Linkage and the Maintenance of Heritable Variation by Mutation-Selection Balance

The role of linkage in influencing heritable variation maintained through a balance between mutation and stabilizing selection is investigated for two different models. In both cases one trait is considered and the interactions within and between loci are assumed to be additive. Contrary to most earlier investigations of this problem no a priori assumptions on the distribution of genotypic values are imposed. For a deterministic two-locus two-allele model with recombination and mutation, related to the symmetric viability model, a complete nonlinear analysis is performed. It is shown that, depending on the recombination rate, multiple stable equilibria may coexist. The equilibrium genetic and genic variances are calculated. For a polygenic trait in a finite population with a possible continuum of allelic effects a simulation study is performed. In both models the equilibrium genetic and genic variances are roughly equal to the house-of-cards prediction or its finite population counterpart as long as the recombination rate is not extremely low. However, negative linkage disequilibrium builds up. If the loci are very closely linked the equilibrium additive genetic variance is slightly lower than the house-of-cards prediction, but the genic variance is much higher. Depending on whether the parameters are in favor of the house-of-cards or the Gaussian approximation, different behavior of the genetic system occurs with respect to linkage.     

A population genetics model for multiple quantitative traits exhibiting pleiotropy and epistasis

We study a population genetics model of an organism with a genome of L(tot)loci that determine the values of T quantitative traits. Each trait is controlled by a subset of L loci assigned randomly from the genome. There is an optimum value for each trait, and stabilizing selection acts on the phenotype as a whole to maintain actual trait values close to their optima. The model contains pleiotropic effects (loci can affect more than one trait) and epistasis in fitness. We use adaptive walk simulations to find high-fitness genotypes and to study the way these genotypes are distributed in sequence space. We then simulate the evolution of haploid and diploid populations on these fitness landscapes and show that the genotypes of populations are able to drift through sequence space despite stabilizing selection on the phenotype. We study the way the rate of drift and the extent of the accessible region of sequence space is affected by mutation rate, selection strength, population size, recombination rate, and the parameters L and T that control the landscape shape. There are three regimes of the model. If LTL(tot), there are many small peaks that can be spread over a wide region of sequence space. Compensatory neutral mutations are important in the population dynamics in this case.     

The relation between the genetic architecture of quantitative traits and long-term genetic response

The genetic architecture of a quantitative trait refers to the number of genetic variants, allele frequencies, and effect sizes of variants that affect a trait and their mode of gene action. This study was conducted to investigate the effect of four shapes of allelic frequency distributions (constant, uniform, L-shaped and U-shaped) and different number of trait-affecting loci (50, 100, 200, 500) on allelic frequency changes, long term genetic response, and maintaining genetic variance. To this end, a population of 440 individuals composed of 40 males and 400 females as well as a genome of 200 cM consisting of two chromosomes and with a mutation rate of 2.5?×?10^?5 per locus was simulated. Selection of superior animals was done using best linear unbiased prediction (BLUP) with assumption of infinitesimal model. Selection intensity was constant over 30 generations of selection. The highest genetic progress obtained when the allelic frequency had L-shaped distribution and number of trait-affecting loci was high (500). Although quantitative genetic theories predict the extinction of genetic variance due to artificial selection in long time, our results showed that under L- and U-shapped allelic frequency distributions, the additive genetic variance is persistent after 30 generations of selection. Further, presence or absence of selection limit can be an indication of low (<50) or high (>100) number of trait-affecting loci, respectively. It was concluded that the genetic architecture of complex traits is an important subject which should be considered in studies concerning long-term response to selection.     

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.     

Near-periodic substitution and the genetic variance induced by environmental change

We investigate a model that describes the evolution of a diploid sexual population in a changing environment. Individuals have discrete generations and are subject to selection on the phenotypic value of a quantitative trait, which is controlled by a finite number of bialleic loci. Environmental change is taken to lead to a uniformly changing optimal phenotypic value. The population continually adapts to the changing environment, by allelic substitution, at the loci controlling the trait. We investigate the detailed interrelation between the process of allelic substitution and the adaptation and variation of the population, via infinite population calculations and finite population simulations. We find a simple relation between the substitution rate and the rate of change of the optimal phenotypic value.     

