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.     

Definition and Estimation of Higher-Order Gene Fixation Indices

Fixation indices summarize the associations between genes that arise from the joint effects of inbreeding and selection. In this paper, fixation indices are derived for pairs, triplets and quadruplets of genes at a single multiallelic locus. The fixation indices are obtained by dividing cumulants by constants; the cumulants describe the statistical distribution of alleles and the constants are functions of gene frequency. The use of cumulants instead of moments is necessary only for four-gene indices, when the fourth cumulant is used. A second type of four-gene index is also required, and this index is based upon the covariation of second-order cumulants. At multiallelic loci, a large number of indices is possible. If alleles are selectively neutral, the number of indices is reduced and the relationship between gene identity and gene cumulants is shown.--Two-gene indices can always be estimated from genotypic frequency data at a single polymorphic locus. Three-gene indices are also estimable except when allele frequency equals one-half. Four-gene indices are not estimable unless selection is assumed to have an equal effect upon each allele (such as under selective neutrality) and the locus contains at least three alleles of unequal frequency. For diallelic or selected loci, an alternative four-gene fixation index is proposed. This index incorporates both types of four-gene associations but cannot be related to gene identity.     

Pleiotropic Models of Polygenic Variation, Stabilizing Selection, and Epistasis

We show that in polymorphic populations many polygenic traits pleiotropically related to fitness are expected to be under apparent ``stabilizing selection' independently of the real selection acting on the population. This occurs, for example, if the genetic system is at a stable polymorphic equilibrium determined by selection and the nonadditive contributions of the loci to the trait value either are absent, or are random and independent of those to fitness. Stabilizing selection is also observed if the polygenic system is at an equilibrium determined by a balance between selection and mutation (or migration) when both additive and nonadditive contributions of the loci to the trait value are random and independent of those to fitness. We also compare different viability models that can maintain genetic variability at many loci with respect to their ability to account for the strong stabilizing selection on an additive trait. Let V(m) be the genetic variance supplied by mutation (or migration) each generation, V(g) be the genotypic variance maintained in the population, and n be the number of the loci influencing fitness. We demonstrate that in mutation (migration)-selection balance models the strength of apparent stabilizing selection is order V(m)/V(g). In the overdominant model and in the symmetric viability model the strength of apparent stabilizing selection is approximately 1/(2n) that of total selection on the whole phenotype. We show that a selection system that involves pairwise additive by additive epistasis in maintaining variability can lead to a lower genetic load and genetic variance in fitness (approximately 1/(2n) times) than an equivalent selection system that involves overdominance. We show that, in the epistatic model, the apparent stabilizing selection on an additive trait can be as strong as the total selection on the whole phenotype.     

Powerful detection of polygenic selection and evidence of environmental adaptation in US beef cattle

Selection on complex traits can rapidly drive evolution, especially in stressful environments. This polygenic selection does not leave intense sweep signatures on the genome, rather many loci experience small allele frequency shifts, resulting in large cumulative phenotypic changes. Directional selection and local adaptation are changing populations; but, identifying loci underlying polygenic or environmental selection has been difficult. We use genomic data on tens of thousands of cattle from three populations, distributed over time and landscapes, in linear mixed models with novel dependent variables to map signatures of selection on complex traits and local adaptation. We identify 207 genomic loci associated with an animal’s birth date, representing ongoing selection for monogenic and polygenic traits. Additionally, hundreds of additional loci are associated with continuous and discrete environments, providing evidence for historical local adaptation. These candidate loci highlight the nervous system’s possible role in local adaptation. While advanced technologies have increased the rate of directional selection in cattle, it has likely been at the expense of historically generated local adaptation, which is especially problematic in changing climates. When applied to large, diverse cattle datasets, these selection mapping methods provide an insight into how selection on complex traits continually shapes the genome. Further, understanding the genomic loci implicated in adaptation may help us breed more adapted and efficient cattle, and begin to understand the basis for mammalian adaptation, especially in changing climates. These selection mapping approaches help clarify selective forces and loci in evolutionary, model, and agricultural contexts.     

Contemporary selection pressures in modern societies? Which factors best explain variance in human reproduction and mating?

