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.     

Pleiotropy and Multilocus Polymorphisms

It is demonstrated that systems of two pleiotropically related characters controlled by additive diallelic loci can maintain under Gaussian stabilizing selection a stable polymorphism in more than two loci. It is also shown that such systems may have multiple stable polymorphic equilibria. Stabilizing selection generates negative linkage disequilibrium, as a result of which the equilibrium phenotypic variances are quite low, even though the level of allelic polymorphisms can be very high. Consequently, large amounts of additive genetic variation can be hidden in populations at equilibrium under stabilizing selection on pleiotropically related characters.     

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.     

Pleiotropic Models of Quantitative Variation

It is widely held that each gene typically affects many characters, and that each character is affected by many genes. Moreover, strong stabilizing selection cannot act on an indefinitely large number of independent traits. This makes it likely that heritable variation in any one trait is maintained as a side effect of polymorphisms which have nothing to do with selection on that trait. This paper examines the idea that variation is maintained as the pleiotropic side effect of either deleterious mutation, or balancing selection. If mutation is responsible, it must produce alleles which are only mildly deleterious (s approximately 10(-3)), but nevertheless have significant effects on the trait. Balancing selection can readily maintain high heritabilities; however, selection must be spread over many weakly selected polymorphisms if large responses to artificial selection are to be possible. In both classes of pleiotropic model, extreme phenotypes are less fit, giving the appearance of stabilizing selection on the trait. However, it is shown that this effect is weak (of the same order as the selection on each gene): the strong stabilizing selection which is often observed is likely to be caused by correlations with a limited number of directly selected traits. Possible experiments for distinguishing the alternatives are discussed.     

Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle

Most quantitative traits in most populations exhibit heritable genetic variation. Lande proposed that high levels of heritable variation may be maintained by mutation in the face of stabilizing selection. Several analyses have appeared of two distinct models with n additive polygenic loci subject to mutation and stabilizing selection. Each is reviewed and a new analysis and model are presented. Lande and Fleming analyzed extensions of a model originally treated by Kimura which assumes a continuum of possible allelic effects at each locus. Latter and Bulmer analyzed a model with diallelic loci. The published analyses of these models lead to qualitatively different predictions concerning the dependence of the equilibrium genetic variance on the underlying biological parameters. A new asymptotic analysis of the Kimura model shows that the different predictions are not consequences of the number of alleles assumed but rather are attributable to assumptions concerning the relative magnitudes of per locus mutation rates, the phenotypic effects of mutation, and the intensity of selection. This conclusion is reinforced by analysis of a model with triallelic loci. None of the approximate analyses presented are mathematically rigorous. To quantify their accuracy and display the domains of validity for alternative approximations, numerically determined equilibria are presented. In addition, empirical estimates of mutation rates and selection intensity are reviewed, revealing weaknesses in both the data and its connection to the models. Although the mathematical results and underlying biological requirements of my analyses are quite different from those of Lande, the results do not refute his hypothesis that considerable additive genetic variance may be maintained by mutation-selection balance. However, I argue that the validity of this hypothesis can only be determined with additional data and mathematics.     

Heterogeneous selection at specific loci in natural environments in Arabidopsis thaliana

Genetic variation for quantitative traits is often greater than that expected to be maintained by mutation in the face of purifying natural selection. One possible explanation for this observed variation is the action of heterogeneous natural selection in the wild. Here we report that selection on quantitative trait loci (QTL) for fitness traits in the model plant species Arabidopsis thaliana differs among natural ecological settings and genetic backgrounds. At one QTL, the allele that enhanced the viability of fall-germinating seedlings in North Carolina reduced the fecundity of spring-germinating seedlings in Rhode Island. Several other QTL experienced strong directional selection, but only in one site and seasonal cohort. Thus, different loci were exposed to selection in different natural environments. Selection on allelic variation also depended upon the genetic background. The allelic fitness effects of two QTL reversed direction depending on the genotype at the other locus. Moreover, alternative alleles at each of these loci caused reversals in the allelic fitness effects of a QTL closely linked to TFL1, a candidate developmental gene displaying nucleotide sequence polymorphism consistent with balancing selection. Thus, both environmental heterogeneity and epistatic selection may maintain genetic variation for fitness in wild plant species.     