Decoupling of differentiation between traits and their underlying genes in response to divergent selection

We dissected the relationship between genetic differentiation (Q(ST)) for a trait and its underlying genes (G(STq), differentiation for a quantitative locus) in an evolutionary context, with the aim of identifying the conditions in which these two measurements are decoupled. We used two parameters (θ(B) and θ(W)) scaling the contributions of inter- and intrapopulation allelic covariation between genes controlling the trait of interest. We monitored the changes in θ(B) and θ(W), Q(ST) and G(STq) over successive generations of divergent and stabilizing selection, in simulations for an outcrossing species with extensive gene flow. The dynamics of these parameters are characterized by two phases. Initially, during the earliest generations, differentiation of the trait increases very rapidly and the principal and immediate driver of Q(ST) is θ(B). During subsequent generations, G(STq) increases steadily and makes an equal contribution to Q(ST). These results show that selection first captures beneficial allelic associations at different loci at different populations, and then targets changes in allelic frequencies. The same patterns are observed when environmental change modifies divergent selection, as shown by the very rapid response of θ(B) to the changes of selection regimes. We compare our results with previous experimental findings and consider their relevance to the detection of molecular signatures of natural selection.     

Clines in polygenic traits

This article outlines theoretical models of clines in additive polygenic traits, which are maintained by stabilizing selection towards a spatially varying optimum. Clines in the trait mean can be accurately predicted, given knowledge of the genetic variance. However, predicting the variance is difficult, because it depends on genetic details. Changes in genetic variance arise from changes in allele frequency, and in linkage disequilibria. Allele frequency changes dominate when selection is weak relative to recombination, and when there are a moderate number of loci. With a continuum of alleles, gene flow inflates the genetic variance in the same way as a source of mutations of small effect. The variance can be approximated by assuming a Gaussian distribution of allelic effects; with a sufficiently steep cline, this is accurate even when mutation and selection alone are better described by the 'House of Cards' approximation. With just two alleles at each locus, the phenotype changes in a similar way: the mean remains close to the optimum, while the variance changes more slowly, and over a wider region. However, there may be substantial cryptic divergence at the underlying loci. With strong selection and many loci, linkage disequilibria are the main cause of changes in genetic variance. Even for strong selection, the infinitesimal model can be closely approximated by assuming a Gaussian distribution of breeding values. Linkage disequilibria can generate a substantial increase in genetic variance, which is concentrated at sharp gradients in trait means.     

COMPUTER PROGRAMS: nessi: a program for numerical estimations in sporophytic self-incompatibility genetic systems

nessi is a computer program generating predictions about allelic and genotypic frequencies at the S-locus in sporophytic self-incompatibility systems under finite and infinite populations. For any pattern of dominance relationships among self-incompatibility alleles, nessi computes deterministic equilibrium frequencies and estimates distributions in samples from finite populations of the number of alleles at equilibrium, allelic and genotypic frequencies at equilibrium and allelic and genotypic frequency changes in a single generation. These predictions can be used to rigorously test the impact of negative frequency-dependent selection on diversity patterns in natural populations.     

The diffusion model for migration and selection in a dioecious population

The diffusion approximation is derived for migration and selection at a multiallelic locus in a dioecious population subdivided into a lattice of panmictic colonies. Generations are discrete and nonoverlapping; autosomal and X-linked loci are analyzed. The relation between juvenile and adult subpopulation numbers is very general and includes both soft and hard selection; the zygotic sex ratio is the same in every colony. All the results hold for both adult and juvenile migration. If ploidy-weighted average selection, drift, and diffusion coefficients are used, then the ploidy-weighted average allelic frequencies satisfy the corresponding partial differential equation for a monoecious population. The boundary conditions and the unidimensional transition conditions for coincident discontinuities in the carrying capacity and migration rate extend identically. The previous unidimensional formulation and analysis of symmetric, nearest-neighbor migration of a monoecious population across a geographical barrier is generalized to symmetric migration of arbitrary finite range, and the transition conditions are shown to hold for a dioecious population. Thus, the entire theory of clines and of the wave of advance of favorable alleles is applicable to dioecious populations.This work was supported by National Science Foundation grant BSR-9006285     

