The effect of repeated cycles of selection on genetic variance,heritability, and response

Summary The genetic variance of a quantitative trait decreases under directional selection due to generation of linkage disequilibrium. After a few cycles of selection on individual phenotype, a limit is reached where there is no further reduction in the genetic variance. Bulmer's model is extended to an animal breeding situation where selection is on information on relatives rather than on the individual's own performance. Algebraic expressions are derived to predict the decrease in genetic variance and associated reductions in heritability and response in the limit. Consequences of the results are discussed in the context of breeding strategies.     

Evolution of branched regulatory genetic pathways: directional selection on pleiotropic loci accelerates developmental system drift

Developmental systems are regulated by a web of interacting loci. One common and useful approach in studying the evolution of development is to focus on classes of interacting elements within these systems. Here, we use individual-based simulations to study the evolution of traits controlled by branched developmental pathways involving three loci, where one locus regulates two different traits. We examined the system under a variety of selective regimes. In the case where one branch was under stabilizing selection and the other under directional selection, we observed "developmental system drift": the trait under stabilizing selection showed little phenotypic change even though the loci underlying that trait showed considerable evolutionary divergence. This occurs because the pleiotropic locus responds to directional selection and compensatory mutants are then favored in the pathway under stabilizing selection. Though developmental system drift may be caused by other mechanisms, it seems likely that it is accelerated by the same underlying genetic mechanism as that producing the Dobzhansky-Muller incompatibilities that lead to speciation in both linear and branched pathways. We also discuss predictions of our model for developmental system drift and how different selective regimes affect probabilities of speciation in the branched pathway system.     

Comparing environmental and genetic variance as adaptive response to fluctuating selection

Phenotypic variation within populations has two sources: genetic variation and environmental variation. Here, we investigate the coevolution of these two components under fluctuating selection. Our analysis is based on the lottery model in which genetic polymorphism can be maintained by negative frequency-dependent selection, whereas environmental variation can be favored due to bet-hedging. In our model, phenotypes are characterized by a quantitative trait under stabilizing selection with the optimal phenotype fluctuating in time. Genotypes are characterized by their phenotypic offspring distribution, which is assumed to be Gaussian with heritable variation for its mean and variance. Polymorphism in the mean corresponds to genetic variance while the width of the offspring distribution corresponds to environmental variance. We show that increased environmental variance is favored whenever fluctuations in the selective optima are sufficiently strong. Given the environmental variance has evolved to its optimum, genetic polymorphism can still emerge if the distribution of selective optima is sufficiently asymmetric or leptokurtic. Polymorphism evolves in a diagonal direction in trait space: one type becomes a canalized specialist for the more common ecological conditions and the other type a de-canalized bet-hedger thriving on the less-common conditions. All results are based on analytical approximations, complemented by individual-based simulations.     

Establishment and maintenance of adaptive genetic divergence under migration, selection, and drift

There is a long tradition in population genetics of exploring the maintenance of variation under migration-selection balance using deterministic models that assume infinite population size. With finite population size, stochastic dynamics can greatly reduce the potential for the maintenance of polymorphism, but this has yet to be explored in detail. Here, classical two-patch models are extended to predict: (1) the probability of a locally beneficial mutation rising in frequency in the patch where it is favored and (2) the critical threshold migration rate above which the maintenance of polymorphism is much less likely. Individual-based simulations show that these approximations provide accurate predictions across a wide range of parameter space.     

The joint effects of kin, multilevel selection and indirect genetic effects on response to genetic selection

Kin and levels-of-selection models are common approaches for modelling social evolution. Indirect genetic effect (IGE) models represent a different approach, specifying social effects on trait values rather than fitness. We investigate the joint effect of relatedness, multilevel selection and IGEs on response to selection. We present a measure for the degree of multilevel selection, which is the natural partner of relatedness in expressions for response. Response depends on both relatedness and the degree of multilevel selection, rather than only one or the other factor. Moreover, response is symmetric in relatedness and the degree of multilevel selection, indicating that both factors have exactly the same effect. Without IGEs, the key parameter is the product of relatedness and the degree of multilevel selection. With IGEs, however, multilevel selection without relatedness can explain evolution of social traits. Thus, next to relatedness and multilevel selection, IGEs are a key element in the genetical theory of social evolution.     

Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.     