The maintenance (or not) of polygenic variation by soft selection in heterogeneous environments

On the basis of single-locus models, spatial heterogeneity of the environment coupled with strong population regulation within each habitat (soft selection) is considered an important mechanism maintaining genetic variation. We studied the capacity of soft selection to maintain polygenic variation for a trait determined by several additive loci, selected in opposite directions in two habitats connected by dispersal. We found three main types of stable equilibria. Extreme equilibria are characterized by extreme specialization to one habitat and loss of polymorphism. They are analogous to monomorphic equilibria in singe-locus models and are favored by similar factors: high dispersal, weak selection, and low marginal average fitness of intermediate genotypes. At the remaining two types of equilibria the population mean is intermediate but variance is very different. At fully polymorphic equilibria all loci are polymorphic, whereas at low-variance equilibria at most one locus remains polymorphic. For most parameters only one type of equilibrium is stable; the transition between the domains of fully polymorphic and low-variance equilibria is typically sharp. Low-variance equilibria are favored by high marginal average fitness of intermediate genotypes, in contrast to single-locus models, in which marginal overdominance is particularly favorable for maintenance of polymorphism. The capacity of soft selection to maintain polygenic variation is thus more limited than extrapolation from single-locus models would suggest, in particular if dispersal is high and selection weak. This is because in a polygenic model, variance can evolve independently of the mean, whereas in the single-locus two-allele case, selection for an intermediate mean automatically leads to maintenance of polymorphism.     

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.     

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.     

Selection for Antimicrobial Peptide Diversity in Frogs Leads to Gene Duplication and Low Allelic Variation

Antimicrobial peptides are highly diverse pathogen-killing molecules. In many taxa, their evolution is characterized by positive selection and frequent gene duplication. It has been proposed that genes encoding antimicrobial peptides might be subject to balancing selection and/or an enhanced mutation rate, but these hypotheses have not been well evaluated because allelic variation has rarely been studied at antimicrobial peptide loci. We present an evolutionary analysis of novel antimicrobial peptide genes from leopard frogs, Rana. Our results demonstrate that a single genome contains multiple homologous copies, among which there is an excess of nonsynonymous nucleotide site divergence relative to that expected from synonymous site divergence. Thus, we confirm the trends of recurrent duplication and positive selection. Allelic variation is quite low relative to interspecies divergence, indicating a recent positive selective sweep with no evidence of balancing selection. Repeated gene duplication, rather than a balanced maintenance of divergent allelic variants at individual loci, appears to be how frogs have responded to selection for a diverse suite of antimicrobial peptides. Our data also support a pattern of enhanced synonymous site substitution in the mature peptide region of the gene, but we cannot conclude that this is due to an elevated mutation rate.     

Environmental heterogeneity and balancing selection in the acorn barnacle Semibalanus balanoides

The northern acorn barnacle Semibalans banlanoides occupies several intertidal microhabitats which vary greatly in their degree of physical stress. This environmental heterogeneity creates distinct selection regimes which can maintain genetic variation in natural populations. Despite considerable attention placed on the link between spatial variation in fitness and balancing selection at specific loci, experimental manipulations and fitness estimates for molecular polymorphisms have rarely been conducted in the wild. The aim of this transplant experiment was to manipulate the level of physical stress experienced by a cohort of barnacles in the field and then investigate the spatial variation in fitness for genotypes at three loci: two candidate allozymes and the mitochondrial DNA control region. The viability of mannose-6-phosphate isomerase (Mpi) genotypes was dependent on the level of physical stress experienced in the various treatments; alternative homozygotes were favoured in alternative high stress-low stress environments. In contrast, the fitness of genotypes at other loci was equivalent among treatments and unaffected by the manipulation. Evaluated in the light of balancing selection models, these data indicate that the presence of multiple environmental niches is sufficient to promote a stable Mpi polymorphism in barnacle populations and that allelic variation at this locus reflects the process of adaptation to the heterogeneous intertidal landscape.     

Spatially and temporally varying selection on intrapopulation quantitative trait loci for a life history trade-off in Mimulus guttatus

Why do populations remain genetically variable despite strong continuous natural selection? Mutation reconstitutes variation eliminated by selection and genetic drift, but theoretical and experimental studies each suggest that mutation‐selection balance insufficient to explain extant genetic variation in most complex traits. The alternative hypothesis of balancing selection, wherein selection maintains genetic variation, is an aggregate of multiple mechanisms (spatial and temporal heterogeneity in selection, frequency‐dependent selection, antagonistic pleiotropy, etc.). Most of these mechanisms have been demonstrated for Mendelian traits, but there is little comparable data for loci affecting quantitative characters. Here, we report a 3‐year field study of selection on intrapopulation quantitative trait loci (QTL) of flower size, a highly polygenic trait in Mimulus guttatus. The QTL exhibit antagonistic pleiotropy: alleles that increase flower size, reduce viability, but increase fecundity. The magnitude and direction of selection fluctuates yearly and on a spatial scale of metres. This study provides direct evidence of balancing selection mechanisms on QTL of an ecologically relevant trait.     

