Pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance

A pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance is investigated. Mutations have effects on a metric trait and deleterious effects on fitness, for which a bivariate gamma distribution is assumed. Equations for calculating the strength of apparent stabilizing selection (V(s)) and the genetic variance maintained in segregating populations (V(G)) were derived. A large population can hold a high genetic variance but the apparent stabilizing selection may or may not be relatively strong, depending on other properties such as the distribution of mutation effects. If the distribution of mutation effects on fitness is continuous such that there are few nearly neutral mutants, or a minimum fitness effect is assumed if most mutations are nearly neutral, V(G) increases to an asymptote as the population size increases. Both V(G) and V(s) are strongly affected by the shape of the distribution of mutation effects. Compared with mutants of equal effect, allowing their effects on fitness to vary across loci can produce a much higher V(G) but also a high V(s) (V(s) in phenotypic standard deviation units, which is always larger than the ratio V(P)/V(m)), implying weak apparent stabilizing selection. If the mutational variance V(m) is approximately 10(-3)V(e) (V(e), environmental variance), the model can explain typical values of heritability and also apparent stabilizing selection, provided the latter is quite weak as suggested by a recent review.     

Deleterious Mutations, Apparent Stabilizing Selection and the Maintenance of Quantitative Variation

Apparent stabilizing selection on a quantitative trait that is not causally connected to fitness can result from the pleiotropic effects of unconditionally deleterious mutations, because as N. Barton noted, "...individuals with extreme values of the trait will tend to carry more deleterious alleles...." We use a simple model to investigate the dependence of this apparent selection on the genomic deleterious mutation rate, U; the equilibrium distribution of K, the number of deleterious mutations per genome; and the parameters describing directional selection against deleterious mutations. Unlike previous analyses, we allow for epistatic selection against deleterious alleles. For various selection functions and realistic parameter values, the distribution of K, the distribution of breeding values for a pleiotropically affected trait, and the apparent stabilizing selection function are all nearly Gaussian. The additive genetic variance for the quantitative trait is kQa2, where k is the average number of deleterious mutations per genome, Q is the proportion of deleterious mutations that affect the trait, and a2 is the variance of pleiotropic effects for individual mutations that do affect the trait. In contrast, when the trait is measured in units of its additive standard deviation, the apparent fitness function is essentially independent of Q and a2; and beta, the intensity of selection, measured as the ratio of additive genetic variance to the "variance" of the fitness curve, is very close to s = U/k, the selection coefficient against individual deleterious mutations at equilibrium. Therefore, this model predicts appreciable apparent stabilizing selection if s exceeds about 0.03, which is consistent with various data. However, the model also predicts that beta must equal Vm/VG, the ratio of new additive variance for the trait introduced each generation by mutation to the standing additive variance. Most, although not all, estimates of this ratio imply apparent stabilizing selection weaker than generally observed. A qualitative argument suggests that even when direct selection is responsible for most of the selection observed on a character, it may be essentially irrelevant to the maintenance of variation for the character by mutation-selection balance. Simple experiments can indicate the fraction of observed stabilizing selection attributable to the pleiotropic effects of deleterious mutations.     

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.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

Influence of dominance, leptokurtosis and pleiotropy of deleterious mutations on quantitative genetic variation at mutation-selection balance

In models of maintenance of genetic variance (V (G)) it has often been assumed that mutant alleles act additively. However, experimental data show that the dominance coefficient varies among mutant alleles and those of large effect tend to be recessive. On the basis of empirical knowledge of mutations, a joint-effect model of pleiotropic and real stabilizing selection that includes dominance is constructed and analyzed. It is shown that dominance can dramatically alter the prediction of equilibrium V (G). Analysis indicates that for the situations where mutations are more recessive for fitness than for a quantitative trait, as supported by the available data, the joint-effect model predicts a significantly higher V (G) than does an additive model. Importantly, for what seem to be realistic distributions of mutational effects (i.e., many mutants may not affect the quantitative trait substantially but are likely to affect fitness), the observed high levels of genetic variation in the quantitative trait under strong apparent stabilizing selection can be generated. This investigation supports the hypothesis that most V (G) comes from the alleles nearly neutral for fitness in heterozygotes while apparent stabilizing selection is contributed mainly by the alleles of large effect on the quantitative trait. Thus considerations of dominance coefficients of mutations lend further support to our previous conclusion that mutation-selection balance is a plausible mechanism of the maintenance of the genetic variance in natural populations.     

