Stable genetic polymorphism in heterogeneous environments: balance between asymmetrical dispersal and selection in the acorn barnacle

Elucidating the processes responsible for maintaining polymorphism at ecologically relevant genes is intimately related to understanding the interplay between selection imposed by habitat heterogeneity and a species' capacity for dispersal in the face of environmental constraints. In this paper, we used a model-based approach to solve equilibria of balanced polymorphism, given values of fitness and larval dispersal among different habitats in the acorn barnacle Semibalanus balanoides from the Gulf of St Lawrence. Our results showed that allele frequencies observed at both MPI* and GPI* loci represented stable equilibria, given empirical estimates of fitness values, and that considerably more larvae dispersed from one region (north) to the other (south) than vice versa. Dispersal conditions were predicted to be similar for the maintenance of polymorphism at both loci. Moreover, the values of asymmetrical dispersal required by the model to reach stable equilibria were compatible with empirical estimates of larval dispersal and oceanic circulation documented in this system. Overall, this study illustrated the usefulness of a modified and computable version of Bulmer's model (1972) in order to test hypotheses of balanced polymorphism resulting from interactions between spatial selection and asymmetrical dispersal.     

Experimental demonstration of the impact of hard and soft selection regimes on polymorphism maintenance in spatially heterogeneous environments†

Predicting and managing contemporary adaption requires a proper understanding of the determinants of genetic variation. Spatial heterogeneity of the environment may stably maintain polymorphism when habitat contribution to the next generation can be considered independent of the degree of adaptation of local populations within habitats (i.e., under soft selection). In contrast, when habitats contribute proportionally to the mean fitness of the populations they host (hard selection), polymorphism is not expected to be maintained by selection. Although mathematically established decades ago, this prediction had never been demonstrated experimentally. Here, we provide an experimental test in which polymorphic populations of Escherichia coli growing in heterogeneous habitats were exposed to hard and soft selection regimes. As predicted by theory, polymorphism was preserved longer under soft selection. Complementary tests established that soft selection slowed fixation processes and could even protect polymorphism in the long term by providing a systematic advantage to rare genotypes.     

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.     

Evolution in fine-grained environments. II. Habitat selection as a homeostatic mechanism

A model of genotype specific habitat selection is developed for an organism subject to within-lifetime environmental fluctuations. Habitat selection is first overlaid upon both hard and soft selection Levene models with either discrete or continuous habitats. It is shown that even if all genotypes have identical physiological and fitness responses within a habitat, habitat selection can still maintain a polymorphism. In other words, physiological divergence is not a necessary prerequisite for divergence in habitat preferences. Within-lifetime environmental variability is then assumed to occur within each chosen habitat. It is shown that habitat selection acts as an evolutionary filter that can enhance the fitness impact of some niches and effectively eliminate the impact of others such that it generally increases the chances for a polymorphism under soft selection. However, density-dependent effects obscure the relationship between physiological fitness and evolutionary outcome. Indeed, it is possible for selection to favor an allele causing its bearers to preferentially go to the niche to which they are least physiologically adapted. Hence, changes in habitat preference can evolve before an organism has completely adapted physiologically to a new habitat. The fitness impact of habitat selection interacts with both homeostatic avoidance mechanisms (i.e., short-term buffering) and with tolerance (long-term) mechanisms. In general, habitat selection will be most favored in those organisms deficient in long-term tolerance. Moreover, habitat selection tends to accentuate selection favoring short-term avoidance mechanisms. Thus, organisms displaying much habitat selection should have poor physiological long-term tolerances but effective physiological short-term avoidance mechanisms. Finally, if the fitness costs associated with habitat selection are too large to be ignored and are comparable for all genotypes, habitat selection directs the selective pressures back onto the physiological homeostatic capabilities. Hence, the very existence and extent of habitat selection depends critically upon the physiological capabilities of the organism.     

Polygenic variation maintained by balancing selection: pleiotropy, sex-dependent allelic effects and G x E interactions

We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G x E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model produces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleiotropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G x E can maintain variation at loci for which the coefficient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The critical value depends on the correlation between substitution effects at different loci. For large positive correlations (e.g., rho(ij)2>3/4), even extreme fluctuations in allelic effects cannot maintain variation. Surprisingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

Polymorphism at two loci through selection for linear metric deviation.

A model of phenotypic stabilising selection in which the fitness of an individual depends solely on its phenotype, and not directly on its genetic constitution, is explored algebraically for a system of two linked loci of unequal effect. It is found that selection for metric deviation gives rise to polymorphic gametefrequency equilibria for a variety of fitness regimes. Stability of non-trivial equilibria occurs for a wide range of parameter sets. Stability is facilitated by close linkage and inequality between gene effects. It is suggested that, in general genetic variation may be maintained under stabilising selection when the fitness of double heterozygotes exceeds that of the phenotypically intermediate homozygotes.     