DRIFT PROMOTES SPECIATION BY SEXUAL SELECTION

Quantitative genetic models of sexual selection have generally failed to provide a direct connection to speciation and to explore the consequences of finite population size. The connection to speciation has been indirect because the models have treated only the evolution of male and female traits and have stopped short of modeling the evolution of sexual isolation. In this article we extend Lande's (1981) model of sexual selection to quantify predictions about the evolution of sexual isolation and speciation. Our results, based on computer simulations, support and extend Lande's claim that drift along a line of equilibria can rapidly lead to sexual isolation and speciation. Furthermore, we show that rapid speciation can occur by drift in populations of appreciable size ( N_e ≥ 1000). These results are in sharp contrast to the opinion of many researchers and textbook writers who have argued that drift does not play an important role in speciation. We argue that drift may be a powerful amplifier of speciation under a wide variety of modeling assumptions, even when selection acts directly on female mating preferences.     

Polygenic variation maintained by balancing selection: pleiotropy, sex-dependent allelic effects and G x E interactions

We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G x E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model produces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleiotropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G x E can maintain variation at loci for which the coefficient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The critical value depends on the correlation between substitution effects at different loci. For large positive correlations (e.g., rho(ij)2>3/4), even extreme fluctuations in allelic effects cannot maintain variation. Surprisingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms.     

Genetic architecture of complex traits and accuracy of genomic prediction: coat colour, milk-fat percentage, and type in Holstein cattle as contrasting model traits

Prediction of genetic merit using dense SNP genotypes can be used for estimation of breeding values for selection of livestock, crops, and forage species; for prediction of disease risk; and for forensics. The accuracy of these genomic predictions depends in part on the genetic architecture of the trait, in particular number of loci affecting the trait and distribution of their effects. Here we investigate the difference among three traits in distribution of effects and the consequences for the accuracy of genomic predictions. Proportion of black coat colour in Holstein cattle was used as one model complex trait. Three loci, KIT, MITF, and a locus on chromosome 8, together explain 24% of the variation of proportion of black. However, a surprisingly large number of loci of small effect are necessary to capture the remaining variation. A second trait, fat concentration in milk, had one locus of large effect and a host of loci with very small effects. Both these distributions of effects were in contrast to that for a third trait, an index of scores for a number of aspects of cow confirmation ("overall type"), which had only loci of small effect. The differences in distribution of effects among the three traits were quantified by estimating the distribution of variance explained by chromosome segments containing 50 SNPs. This approach was taken to account for the imperfect linkage disequilibrium between the SNPs and the QTL affecting the traits. We also show that the accuracy of predicting genetic values is higher for traits with a proportion of large effects (proportion black and fat percentage) than for a trait with no loci of large effect (overall type), provided the method of analysis takes advantage of the distribution of loci effects.     

