The effect of non-additive genetic interactions on selection in multi-locus genetic models

Additive genetic variance might usually be expected to decrease in a finite population because of genetic drift. However, both theoretical and empirical studies have shown that the additive genetic variance of a population could, in some cases, actually increase owing to the action of genetic drift in presence of non-additive effects. We used Monte-Carlo simulations to address a less-well-studied issue: the effects of directional truncation selection on a trait affected by non-additive genetic variation. We investigated the effects on genetic variance and the response to selection. We compared two different genetic models, representing various numbers of loci. We found that the additive genetic variance could also increase in the case of truncation selection, when dominance and epistasis was present. Additive-by-additive epistatic effects generally gave a higher increase in additive variance compared to dominance. However, the magnitude of the increase differed depending on the particular model and on the number of loci.     

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.     

Effect on linkage disequilibrium of selection for a quantitative character with epistasis

Summary Selection for a character controlled by additive genes induces linkage disequilibrium which reduces the additive genetic variance usable for further selective gains. Additive x additive epistasis contributes to selection response through development of linkage disequilibrium between interacting loci. To investigate the relative importance of the two effects of linkage disequilibrium, formulae are presented and results are reported of simulations using models involving additive, additive x additive and dominance components. The results suggest that so long as epistatic effects are not large relative to additive effects, and the proportion of pairs of loci which show epistasis is not very high, the predominant effect of linkage disequilibrium will be to reduce the rate of selection response.     

Effects of genetic drift on variance components under a general model of epistasis

We analyze the changes in the mean and variance components of a quantitative trait caused by changes in allele frequencies, concentrating on the effects of genetic drift. We use a general representation of epistasis and dominance that allows an arbitrary relation between genotype and phenotype for any number of diallelic loci. We assume initial and final Hardy-Weinberg and linkage equilibrium in our analyses of drift-induced changes. Random drift generates transient linkage disequilibria that cause correlations between allele frequency fluctuations at different loci. However, we show that these have negligible effects, at least for interactions among small numbers of loci. Our analyses are based on diffusion approximations that summarize the effects of drift in terms of F, the inbreeding coefficient, interpreted as the expected proportional decrease in heterozygosity at each locus. For haploids, the variance of the trait mean after a population bottleneck is var(delta(z)) = sigma(n)k=1 FkV(A(k)), where n is the number of loci contributing to the trait variance, V(A(1)) = V(A) is the additive genetic variance, and V(A(k)) is the kth-order additive epistatic variance. The expected additive genetic variance after the bottleneck, denoted (V*(A)), is closely related to var(delta(z)); (V*(A)) = (1 - F) sigma(n)k=1 kFk-1V(A(k)). Thus, epistasis inflates the expected additive variance above V(A)(1 - F), the expectation under additivity. For haploids (and diploids without dominance), the expected value of every variance component is inflated by the existence of higher order interactions (e.g., third-order epistasis inflates (V*(AA. This is not true in general with diploidy, because dominance alone can reduce (V*(A)) below V(A)(1 - F) (e.g., when dominant alleles are rare). Without dominance, diploidy produces simple expressions: var(delta(z)) = sigma(n)k=1 (2F)kV(A(k)) and (V(A)) = (1 - F) sigma(n)k=1 k(2F)k-1V(A(k)). With dominance (and even without epistasis), var(delta(z)) and (V*(A)) no longer depend solely on the variance components in the base population. For small F, the expected additive variance simplifies to (V*(A)) approximately equal to (1 - F)V(A) + 4FV(AA) + 2FV(D) + 2FC(AD), where C(AD) is a sum of two terms describing covariances between additive effects and dominance and additive X dominance interactions. Whether population bottlenecks lead to expected increases in additive variance depends primarily on the ratio of nonadditive to additive genetic variance in the base population, but dominance precludes simple predictions based solely on variance components. We illustrate these results using a model in which genotypic values are drawn at random, allowing extreme and erratic epistatic interactions. Although our analyses clarify the conditions under which drift is expected to increase V(A), we question the evolutionary importance of such increases.     

Genomic selection of purebred animals for crossbred performance in the presence of dominant gene action

Background

Genomic selection is an appealing method to select purebreds for crossbred performance. In the case of crossbred records, single nucleotide polymorphism (SNP) effects can be estimated using an additive model or a breed-specific allele model. In most studies, additive gene action is assumed. However, dominance is the likely genetic basis of heterosis. Advantages of incorporating dominance in genomic selection were investigated in a two-way crossbreeding program for a trait with different magnitudes of dominance. Training was carried out only once in the simulation.