Pleiotropic Stabilizing Selection Limits the Number of Polymorphic Loci to at Most the Number of Characters

We demonstrate that, in a model incorporating weak Gaussian stabilizing selection on n additively determined characters, at most n loci are polymorphic at a stable equilibrium. The number of characters is defined to be the number of independent components in the Gaussian selection scheme. We also assume linkage equilibrium, and that either the number of loci is large enough that the phenotypic distribution in the population can be approximated as multivariate Gaussian or that selection is weak enough that the mean fitness of the population can be approximated using only the mean and the variance of the characters in the population. Our results appear to rule out antagonistic pleiotropy without epistasis as a major force in maintaining additive genetic variation in a uniform environment. However, they are consistent with the maintenance of variability by genotype-environment interaction if a trait in different environments corresponds to different characters and the number of different environments exceeds the number of polymorphic loci that affect the trait.     

Joint effects of pleiotropic selection and stabilizing selection on the maintenance of quantitative genetic variation at mutation-selection balance

In quantitative genetics, there are two basic "conflicting" observations: abundant polygenic variation and strong stabilizing selection that should rapidly deplete that variation. This conflict, although having attracted much theoretical attention, still stands open. Two classes of model have been proposed: real stabilizing selection directly on the metric trait under study and apparent stabilizing selection caused solely by the deleterious pleiotropic side effects of mutations on fitness. Here these models are combined and the total stabilizing selection observed is assumed to derive simultaneously through these two different mechanisms. Mutations have effects on a metric trait and on fitness, and both effects vary continuously. The genetic variance (V(G)) and the observed strength of total stabilizing selection (V(s,t)) are analyzed with a rare-alleles model. Both kinds of selection reduce V(G) but their roles in depleting it are not independent: The magnitude of pleiotropic selection depends on real stabilizing selection and such dependence is subject to the shape of the distributions of mutational effects. The genetic variation maintained thus depends on the kurtosis as well as the variance of mutational effects: All else being equal, V(G) increases with increasing leptokurtosis of mutational effects on fitness, while for a given distribution of mutational effects on fitness, V(G) decreases with increasing leptokurtosis of mutational effects on the trait. The V(G) and V(s,t) are determined primarily by real stabilizing selection while pleiotropic effects, which can be large, have only a limited impact. This finding provides some promise that a high heritability can be explained under strong total stabilizing selection for what are regarded as typical values of mutation and selection parameters.     

HERITABILITY OF FITNESS COMPONENTS IN A WILD BIRD POPULATION

Consistently with the prediction that selection should deplete additive genetic variance ( V_A ) in fitness, traits closely associated to fitness have been shown to exhibit low heritabilities ( h ²= V_A /( V_A + V_R )). However, empirical data from the wild indicate that this is in fact due to increased residual variance ( V_R ), rather than due to decreased additive genetic variance, but the studies in this topic are still rare. We investigated relationships between trait heritabilities, additive genetic variances, and traits' contribution to lifetime reproductive success (≈fitness) in a red-billed gull ( Larus novaehollandiae ) population making use of animal model analyses as applied to 15 female and 13 male traits. We found that the traits closely associated with fitness tended to have lower heritabilities than traits less closely associated with fitness. However, in contrast with the results of earlier studies in the wild, the low heritability of the fitness-related traits was not only due to their high residual variance, but also due to their low additive genetic variance. Permanent environment effects—integrating environmental effects experienced in early life as well as nonadditive genetic effects—on many traits were large, but unrelated to traits' importance for fitness.     

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.     

Dynamics of Genetic Variability in Two-Locus Models of Stabilizing Selection

We study a two locus model, with additive contributions to the phenotype, to explore the dynamics of different phenotypic characteristics under stabilizing selection and recombination. We demonstrate that the interaction of selection and recombination results in constraints on the mode of phenotypic evolution. Let V(g) be the genic variance of the trait and C(L) be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that, independent of the initial conditions, the dynamics of the system on the plane (V(g), C(L)) are typically characterized by a quick approach to a straight line with slow evolution along this line afterward. We analyze how the mode and the rate of phenotypic evolution depend on the strength of selection relative to recombination, on the form of fitness function, and the difference in allelic effect. We argue that if selection is not extremely weak relative to recombination, linkage disequilibrium generated by stabilizing selection influences the dynamics significantly. We demonstrate that under these conditions, which are plausible in nature and certainly the case in artificial stabilizing selection experiments, the model can have a polymorphic equilibrium with positive linkage disequilibrium that is stable simultaneously with monomorphic equilibria.     