The genetic differentiation at quantitative trait loci under local adaptation

Most adaptive traits are controlled by large number of genes that may all together be the targets of selection. Adaptation may thus involve multiple but not necessarily substantial allele frequency changes. This has important consequences for the detection of selected loci and implies that a quantitative genetics framework may be more appropriate than the classical 'selective sweep' paradigm. Preferred methods to detect loci involved in local adaptation are based on the detection of 'outlier' values of the allelic differentiation F(ST) . A quantitative genetics framework is adopted here to review theoretical expectations for how allelic differentiation at quantitative trait loci (F(STQ) ) relates to (i), neutral genetic differentiation (F(ST) ) and (ii), phenotypic differentiation (Q(ST) ). We identify cases where results of outlier-based methods are likely to be poor and where differentiation at selected loci conveys little information regarding local adaptation. A first case is when neutral differentiation is high, so that local adaptation does not necessitate increased differentiation. A second case is when local adaptation is reached via an increased covariance of allelic effects rather than via allele frequency changes, which is more likely under high gene flow when the number of loci is high and selection is recent. The comparison of theoretical predictions with observed data from the literature suggests that polygenic local adaptation involving only faint allele frequency changes are very likely in some species such as forest trees and for climate-related traits. Recent methodological improvements that may alleviate the weakness of F(ST) -based detection methods are presented.     

Intraspecific trait variation across elevation predicts a widespread tree species' climate niche and range limits

Global change is widely altering environmental conditions which makes accurately predicting species range limits across natural landscapes critical for conservation and management decisions. If climate pressures along elevation gradients influence the distribution of phenotypic and genetic variation of plant functional traits, then such trait variation may be informative of the selective mechanisms and adaptations that help define climatic niche limits. Using extensive field surveys along 16 elevation transects and a large common garden experiment, we tested whether functional trait variation could predict the climatic niche of a widespread tree species (Populus angustifolia) with a double quantile regression approach. We show that intraspecific variation in plant size, growth, and leaf morphology corresponds with the species' total climate range and certain climatic limits related to temperature and moisture extremes. Moreover, we find evidence of genetic clines and phenotypic plasticity at environmental boundaries, which we use to create geographic predictions of trait variation and maximum values due to climatic constraints across the western US. Overall, our findings show the utility of double quantile regressions for connecting species distributions and climate gradients through trait‐based mechanisms. We highlight how new approaches like ours that incorporate genetic variation in functional traits and their response to climate gradients will lead to a better understanding of plant distributions as well as identifying populations anticipated to be maladapted to future environments.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Long-term impacts of genome-enabled selection

The objective was to evaluate the effects of directional selection based on estimated genomic breeding values (GEBVs) for a quantitative trait. Selection affects GEBV prediction accuracy as well as genetic architecture via changes in allelic frequencies and linkage disequilibrium (LD), and the resulting changes are different from those in the absence of selection. How marker density affects long-term GEBV accuracy and selection response needs to be understood as well. Simulations were used to characterize the impact of selection based on GEBVs over generations. Single-nucleotide polymorphism (SNP) marker effects were estimated with the Bayesian Lasso method in the base generation, and these estimates were used to calculate the GEBVs in subsequent generations. GEBV accuracy decreased over generations of selection, and it was lower than under random selection, where a decay took place as well. In the long term, selection response tended to reach a plateau, but, at higher marker density, both the magnitude and duration of the response were larger. Selection changed quantitative trait loci (QTL) allele frequencies and generated new but unfavorable LD for prediction. Family effects had a considerable contribution to GEBV accuracy in early generations of selection.     

Polygenic score accuracy in ancient samples: Quantifying the effects of allelic turnover

Polygenic scores link the genotypes of ancient individuals to their phenotypes, which are often unobservable, offering a tantalizing opportunity to reconstruct complex trait evolution. In practice, however, interpretation of ancient polygenic scores is subject to numerous assumptions. For one, the genome-wide association (GWA) studies from which polygenic scores are derived, can only estimate effect sizes for loci segregating in contemporary populations. Therefore, a GWA study may not correctly identify all loci relevant to trait variation in the ancient population. In addition, the frequencies of trait-associated loci may have changed in the intervening years. Here, we devise a theoretical framework to quantify the effect of this allelic turnover on the statistical properties of polygenic scores as functions of population genetic dynamics, trait architecture, power to detect significant loci, and the age of the ancient sample. We model the allele frequencies of loci underlying trait variation using the Wright-Fisher diffusion, and employ the spectral representation of its transition density to find analytical expressions for several error metrics, including the expected sample correlation between the polygenic scores of ancient individuals and their true phenotypes, referred to as polygenic score accuracy. Our theory also applies to a two-population scenario and demonstrates that allelic turnover alone may explain a substantial percentage of the reduced accuracy observed in cross-population predictions, akin to those performed in human genetics. Finally, we use simulations to explore the effects of recent directional selection, a bias-inducing process, on the statistics of interest. We find that even in the presence of bias, weak selection induces minimal deviations from our neutral expectations for the decay of polygenic score accuracy. By quantifying the limitations of polygenic scores in an explicit evolutionary context, our work lays the foundation for the development of more sophisticated statistical procedures to analyze both temporally and geographically resolved polygenic scores.     

