Genetic and Statistical Analyses of Strong Selection on Polygenic Traits: What,Me Normal?

We develop a general population genetic framework for analyzing selection on many loci, and apply it to strong truncation and disruptive selection on an additive polygenic trait. We first present statistical methods for analyzing the infinitesimal model, in which offspring breeding values are normally distributed around the mean of the parents, with fixed variance. These show that the usual assumption of a Gaussian distribution of breeding values in the population gives remarkably accurate predictions for the mean and the variance, even when disruptive selection generates substantial deviations from normality. We then set out a general genetic analysis of selection and recombination. The population is represented by multilocus cumulants describing the distribution of haploid genotypes, and selection is described by the relation between mean fitness and these cumulants. We provide exact recursions in terms of generating functions for the effects of selection on non-central moments. The effects of recombination are simply calculated as a weighted sum over all the permutations produced by meiosis. Finally, the new cumulants that describe the next generation are computed from the non-central moments. Although this scheme is applied here in detail only to selection on an additive trait, it is quite general. For arbitrary epistasis and linkage, we describe a consistent infinitesimal limit in which the short-term selection response is dominated by infinitesimal allele frequency changes and linkage disequilibria. Numerical multilocus results show that the standard Gaussian approximation gives accurate predictions for the dynamics of the mean and genetic variance in this limit. Even with intense truncation selection, linkage disequilibria of order three and higher never cause much deviation from normality. Thus, the empirical deviations frequently found between predicted and observed responses to artificial selection are not caused by linkage-disequilibrium-induced departures from normality. Disruptive selection can generate substantial four-way disequilibria, and hence kurtosis; but even then, the Gaussian assumption predicts the variance accurately. In contrast to the apparent simplicity of the infinitesimal limit, data suggest that changes in genetic variance after 10 or more generations of selection are likely to be dominated by allele frequency dynamics that depend on genetic details.     

Pleiotropic Stabilizing Selection Limits the Number of Polymorphic Loci to at Most the Number of Characters

We demonstrate that, in a model incorporating weak Gaussian stabilizing selection on n additively determined characters, at most n loci are polymorphic at a stable equilibrium. The number of characters is defined to be the number of independent components in the Gaussian selection scheme. We also assume linkage equilibrium, and that either the number of loci is large enough that the phenotypic distribution in the population can be approximated as multivariate Gaussian or that selection is weak enough that the mean fitness of the population can be approximated using only the mean and the variance of the characters in the population. Our results appear to rule out antagonistic pleiotropy without epistasis as a major force in maintaining additive genetic variation in a uniform environment. However, they are consistent with the maintenance of variability by genotype-environment interaction if a trait in different environments corresponds to different characters and the number of different environments exceeds the number of polymorphic loci that affect the trait.     

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.     

Quantitative genetic models for the balance between migration and stabilizing selection

The evolution of a quantitative trait subject to stabilizing selection and immigration, with the immigrants deviating from the local optimum, is considered under a number of different models of the underlying genetic basis of the trait. By comparing exact predictions under the infinitesimal model obtained using numerical methods with predictions of a simplified approximate model based on ignoring linkage disequilibrium, the increase in the expressed genetic variance as a result of linkage disequilibrium generated by migration is shown to be relatively small and negligible, provided that the genetic variance relative to the squared deviation of immigrants from the local optimum is sufficiently large or selection and migration is sufficiently weak. Deviation from normality is shown to be less important by comparing predictions of the infinitesimal model with a model presupposing normality. For a more realistic symmetric model, involving a finite number of loci only, no linkage and equal effects and frequencies across loci, additional changes in the genetic variance arise as a result of changes in underlying allele frequencies. Again, provided that the genetic variance relative to the squared deviation of the immigrants from the local optimum is small, the difference between the predictions of infinitesimal and the symmetric model are small unless the number of loci is very small. However, if the genetic variance relative to the squared deviation of the immigrants from the local optimum is large, or if selection and migration are strong, both linkage disequilibrium and changes in the genetic variance as a result of changes in underlying allele frequencies become important.     

Evolution of recombination due to random drift

In finite populations subject to selection, genetic drift generates negative linkage disequilibrium, on average, even if selection acts independently (i.e., multiplicatively) upon all loci. Negative disequilibrium reduces the variance in fitness and hence, by Fisher's (1930) fundamental theorem, slows the rate of increase in mean fitness. Modifiers that increase recombination eliminate the negative disequilibria that impede selection and consequently increase in frequency by "hitchhiking." Thus, stochastic fluctuations in linkage disequilibrium in finite populations favor the evolution of increased rates of recombination, even in the absence of epistatic interactions among loci and even when disequilibrium is initially absent. The method developed within this article allows us to quantify the strength of selection acting on a modifier allele that increases recombination in a finite population. The analysis indicates that stochastically generated linkage disequilibria do select for increased recombination, a result that is confirmed by Monte Carlo simulations. Selection for a modifier that increases recombination is highest when linkage among loci is tight, when beneficial alleles rise from low to high frequency, and when the population size is small.     

