias,and the evolution of sex*

Stabilizing selection around a fixed phenotypic optimum is expected to disfavor sexual reproduction, since asexually reproducing organisms can maintain a higher fitness at equilibrium, while sex disrupts combinations of compensatory mutations. This conclusion rests on the assumption that mutational effects on phenotypic traits are unbiased, that is, mutation does not tend to push phenotypes in any particular direction. In this article, we consider a model of stabilizing selection acting on an arbitrary number of polygenic traits coded by bialellic loci, and show that mutational bias may greatly reduce the mean fitness of asexual populations compared with sexual ones in regimes where mutations have weak to moderate fitness effects. Indeed, mutation and drift tend to push the population mean phenotype away from the optimum, this effect being enhanced by the low effective population size of asexual populations. In a second part, we present results from individual‐based simulations showing that positive rates of sex are favored when mutational bias is present, while the population evolves toward complete asexuality in the absence of bias. We also present analytical (QLE) approximations for the selective forces acting on sex in terms of the effect of sex on the mean and variance in fitness among offspring.     

Inbreeding Depression and Inferred Deleterious-Mutation Parameters in Daphnia

DENG and LYNCH recently proposed a method for estimating deleterious genomic mutation parameters from changes in the mean and genetic variance of fitness traits upon inbreeding in outcrossing populations. Such observations are readily acquired in cyclical parthenogens. Selfing and life-table experiments were performed for two such Daphnia populations. We observed a significant inbreeding depression and an increase of genetic variance for all traits analyzed. DENG and LYNCH's original procedures were extended to estimate genomic mutation rate (U), mean dominance coefficient (h), mean selection coefficient (s), and scaled genomic mutational variance (V(m)/V(e)). On average, U, h, s and V(m)/V(e) (^ indicates an estimate) are 0.74, 0.30, 0.14 and 4.6E-4, respectively. For the true values, the U and h are lower bounds, and s and V(m)/V(e) upper bounds. The present U, h and V(m)/V(e) are in general concordance with earlier results. The discrepancy between the present s and that from mutation-accumulation experiments in Drosophila (~0.04) is discussed. It is shown that different reproductive modes do not affect gene frequency at mutation-selection equilibrium if mutational effects on fitness are multiplicative and not completely recessive.     

Pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance

A pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance is investigated. Mutations have effects on a metric trait and deleterious effects on fitness, for which a bivariate gamma distribution is assumed. Equations for calculating the strength of apparent stabilizing selection (V(s)) and the genetic variance maintained in segregating populations (V(G)) were derived. A large population can hold a high genetic variance but the apparent stabilizing selection may or may not be relatively strong, depending on other properties such as the distribution of mutation effects. If the distribution of mutation effects on fitness is continuous such that there are few nearly neutral mutants, or a minimum fitness effect is assumed if most mutations are nearly neutral, V(G) increases to an asymptote as the population size increases. Both V(G) and V(s) are strongly affected by the shape of the distribution of mutation effects. Compared with mutants of equal effect, allowing their effects on fitness to vary across loci can produce a much higher V(G) but also a high V(s) (V(s) in phenotypic standard deviation units, which is always larger than the ratio V(P)/V(m)), implying weak apparent stabilizing selection. If the mutational variance V(m) is approximately 10(-3)V(e) (V(e), environmental variance), the model can explain typical values of heritability and also apparent stabilizing selection, provided the latter is quite weak as suggested by a recent review.     

Perspective: Sign epistasis and genetic constraint on evolutionary trajectories

Epistasis for fitness means that the selective effect of a mutation is conditional on the genetic background in which it appears. Although epistasis is widely observed in nature, our understanding of its consequences for evolution by natural selection remains incomplete. In particular, much attention focuses only on its influence on the instantaneous rate of changes in frequency of selected alleles via epistatic contribution to the additive genetic variance for fitness. Thus, in this framework epistasis only has evolutionary importance if the interacting loci are simultaneously segregating in the population. However, the selective accessibility of mutational trajectories to high fitness genotypes may depend on the genetic background in which novel mutations appear, and this effect is independent of population polymorphism at other loci. Here we explore this second influence of epistasis on evolution by natural selection. We show that it is the consequence of a particular form of epistasis, which we designate sign epistasis. Sign epistasis means that the sign of the fitness effect of a mutation is under epistatic control; thus, such a mutation is beneficial on some genetic backgrounds and deleterious on others. Recent experimental innovations in microbial systems now permit assessment of the fitness effects of individual mutations on multiple genetic backgrounds. We review this literature and identify many examples of sign epistasis, and we suggest that the implications of these results may generalize to other organisms. These theoretical and empirical considerations imply that strong genetic constraint on the selective accessibility of trajectories to high fitness genotypes may exist and suggest specific areas of investigation for future research.     

