Predicting cumulated response to directional selection in finite panmictic populations

Summary Accurate prediction of the cumulated genetic gain requires predicting genetic variance over time under the joint effects of selection and limited population size. An algorithm is proposed to quantify at each generation the effects of these factors on average coefficient of inbreeding, genetic variance, and genetic mean, under a purely additive polygenic model, with no mutation, and under the assumption of absence of inbreeding depression on viability affecting selection differentials. This algorithm is relevant to populations where mating is at random and generations do not overlap. It was tested via Monte Carlo simulation on a population of 3 males and 25 females mass selected out of 50 candidates of each sex, over 30 generations. For two values of the initial heritability of the selected trait, 0.5 and 0.9 (to represent high accuracy in index selection), predicted values of the genetic variance are in agreement with observed results up to the 12th and 19th generations, respectively. Beyond these generations, the variance is overestimated, due to an underestimation of the effect of selection on the rate of inbreeding. Finally, the algorithm provides predictions of the cumulated responses close to the observed values in both selected populations. It is concluded that, as regards the hypotheses of the study, the proposed algorithm is satisfactory, and could be used to optimize selection methods with respect to the cumulated genetic gain in the mid- or long-term. Possible extensions of the algorithm to more realistic situations are discussed.     

Bias in the prediction of genetic gain due to mass and half-sib selection in random mating populations

The prediction of gains from selection allows the comparison of breeding methods and selection strategies, although these estimates may be biased. The objective of this study was to investigate the extent of such bias in predicting genetic gain. For this, we simulated 10 cycles of a hypothetical breeding program that involved seven traits, three population classes, three experimental conditions and two breeding methods (mass and half-sib selection). Each combination of trait, population, heritability, method and cycle was repeated 10 times. The predicted gains were biased, even when the genetic parameters were estimated without error. Gain from selection in both genders is twice the gain from selection in a single gender only in the absence of dominance. The use of genotypic variance or broad sense heritability in the predictions represented an additional source of bias. Predictions based on additive variance and narrow sense heritability were equivalent, as were predictions based on genotypic variance and broad sense heritability. The predictions based on mass and family selection were suitable for comparing selection strategies, whereas those based on selection within progenies showed the largest bias and lower association with the realized gain.     

Predicted rates of inbreeding with additive maternal effects

Maternal effects play an important role in fitness and other aspects of individual performance in many species, particularly mammalian, yet their impact on genetic variation within species and its rate of loss during selection has been neglected. In this paper we extend the theory of expected long-term genetic contributions to include maternal effects, and tested the accuracy of predicted rates of inbreeding for populations under mass selection by comparison with simulations. The model includes selective advantages of direct and maternal additive genetic effects, and also the selective advantage of a common maternal environmental effect. The population structures investigated had a fixed number of dams per sire and fixed family size. Most prediction errors of the rate of inbreeding (deltaF) were less than 8% of the simulated means and were lower in magnitude than the prediction errors of genetic gain (deltaG). The predictions of deltaG from contributions equalled previously published predictions. A variation in maternal genetic effects resulted in a much larger deltaF than for an equally sized variation in common maternal environmental effects. For a fixed genetic gain, deltaF increased as the maternal heritability increased. The influence of family size, mating ratio and age structure on deltaF was greater with maternal effects than with only direct genetic effects included. In conclusion, maternal effects can be a very important aspect to consider when predicting deltaF in populations under selection, and the developed methodology gives good predictions.     

On the Distribution of the Mean and Variance of a Quantitative Trait under Mutation-Selection-Drift Balance

The distributions of the mean phenotype and of the genetic variance of a polygenic trait under a balance between mutation, stabilizing selection and genetic drift are investigated. This is done by stochastic simulations in which each individual and each gene are represented. The results are compared with theoretical predictions. Some aspects of the existing theories for the evolution of quantitative traits are discussed. The maintenance of genetic variance and the average dynamics of phenotypic evolution in finite populations (with N(e) < 1000) are generally simpler than those suggested by some recent deterministic theories for infinite populations.     

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.     