Dynamics of a quantitative character subject only to stabilising selection

The implications of stabilising selection on a quantitative trait, in the absence of other evolutionary forces, are theoretically investigated in a randomly mating population. The dynamics of various statistics that describe the alleles contributing to the trait are determined and used to infer the behaviour of the trait. Dynamical solutions of the distribution of allelic effects and the distribution of the trait are found when all initial distributions of allelic effects are Gaussian and linkage disequilibria are neglected. Some results for the behaviour of the mean and the variance of genotypic effects of the population, when subject to a moving optimum, are derived. When the initial distributions of allelic effects are not Gaussian, but possess a small asymmetry, the mean and the variance of the allelic effects differ only slightly from the Gaussian results. By contrast, the third central moments of allelic effects, are, at all loci, strictly zero in the Gaussian case but are generally non-zero for non-symmetric initial distributions. To leading order in a quantitative measure of the asymmetry of the distribution, we determine the third central moment of allelic effects.     

COMPARISON OF NON‐GAUSSIAN QUANTITATIVE GENETIC MODELS FOR MIGRATION AND STABILIZING SELECTION

The balance between stabilizing selection and migration of maladapted individuals has formerly been modeled using a variety of quantitative genetic models of increasing complexity, including models based on a constant expressed genetic variance and models based on normality. The infinitesimal model can accommodate nonnormality and a nonconstant genetic variance as a result of linkage disequilibrium. It can be seen as a parsimonious one‐parameter model that approximates the underlying genetic details well when a large number of loci are involved. Here, the performance of this model is compared to several more realistic explicit multilocus models, with either two, several or a large number of alleles per locus with unequal effect sizes. Predictions for the deviation of the population mean from the optimum are highly similar across the different models, so that the non‐Gaussian infinitesimal model forms a good approximation. It does, however, generally estimate a higher genetic variance than the multilocus models, with the difference decreasing with an increasing number of loci. The difference between multilocus models depends more strongly on the effective number of loci, accounting for relative contributions of loci to the variance, than on the number of alleles per locus.     

The evolution of genetic canalization under fluctuating selection

Abstract.— If the direction of selection changes from generation to generation, the ability to respond to selection is maladaptive: the response to selection in one generation leads to reduced fitness in the next. Because the response is determined by the amount of genetic variance expressed at the phenotypic level, rapidly fluctuating selection should favor modifier genes that reduce the phenotypic effect of alleles segregating at structural loci underlying the trait. Such reduction in phenotypic expression of genetic variation has been named "genetic canalization." I support this argument with a series of two- and multilocus models with alternating linear selection and Gaussian selection with fluctuating optimum. A canalizing modifier gene affects the fitness of its carriers in three ways: (1) it reduces the phenotypic consequences of genetic response to previous selection; (2) it reduces the genetic response to selection, which is manifested as linkage disequilibrium between the modifier and structural loci; and (3) it reduces the phenotypic variance. The first two effects reduce fitness under directional selection sustained for several generations, but improve fitness when the direction of selection has just been reversed. The net effect tends to favor a canalizing modifier under rapidly fluctuating selection regimes (period of eight generations or less). The third effect improves fitness of the modifier allele if the fitness function is convex and reduces it if the function is concave. Under fluctuating Gaussian selection, the population is more likely to experience the concave portion of the fitness function when selection is stronger. Therefore, only weak to moderately strong fluctuating Gaussian selection favors genetic canalization. This paper considerably broadens the conditions that favor genetic canalization, which so far has only been postulated to evolve under long-term stabilizing selection.     

The divergence of a polygenic system subject to stabilizing selection, mutation and drift

Polygenic variation can be maintained by a balance between mutation and stabilizing selection. When the alleles responsible for variation are rare, many classes of equilibria may be stable. The rate at which drift causes shifts between equilibria is investigated by integrating the gene frequency distribution W2N II (pq)4N mu-1. This integral can be found exactly, by numerical integration, or can be approximated by assuming that the full distribution of allele frequencies is approximately Gaussian. These methods are checked against simulations. Over a wide range of population sizes, drift will keep the population near an equilibrium which minimizes the genetic variance and the deviation from the selective optimum. Shifts between equilibria in this class occur at an appreciable rate if the product of population size and selection on each locus is small (Ns alpha 2 less than 10). The Gaussian approximation is accurate even when the underlying distribution is strongly skewed. Reproductive isolation evolves as populations shift to new combinations of alleles: however, this process is slow, approaching the neutral rate (approximately mu) in small populations.     

