Equilibrium Genetic Variance Research Articles

Polygenic Adaptation in a Population of Finite Size.

Polygenic adaptation in response to selection on quantitative traits has become an important topic in evolutionary biology. Here we review the recent literature on models of polygenic adaptation. In particular, we focus on a model that includes mutation and both directional and stabilizing selection on a highly polygenic trait in a population of finite size (thus experiencing random genetic drift). Assuming that a sudden environmental shift of the fitness optimum occurs while the population is in a stochastic equilibrium, we analyze the adaptation of the trait to the new optimum. When the shift is not too large relative to the equilibrium genetic variance and this variance is determined by loci with mostly small effects, the approach of the mean phenotype to the optimum can be approximated by a rapid exponential process (whose rate is proportional to the genetic variance). During this rapid phase the underlying changes to allele frequencies, however, may depend strongly on genetic drift. While trait-increasing alleles with intermediate equilibrium frequencies are dominated by selection and contribute positively to changes of the trait mean (i.e., are aligned with the direction of the optimum shift), alleles with low or high equilibrium frequencies show more of a random dynamics, which is expected when drift is dominating. A strong effect of drift is also predicted for population size bottlenecks. Our simulations show that the presence of a bottleneck results in a larger deviation of the population mean of the trait from the fitness optimum, which suggests that more loci experience the influence of drift.

Important role of genetic drift in rapid polygenic adaptation.

We analyzed a model to determine the factors that facilitate or limit rapid polygenic adaptation. This model includes population genetic terms of mutation and both directional and stabilizing selection on a highly polygenic trait in a diploid population of finite size. First, we derived the equilibrium distribution of the allele frequencies of the multilocus model by diffusion approximation. This formula describing the equilibrium allele frequencies as a mutation‐selection‐drift balance was examined by computer simulation using parameter values inferred for human height, a well‐studied polygenic trait. Second, assuming that a sudden environmental shift of the fitness optimum occurs while the population is in equilibrium, we analyzed the adaptation of the trait to the new optimum. The speed at which the trait mean approaches the new optimum increases with the equilibrium genetic variance. Thus, large population size and/or large mutation rate may facilitate rapid adaptation. Third, the contribution of an individual locus i to polygenic adaptation depends on the compound parameter γipi0qi0, where γi is the effect size, pi0 the equilibrium frequency of the trait‐increasing allele of this locus, and qi0=1-pi0. Thus, only loci with large values of this parameter contribute coherently to polygenic adaptation. Given that mutation rates are relatively small, this is more likely in large populations, in which the effects of drift are limited.

Effects of partial selfing on the equilibrium genetic variance, mutation load, and inbreeding depression under stabilizing selection.

The mating system of a species is expected to have important effects on its genetic diversity. In this article, we explore the effects of partial selfing on the equilibrium genetic variance Vg , mutation load L, and inbreeding depression δ under stabilizing selection acting on a arbitrary number n of quantitative traits coded by biallelic loci with additive effects. When the U/n ratio is low (where U is the total haploid mutation rate on selected traits) and effective recombination rates are sufficiently high, genetic associations between loci are negligible and the genetic variance, mutation load, and inbreeding depression are well predicted by approximations based on single-locus models. For higher values of U/n and/or lower effective recombination, moderate genetic associations generated by epistasis tend to increase Vg , L, and δ, this regime being well predicted by approximations including the effects of pairwise associations between loci. For yet higher values of U/n and/or lower effective recombination, a different regime is reached under which the maintenance of coadapted gene complexes reduces Vg , L, and δ. Simulations indicate that the values of Vg , L, and δ are little affected by assumptions regarding the number of possible alleles per locus.

Understanding genetic variance, load, and inbreeding depression with selfing

A recommendation of: Diala Abu Awad and Denis Roze Effects of partial selfing on the equilibrium genetic variance, mutation load and inbreeding depression under stabilizing selection 10.1101/180000

Maintenance of Quantitative Genetic Variance Under Partial Self-Fertilization, with Implications for Evolution of Selfing.

