Multilocus selection in subdivided populations II. Maintenance of polymorphism under weak or strong migration

The potential of maintaining multilocus polymorphism by migration-selection balance is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, selection may vary across demes, and loci are multiallelic. It is shown that if migration and recombination are strong relative to selection, then with weak or no epistasis and intermediate dominance at every locus and in every deme, arbitrarily many alleles can be maintained at arbitrarily many loci at a stable equilibrium. If migration is weak relative to selection and recombination, then with weak or no epistasis and intermediate dominance at every locus and in every deme, as many alleles as there are demes can be maintained at arbitrarily many loci at equilibrium. In both cases open sets of such parameter combinations are constructed, thus the results are robust with respect to small, but arbitrary, perturbations in the parameters. For weak migration, the number of demes is, in fact, a generic upper bound to the number of alleles that can be maintained at any locus. Thus, several scenarios are identified under which multilocus polymorphism can be maintained by migration-selection balance when this is impossible in a panmictic population.      

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.     

Spatial heterogeneity in the strength of selection against deleterious alleles and the mutation load

According to current estimates of genomic deleterious mutation rates (which are often of the order 0.1-1) the mutation load (defined as a reduction in the average fitness of a population due to the presence of deleterious alleles) may be important in many populations. In this paper, I use multilocus simulations to explore the effect of spatial heterogeneity in the strength of selection against deleterious alleles on the mutation load (for example, it has been suggested that stressful environments may increase the strength of selection). These simulations show contrasted results: in some situations, spatial heterogeneity may greatly reduce the mutation load, due to the fact that migrants coming from demes under stronger selection carry relatively few deleterious alleles, and benefit from a strong advantage within demes under weaker selection (where individuals carry many more deleterious alleles); in other situations, however, deleterious alleles accumulate within demes under stronger selection, due to migration pressure from demes under weaker selection, leading to fitness erosion within those demes. This second situation is more frequent when the productivity of the different demes is proportional to their mean fitness. The effect of spatial heterogeneity is greatly reduced, however, when the response to environmental differences is inconsistent across loci.     

Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration

The dynamics and equilibrium structure of a deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is irreducible and aperiodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Various limiting cases are derived in which one or more of the three evolutionary forces, selection, migration, and recombination, are weak relative to the others. Two are particularly interesting. If migration and recombination are strong relative to selection, the dynamics can be conceived as a perturbation of the so-called weak-selection limit, a simple dynamical system for suitably averaged allele frequencies. Under nondegeneracy assumptions on this weak-selection limit which are generic, every equilibrium of the full dynamics is a perturbation of an equilibrium of the weak-selection limit and has the same stability properties. The number of equilibria is the same in both systems, equilibria in the full (perturbed) system are in quasi-linkage equilibrium, and differences among allele frequencies across demes are small. If migration is weak relative to recombination and epistasis is also weak, then every equilibrium is a perturbation of an equilibrium of the corresponding system without migration, has the same stability properties, and is in quasi-linkage equilibrium. In both cases, every trajectory converges to an equilibrium, thus no cycling or complicated dynamics can occur.      

INTEGRATING LANDSCAPE GENOMICS AND SPATIALLY EXPLICIT APPROACHES TO DETECT LOCI UNDER SELECTION IN CLINAL POPULATIONS

Uncovering the genetic basis of adaptation hinges on the ability to detect loci under selection. However, population genomics outlier approaches to detect selected loci may be inappropriate for clinal populations or those with unclear population structure because they require that individuals be clustered into populations. An alternate approach, landscape genomics, uses individual‐based approaches to detect loci under selection and reveal potential environmental drivers of selection. We tested four landscape genomics methods on a simulated clinal population to determine their effectiveness at identifying a locus under varying selection strengths along an environmental gradient. We found all methods produced very low type I error rates across all selection strengths, but elevated type II error rates under “weak” selection. We then applied these methods to an AFLP genome scan of an alpine plant, Campanula barbata, and identified five highly supported candidate loci associated with precipitation variables. These loci also showed spatial autocorrelation and cline patterns indicative of selection along a precipitation gradient. Our results suggest that landscape genomics in combination with other spatial analyses provides a powerful approach for identifying loci potentially under selection and explaining spatially complex interactions between species and their environment.     