Phenotypic variation and natural selection at catsup, a pleiotropic quantitative trait gene in Drosophila

Quantitative traits are shaped by networks of pleiotropic genes . To understand the mechanisms that maintain genetic variation for quantitative traits in natural populations and to predict responses to artificial and natural selection, we must evaluate pleiotropic effects of underlying quantitative trait genes and define functional allelic variation at the level of quantitative trait nucleotides (QTNs). Catecholamines up (Catsup), which encodes a negative regulator of tyrosine hydroxylase , the rate-limiting step in the synthesis of the neurotransmitter dopamine, is a pleiotropic quantitative trait gene in Drosophila melanogaster. We used association mapping to determine whether the same or different QTNs at Catsup are associated with naturally occurring variation in multiple quantitative traits. We sequenced 169 Catsup alleles from a single population and detected 33 polymorphisms with little linkage disequilibrium (LD). Different molecular polymorphisms in Catsup are independently associated with variation in longevity, locomotor behavior, and sensory bristle number. Most of these polymorphisms are potentially functional variants in protein coding regions, have large effects, and are not common. Thus, Catsup is a pleiotropic quantitative trait gene, but individual QTNs do not have pleiotropic effects. Molecular population genetic analyses of Catsup sequences are consistent with balancing selection maintaining multiple functional polymorphisms.     

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.     

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.     

Selective pressures on MHC class II genes in the guppy (Poecilia reticulata) as inferred by hierarchical analysis of population structure

Genes of the major histocompatibility complex, which are the most polymorphic of all vertebrate genes, are a pre‐eminent system for the study of selective pressures that arise from host–pathogen interactions. Balancing selection capable of maintaining high polymorphism should lead to the homogenization of MHC allele frequencies among populations, but there is some evidence to suggest that diversifying selection also operates on the MHC. However, the pattern of population structure observed at MHC loci is likely to depend on the spatial and/or temporal scale examined. Here, we investigated selection acting on MHC genes at different geographic scales using Venezuelan guppy populations inhabiting four regions. We found a significant correlation between MHC and microsatellite allelic richness across populations, which suggests the role of genetic drift in shaping MHC diversity. However, compared to microsatellites, more MHC variation was explained by differences between populations within larger geographic regions and less by the differences between the regions. Furthermore, among proximate populations, variation in MHC allele frequencies was significantly higher compared to microsatellites, indicating that selection acting on MHC may increase population structure at small spatial scales. However, in populations that have significantly diverged at neutral markers, the population‐genetic signature of diversifying selection may be eradicated in the long term by that of balancing selection, which acts to preserve rare alleles and thus maintain a common pool of MHC alleles.     

The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

Functional and geographical differentiation of candidate balanced polymorphisms in Arabidopsis thaliana

Molecular population genetic analysis of three chromosomal regions in Arabidopsis thaliana suggested that balancing selection might operate to maintain variation at three novel candidate adaptive trait genes, including SOLUBLE STARCH SYNTHASE I (SSI) , PLASTID TRANSCRIPTIONALLY ACTIVE 7(PTAC7) , and BELL-LIKE HOMEODOMAIN 10 (BLH10). If balanced polymorphisms are indeed maintained at these loci, then we would expect to observe functional variation underlying the previously detected signatures of selection. We observe multiple replacement polymorphisms within and in the 32 amino acids just upstream of the protein–protein interacting BELL domain at the BLH10 locus. While no clear protein sequence differences are found between allele types in SSI and PTAC7, these two genes show evidence for allele-specific variation in expression levels. Geographical patterns of allelic differentiation seem consistent with population stratification in this species and a significant longitudinal cline was observed at all three candidate loci. These data support a hypothesis of balancing selection at all three candidate loci and provide a basis for more detailed functional work by identifying possible functional differences that might be selectively maintained.     

Dependence of expected heterozygosity on locus number with stabilizing selection and drift

I determine expected levels of heterozygosity in two allele multilocus models with mutation, stabilizing selection and drift. In the range 2 to 32 loci, the per locus heterozygosity can depend on the locus number. The per locus heterozygosity for ten loci can be as low as three fourths of the per locus heterozygosity in the limit, as the number of loci gets large. Simulations indicate that this dependence on locus number is not due to the population approaching equilibria at which the mean differs from the optimum, but is due to changes in the substitution rate as a function of the number of loci.     

DIFFERENTIAL DOMINANCE OF PLEIOTROPIC LOCI FOR MOUSE SKELETAL TRAITS

The term "differential dominance" describes the situation in which the dominance effects at a pleiotropic locus vary between traits. Directional selection on the phenotype can lead to balancing selection on differentially dominant pleiotropic loci. Even without any individual overdominant traits, some linear combination of traits will display overdominance at a locus displaying differential dominance. Multivariate overdominance may be responsible, in part, for high levels of heterozygosity found in natural populations. We examine differential dominance of 70 mouse skeletal traits at 92 quantitative trait loci (QTL). Our results indicate moderate to strong additive and dominance effects at pleiotropic loci, low levels of individual-trait overdominance, and universal multivariate overdominance. Multivariate overdominance affects a range of 6% to 81% of morphospace, with a mean of 32%. Multivariate overdominance tends to affect a larger percentage of morphospace at pleiotropic loci with antagonistic effects on multiple traits (42%). We conclude that multivariate overdominance is common and should be considered in models and in empirical studies of the role of genetic variation in evolvability.