Migration, selection and histocompatibility

The extreme polymorphism of the HLA system allows a very precise definition of populations by gene frequencies. The existence of several linked loci (HLA-A, B, C, D, DR) induces the possibility of detection of preferential chromosomes in populations (particular associations of alleles or linkage disequilibria). These linkage disequilibria indicate that the conditions required for Hardy-Weinberg equilibrium are not met. The linkage disequilibria can be used as indicators for the evaluation of genetic dift, founder effect and inbreeding in small populations or for the detection of selection and migration in larger populations.     

The Effects of Overdominance on Linkage in a Multilocus System

Computer simulations were performed with overdominant multiple alleles among tightly linked multiple loci under a multiplicative fitness model. The quantity X²/N(n — 1) was introduced as a new measure of linkage disequilibrium which, unlike previously available measures, can be applied to multiple allele models, where N is the sample size, and n is the number of alleles at the locus possessing fewest alleles. Simulations showed that (1) With multiple (three or four) alleles, the approach to stable disequilibrium is slower and the amount of disequilibrium established is weaker than in a two allele system. (2) The number of complementary chromosomes is a function of number of alleles and of population size. (3) As population size increases, the rate of the approach to stable disequilibrium is slower. (4) There is an optimum selection coefficient which minimizes the transient fixation probability of alleles when linkage is present. (5) The absence of linkage disequilibrium is in most cases not a practical method of testing the hypothesis of balancing selection of genetic polymorphisms because it depends strongly on population size in determining linkage disequilibria.     

RESPONSE TO SELECTION IN PARTIALLY SELF-FERTILIZING POPULATIONS. I. SELECTION ON A SINGLE TRAIT

Self-fertilization is a common form of reproduction in plants and it has important implications for quantitative trait evolution. Here, I present a model of selection on quantitative traits that can accommodate any level of self-fertilization. The “structured linear model” (SLM) predicts the evolution of the mean phenotype as a function of three distinct quantities: the mean additive genetic value, the directional dominance, and the mean inbreeding coefficient. Stochastic simulations of truncation selection demonstrate the accuracy of the SLM in predicting changes in the mean and variance of a quantitative trait over the full range of selfing rates. They also illustrate how complex interactions between selection and mating system determine the population distribution of inbreeding coefficients and also the amount of linkage disequilibrium. Changes in the genetic variance due to linkage disequilibria, which are commonly referred to as the “Bulmer effect,” are greatly magnified by selfing. This complicates the relationship between selfing rate and response to selection. Like the random mating theory, the parameters of the SLM can be estimated from phenotypic data.     

ON THE EVOLUTION OF PHENOTYPIC PLASTICITY IN A SPATIALLY HETEROGENEOUS ENVIRONMENT

A genetic model for the dynamics of a quantitative trait is analyzed in terms of gene frequencies, linkage disequilibria, and environmental effects on the trait. In a randomly mating population, at each generation progeny move to niches where they are subject to weak Gaussian selection on the trait, with different fitness levels in the different niches. Initially, the variability of the trait is due to additive loci with heterozygous homeostasis. The evolution of plasticity is then described in terms of the invasion of the population by genetic modifiers that may epistatically affect the trait, its optimum in each niche, the strengths of selection, and other parameters characteristic of the niches. We show that the evolution of trait means within niches depends on the overall evolution in the whole system, and in general, optimum phenotypic values are not attained. The reaction norm and genotype-environment interaction may evolve even if the only effects of the modifier are on individual rates of dispersal, or on fitness effects resulting from the different environments in the different niches; this evolution does not require that the modifier affect parameters that influence the values of the trait. It is conjectured that in the least frequently reached niches with low fitness levels, the deviations from the trait optima should be larger than those in more commonly experienced and less stringent niches. Our analysis makes explicit the different contribution of between- and within-niche effects on the evolutionary dynamics of phenotypic plasticity in heterogeneous environments.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

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.     

Allozyme Frequency Changes Associated with Selection for Increased Grain Yield in Maize (ZEA MAYS L.)

Frequency changes of alleles at eight enzyme loci were monitored in four long-term maize selection experiments. The results indicate that changes in frequencies of the alleles at these loci are associated with changes due to selection for improved grain yield. The frequencies changed more than is consistent with the hypothesis of selective neutrality. In addition, significant deviations from a random-drift model were nearly always accompanied by significant linear trends as would result if allozyme frequencies respond to directional selection. Evaluations of linkages and linkage disequilibria in the selected populations indicate that the eight enzyme loci responded independently as selection progressed.     