Epistasis in polygenic traits and the evolution of genetic architecture under stabilizing selection

We consider the effects of epistasis in a polygenic trait in the balance of mutation and stabilizing selection. The main issues are the genetic variation maintained in equilibrium and the evolution of the mutational effect distribution. The model assumes symmetric mutation and a continuum of alleles at all loci. Epistasis is modeled proportional to pairwise products of the single-locus effects. A general analytical formalism is developed. Assuming linkage equilibrium, we derive results for the equilibrium mutation load and the genetic and mutational variance in the house of cards and the Gaussian approximation. The additive genetic variation maintained in mutation-selection balance is reduced by any pattern of the epistatic interactions. The mutational variance, in contrast, is often increased. Large differences in mutational effects among loci emerge, and a negative correlation among (standard mean) locus mutation effects and mutation rates is predicted. Contrary to the common view since Waddington, we find that stabilizing selection in general does not lead to canalization of the trait. We propose that canalization as a target of selection instead occurs at the genic level. Here, primarily genes with a high mutation rate are buffered, often at the cost of decanalization of other genes. An intuitive interpretation of this view is given in the discussion.     

Dominance Genetic Variance for Traits Under Directional Selection in Drosophila serrata

In contrast to our growing understanding of patterns of additive genetic variance in single- and multi-trait combinations, the relative contribution of nonadditive genetic variance, particularly dominance variance, to multivariate phenotypes is largely unknown. While mechanisms for the evolution of dominance genetic variance have been, and to some degree remain, subject to debate, the pervasiveness of dominance is widely recognized and may play a key role in several evolutionary processes. Theoretical and empirical evidence suggests that the contribution of dominance variance to phenotypic variance may increase with the correlation between a trait and fitness; however, direct tests of this hypothesis are few. Using a multigenerational breeding design in an unmanipulated population of Drosophila serrata, we estimated additive and dominance genetic covariance matrices for multivariate wing-shape phenotypes, together with a comprehensive measure of fitness, to determine whether there is an association between directional selection and dominance variance. Fitness, a trait unequivocally under directional selection, had no detectable additive genetic variance, but significant dominance genetic variance contributing 32% of the phenotypic variance. For single and multivariate morphological traits, however, no relationship was observed between trait–fitness correlations and dominance variance. A similar proportion of additive and dominance variance was found to contribute to phenotypic variance for single traits, and double the amount of additive compared to dominance variance was found for the multivariate trait combination under directional selection. These data suggest that for many fitness components a positive association between directional selection and dominance genetic variance may not be expected.     

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     

Reconciling strong stabilizing selection with the maintenance of genetic variation in a natural population of black field crickets (Teleogryllus commodus)

Genetic variation in single traits, including those closely related to fitness, is pervasive and generally high. By contrast, theory predicts that several forms of selection, including stabilizing selection, will eliminate genetic variation. Stabilizing selection in natural populations tends to be stronger than that assumed in theoretical models of the maintenance of genetic variation. The widespread presence of genetic variation in the presence of strong stabilizing selection is a persistent problem in evolutionary genetics that currently has no compelling explanation. The recent insight that stabilizing selection often acts most strongly on trait combinations via correlational selection may reconcile this problem. Here we show that for a set of male call properties in the cricket Teleogryllus commodus, the pattern of multivariate stabilizing sexual selection is closely associated with the degree of additive genetic variance. The multivariate trait combinations experiencing the strongest stabilizing selection harbored very little genetic variation while combinations under weak selection contained most of the genetic variation. Our experiment provides empirical support for the prediction that a small number of trait combinations experiencing strong stabilizing selection will have reduced genetic variance and that genetically independent trait combinations experiencing weak selection can simultaneously harbor much higher levels of genetic variance.     