We analyze two models of the maintenance of quantitative genetic variance in a mixed-mating system of self-fertilization and outcrossing. In both models purely additive genetic variance is maintained by mutation and recombination under stabilizing selection on the phenotype of one or more quantitative characters. The Gaussian allele model (GAM) involves a finite number of unlinked loci in an infinitely large population, with a normal distribution of allelic effects at each locus within lineages selfed for τ consecutive generations since their last outcross. The infinitesimal model for partial selfing (IMS) involves an infinite number of loci in a large but finite population, with a normal distribution of breeding values in lineages of selfing age τ. In both models a stable equilibrium genetic variance exists, the outcrossed equilibrium, nearly equal to that under random mating, for all selfing rates, r, up to critical value, [Formula: see text], the purging threshold, which approximately equals the mean fitness under random mating relative to that under complete selfing. In the GAM a second stable equilibrium, the purged equilibrium, exists for any positive selfing rate, with genetic variance less than or equal to that under pure selfing; as r increases above [Formula: see text] the outcrossed equilibrium collapses sharply to the purged equilibrium genetic variance. In the IMS a single stable equilibrium genetic variance exists at each selfing rate; as r increases above [Formula: see text] the equilibrium genetic variance drops sharply and then declines gradually to that maintained under complete selfing. The implications for evolution of selfing rates, and for adaptive evolution and persistence of predominantly selfing species, provide a theoretical basis for the classical view of Stebbins that predominant selfing constitutes an "evolutionary dead end."

Effects of epistasis and the evolution of genetic architecture: Exact results for a 2-locus model

We study a two-locus model of a quantitative trait with a continuum-of alleles and multilinear epistasis that evolves under mutation, selection, and genetic drift. We derive analytical results based on the so-called House of Gauss approximation for the genetic variance, the mean phenotype, and the mutational variance in the balance of the evolutionary forces. The analytical work is complemented by extensive individual-based computer simulations. We find that (1) analytical results are accurate in a large parameter space; (2) epistasis always reduces the equilibrium genetic variance, as predicted in earlier studies that exclude drift; (3) large-scale stochastic fluctuations and non-equilibrium phenomena like adaptive inertia can strongly influence the evolution of the genetic architecture of the trait.

Sexually antagonistic genetic variation for fitness in red deer

Evolutionary theory predicts the depletion of genetic variation in natural populations as a result of the effects of selection, but genetic variation is nevertheless abundant for many traits that are under directional or stabilizing selection. Evolutionary geneticists commonly try to explain this paradox with mechanisms that lead to a balance between mutation and selection. However, theoretical predictions of equilibrium genetic variance under mutation-selection balance are usually lower than the observed values, and the reason for this is unknown. The potential role of sexually antagonistic selection in maintaining genetic variation has received little attention in this debate, surprisingly given its potential ubiquity in dioecious organisms. At fitness-related loci, a given genotype may be selected in opposite directions in the two sexes. Such sexually antagonistic selection will reduce the otherwise-expected positive genetic correlation between male and female fitness. Both theory and experimental data suggest that males and females of the same species may have divergent genetic optima, but supporting data from wild populations are still scarce. Here we present evidence for sexually antagonistic fitness variation in a natural population, using data from a long-term study of red deer (Cervus elaphus). We show that male red deer with relatively high fitness fathered, on average, daughters with relatively low fitness. This was due to a negative genetic correlation between estimates of fitness in males and females. In particular, we show that selection favours males that carry low breeding values for female fitness. Our results demonstrate that sexually antagonistic selection can lead to a trade-off between the optimal genotypes for males and females; this mechanism will have profound effects on the operation of selection and the maintenance of genetic variation in natural populations.

The frequency of the perfect genotype in a population subject to pleiotropic mutation

We consider a large population of asexual organisms characterised by a number of quantitative traits that are subject to stabilising selection. Mutation is taken to act pleiotropically, with every mutation generally changing all of the traits under selection. We focus on the equilibrium distribution of the population, where mutation and selection are in balance. It has been previously established that the equilibrium distribution of genotypic effects may be anomalous, as it may contain a singular spike—a Dirac delta function—corresponding to a non-zero proportion of the population having exactly optimal genotypic values. In the present work, we present exact results for the case where three traits are under selection. These results give the equilibrium genetic variance of the population, and the proportion of the population that have the optimal genotype. This is achieved for two different spherically symmetric distributions of mutant effects. Additionally, a simple and robust numerical approach is also presented that allows the treatment of some other mutation distributions, where there are an arbitrary number of selected traits.

Impact of nonrandom mating on genetic variance and gene flow in populations with mass selection.

The mechanisms by which nonrandom mating affects selected populations are not completely understood and remain a subject of scientific debate in the development of tractable predictors of population characteristics. The main objective of this study was to provide a predictive model for the genetic variance and covariance among mates for traits subjected to directional selection in populations with nonrandom mating based on the pedigree. Stochastic simulations were used to check the validity of this model. Our predictions indicate that the positive covariance among mates that is expected to result with preferential mating of relatives can be severely overpredicted from neutral expectations. The covariance expected from neutral theory is offset by an opposing covariance between the genetic mean of an individual's family and the Mendelian sampling term of its mate. This mechanism was able to predict the reduction in covariance among mates that we observed in the simulated populations and, in consequence, the equilibrium genetic variance and expected long-term genetic contributions. Additionally, this study provided confirmatory evidence on the postulated relationships of long-term genetic contributions with both the rate of genetic gain and the rate of inbreeding (deltaF) with nonrandom mating. The coefficient of variation of the expected gene flow among individuals and deltaF was sensitive to nonrandom mating when heritability was low, but less so as heritability increased, and the theory developed in the study was sufficient to explain this phenomenon.