Phenotypic traits in humans are under selection pressure and are still evolving, but the relative importance of these traits remains to be investigated. We therefore analyzed jointly phenotypic traits associated with number of children and having ever been married. This provides insights into the relative contribution of each trait and indicates the potential selection pressure induced by a specific trait relative to others. To shed light on potential selection on the genome level, all analyses include a multivariate polygenic risk score of general cognitive ability. We used the data from the Wisconsin Longitudinal Study (WLS), a dataset consisting of 4991 men and 5326 women almost all whites, educated at least at A-level. The focus was on the association between age, education level, wages, religious intensity, fathers' age at child's birth, ratings of facial attractiveness, number of siblings of the respondent, as well as the polygenic risk score of general cognitive ability on the following dependent variables: i) number of children, ii) ever being married, and iii) age at first birth. For each factor we additionally examined the relative contribution to the overall variance explained of the dependent variable. Having been married and, thus, mate selection, is the most important determinant for the number of children for both men and women. Wages explain most of the total variance for “ever married”, yet in different directions for men and women, as is also the case for the association between wages and number of children. In both women and men, education explains most of the variance in age at first birth, and the effect is postponing. Furthermore, although the phenotype education is negatively associated with the number of children in both sexes, this holds true for the polygenic risk score for cognitive ability only in men. In addition, in men, the polygenic risk score for cognitive ability also has a positive effect on reproduction due to its positive interaction with wages. Anyhow, with the exception of having ever been married, all other variables explain only a small proportion of the variation in fertility outcomes. Although our results are consistent with the hypothesis that there is selection pressure for rather recently arising traits as education and income, on the basis of our results we are not able to draw any final conclusion on selection.     

On flexible finite polygenic models for multiple-trait evaluation

Finite polygenic models (FPM) might be an alternative to the infinitesimal model (TIM) for the genetic evaluation of pedigreed multiple-generation populations for multiple quantitative traits. I present a general flexible Bayesian method that includes the number of genes in the FPM as an additional random variable. Markov-chain Monte Carlo techniques such as Gibbs sampling and the reversible jump sampler are used for implementation. Sampling of genotypes of all genes in the FPM is done via the use of segregation indicators. A broad range of FPM models, some combined with TIM, are empirically tested for the estimation of variance components and the number of genes in the FPM. Four simulation scenarios were studied, including genetic models with 5 or 50 additive independent diallelic genes affecting the traits, and random selection or selection on one of the traits was performed. The results in this study were based on ten replicates per simulation scenario. In the case of random selection, uniform priors on additive gene effects led to posterior mean estimates of genetic variance that were positively correlated with the number of genes fitted in the FPM. In the case of trait selection, assuming normal priors on gene effects also led to genetic variance estimates for the selected trait that were negatively correlated with the number of genes in the FPM. This negative correlation was not observed for the unselected trait. Treating the number of genes in the FPM as random revealed a positive correlation between prior and posterior mean estimates of this number, but the prior hardly affected the posterior estimates of genetic variance. Posterior inferences about the number of genes should be considered to be indicative where trait selection seems to improve the power of distinguishing between TIM and FPM. Based on the results of this study, I suggest not replacing TIM by the FPM, but combining TIM and FPM with the number of genes treated as random, to facilitate a highly flexible and thereby robust method for variance component estimation in pedigreed populations. Further study is required to explore the full potential of these models under different genetic model assumptions.     

Multivariate stabilizing selection and pleiotropy in the maintenance of quantitative genetic variation

Abstract. We investigate maintenance of quantitative genetic variation at mutation-selection balance for multiple traits. The intrinsic strength of real stabilizing selection on one of these traits denoted the "target trait" and the observed strength of apparent stabilizing selection on the target trait can be quite different: the latter, which is estimable, is much smaller (i.e., implying stronger selection) than the former. Distinguishing them may enable the mutation load to be relaxed when considering multivariate stabilizing selection. It is shown that both correlations among mutational effects and among strengths of real stabilizing selection on the traits are not important unless they are high. The analysis for independent situations thus provides a good approximation to the case where mutant and stabilizing selection effects are correlated. Multivariate stabilizing selection can be regarded as a combination of stabilizing selection on the target trait and the pleiotropic direct selection on fitness that is solely due to the effects of real stabilizing selection on the hidden traits. As the overall fitness approaches a constant value as the number of traits increases, multivariate stabilizing selection can maintain abundant genetic variance only under quite weak selection. The common observations of high polygenic variance and strong stabilizing selection thus imply that if the mutation-selection balance is the true mechanism of maintenance of genetic variation, the apparent stabilizing selection cannot arise solely by real stabilizing selection simultaneously on many metric traits.     