Genetic response from marker assisted selection in an outbred population for differing marker bracket sizes and with two identified quantitative trait loci.

Effect of flanking quantitative trait loci (QTL)-marker bracket size on genetic response to marker assisted selection in an outbred population was studied by simulation of a nucleus breeding scheme. In addition, genetic response with marker assisted selection (MAS) from two quantitative trait loci on the same and different chromosome(s) was investigated. QTL that explained either 5% or 10% of phenotypic variance were simulated. A polygenic component was simulated in addition to the quantitative trait loci. In total, 35% of the phenotypic variance was due to genetic factors. The trait was measured on females only. Having smaller marker brackets flanking the QTL increased the genetic response from MAS selection. This was due to the greater ability to trace the QTL transmission from one generation to the next with the smaller flanking QTL-marker bracket, which increased the accuracy of estimation of the QTL allelic effects. Greater negative covariance between effects at both QTL was observed when two QTL were located on the same chromosome compared to different chromosomes. Genetic response with MAS was greater when the QTL were on the same chromosome in the early generations and greater when they were on different chromosomes in the later generations of MAS.     

Genotypic distribution at the limits to natural and artificial selection with mutation

A general procedure for analysing the change of genotypic distributions under stabilizing and truncation selection is described here and used to investigate the genotypic distribution at the limits to selection. For comparison, a simple approximate procedure using a normal distribution is also presented. It is clear that in the long term truncation introduces departures from normality mainly through gene frequency change, rather than through the generation of linkage disequilibrium under random mating. The Gaussian approximation performs reasonably well for additive gene effects unless the mean gene frequency is very extreme (say, outside the range of 0.05 to 0.95) and the number of loci is small (say, less then 50) regardless of the type of selection in operation. The genotypic distribution at the limits to selection largely depends on the type of limit reached. If a limit is obtained due to the action of natural selection before the exhaustion of existing variation, the distribution will normally not be very skew, but if a limit is reached at which mutation plays a central role in the maintenance of genetic variability, it could have high coefficients of skewness and kurtosis. The role of mutation on the long-term response is also discussed.     

Population structure and strong divergent selection shape phenotypic diversification in maize landraces

To conserve the long-term selection potential of maize, it is necessary to investigate past and present evolutionary processes that have shaped quantitative trait variation. Understanding the dynamics of quantitative trait evolution is crucial to future crop breeding. We characterized population differentiation of maize landraces from the State of Oaxaca, Mexico for quantitative traits and molecular markers. Qst values were much higher than Fst values obtained for molecular markers. While low values of Fst (0.011 within-village and 0.003 among-villages) suggest that considerable gene flow occurred among the studied populations, high levels of population differentiation for quantitative traits were observed (ie an among-village Qst value of 0.535 for kernel weight). Our results suggest that although quantitative traits appear to be under strong divergent selection, a considerable amount of gene flow occurs among populations. Furthermore, we characterized nonproportional changes in the G matrix structure both within and among villages that are consequences of farmer selection. As a consequence of these differences in the G matrix structure, the response to multivariate selection will be different from one population to another. Large changes in the G matrix structure could indicate that farmers select for genes of major and pleiotropic effect. Farmers' decision and selection strategies have a great impact on phenotypic diversification in maize landraces.