How does selfing affect the genetic variance of quantitative traits? An updated meta‐analysis on empirical results in angiosperm species

Most theoretical works predict that selfing should reduce the level of additive genetic variance available for quantitative traits within natural populations. Despite a growing number of quantitative genetic studies undertaken during the last two decades, this prediction is still not well supported empirically. To resolve this issue and confirm or reject theoretical predictions, we reviewed quantitative trait heritability estimates from natural plant populations with different rates of self‐fertilization and carried out a meta‐analysis. In accordance with models of polygenic traits under stabilizing selection, we found that the fraction of additive genetic variance is negatively correlated with the selfing rate. Although the mating system explains a moderate fraction of the variance, the mean reduction of narrow‐sense heritability values between strictly allogamous and predominantly selfing populations is strong, around 60%. Because some nonadditive components of genetic variance become selectable under inbreeding, we determine whether self‐fertilization affects the relative contribution of these components to genetic variance by comparing narrow‐sense heritability estimates from outcrossing populations with broad‐sense heritability estimated in autogamous populations. Results suggest that these nonadditive components of variance may restore some genetic variance in predominantly selfing populations; it remains, however, uncertain how these nonadditive components will contribute to adaptation.     

MHC class IIB alleles contribute to both additive and nonadditive genetic effects on survival in Chinook salmon

The genes of the major histocompatibility complex (MHC) are found in all vertebrates and are an important component of individual fitness through their role in disease and pathogen resistance. These genes are among the most polymorphic in genomes and the mechanism that maintains the diversity has been actively debated with arguments for natural selection centering on either additive or nonadditive genetic effects. Here, we use a quantitative genetics breeding design to examine the genetic effects of MHC class IIB alleles on offspring survivorship in Chinook salmon (Oncorhynchus tshawytscha). We develop a novel genetic algorithm that can be used to assign values to specific alleles or genotypes. We use this genetic algorithm to show simultaneous additive and nonadditive effects of specific MHC class IIB alleles and genotypes on offspring survivorship. The additive effect supports the rare-allele hypothesis as a potential mechanism for maintaining genetic diversity at the MHC. However, contrary to the overdominance hypothesis, the nonadditive effect led to underdominance at one heterozygous genotype, which could instead reduce variability at the MHC. Our algorithm is an advancement over traditional animal models that only partition variance in fitness to additive and nonadditive genetic effects, but do not allocate these effects to specific alleles and genotypes. Additionally, we found evidence of nonrandom segregation during meiosis in females that promotes an MHC allele that is associated with higher survivorship. Such nonrandom segregation could further reduce variability at the MHC and may explain why Chinook salmon has one of the lowest levels of MHC diversity of all vertebrates.     

The effects of intraspecific competition and stabilizing selection on a polygenic trait

The equilibrium properties of an additive multilocus model of a quantitative trait under frequency- and density-dependent selection are investigated. Two opposing evolutionary forces are assumed to act: (i) stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and (ii) intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the equilibrium structure, in particular, number, degree of polymorphism, and genetic variance of stable equilibria, is affected by the strength of frequency dependence, and what role the number of loci, the amount of recombination, and the demographic parameters play. To this end, we employ a statistical and numerical approach, complemented by analytical results, and explore how the equilibrium properties averaged over a large number of genetic systems with a given number of loci and average amount of recombination depend on the ecological and demographic parameters. We identify two parameter regions with a transitory region in between, in which the equilibrium properties of genetic systems are distinctively different. These regions depend on the strength of frequency dependence relative to pure stabilizing selection and on the demographic parameters, but not on the number of loci or the amount of recombination. We further study the shape of the fitness function observed at equilibrium and the extent to which the dynamics in this model are adaptive, and we present examples of equilibrium distributions of genotypic values under strong frequency dependence. Consequences for the maintenance of genetic variation, the detection of disruptive selection, and models of sympatric speciation are discussed.     

Very low levels of direct additive genetic variance in fitness and fitness components in a red squirrel population

A trait must genetically correlate with fitness in order to evolve in response to natural selection, but theory suggests that strong directional selection should erode additive genetic variance in fitness and limit future evolutionary potential. Balancing selection has been proposed as a mechanism that could maintain genetic variance if fitness components trade off with one another and has been invoked to account for empirical observations of higher levels of additive genetic variance in fitness components than would be expected from mutation–selection balance. Here, we used a long‐term study of an individually marked population of North American red squirrels (Tamiasciurus hudsonicus) to look for evidence of (1) additive genetic variance in lifetime reproductive success and (2) fitness trade‐offs between fitness components, such as male and female fitness or fitness in high‐ and low‐resource environments. “Animal model” analyses of a multigenerational pedigree revealed modest maternal effects on fitness, but very low levels of additive genetic variance in lifetime reproductive success overall as well as fitness measures within each sex and environment. It therefore appears that there are very low levels of direct genetic variance in fitness and fitness components in red squirrels to facilitate contemporary adaptation in this population.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.