Effects of Pleiotropy on Predictions concerning Mutation-Selection Balance for Polygenic Traits

Previous mathematical analyses of mutation-selection balance for metric traits assume that selection acts on the relevant loci only through the character(s) under study. Thus, they implicitly assume that all of the relevant mutation and selection parameters are estimable. A more realistic analysis must recognize that many of the pleiotropic effects of loci contributing variation to a given character are not known. To explore the consequences of these hidden effects, I analyze models of two pleiotropically connected polygenic traits, denoted P1 and P2. The actual equilibrium genetic variance for P1, based on complete knowledge of all mutation and selection parameters for both P1 and P2, can be compared to a prediction based solely on observations of P1. This extrapolation mimics empirically obtainable predictions because of the inevitability of unknown pleiotropic effects. The mutation parameters relevant to P1 are assumed to be known, but selection intensity is estimated from the within-generation reduction of phenotypic variance for P1. The extrapolated prediction is obtained by substituting these parameters into formulas based on single-character analyses. Approximate analytical and numerical results show that the level of agreement between these univariate extrapolations and the actual equilibrium variance depends critically on both the genetic model assumed and the relative magnitudes of the mutation and selection parameters. Unless per locus mutation rates are extremely high, i.e., generally greater than 10(-4), the widely used gaussian approximation for genetic effects at individual loci is not applicable. Nevertheless, the gaussian approximations predict that the true and extrapolated equilibria are in reasonable agreement, i.e., within a factor of two, over a wide range of parameter values. In contrast, an alternative approximation that applies for moderate and low per locus mutation rates predicts that the extrapolation will generally overestimate the true equilibrium variance unless there is little selection associated with hidden effects. The tendency to overestimate is understandable because selection acts on all of the pleiotropic manifestations of a new mutation, but equilibrium covariances among the characters affected may not reveal all of this selection. This casts doubt on the proposal that much of the additive polygenic variance observed in natural populations can be explained by mutation-selection balance. It also indicates the difficulty of critically evaluating this hypothesis.     

Body weight and tail length divergence in mice selected for rate of development

A series of mouse lines has been produced by 19 generations of restricted index selection for rate of development during early and late ontogeny. The selection program was based on an index with the following four replicated selection treatments: E(+) and E(-) were selected to alter birth to 10-day body weight gain while holding late gain for both selection lines constant; correspondingly, L(+) and L(-) were selected to alter 28- to 56-day body weight gain holding early gain for both lines constant. Herein, we characterize response to selection for growth rate by analyzing age-specific mouse body weight and tail lengths and for growth curves using a logistics model. Selection on developmental rate has resulted in divergence in both age-specific and growth curve traits. E(+) and L(+) lines reached identical weights during the late selection interval, then diverged to unique mature weights. E(-) and L(-) lines similarly achieved identical weights during late selection and diverged to unique mature weights. However, the shapes of early and late growth curves were significantly divergent, and at least two distinct growth patterns are shown to result from selection. Response in body weight gain was accompanied by similar, though less pronounced, change in tail length traits. Significant response during intervals of restricted growth was also found, especially in lines selected for late gain. The evolution of the growth trajectory under restricted index selection is discussed in terms of drift and available additive genetic variation and covariation.     

An experimental comparison of selection alternatives to plateaued response

Three alternative selection methods for extending selection limits or breaking response plateaus were compared over ten generations in a replicated model experiment using two unrelated populations of Drosophila melanogaster that no longer responded to purebred selection for high egg number, a heterotic polygenic trait. The three methods were: (1) reciprocal recurrent selection (RRS) with selection within each of the plateaued populations based solely on crossbred performance, (2) a modification of reciprocal recurrent selection (MRRS) with selection within each population based on both purebred and crossbred performance, and (3) purebred selection within a new synthetic population formed by crossing the two plateaued populations.--Conflicting estimates were obtained for heritability of purebred egg number in each of the plateaued populations. The realized heritability values and estimates from diallel analyses indicated an absence of additive genetic variation for both populations; however, estimates from conventional intraclass correlation methods were positive. The diallel analyses revealed significant amounts of nonadditive gene effects for purebred egg number in each population, while the significant gene effects for crossbred egg numbers were additive. Estimates of the genetic correlation between purebred and crossbred egg number were negative (-0.85 +/- 0.68 and -0.32 +/- 0.25) for the two base populations.--All three alternatives to continued purebred selection gave significant responses, with the average gain per generation from MRRS being significantly superior to the other two methods. Observed purebred and crossbred responses under RRS were in agreement with quantitative genetic theory. Such was not the case for MRRS, which suggested the possibility of major gene segregation.--Evidence supporting a negative genetic correlation between purebred and crossbred performance and the possibility of overdominance is presented and discussed.     