Molecular markers to study genetic drift and selection in wheat populations

Studying the heterogeneity in variation of gene frequency among populations or between generations may be a possible way to detect genomic regions experiencing selection. In order to evaluate this approach, RFLP markers were used to compare the allelic frequencies in wheat populations that had been submitted to natural selection. In 1984, samples of two composite cross populations were distributed in the French network for dynamic management of genetic resources. Since then, all the sub-populations have been cultivated in the same sites with no human selection. The strong differentiation between populations found for agro-morphological traits (earliness, resistance to pathogens, ...) provided evidence of their adaptation to local conditions. The two initial populations and six derived sub-populations cultivated for 10 years in four contrasted sites were studied with RFLP markers. Differentiation between sub-populations based on RFLP diversity was highly significant. Variations on allelic frequencies of the 30 loci scored were found to be much greater than expected under genetic drift only. This led us to conclude that selection greatly influenced the evolution of the populations. Some of the loci clearly presented a higher differentiation than the others. This might indicate that they were genetically linked to other loci polymorphic in the populations and involved in adaptation. However, the effect of one selected gene on a marker, even located very close to the gene, could not be predicted with certainty. Hence, though the populations were predominantly selfing, it seems that initial linkage disequilibriums between markers and selected genes were not strong enough to control closely the evolution of allelic frequencies at the markers.     

On the relative roles of selection and genetic drift in shaping MHC variation

Genes of the major histocompatibility complex (MHC) have provided some of the clearest examples of how natural selection generates discordances between adaptive and neutral variation in natural populations. The type and intensity of selection as well as the strength of genetic drift are believed to be important in shaping the resulting pattern of MHC diversity. However, evaluating the relative contribution of multiple microevolutionary forces is challenging, and empirical studies have reported contrasting results. For instance, balancing selection has been invoked to explain high levels of MHC diversity and low population differentiation in comparison with other nuclear markers. Other studies have shown that genetic drift can sometimes overcome selection and then patterns of genetic variation at adaptive loci cannot be discerned from those occurring at neutral markers. Both empirical and simulated data also indicate that loss of genetic diversity at adaptive loci can occur faster than at neutral loci when selection and population bottlenecks act simultaneously. Diversifying selection, on the other hand, explains accelerated MHC divergence as the result of spatial variation in pathogen‐mediated selective regimes. Because of all these possible scenarios and outcomes, collecting information from as many study systems as possible, is crucial to enhance our understanding about the evolutionary forces driving MHC polymorphism. In this issue, Miller and co‐workers present an illuminating contribution by combining neutral markers (microsatellites) and adaptive MHC class I loci during the investigation of genetic differentiation across island populations of tuatara Sphenodon punctatus. Their study of geographical variation reveals a major role of genetic drift in shaping MHC variation, yet they also discuss some support for diversifying selection.     

Genetic steady-state under BLUP selection for an infinite and homogeneous population with discrete generations

A matrix derivation is proposed to analytically calculate the asymptotic genetic variance-covariance matrix under BLUP selection according to the initial genetic parameters in a large population with discrete generations. The asymptotic genetic evolution of a homogeneous population with discrete generations is calculated for a selection operating on an index including all information (pedigree and records) from a non-inbred and unselected base population (BLUP selection) or on an index restricted to records of a few ancestral generations. Under the first hypothesis, the prediction error variance of the selection index is independent of selection and is calculated from the genetic parameters of the base population. Under the second hypothesis, the prediction error variance depends on selection. Furthermore, records of several generations of ancestors of the candidates for selection must be used to maintain a constant prediction error variance over time. The number of ancestral generations needed depends on the population structure and on the occurrence of fixed effects. Without fixed effects to estimate, accounting for two generations of ancestors is sufficient to estimate the asymptotic prediction error variance. The amassing of information from an unselected base population proves to be important in order not to overestimate the asymptotic genetic gains and not to underestimate the asymptotic genetic variances.     

Joint effect of selection, linkage and partial inbreeding on additive genetic variance in an infinite population

Under the inifinitesimal model of gene effects, selection reduces the additive genetic variance by inducing negative linkage disequilibrium among selected genes. If the selected genes are linked, the decay of linkage disequilibrium is delayed, and the reduction of additive genetic variance is enhanced. Inbreeding in an infinite population also alters the additive genetic variance through the generation of positive association among genes within a locus. In the present study, the joint effect of selection, linkage and partial inbreeding (partial selfing or partial full-sib mating) on the additive genetic variance was modeled. The recurrence relations of the additive genetic variance between successive generations and the prediction equation of the asymptotic additive genetic variance were derived. Numerical computation showed that although partially inbred populations initially maintain larger genetic variances, the accumulated effect of selection overrides the effect of inbreeding. Stochastic simulation was carried out to check the precision of prediction, showing that the obtained equations give a satisfactory prediction during initial generations. However, the predicted values always overestimate the simulated values, especially in later generations. Based on these results, possible extensions and perspectives of the assumed model were discussed.     