Fisher'S geometrical model of fitness landscape and variance in fitness within a changing environment

The fitness of an individual can be simply defined as the number of its offspring in the next generation. However, it is not well understood how selection on the phenotype determines fitness. In accordance with Fisher's fundamental theorem, fitness should have no or very little genetic variance, whereas empirical data suggest that is not the case. To bridge these knowledge gaps, we follow Fisher's geometrical model and assume that fitness is determined by multivariate stabilizing selection toward an optimum that may vary among generations. We assume random mating, free recombination, additive genes, and uncorrelated stabilizing selection and mutational effects on traits. In a constant environment, we find that genetic variance in fitness under mutation-selection balance is a U-shaped function of the number of traits (i.e., of the so-called "organismal complexity"). Because the variance can be high if the organism is of either low or high complexity, this suggests that complexity has little direct costs. Under a temporally varying optimum, genetic variance increases relative to a constant optimum and increasingly so when the mutation rate is small. Therefore, mutation and changing environment together can maintain high genetic variance. These results therefore lend support to Fisher's geometric model of a fitness landscape.     

GENETIC REGULATORY NETWORK MOTIFS CONSTRAIN ADAPTATION THROUGH CURVATURE IN THE LANDSCAPE OF MUTATIONAL (CO)VARIANCE

Systems biology is accumulating a wealth of understanding about the structure of genetic regulatory networks, leading to a more complete picture of the complex genotype–phenotype relationship. However, models of multivariate phenotypic evolution based on quantitative genetics have largely not incorporated a network‐based view of genetic variation. Here we model a set of two‐node, two‐phenotype genetic network motifs, covering a full range of regulatory interactions. We find that network interactions result in different patterns of mutational (co)variance at the phenotypic level (the M ‐matrix), not only across network motifs but also across phenotypic space within single motifs. This effect is due almost entirely to mutational input of additive genetic (co)variance. Variation in M has the effect of stretching and bending phenotypic space with respect to evolvability, analogous to the curvature of space–time under general relativity, and similar mathematical tools may apply in each case. We explored the consequences of curvature in mutational variation by simulating adaptation under divergent selection with gene flow. Both standing genetic variation (the G ‐matrix) and rate of adaptation are constrained by M , so that G and adaptive trajectories are curved across phenotypic space. Under weak selection the phenotypic mean at migration‐selection balance also depends on M .     

Measures of Variation at DNA Repeat Loci under a General Stepwise Mutation Model

Polymorphisms at tandem repeat loci are caused by mutations with allele sizes occasionally altered by more than one repeat unit in both forward and backward directions. Such mutational changes may occur with asymmetric probabilities. Therefore, a one-step symmetric stepwise mutation model may not be appropriate for studying the population dynamics at all repeat loci. In this work, we evaluated the expectation and variance of the within-population variance of the allele size distribution in a finite population, and the expected homozygosity at a locus by the coalescence approach under a general stepwise mutation model, where mutational transitions of allele sizes can be arbitrary, including being asymmetric. Under the special cases of symmetric one-step, two-step, and multi-step geometric distributions of mutations, our general results reduce to the corresponding results obtained by earlier investigators. The general results indicate that in a finite population, which has reached a steady state under the (general stepwise) mutation and drift balance, the within-population variance of allele sizes has a simple expectation (i.e., proportional toNν, the product of the mutation rate,ν, and effective population size,N). However, its stochastic variance is a quadratic function of this composite parameter,Nν. Furthermore, this second-order variance does not decay with the number of alleles sampled from a population. Application of this theory to data on allele size distributions in unrelated Caucasians from the CEPH pedigree (obtained from the Genome Data Base) shows that the relationship of the variance and mean of within-population variance of allele sizes at tandem repeat loci, grouped by their chromosomal assignment, has a trend compatible with the theory. However, there is an indication that the second-order variance is generally underestimated. One reason for this departure might be that the CEPH sample may not represent a single homogeneous population that reached equilibrium at all tandem repeat loci.     