Effect of mass selection on the within-family genetic variance in finite populations

Summary The adequacy of an expression for the withinfamily genetic variance under pure random drift in an additive infinitesimal model was tested via simulation in populations undergoing mass selection. Two hundred or one thousand unlinked loci with two alleles at initial frequencies of 1/2 were considered. The size of the population was 100 (50 males and 50 females). Full-sib matings were carried out for 15 generations with only one male and one female chosen as parents each generation, either randomly or on an individual phenotypic value. In the unselected population, results obtained from 200 replicates were in agreement with predictions. With mass selection, within-family genetic variance was overpredicted by theory from the 12th and 4th generations for the 1,000 and 200 loci cases, respectively. Taking into account the observed change in gene frequencies in the algorithm led to a much better agreement with observed values. Results for the distribution of gene frequencies and the withinlocus genetic covariance are presented. It is concluded that the expression for the within-family genetic variance derived for pure random drift holds well for mass selection within the limits of an additive infinitesimal model.     

Simultaneous selection on two fitness-related traits in the butterfly Bicyclus anynana

Abstract. Theory about the role of constraints in evolution is abundant, but few empirical data exist to describe the consequences a bias in phenotypic variation has for micro evolution. Responses to natural selection can be severely hampered by a genetic correlation among a suite of traits. Constraints can be studied using antagonistic selection experiments, that is, two-trait selection in opposition to this correlation. The two traits studied here were development time and wing pattern (eyespot size) in the butterfly Bicyclus anynana , both of which have a clear adaptive significance. Rates of response were higher for eyespot size than for development time, but were independent of the concurrent selection (either in the same direction as the correlation or perpendicular to it). Regimes differed in both traits in all directions after 11 generations of selection. The uncoupling lines had higher relative responses than the synergistic lines in development time and equal relative responses in eyespot size. The patterns for eyespot size (reaction norms) were consistent across different rearing temperatures. Differences in lines selected for fast and slow development time were more pronounced at lower temperatures, irrespective of the direction of joint wing pattern selection. Furthermore, correlated responses in pupal weight and growth rate were observed; lines selected for a slower development had higher pupal weights, especially at lower temperatures. The response of the uncoupling lines was not hampered by a lack of selectable genetic variation, and the relative response in the development time was larger than expected based on response in the coupled direction and quantitative genetic predictions. This suggests that the structure of the genetic architecture does not constrain the short-term, independent evolution of both wing pattern and development time.     

Quantitative Genetics of Response to Competitors in Nemophila Menziesii: A Field Experiment

Recent investigations of evolution in heterogeneous environments have begun to accommodate genetic and environmental complexity typical of natural populations. Theoretical studies demonstrate that evolution of polygenic characters depends heavily on the genetic interdependence of the expression of traits in the different environments in which selection occurs, but information concerning this issue is scarce. We conducted a field experiment to assess the genetic variability of the annual plant Nemophila menziesii in five biotic regimes differing in plant density and composition. Significant, though modest, additive genetic variance in plant size was expressed in particular treatments. Evidence of additive genetic tradeoffs between interspecific and intraspecific competitive performance was found, but this result was not consistent throughout the experiment. Two aspects of experimental design may tend to obscure genetically based tradeoffs across environments in many previously published experiments: (1) inability to isolate additive genetic from other sources of variation and (2) use of novel (e.g., laboratory) environments.     