Parallel selection: evolution's surprising predictability

The mechanistic basis of how polygenic traits respond to selection is not well understood. New research provides compelling evidence for widespread parallel selection in independent mouse strains selected for extreme body weight.     

SELECTION, PREDICTION AND RESPONSE

1. The biometric approach to selection experiments has been outlined, and found to be rather deficient because it is based on excessively restrictive models which cannot take into account the complex architectures of quantitative traits as are being revealed today. 2. The nature of polygenes is discussed in detail from the theoretical point of view. In out breeding species, some form of the balanced polygenic complex is likely, showing polymorphism for the constituent genes. Although polymorphism is implicit in the argument, definitive evidence for poiymorphisms has only just appeared. 3. There is no evidence that polygenes differ from any other gene. 4. Several artificial selection experiments are described, in particular in Drosophila. By means of appropriate breeding techniques after obtaining responses to selection, genetic activity controlling quantitative traits can be located to chromosomes, and even specific loci found. Such few studies as have been carried out reveal, in general, the types of genetic architecture predicted on theoretical grounds. 5. Selection for behavioural traits is considered briefly and it appears that no new principles are needed, except that careful environmental control and objective measurement present problems. 6. The results of selection for quantitative traits in micro-organisms reveal similarities to results in higher organisms in the few cases where definitive work has been carried out. 7. Work on the simulation of models by computers has not greatly advanced selection experiment theory, mainly because, with few exceptions, linkage has been ignored. 8. The existing theory on which selection experiments are based is inadequate for several reasons. It cannot predict the rate of response to selection nor the ultimate limits to selection, the nature of correlated responses to selection, nor the nature of gene segregation underlying the observed variability. 9. Strains set up from single inseminated founder females from the same population of Drosophila have been shown to differ genetically for several quantitative traits. Therefore the base population is polymorphic for genes controlling these traits. This was exploited by carrying out directional selection on lines derived from those strains showing a high incidence of scutellar chaetae. This led to far more rapid responses to selection than lines derived from strains where the incidence of scutellar chaetae was lower. 10. Ultimately, one can envisage the selection experiment as it is known today being partly replaced by the manipulation of located genes controlling quantitative traits into certain combinations.     

Spatial patterns of polygenic variation in Impatiens capensis,a species with an environmentally controlled mixed mating system

Impatiens capensis displays a mixed mating system in which individual out-crossing rate is expected to increase with light and resource availability. We investigated the amount and spatial distribution of polygenic variation for 15 morphological traits within and among six natural populations of I. capensis growing in three distinct light habitats (shaded, mixed, full sun). We grew individuals from each population in uniform greenhouse conditions and detected significant genetic variation among families within populations for all the quantitative traits examined. However, only the features related to the vegetative characteristics of seedlings and sexually mature plants show also differentiation at the population level. Surprisingly, even though light availability is likely to be the most important factor affecting the mating system of I. capensis, we find that: (1) trait means of individuals from similar light environments are not more similar than those from different light environments; (2) partitioning of polygenic variance within and among families differs both among populations from the same light habitat and among characters within each population. If natural selection is maintaining such variation, it must operate primarily through heterogeneous selection pressure within, rather than between, populations.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

Identification of ISSR markers associated with productivity traits in silkworm, Bombyx moni L.