Non-equivalent loci and mutation–selection balance

We consider the implications of mutationally non-equivalent loci for large populations of randomly mating diploid organisms under mutation–selection balance. Variation, across loci, of parameters such as the allelic mutational variance and the mutation rate, is shown to reduce the equilibrium genetic variance. This is proved to follow from the genetic variance contributed by a single locus having an underlying convexity. We give approximate results indicating the way small deviations of the mutational parameters, from their mean values, reduce the genetic variance. Numerical estimates of the size of the effect are given for more general variations of the parameters. Variation in the mutation rates has a significantly smaller effect than variation in the mutational variances. Under accepted parameter values, the reduction in genetic variance can be substantial.

On a Genetic Model of Intraspecific Competition and Stabilizing Selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.

On a Genetic Model of Intraspecific Competition and Stabilizing Selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency‐independent component describing stabilizing selection and a frequency‐ and density‐dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.

Mathematical properties of mutation-selection models

This article provides an overview of the mathematical properties of various deterministic mutation-selection models. Many, even basic, results about models with more than two alleles per locus have only been obtained in the past two decades and are scattered throughout the literature. I review the fundamental results about existence and stability of equilibria for classical mutation-selection models with a finite number of alleles, for models like the stepwise-mutation model, and for the continuum-of-alleles model. For all these models, first- and second-order approximations for the equilibrium mean fitness and the mutation load are presented. Then I examine mutation-stabilizing selection balance. As an application of the results on the mutation load, second-order approximations for the equilibrium genetic variance can be derived. Various approximations for this equilibrium variance, which have been obtained in the literature, are concisely reviewed and their ranges of validity are compared. In particular, the dependence of the equilibrium variance on the parameters of the underlying genetic system, such as the form of the mutation distribution or the linkage map, is investigated.

The Genetic Variance Maintained by Pleiotropic Mutation

The equilibrium genetic variance of a quantitative trait under mutation–selection balance is analysed using a population genetic model. I assume that mutations have both phenotypic effects on a quantitative trait and deleterious effects on fitness itself. Hence, genetic variance is generated by pleiotropic effects of such deleterious mutations. If the deleterious effect is large, the predicted genetic variance is notably smaller than the equilibrium genetic variance predicted by the mutation–selection balance model with the rare-alleles approximation. The model further suggests that the ratio of the equilibrium genetic variance to the variance of allelic effects of new mutations is proportional to the ratio of total mutation load to the net selection coefficient on new mutations. It is also suggested that the net selection coefficient is equal to the per-generation proportional increment of genetic variance by new mutations.

Speciation: Founder Events and Their Effects on X-Linked and Autosomal Genes

We analyze the role that random genetic drift and founder events might play in speciation and the evolution of reproductive isolation by examining the relative probabilities that X-linked and autosomal genes affecting reproductive isolation will be involved in a drift-mediated peak shift. X chromosomes are more likely than autosomes to be affected by drift. However, the smaller effective population size and increased drift variance of X-linked genes are exactly offset by their lower, within-population, equilibrium genetic variance at mutation-selection balance. Thus, the among-deme variance of X-linked genes is equivalent to that of autosomal genes. However, the third and higher moments of the distribution of gene frequencies among populations do differentially affect the relative probabilities of peak shifts by founder event processes, such that traits controlled by a single additive X-linked gene are more likely to shift than with a similar autosomal gene. Genes affecting only one sex are more likely to...

The distribution of mutation effects on viability in Drosophila melanogaster.