The Genetic Covariance between Characters Maintained by Pleiotropic Mutations

A statistical genetic model of a multivariate phenotype is derived to investigate the covariation of pleiotropic mutations with additive effects under the combined action of phenotypic selection, linkage and the mating system. Equilibrium formulas for large, randomly mating populations demonstrate that, when selection on polygenic variation is much smaller than twice the harmonic mean recombination rate between loci with interacting fitnesses, linkage disequilibrium is negligible and pleiotropy is the main cause of genetic correlations between characters. Under these conditions, approximate expressions for the dynamics of the genetic covariances due to pleiotropic mutations are obtained. Patterns of genetic covariance between characters and their evolution are discussed with reference to data on polygenic mutation, chromosomal organization and morphological integration.     

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.     

A Population Genetics Model of Marker-Assisted Selection

A deterministic two-loci model was developed to predict genetic response to marker-assisted selection (MAS) in one generation and in multiple generations. Formulas were derived to relate linkage disequilibrium in a population to the proportion of additive genetic variance used by MAS, and in turn to an extra improvement in genetic response over phenotypic selection. Predictions of the response were compared to those predicted by using an infinite-loci model and the factors affecting efficiency of MAS were examined. Theoretical analyses of the present study revealed the nonlinearity between the selection intensity and genetic response in MAS. In addition to the heritability of the trait and the proportion of the marker-associated genetic variance, the frequencies of the selectively favorable alleles at the two loci, one marker and one quantitative trait locus, were found to play an important role in determining both the short- and long-term efficiencies of MAS. The evolution of linkage disequilibrium and thus the genetic response over several generations were predicted theoretically and examined by simulation. MAS dissipated the disequilibrium more quickly than drift alone. In some cases studied, the rate of dissipation was as large as that to be expected in the circumstance where the true recombination fraction was increased by three times and selection was absent.     

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.     

Estimating Additive and Non-Additive Genetic Variances and Predicting Genetic Merits Using Genome-Wide Dense Single Nucleotide Polymorphism Markers

Non-additive genetic variation is usually ignored when genome-wide markers are used to study the genetic architecture and genomic prediction of complex traits in human, wild life, model organisms or farm animals. However, non-additive genetic effects may have an important contribution to total genetic variation of complex traits. This study presented a genomic BLUP model including additive and non-additive genetic effects, in which additive and non-additive genetic relation matrices were constructed from information of genome-wide dense single nucleotide polymorphism (SNP) markers. In addition, this study for the first time proposed a method to construct dominance relationship matrix using SNP markers and demonstrated it in detail. The proposed model was implemented to investigate the amounts of additive genetic, dominance and epistatic variations, and assessed the accuracy and unbiasedness of genomic predictions for daily gain in pigs. In the analysis of daily gain, four linear models were used: 1) a simple additive genetic model (MA), 2) a model including both additive and additive by additive epistatic genetic effects (MAE), 3) a model including both additive and dominance genetic effects (MAD), and 4) a full model including all three genetic components (MAED). Estimates of narrow-sense heritability were 0.397, 0.373, 0.379 and 0.357 for models MA, MAE, MAD and MAED, respectively. Estimated dominance variance and additive by additive epistatic variance accounted for 5.6% and 9.5% of the total phenotypic variance, respectively. Based on model MAED, the estimate of broad-sense heritability was 0.506. Reliabilities of genomic predicted breeding values for the animals without performance records were 28.5%, 28.8%, 29.2% and 29.5% for models MA, MAE, MAD and MAED, respectively. In addition, models including non-additive genetic effects improved unbiasedness of genomic predictions.     