Epistasis,inbreeding depression,and the evolution of self-fertilization

Inbreeding depression resulting from partially recessive deleterious alleles is thought to be the main genetic factor preventing self-fertilizing mutants from spreading in outcrossing hermaphroditic populations. However, deleterious alleles may also generate an advantage to selfers in terms of more efficient purging, while the effects of epistasis among those alleles on inbreeding depression and mating system evolution remain little explored. In this article, we use a general model of selection to disentangle the effects of different forms of epistasis (additive-by-additive, additive-by-dominance, and dominance-by-dominance) on inbreeding depression and on the strength of selection for selfing. Models with fixed epistasis across loci, and models of stabilizing selection acting on quantitative traits (generating distributions of epistasis) are considered as special cases. Besides its effects on inbreeding depression, epistasis may increase the purging advantage associated with selfing (when it is negative on average), while the variance in epistasis favors selfing through the generation of linkage disequilibria that increase mean fitness. Approximations for the strengths of these effects are derived, and compared with individual-based simulation results.     

Genetic differentiation of neutral markers and quantitative traits in predominantly selfing metapopulations: confronting theory and experiments with Arabidopsis thaliana

The comparison of the genetic differentiation of quantitative traits (QST) and molecular markers (FST) can inform on the strength and spatial heterogeneity of selection in natural populations, provided that markers behave neutrally. However, selection may influence the behaviour of markers in selfing species with strong linkage disequilibria among loci, therefore invalidating this test of detection of selection. We address this issue by monitoring the genetic differentiation of five microsatellite loci (FST) and nine quantitative traits (QST) in experimental metapopulations of the predominantly selfing species Arabidopsis thaliana, that evolved during eight generations. Metapopulations differed with respect to population size and selection heterogeneity. In large populations, the genetic differentiation of neutral microsatellites was much larger under heterogeneous selection than under uniform selection. Using simulations, we show that this influence of selection heterogeneity on FST can be attributable to initial linkage disequilibria among loci, creating stronger genetic differentiation of QTL than expected under a simple additive model with no initial linkage. We found no significant differences between FST and QST regardless of selection heterogeneity, despite a demonstrated effect of selection on QST values. Additional data are required to validate the role of mating system and linkage disequilibria in the joint evolution of neutral and selected genetic differentiation, but our results suggest that FST/QST comparisons can be conservative tests to detect selection in selfing species.     

Deleterious mutations in a hybrid zone: can mutational load decrease the barrier to gene flow?

The aim of this paper is to investigate the effect of deleterious mutations in a hybrid zone maintained by selection against hybrids. In such zones, linkage disequilibria among hybrid depression loci, resulting from a balance between migration and selection, are crucial in maintaining the barrier because they allow each locus, in addition to its own selection coefficient, to cumulate indirect selective effects from other loci. Deleterious alleles produce heterosis and increase by this means the effective migration rate in structured populations. In a hybrid zone, they therefore contribute to decrease linkage disequilibria as well as the barrier to gene flow imposed by hybrid depression. However, deleterious mutations have no effect: (i) when selection against hybrids is weak, because linkage disequilibria are small even without heterosis in this case, or (ii) when selection against hybrids is so strong that it overwhelms heterosis. On the other hand, with moderate selection against hybrids, the decrease in the strength of the barrier due to heterosis may reach detectable levels, although it requires relatively small population sizes and/or migration rates. The effect is expected to be small and only within small genomes where loci are tightly linked can it become strong. Nevertheless, neglecting mutational load may to some extent obscure the estimations of selective parameters based either on artificial F1 crosses or on cline characteristics.     

Quantitative genetic variation in Daphnia: temporal changes in genetic architecture

Nonadditive genetic variation and genetic disequilibrium are two important factors that influence the evolutionary trajectory of natural populations. We assayed quantitative genetic variation in a temporary-pond-dwelling population of Daphnia pulex over a full season to examine the role of nonadditive genetic variation and genetic disequilibrium in determining the short-term evolutionary trajectory of a cyclic parthenogen. Quantitative traits were influenced by three factors: (1) clonal selection significantly changed the population mean phenotype during the course of the growing season; (2) sexual reproduction and recombination led to significant changes in life-history trait means and the levels of expressed genetic variation, implying the presence of substantial nonadditive genetic variation and genetic disequilibrium; and (3) Egg-bank effects were found to be an important component of the realized year-to-year change. Additionally, we examined the impact of genetic disequilibria induced by clonal selection on the genetic (co)variance structure with a common principal components model. Clonal selection caused significant changes in the (co)variance structure that were eliminated by a single bout of random mating, suggesting that a build-up of disequilibria was the primary source of changes in the (co)variance structure. The results of this study highlight the complexity of natural selection operating on populations that undergo alternating phases of sexual and asexual reproduction.