Results

When the dominance variance and heterosis were large and overdominance was present, a dominance model including both additive and dominance SNP effects gave substantially greater cumulative response to selection than the additive model. Extra response was the result of an increase in heterosis but at a cost of reduced purebred performance. When the dominance variance and heterosis were realistic but with overdominance, the advantage of the dominance model decreased but was still significant. When overdominance was absent, the dominance model was slightly favored over the additive model, but the difference in response between the models increased as the number of quantitative trait loci increased. This reveals the importance of exploiting dominance even in the absence of overdominance. When there was no dominance, response to selection for the dominance model was as high as for the additive model, indicating robustness of the dominance model. The breed-specific allele model was inferior to the dominance model in all cases and to the additive model except when the dominance variance and heterosis were large and with overdominance. However, the advantage of the dominance model over the breed-specific allele model may decrease as differences in linkage disequilibrium between the breeds increase. Retraining is expected to reduce the advantage of the dominance model over the alternatives, because in general, the advantage becomes important only after five or six generations post-training.

Conclusion

Under dominance and without retraining, genomic selection based on the dominance model is superior to the additive model and the breed-specific allele model to maximize crossbred performance through purebred selection.     

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

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

Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

Genetics of tassel branch number in maize and its implications for a selection program for small tassel size

Summary Tassel branch numbers of six crosses of maize (Zea mays L.) were analyzed to determine inheritance of this trait. Generation mean analyses were used to estimate genetic effects, and additive and nonadditive components of variance were calculated and evaluated for bias due to linkage. Both narrow-sense and broad-sense heritabilities were estimated. Additive genetic variance estimates were significant in five of the six crosses, whereas estimates of variance due to nonadditive components were significant in only three crosses. Additionally, estimates of additive variance components usually were larger than corresponding nonadditive components. There was no evidence for linkage bias in these estimates. Estimates of additive genetic effects were significant in four of six crosses, but significant dominance, additive × additive and additive × dominance effects also were detected. Additive, dominance, and epistatic gene action, therefore, all influenced the inheritance of tassel branch number, but additive gene action was most important. Both narrow-sense and broadsense heritability estimates were larger than those reported for other physiological traits of maize and corroborated conclusions concerning the importance of additive gene action inferred from analyses of genetic effects and variances. We concluded that selection for smalltasseled inbreds could be accomplished most easily through a mass-selection and/or pedigree-selection system. Production of a small-tasseled hybrid would require crossing of two small-tasseled inbreds. We proposed two genetic models to explain unexpected results obtained for two crosses. One model involved five interacting loci and the other employed two loci displaying only additive and additive × additive gene action.Journal Paper No. J-9231 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa 50011. Project No. 2152     

The Contribution of Quantitative Trait Loci and Neutral Marker Loci to the Genetic Variances and Covariances among Quantitative Traits in Random Mating Populations

Using Cockerham's approach of orthogonal scales, we develop genetic models for the effect of an arbitrary number of multiallelic quantitative trait loci (QTLs) or neutral marker loci (NMLs) upon any number of quantitative traits. These models allow the unbiased estimation of the contributions of a set of marker loci to the additive and dominance variances and covariances among traits in a random mating population. The method has been applied to an analysis of allozyme and quantitative data from the European oyster. The contribution of a set of marker loci may either be real, when the markers are actually QTLs, or apparent, when they are NMLs that are in linkage disequilibrium with hidden QTLs. Our results show that the additive and dominance variances contributed by a set of NMLs are always minimum estimates of the corresponding variances contributed by the associated QTLs. In contrast, the apparent contribution of the NMLs to the additive and dominance covariances between two traits may be larger than, equal to or lower than the actual contributions of the QTLs. We also derive an expression for the expected variance explained by the correlation between a quantitative trait and multilocus heterozygosity. This correlation explains only a part of the genetic variance contributed by the markers, i.e., in general, a combination of additive and dominance variances and, thus, provides only very limited information relative to the method supplied here.     

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.     

Estimates of genetic variance in an F2 maize population

Maize (Zea mays L.) breeders have used several genetic-statistical models to study the inheritance of quantitative traits. These models provide information on the importance of additive, dominance, and epistatic genetic variance for a quantitative trait. Estimates of genetic variances are useful in understanding heterosis and determining the response to selection. The objectives of this study were to estimate additive and dominance genetic variances and the average level of dominance for an F2 population derived from the B73 x Mo17 hybrid and use weighted least squares to determine the importance of digenic epistatic variances relative to additive and dominance variances. Genetic variances were estimated using Design III and weighted least squares analyses. Both analyses determined that dominance variance was more important than additive variance for grain yield. For other traits, additive genetic variance was more important than dominance variance. The average level of dominance suggests either overdominant gene effects were present for grain yield or pseudo-overdominance because of linkage disequilibrium in the F2 population. Epistatic variances generally were not significantly different from zero and therefore were relatively less important than additive and dominance variances. For several traits estimates of additive by additive epistatic variance decreased estimates of additive genetic variance, but generally the decrease in additive genetic variance was not significant.     