Bombyx mori L., commonly recognised around the world as the mulberry silkworm, is characterized by a wide variability in yield and developmental traits, which have been proven through conventional genetic analysis to be of polygenic nature. A large number of morpho-biochemical traits and RFLP and RAPD markers are mapped on different linkage groups, but to this point very little attention has been given to unravelling the genetics of yield traits. To address this issue, polymorphic profiles of 147 markers generated with 12 ISSR primers on the genomic DNA of 20 silkworm stocks of diverse yield status were subjected to multiple regression and discriminant function analyses (DFA). This led to the identification of eight markers generated by six primers, which demonstrated high beta-coefficient indices of -0.451 to -0.940. Furthermore, a significant difference between the yield traits for stocks with and without the specific marker could also be established. The inheritance pattern of one marker, L13800bp, identified at the first step of selection of markers through stepwise regression analyses for five yield parameters is discussed in the context of applying multiple regression analysis for establishing association, if not linkage, between a group of DNA markers and a particular yield trait of polygenic nature and using such markers in molecular marker-assisted breeding programs.     

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.     

Levels of selection on threshold characters

Threshold models are useful for understanding the evolution of dimorphic traits with polygenic bases. Selection for threshold characters on individuals is expected to be frequency dependent because of the peculiar way that selection views underlying genetic and environmental factors. Selection among individuals is inefficient because individual phenotypes fall into only two discrete categories that map imperfectly to the underlying genes. Incidence, however, can be continuously distributed among groups, making among-group selection relatively more efficient. Differently put, the group-mean phenotype can be a better predictor of an individual's genotype than that individual's own phenotype. Because evolution in group-structured populations is governed by the balance of selection within and between groups, we can expect threshold traits to evolve in fundamentally different ways when group mean fitness is a function of morph frequency. We extend the theory of selection on threshold traits to include group selection using contextual analysis. For the simple case of linear group-fitness functions, we show that the group-level component of selection, like the individual-level component, is frequency dependent. However, the conditions that determine which component dominates when levels of selection are in conflict (as described by Hamilton's rule) are not frequency dependent. Thus, enhanced group selection is not an inherent property of threshold characters. Nevertheless, we show that predicting the effects of multiple levels of selection on dimorphic traits requires special considerations of the threshold model.     

Towards genomic selection in apple (Malus ×domestica Borkh.) breeding programmes: Prospects, challenges and strategies

The apple genome sequence and the availability of high-throughput genotyping technologies have initiated a new era where SNP markers are abundant across the whole genome. Genomic selection (GS) is a statistical approach that utilizes all available genome-wide markers simultaneously to estimate breeding values or total genetic values. For breeding programmes, GS is a promising alternative to the traditional marker-assisted selection for manipulating complex polygenic traits often controlled by many small-effect genes. Various factors, such as genetic architecture of selection traits, population size and structure, genetic evaluation systems, density of SNP markers and extent of linkage disequilibrium, have been shown to be the key drivers of the accuracy of GS. In this paper, we provide an overview of the status of these aspects in current apple-breeding programmes. Strategies for GS for fruit quality and disease resistance are discussed, and an update on an empirical genomic selection study in a New Zealand apple-breeding programme is provided, along with a foresight of expected accuracy from such selection.     

Joint Analysis of Psychiatric Disorders Increases Accuracy of Risk Prediction for Schizophrenia,Bipolar Disorder,and Major Depressive Disorder

Genetic risk prediction has several potential applications in medical research and clinical practice and could be used, for example, to stratify a heterogeneous population of patients by their predicted genetic risk. However, for polygenic traits, such as psychiatric disorders, the accuracy of risk prediction is low. Here we use a multivariate linear mixed model and apply multi-trait genomic best linear unbiased prediction for genetic risk prediction. This method exploits correlations between disorders and simultaneously evaluates individual risk for each disorder. We show that the multivariate approach significantly increases the prediction accuracy for schizophrenia, bipolar disorder, and major depressive disorder in the discovery as well as in independent validation datasets. By grouping SNPs based on genome annotation and fitting multiple random effects, we show that the prediction accuracy could be further improved. The gain in prediction accuracy of the multivariate approach is equivalent to an increase in sample size of 34% for schizophrenia, 68% for bipolar disorder, and 76% for major depressive disorders using single trait models. Because our approach can be readily applied to any number of GWAS datasets of correlated traits, it is a flexible and powerful tool to maximize prediction accuracy. With current sample size, risk predictors are not useful in a clinical setting but already are a valuable research tool, for example in experimental designs comparing cases with high and low polygenic risk.     