Multivariate selection under adverse genetic correlations: impacts of population sizes and selection strategies on gains and coancestry in forest tree breeding

We examined gene models for two traits with and without antagonistic pleiotropy using a locus-based simulation model to investigate the effects of different population sizes, heritabilities and economic weights, using index selection, and index selection with optimum selection (OS), over 10 generations. Gene models included additive and dominance gene action, with equal and varying initial allele frequencies with and without pleiotropy for a fixed level of resources (i.e. founder sizes each generation of 40, 80 and 160 with progeny arrays that totaled 800 per generation). Pleiotropy (with an initial r _g of −0.5), reduced gain by ~8–10% when heritabilities for both traits were the same (0.2), relative to non-pleiotropic cases. When traits had different heritabilities (i.e. 0.2 and 0.4), gains in the lower heritability trait were substantially lower, especially with pleiotropy present. In general, OS with slightly larger population sizes could offset losses in gain, but rarely overrode the large effects of different heritabilities or economic weights. Pleiotropy increased response variance among traits, which was magnified when heritabilities were different. Identifying an appropriate weight on relatedness in the OS process is important to manage coancestry expectations around the loss of alleles (or fixation of recessive alleles) and to minimise response variance. The dynamics of selection intensity, drift, rate of coancestry build-up, response variance, etc. are complex for multi-trait selection; however, a few economically viable strategies could reduce the adverse effects of selecting against genetic correlations without drastically impairing gain.     

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.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.     

An equivalence between models of restricted selection and genetic groups

Summary An equivalence between a model of restricted selection and a model of genetic groups is presented. This correspondence leads to a realization of how genetic groups account for selection. Specifically, genetic groups act to remove the covariance between predictions of sire merit and functions of the true selection differentials. Further results illustrate a correspondence between models of selection on random effects and models of selection on residuals. Application of the results is useful, not in establishing concrete definitions for the structure of genetic groups, but in the analysis of how groups account for selection.     

A test of the hypothesis that correlational selection generates genetic correlations

Theory predicts that correlational selection on two traits will cause the major axis of the bivariate G matrix to orient itself in the same direction as the correlational selection gradient. Two testable predictions follow from this: for a given pair of traits, (1) the sign of correlational selection gradient should be the same as that of the genetic correlation, and (2) the correlational selection gradient should be positively correlated with the value of the genetic correlation. We test this hypothesis with a meta-analysis utilizing empirical estimates of correlational selection gradients and measures of the correlation between the two focal traits. Our results are consistent with both predictions and hence support the underlying hypothesis that correlational selection generates a genetic correlation between the two traits and hence orients the bivariate G matrix.     

On the evolution of epistasis I: diploids under selection

One interpretation of recent literature on the evolution of phenotypic modularity is that evolution should act to decrease the degree of interaction between genes that contribute to different phenotypes. This issue is addressed directly here using a fitness scheme determined by two genetic loci and a third locus which modifies a measure of statistical interaction between the fitnesses due to the first two. The equilibrium structure of such an epistasis-modifying locus is studied. It is shown that under well-specified conditions a modifying allele that increases epistasis succeeds. In other words, genetic interactions tend to become stronger. It is speculated that this occurs because the mean fitness in such models is locally increasing as a function of the degree of epistasis.     

Prediction of effects of genetic drift on variance components under a general model of epistasis

For a model of diallelic loci with arbitrary epistasis, Barton and Turelli [2004. Effects of genetic drift on variance components under a general model of epistasis. Evolution 58, 2111-2132] gave results for variances among and within replicate lines obtained by inbreeding without selection. Here, we discuss the relation between their population genetic methods and classical quantitative genetic arguments. In particular, we consider the case of no dominance using classical identity by descent arguments, which generalizes their results from two alleles to multiple alleles. To clarify the connections between the alternative methods, we obtain the same results using an intermediate method, which explicitly identifies the statistical effects of sets of loci. We also discuss the effects of population bottlenecks on covariances among relatives.