The long-term evolution of multilocus traits under frequency-dependent disruptive selection

Frequency-dependent disruptive selection is widely recognized as an important source of genetic variation. Its evolutionary consequences have been extensively studied using phenotypic evolutionary models, based on quantitative genetics, game theory, or adaptive dynamics. However, the genetic assumptions underlying these approaches are highly idealized and, even worse, predict different consequences of frequency-dependent disruptive selection. Population genetic models, by contrast, enable genotypic evolutionary models, but traditionally assume constant fitness values. Only a minority of these models thus addresses frequency-dependent selection, and only a few of these do so in a multilocus context. An inherent limitation of these remaining studies is that they only investigate the short-term maintenance of genetic variation. Consequently, the long-term evolution of multilocus characters under frequency-dependent disruptive selection remains poorly understood. We aim to bridge this gap between phenotypic and genotypic models by studying a multilocus version of Levene's soft-selection model. Individual-based simulations and deterministic approximations based on adaptive dynamics theory provide insights into the underlying evolutionary dynamics. Our analysis uncovers a general pattern of polymorphism formation and collapse, likely to apply to a wide variety of genetic systems: after convergence to a fitness minimum and the subsequent establishment of genetic polymorphism at multiple loci, genetic variation becomes increasingly concentrated on a few loci, until eventually only a single polymorphic locus remains. This evolutionary process combines features observed in quantitative genetics and adaptive dynamics models, and it can be explained as a consequence of changes in the selection regime that are inherent to frequency-dependent disruptive selection. Our findings demonstrate that the potential of frequency-dependent disruptive selection to maintain polygenic variation is considerably smaller than previously expected.     

Polymorphism in multilocus host parasite coevolutionary interactions

Numerous loci in host organisms are involved in parasite recognition, such as major histocompatibility complex (MHC) genes in vertebrates or genes involved in gene-for-gene (GFG) relationships in plants. Diversity is commonly observed at such loci and at corresponding loci encoding antigenic molecules in parasites. Multilocus theoretical models of host-parasite coevolution predict that polymorphism is more likely than in single-locus interactions because recurrent coevolutionary cycles are sustained by indirect frequency-dependent selection as rare genotypes have a selective advantage. These cycles are stabilized by direct frequency-dependent selection, resulting from repeated reinfection of the same host by a parasite, a feature of most diseases. Here, it is shown that for realistically small costs of resistance and virulence, polycyclic disease and high autoinfection rates, stable polymorphism of all possible genotypes is obtained in parasite populations. Two types of epistatic interactions between loci tend to increase the parameter space in which stable polymorphism can occur with all possible host and parasite genotypes. In the parasite, the marginal cost of each additional virulence allele should increase, while in the host, the marginal cost of each additional resistance allele should decrease. It is therefore predicted that GFG polymorphism will be stable (and hence detectable) when there is partial complementation of avirulence genes in the parasite and of resistance genes in the host.     

Sex-biased dispersal and adaptation to marginal habitats

If gene flow occurs through both sexes but only females contribute to population growth, adaptation to marginal (sink) habitats should be differentially affected by male versus female dispersal. Here I address this problem with two models. First, I consider the fate of a rare allele that improves fitness in the marginal habitat but reduces fitness in the core (source) habitat. Then I study the evolution of a polygenic character mediating a trade-off in fitness between the habitats. Both approaches led to qualitatively similar predictions. The effect of a difference in the dispersal rate between the sexes depends on the degree to which immigration from the core habitat boosts the reproductive output from the marginal habitat. This boost is slight if the marginal habitat is able to sustain well a population without immigration. In that case, both female- and male-biased dispersal is more favorable for adaptation to marginal habitats than equal dispersal of both sexes (assuming that the dispersal rate averaged over the sexes is kept constant). In contrast, if the marginal habitat is an absolute sink unable to sustain a population without immigration, the conditions for adaptation to that habitat are least favorable under highly male-biased dispersal and most favorable under highly female-biased dispersal. Under some circumstances, high average (male+female) dispersal is more favorable than low dispersal. Thus, gene flow should not be seen solely as thwarting adaptation to marginal habitats. The results are interpreted in terms of how male and female dispersal affects the relative rate of gene flow from the source to the sink habitat and in the opposite direction. This study predicts that ecological niches of taxa with female-biased dispersal should tend to be broader and more evolutionarily flexible.     

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.     