Marker-assisted prediction of non-additive genetic values

It has become increasingly clear from systems biology arguments that interaction and non-linearity play an important role in genetic regulation of phenotypic variation for complex traits. Marker-assisted prediction of genetic values assuming additive gene action has been widely investigated because of its relevance in artificial selection. On the other hand, it has been less well-studied when non-additive effects hold. Here, we explored a nonparametric model, radial basis function (RBF) regression, for predicting quantitative traits under different gene action modes (additivity, dominance and epistasis). Using simulation, it was found that RBF had better ability (higher predictive correlations and lower predictive mean square errors) of predicting merit of individuals in future generations in the presence of non-additive effects than a linear additive model, the Bayesian Lasso. This was true for populations undergoing either directional or random selection over several generations. Under additive gene action, RBF was slightly worse than the Bayesian Lasso. While prediction of genetic values under additive gene action is well handled by a variety of parametric models, nonparametric RBF regression is a useful counterpart for dealing with situations where non-additive gene action is suspected, and it is robust irrespective of mode of gene action.     

Estimates of Genetic Parameters of Body Weight in Descendants of X-Irradiated Rat Spermatogonia

Effects of nine generations of 450r per generation of ancestral spermatogonial X irradiation of inbred rats on genetic parameters of body weight at 3, 6 and 10 weeks of age and of weight gains between these periods were studied. Covariances among relatives were estimated by mixed model and regression techniques in randomly selected lines with (R) and without (C) radiation history. Analyses of the data were based on five linear genetic models combining additive direct, additive indirect (maternal), dominance and environmental effects. Parameters in these models were estimated by generalized least-squares. A model including direct and indirect genetic effects fit more closely to the data in both R and C lines. Overdominance of induced mutations did not seem to be present. Ancestral irradiation increased maternal additive genetic variances of body weights and gains but not direct genetic variances. Theoretically, due to a negative direct-maternal genetic correlation, within full-sib family selection would be ineffective in increasing body weight at six weeks in both R and C lines. However, progress from mass selection would be expected to be faster in the R lines.     

Adaptation of Maize to Temperate Climates: Mid-Density Genome-Wide Association Genetics and Diversity Patterns Reveal Key Genomic Regions,with a Major Contribution of the Vgt2 (ZCN8) Locus

The migration of maize from tropical to temperate climates was accompanied by a dramatic evolution in flowering time. To gain insight into the genetic architecture of this adaptive trait, we conducted a 50K SNP-based genome-wide association and diversity investigation on a panel of tropical and temperate American and European representatives. Eighteen genomic regions were associated with flowering time. The number of early alleles cumulated along these regions was highly correlated with flowering time. Polymorphism in the vicinity of the ZCN8 gene, which is the closest maize homologue to Arabidopsis major flowering time (FT) gene, had the strongest effect. This polymorphism is in the vicinity of the causal factor of Vgt2 QTL. Diversity was lower, whereas differentiation and LD were higher for associated loci compared to the rest of the genome, which is consistent with selection acting on flowering time during maize migration. Selection tests also revealed supplementary loci that were highly differentiated among groups and not associated with flowering time in our panel, whereas they were in other linkage-based studies. This suggests that allele fixation led to a lack of statistical power when structure and relatedness were taken into account in a linear mixed model. Complementary designs and analysis methods are necessary to unravel the architecture of complex traits. Based on linkage disequilibrium (LD) estimates corrected for population structure, we concluded that the number of SNPs genotyped should be at least doubled to capture all QTLs contributing to the genetic architecture of polygenic traits in this panel. These results show that maize flowering time is controlled by numerous QTLs of small additive effect and that strong polygenic selection occurred under cool climatic conditions. They should contribute to more efficient genomic predictions of flowering time and facilitate the dissemination of diverse maize genetic resources under a wide range of environments.     

Prediction of Selection Response for Threshold Dichotomous Traits

This paper presents a formula to predict expected response to one generation of truncation selection for a dichotomous trait under polygenic additive inheritance. The derivation relies on the threshold liability concept and on the normality assumption of the joint distribution of additive genetic values and their predictors used as selection criteria. This formula accounts for asymmetry of response when both the prevalence of the trait and the selection rate differ from 1/2 via a bivariate normal integral term. The relationship with the classical formula R = iota rho sigma G is explained with a Taylor expansion about a zero value of the correlation factor. Properties are illustrated with an example of sire selection based on progeny test performance which shows a departure from usual predictions up to 15-20% at low (0.05) or high (0.95) selection rates. Univariate approximations and extensions to several paths of genetic change are also discussed.