Epistatic selection opposed by immigration in multiple locus genetic systems

A model of “complete” epistatis is considered in which all “plus” alleles must be present in an individual before the adaptive phenotype is expressed. The conditions under which the plus alleles and hence the adaptive phenotype can increase and reach a stable equilibrium in the presence of immigration of gametes carrying minus alleles are found. In haploids and diploids in which the plus alleles are recessive, frequencies of the plus alleles are the same at all loci, regardless of the linkage relationships. Tight linkage favors the existence of a locally stable polymorphic equilibrium, but the equilibrium with only minus alleles is locally stable unless there is very tight linkage or very strong selection. Thus, this kind of epistasis, which provides a simple model for a character that requires several components to be present at the same time, is very sensitive to even a small amount of immigration. Hence, the evolution of such characters is likely only in completely rather than partially isolated populations.     

Consequences of population structure on genes under balancing selection

This paper describes a new approach to modeling population structure for genes under strong balancing selection of the type seen in plant self-incompatibility systems and the major histocompatibility complex (MHC) system of vertebrates. Simple analytic solutions for the number of alleles maintained at equilibrium and the expected proportion of alleles shared between demes at various levels are derived and checked against simulation results. The theory accurately captures the dynamics of allele number in a subdivided population and identifies important values of m (migration rate) at which allele number and distribution change qualitatively. Starting from a panmictic population, as migration among demes decreases a qualitative change in dynamics is seen at approximately m(crit) approximately equal to the square root of(s/4piNT) where NT is the total population size and s is a measure of the strength of selection. At this point, demes can no longer maintain their panmictic allele number, due to increasing isolation from the total population. Another qualitative change occurs at a migration rate on the same order of magnitude as the mutation rate, mu. At this point, the demes are highly differentiated for allele complement, and the total number of alleles in the population is increased. Because in general u < m<(crit) at intermediate migration rates slightly fewer alleles may be maintained in the total population than are maintained at panmixia. Within this range, total allele number may not be the best indicator of whether a population is effectively panmictic, and some caution should be used when interpreting samples from such populations. The theory presented here can help to analyze data from genes under balancing selection in subdivided populations.     

Genomic islands of speciation or genomic islands and speciation?

Populations of the malaria mosquito, Anopheles gambiae, are comprised of at least two reproductively isolated, sympatric populations. In this issue, White et al. (2010) use extensive sampling, high‐density tiling microarrays, and an updated reference genome to clarify and expand our knowledge of genomic differentiation between these populations. It is now clear that DNA near the centromeres of all three chromosomes are in near‐perfect disequilibrium with each other. This is in stark contrast to the remaining 97% of the assembled genome, where fixed differences between populations have not been found, and many polymorphisms are shared. This pattern, coupled with direct evidence of hybridization in nature, supports models of “mosaic” speciation, where ongoing hybridization homogenizes variation in most of the genome while loci under strong selection remain in disequilibrium with each other. However, unambiguously demonstrating that selection maintains the association of these pericentric “speciation islands” in the face of gene flow is difficult. Low recombination at all three loci complicates the issue, and increases the probability that selection unrelated to the speciation process alters patterns of variation in these loci. Here, we discuss these different scenarios in light of this new data.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

The effect of subdivision on variation at multi-allelic loci under balancing selection

Simulations are used to investigate the expected pattern of variation at loci under different forms of multi-allelic balancing selection in a finite island model of a subdivided population. The objective is to evaluate the effect of restricted migration among demes on the distribution of polymorphism at the selected loci at equilibrium, and to compare the results with those expected for a neutral locus. The results show that the expected number of alleles maintained, and numbers of nucleotide differences between alleles, are relatively insensitive to the migration rate, and differentiation remains low even under very restricted migration. However, nucleotide divergence between copies of functionally identical alleles increases sharply when migration decreases. These results are discussed in relation to published surveys of allelic diversity in MHC and plant self-incompatibility systems, and to the possibility of inferring ancient population genetic events and processes. In addition, it is shown that, for sporophytic self-incompatibility systems, it is not necessarily true in a subdivided population that recessive alleles are more frequent than dominant ones.     