Quantitative genetic models for the balance between migration and stabilizing selection

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

QUANTITATIVE GENETIC VARIANCE MAINTAINED BY FLUCTUATING SELECTION WITH OVERLAPPING GENERATIONS: VARIANCE COMPONENTS AND COVARIANCES

The quantitative genetic variance-covariance that can be maintained in a random environment is studied, assuming overlapping generations and Gaussian stabilizing selection with a fluctuating optimum. The phenotype of an individual is assumed to be determined by additive contributions from each locus on paternal and maternal gametes (i.e., no epistasis and no dominance). Recurrent mutation is ignored, but linkage between loci is arbitrary. The genotype distribution in the evolutionarily stable population is generically discrete: only a finite number of polymorphic alleles with distinctly different effects are maintained, even though we allow a continuum of alleles with arbitrary phenotypic contributions to invade. Fluctuating selection maintains nonzero genetic variance in the evolutionarily stable population if the environmental heterogeneity is larger than a certain threshold. Explicit asymptotic expressions for the standing variance-covariance components are derived for the population near the threshold, or for large generational overlap, as a function of environmental variability and genetic parameters (i.e., number of loci, recombination rate, etc.), using the fact that the genotype distribution is discrete. Above the threshold, the population maintains considerable genetic variance in the form of positive linkage disequilibrium and positive gamete covariance (Hardy-Weinberg disequilibrium) as well as allelic variance. The relative proportion of these disequilibrium variances in the total genetic variance increases with the environmental variability.     

The additive genetic variance after bottlenecks is affected by the number of loci involved in epistatic interactions

Abstract We investigated the role of the number of loci coding for a neutral trait on the release of additive variance for this trait after population bottlenecks. Different bottleneck sizes and durations were tested for various matrices of genotypic values, with initial conditions covering the allele frequency space. We used three different types of matrices. First, we extended Cheverud and Routman's model by defining matrices of "pure" epistasis for three and four independent loci; second, we used genotypic values drawn randomly from uniform, normal, and exponential distributions; and third we used two models of simple metabolic pathways leading to physiological epistasis. For all these matrices of genotypic values except the dominant metabolic pathway, we find that, as the number of loci increases from two to three and four, an increase in the release of additive variance is occurring. The amount of additive variance released for a given set of genotypic values is a function of the inbreeding coefficient, independently of the size and duration of the bottleneck. The level of inbreeding necessary to achieve maximum release in additive variance increases with the number of loci. We find that additive-by-additive epistasis is the type of epistasis most easily converted into additive variance. For a wide range of models, our results show that epistasis, rather than dominance, plays a significant role in the increase of additive variance following bottlenecks.     

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.     

Genetic correlation and response to selection in simulated populations

Summary Effects of truncation selection of a primary trait upon genetic correlation with a secondary trait were examined over 30 generations in genetic populations simulated by computer. Populations were 24 males and 24 females mated randomly with replacement; number of offspring was determined by intensity of selection. Each trait was controlled by 48 loci segregating independently, effects were equal at every locus, and gene frequency was arbitrarily set at 0.5 at each locus in the initial generation. All combinations of three genetic correlations, three intensities of selection, and three environmental variances were simulated. Gene action was additive. Genetic correlation was set by number of loci which affected both traits and was measured each generation as the product-moment correlation of genotypic values and estimated by two methods of combining phenotypic covariances between parent and offspring.Genetic correlations in each offspring generation remained consistently near initial correlations for all environmental variances when fraction of offspring saved as parents was as large as one-half. When the fraction of offspring saved was as small as one-fifth, genetic correlations decreased but most rapidly with heritability high and after the 15th generation of selection. Truncation selection caused genetic correlation to decrease in those offspring selected to become parents of the next generation. Amount of reduction depended on heritability of the selected trait rather than on degree of truncation selection. Estimates of genetic correlation from phenotypic covariances between parent and offspring fluctuated markedly from real correlations in the small populations simulated.Michigan Agricultural Experiment Station Journal Article 4836. Part of North Central Regional Project NC-2.     

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.     

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.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.