QUANTITATIVE GENETIC VARIANCE MAINTAINED BY FLUCTUATING SELECTION WITH OVERLAPPING GENERATIONS: VARIANCE COMPONENTS AND COVARIANCES

The quantitative genetic variance-covariance that can be maintained in a random environment is studied, assuming overlapping generations and Gaussian stabilizing selection with a fluctuating optimum. The phenotype of an individual is assumed to be determined by additive contributions from each locus on paternal and maternal gametes (i.e., no epistasis and no dominance). Recurrent mutation is ignored, but linkage between loci is arbitrary. The genotype distribution in the evolutionarily stable population is generically discrete: only a finite number of polymorphic alleles with distinctly different effects are maintained, even though we allow a continuum of alleles with arbitrary phenotypic contributions to invade. Fluctuating selection maintains nonzero genetic variance in the evolutionarily stable population if the environmental heterogeneity is larger than a certain threshold. Explicit asymptotic expressions for the standing variance-covariance components are derived for the population near the threshold, or for large generational overlap, as a function of environmental variability and genetic parameters (i.e., number of loci, recombination rate, etc.), using the fact that the genotype distribution is discrete. Above the threshold, the population maintains considerable genetic variance in the form of positive linkage disequilibrium and positive gamete covariance (Hardy-Weinberg disequilibrium) as well as allelic variance. The relative proportion of these disequilibrium variances in the total genetic variance increases with the environmental variability.     

Evidence for overdominant selection maintaining X-linked fitness variation in Drosophila melanogaster

The role of balancing selection in maintaining genetic variation for fitness is largely unresolved. This reflects the inherent difficulty in distinguishing between models of recurrent mutation versus selection, which produce similar patterns of inbreeding depression, as well as the limitations of testing such hypotheses when fitness variation is averaged across the genome. Signatures of X-linked overdominant selection are less likely to be obscured by mutational variation because X-linked mutations are rapidly eliminated by purifying selection in males. Although models maintaining genetic variation for fitness are not necessarily mutually exclusive, a series of predictions for identifying X-linked overdominant selection can be used to separate its contribution from other underlying processes. We consider the role of overdominant selection in maintaining fitness variation in a sample of 12 X chromosomes from a population of Drosophila melanogaster. Substantial variation was observed for male reproductive success and female fecundity, with heterozygous-X genotypes exhibiting the greatest degree of variance, a finding that agrees well with predictions of the overdominance model. The importance of X-linked overdominant selection is discussed along with models of recurrent mutation and sexually antagonistic selection.     

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.     

Mutation and evolution of microsatellite loci in Neurospora

The patterns of mutation and evolution at 13 microsatellite loci were studied in the filamentous fungal genus Neurospora. First, a detailed investigation was performed on five microsatellite loci by sequencing each microsatellite, together with its nonrepetitive flanking regions, from a set of 147 individuals from eight species of Neurospora. To elucidate the genealogical relationships among microsatellite alleles, repeat number was mapped onto trees constructed from flanking-sequence data. This approach allowed the potentially convergent microsatellite mutations to be placed in the evolutionary context of the less rapidly evolving flanking regions, revealing the complexities of the mutational processes that have generated the allelic diversity conventionally assessed in population genetic studies. In addition to changes in repeat number, frequent substitution mutations within the microsatellites were detected, as were substitutions and insertion/deletions within the flanking regions. By comparing microsatellite and flanking-sequence divergence, clear evidence of interspecific allele length homoplasy and microsatellite mutational saturation was observed, suggesting that these loci are not appropriate for inferring phylogenetic relationships among species. In contrast, little evidence of intraspecific mutational saturation was observed, confirming the utility of these loci for population-level analyses. Frequency distributions of alleles within species were generally consistent with the stepwise mutational model. By comparing variation within species at the microsatellites and the flanking-sequence, estimated microsatellite mutation rates were approximately 2500 times greater than mutation rates of flanking DNA and were consistent with estimates from yeast and fruit flies. A positive relationship between repeat number and variance in repeat number was significant across three genealogical depths, suggesting that longer microsatellite alleles are more mutable than shorter alleles. To test if the observed patterns of microsatellite variation and mutation could be generalized, an additional eight microsatellite loci were characterized and sequenced from a subset of the same Neurospora individuals.     

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.