Parameters of continuous distributions of effects and rates of spontaneous mutation for relative viability in Drosophila are estimated by maximum likelihood from data of two published experiments on accumulation of mutations on protected second chromosomes. A model of equal mutant effects gives a poor fit to the data of the two experiments; higher likelihoods are obtained with leptokurtic distributions or for models in which there is more than one class of mutation effect. Minimum estimates of mutation rates (events per generation) at polygenes affecting viability on chromosome 2 are 0.14 and 0.068, but estimates are strongly confounded with other parameters in the model. Separate information on rates of molecular divergence between Drosophila species and from rates of movement of transposable elements is used to infer the overall genomic mutation rate in Drosophila, and the viability data are analyzed with mutation rate as a known parameter. If, for example, a mutation rate for chromosome 2 of 0.4 is assumed, maximum likelihood estimates of mean mutant effect on relative viability are 0.4% and 1%, but the majority of mutations have very much smaller effects than these values as distributions are highly leptokurtic. The methodology is applied to estimate viability effects of single P element insertional mutations. The mean effect per insertion is found to be higher, and their distribution is found to be less leptokurtic than for spontaneous mutations. The equilibrium genetic variance of viability predicted by a mutation-selection balance model with parameters estimated from the mutation accumulation experiments is similar to laboratory estimates of genetic variance of viability from natural populations of Drosophila.

A pleiotropic nonadditive model of variation in quantitative traits.

A model of mutation-selection-drift balance incorporating pleiotropic and dominance effects of new mutations on quantitative traits and fitness is investigated and used to predict the amount and nature of genetic variation maintained in segregating populations. The model is based on recent information on the joint distribution of mutant effects on bristle traits and fitness in Drosophila melanogaster from experiments on the accumulation of spontaneous and P element-induced mutations. These experiments suggest a leptokurtic distribution of effects with an intermediate correlation between effects on the trait and fitness. Mutants of large effect tend to be partially recessive while those with smaller effect are on average additive, but apparently with very variable gene action. The model is parameterized with two different sets of information derived from P element insertion and spontaneous mutation data, though the latter are not fully known. They differ in the number of mutations per generation which is assumed to affect the trait. Predictions of the variance maintained for bristle number assuming parameters derived from effects of P element insertions, in which the proportion of mutations with an effect on the trait is small, fit reasonably well with experimental observations. The equilibrium genetic variance is nearly independent of the degree of dominance of new mutations. Heritabilities of between 0.4 and 0.6 are predicted with population sizes from 10(4) to 10(6), and most of the variance for the metric trait in segregating population is due to a small proportion of mutations (about 1% of the total number) with neutral or nearly neutral effects on fitness and intermediate effects on the trait (0.1-0.5 sigma P).(ABSTRACT TRUNCATED AT 250 WORDS)

Filling a Gap in the Prediction of the Equilibrium Genetic Variance

We derive analytical predictions for the variance and kurtosis of the equilibrium distribution of allelic effects under stabilizing selection and drift, and for any value of the kurtosis of the distribution of the mutational effects. Numerical results relative to the equilibrium genetic variance are compared with other analytical predictions and with simulation results from the literature. Our prediction is superior for relatively weak selection (or relatively low variance of the mutational effects) and not to small population sizes, where previous methods overestimate the equilibrium genetic variance. The behavior of the equilibrium kurtosis is also illustrated.

The distribution of allelic effects under mutation and selection

The Price (1970, 1972) equation is applied to the problem of describing the changes in the moments of allelic effects caused by selection, mutation and recombination at loci governing a quantitative genetic character. For comparable assumptions the resulting equations are the same as those obtained by different means by Barton & Turelli (1987; Turelli & Barton, 1989). The Price equation provides a natural framework within which to examine certain kinds of non-additive allelic effects, recombination and assortative mating. The use of the Price equation is illustrated by finding the equilibrium genetic variance under multiplicative dominance and epistasis and under assortative mating at an additive locus. The limitations of the use of recursion equations for the moments of allelic effects are also discussed.

THE MAINTENANCE OF POLYGENIC VARIATION IN FINITE POPULATIONS.

Models of the maintenance of genetic variance in a polygenic trait have usually assumed that population size is infinite and that selection is weak. Consequently, they will overestimate the amount of variation maintained in finite populations. I derive approximations for the equilibrium genetic variance, V^G in finite populations under weak stabilizing selection for triallelic loci and for an infinite "rare alleles" model. These are compared to results for neutral characters, to the "Gaussian allelic" model, and to Wright's approximation for a biallelic locus under arbitrary selection pressures. For a variety of parameter values, the three-allele, Gaussian, and Wrightian approximations all converge on the neutral model when population size is small. As expected, far less equilibrium genetic variance can be maintained if effective population size, N, is on the order of a few hundred than if N is infinite. All of the models predict that comparisons among populations with N less than about 104 should show substantial differences in V^G. While it is easier to maintain absolute V^G when alleles interact to yield dominance or overdominance for fitness, less additivity also makes V^G more susceptible to differences in N. I argue that experimental data do not seem to reflect the predicted degree of relationship between N and V^G. This calls into question the ability of mutation-selection balance or simple balancing selection to explain observed V^G. The dependence of V^G on N could be used to test the adequacy of mutation-selection balance models.