Stochastic selection in large and small populations

We describe two models of stochastic variation in selection intensity. In both models the arithmetic mean fitness of all genotypes is equal; in both models the geometric mean fitness of the heterozygous genotype is greater than that of both homozygous genotypes. In one model the correlation between the fitnesses of the homozygous genotypes is +1; in the other it is −1. We show that the expected time to absorption of an allele in a finite population is significantly retarded for all initial gene frequencies in the former model. The expected time to absorption of an allele in the latter model is retarded only at extreme initial gene frequencies; at intermediate initial gene frequencies the expected time to absorption is accelerated. We conclude that the criterion for polymorphism based on the geometric mean of the heterozygote being greater than that of both homozygotes provides only limited information about the fate of gene frequency.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

The effect of epistasis on the structure of hybrid zones

Within hybrid zones that are maintained by a balance between selection and dispersal, linkage disequilibrium is generated by the mixing of divergent populations. This linkage disequilibrium causes selection on each locus to act on all other loci, thereby steepening clines, and generating a barrier to gene flow. Diffusion models predict simple relations between the strength of linkage disequilibrium and the dispersal rate, sigma, and between the barrier to gene flow, B, and the reduction in mean fitness, W. The aim of this paper is to test the accuracy of these predictions by comparison with an exact deterministic model of unlinked loci (r = 0.5). Disruptive selection acts on the proportion of alleles from the parental populations (p,q): W = exp[-S(4pq)beta], such that the least fit genotype has fitness e-s. Where beta < 1, fitness is reduced for a wide range of intermediate genotypes; where beta > 1, fitness is only reduced for those genotypes close to p = 0.5. Even with strong epistasis, linkage disequilibria are close to sigma 2p'ip'j/rij, where p'i, p'j are the gradients in allele frequency at loci i, j. The barrier to gene flow, which is reflected in the steepening of neutral clines, is given by [formula: see text] where r, the harmonic mean recombination rate between the neural and selected loci, is here 0.5. This is a close approximation for weak selection, but underestimates B for strong selection. The barrier is stronger for small beta, because hybrid fitness is then reduced over a wider range of p. The widths of the selected clines are harder to predict: though simple approximations are accurate for beta = 1, they become inaccurate for extreme beta because, then, fitness changes sharply with p. Estimates of gene number, made from neutral clines on the assumption that selection acts against heterozygotes, are accurate for weak selection when beta = 1; however, for strong selection, gene number is overestimated. For beta > 1, gene number is systematically overestimated and, conversely, when beta < 1, it is underestimated.     

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.     

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.     

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.     

The effect of variable environments on polymorphism at loci with several alleles I. A symmetric model

Much of the extant polymorphism has been attributed to spatial and temporal variation among selection regimes. Analysis of models entailing two alleles at a single locus has demonstrated that polymorphism may result from variation among selection regimes which prescribe monomorphism if constant. This relationship is studied in the context of several alleles at a locus.One result which is not valid with only two alleles is that variation among selection regimes which specify polymorphic equilibria may lead to a stable monomorphic equilibrium. The analyses of temporal variation and total panmixia spatial variation among environments show that temporal variation allows the simultaneous stability of equilibrium configurations which cannot be simultaneously stable under total panmixia spatial variation (hard or soft selection). The principle that polymorphism is more readily maintained with spatial than temporal variation is invalidated.Supported in part by Purdue Research Foundation and National Science Foundation (USA) grant MCS-8002227     

Linking the spatial scale of environmental variation and the evolution of phenotypic plasticity: selection favors adaptive plasticity in fine-grained environments

Adaptive phenotypic plasticity and adaptive genetic differentiation enable plant lineages to maximize their fitness in response to environmental heterogeneity. The spatial scale of environmental variation relative to the average dispersal distance of a species determines whether selection will favor plasticity, local adaptation, or an intermediate strategy. Habitats where the spatial scale of environmental variation is less than the dispersal distance of a species are fine grained and should favor the expression of adaptive plasticity, while coarse-grained habitats, where environmental variation occurs on spatial scales greater than dispersal, should favor adaptive genetic differentiation. However, there is relatively little information available characterizing the link between the spatial scale of environmental variation and patterns of selection on plasticity measured in the field. I examined patterns of spatial environmental variation within a serpentine mosaic grassland and selection on an annual plant (Erodium cicutarium) within that landscape. Results indicate that serpentine soil patches are a significantly finer-grained habitat than non-serpentine patches. Additionally, selection generally favored increased plasticity on serpentine soils and diminished plasticity on non-serpentine soils. This is the first empirical example of differential selection for phenotypic plasticity in the field as a result of strong differences in the grain of environmental heterogeneity within habitats.     

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.