Parallel selection mapping using artificially selected mice reveals body weight control loci

Understanding how polygenic traits evolve under selection is an unsolved problem, because challenges exist for identifying genes underlying a complex trait and understanding how multilocus selection operates in the genome. Here we study polygenic response to selection using artificial selection experiments. Inbred strains from seven independent long-term selection experiments for extreme mouse body weight ("high" lines weigh 42-77 g versus 16-40 g in "control" lines) were genotyped at 527,572 SNPs to identify loci controlling body weight. We identified 67 parallel selected regions (PSRs) where high lines share variants rarely found among the controls. By comparing allele frequencies in one selection experiment against its unselected control, we found classical selective sweeps centered on the PSRs. We present evidence supporting two G protein-coupled receptors GPR133 and Prlhr as positional candidates controlling body weight. Artificial selection may mimic natural selection in the wild: compared to control loci, we detected reduced heterozygosity in PSRs in unusually large wild mice on islands. Many PSRs overlap loci associated with human height variation, possibly through evolutionary conserved functional pathways. Our data suggest that parallel selection on complex traits may evoke parallel responses at many genes involved in diverse but relevant pathways.     

Multivariate mixed inheritance models for QTL detection on porcine chromosome 6

A series of multivariate mixed-inheritance models is fitted to the data from an outbred-line pig cross commercially used in Norway. Each model accommodates information on polygenic (co)variances between F2 individuals and their F1 parents across the five traits through incorporation of a random animal effect. Considered traits relate to meat quality and are chosen following up the results from a previous evaluation, in which a putative quantitative trait locus (QTL) was identified on chromosome six that affects the amount of intramuscular fat (IMF), meat percentage, meat tenderness and smell intensity (Grindflek et al., 2001). An additional trait included in the model, based on results of other studies, is the backfat thickness. The analysed material comprises data scored for 305 F2 individuals, whereas marker information is available for F1 and F2 generations. Based on the results of the multivariate analysis with the mixed-inheritance model, it was possible to conclude that the evidence for QTLs for meat percentage, meat tenderness and smell intensity in the study of Grindflek et al. (2001) do not represent separate QTLs, but is caused by the fact that the applied pre-adjustment of trait values for polygenic effects failed properly to remove the polygenic variation. The QTL effect on IMF on chromosome six was confirmed.     

A method to optimize selection on multiple identified quantitative trait loci

A mathematical approach was developed to model and optimize selection on multiple known quantitative trait loci (QTL) and polygenic estimated breeding values in order to maximize a weighted sum of responses to selection over multiple generations. The model allows for linkage between QTL with multiple alleles and arbitrary genetic effects, including dominance, epistasis, and gametic imprinting. Gametic phase disequilibrium between the QTL and between the QTL and polygenes is modeled but polygenic variance is assumed constant. Breeding programs with discrete generations, differential selection of males and females and random mating of selected parents are modeled. Polygenic EBV obtained from best linear unbiased prediction models can be accommodated. The problem was formulated as a multiple-stage optimal control problem and an iterative approach was developed for its solution. The method can be used to develop and evaluate optimal strategies for selection on multiple QTL for a wide range of situations and genetic models.     

HOW MUCH HERITABLE VARIATION CAN BE MAINTAINED IN FINITE POPULATIONS BY MUTATION–SELECTION BALANCE?

The joint effects of stabilizing selection, mutation, recombination, and random drift on the genetic variability of a polygenic character in a finite population are investigated. A simulation study is performed to test the validity of various analytical predictions on the equilibrium genetic variance. A new formula for the expected equilibrium variance is derived that approximates the observed equilibrium variance very closely for all parameter combinations we have tested. The computer model simulates the continuum-of-alleles model of Crow and Kimura. However, it is completely stochastic in the sense that it models evolution as a Markov process and does not use any deterministic evolution equations. The theoretical results are compared with heritability estimates from laboratory and natural populations. Heritabilities ranging from 20% to 50%, as observed even in lab populations under a constant environment, can only be explained by a mutation-selection balance if the phenotypic character is neutral or the number of genes contributing to the trait is sufficiently high, typically several hundred, or if there are a few highly variable loci that influence quantitative traits.