Polymorphism and divergence for island-model species

Estimates of the scaled selection coefficient, gamma of Sawyer and Hartl, are shown to be remarkably robust to population subdivision. Estimates of mutation parameters and divergence times, in contrast, are very sensitive to subdivision. These results follow from an analysis of natural selection and genetic drift in the island model of subdivision in the limit of a very large number of subpopulations, or demes. In particular, a diffusion process is shown to hold for the average allele frequency among demes in which the level of subdivision sets the timescale of drift and selection and determines the dynamic equilibrium of allele frequencies among demes. This provides a framework for inference about mutation, selection, divergence, and migration when data are available from a number of unlinked nucleotide sites. The effects of subdivision on parameter estimates depend on the distribution of samples among demes. If samples are taken singly from different demes, the only effect of subdivision is in the rescaling of mutation and divergence-time parameters. If multiple samples are taken from one or more demes, high levels of within-deme relatedness lead to low levels of intraspecies polymorphism and increase the number of fixed differences between samples from two species. If subdivision is ignored, mutation parameters are underestimated and the species divergence time is overestimated, sometimes quite drastically. Estimates of the strength of selection are much less strongly affected and always in a conservative direction.     

The Effects of Background and Interference Selection on Patterns of Genetic Variation in Subdivided Populations

It is well known that most new mutations that affect fitness exert deleterious effects and that natural populations are often composed of subpopulations (demes) connected by gene flow. To gain a better understanding of the joint effects of purifying selection and population structure, we focus on a scenario where an ancestral population splits into multiple demes and study neutral diversity patterns in regions linked to selected sites. In the background selection regime of strong selection, we first derive analytic equations for pairwise coalescent times and F_ST as a function of time after the ancestral population splits into two demes and then construct a flexible coalescent simulator that can generate samples under complex models such as those involving multiple demes or nonconservative migration. We have carried out extensive forward simulations to show that the new methods can accurately predict diversity patterns both in the nonequilibrium phase following the split of the ancestral population and in the equilibrium between mutation, migration, drift, and selection. In the interference selection regime of many tightly linked selected sites, forward simulations provide evidence that neutral diversity patterns obtained from both the nonequilibrium and equilibrium phases may be virtually indistinguishable for models that have identical variance in fitness, but are nonetheless different with respect to the number of selected sites and the strength of purifying selection. This equivalence in neutral diversity patterns suggests that data collected from subdivided populations may have limited power for differentiating among the selective pressures to which closely linked selected sites are subject.     

A genomic scan for divergent selection in a secondary contact zone between Atlantic and Gulf of Mexico oysters, Crassostrea virginica

The degree of population structure within species often varies considerably among loci. This makes it difficult to determine whether observed variance reflects neutral-drift stochasticity or locus-specific selection at one or more loci. This uncertainty is exacerbated when evolutionary equilibrium cannot be assumed and/or admixture potentially inflates genomic variance. Thus, the value of a 'genome scan', where locus-specific summary statistics are compared with a simulated neutral distribution among loci, may be limited in secondary contact zones if the null distribution is sensitive to the timing of secondary contact. Of particular interest here is the wide variance previously observed in locus-specific patterns of population structure between Atlantic and Gulf of Mexico populations of eastern oyster, Crassostrea virginica. To test the robustness of an equilibrium null model, we compared among-locus distributions of F(ST) simulated under migration-drift equilibrium and several nonequilibrium secondary contact histories. We then tested for evidence of divergent selection between two oyster populations on either side of a secondary contact zone using 215 amplified fragment length polymorphism (AFLP) loci. Constant-migration equilibrium and nonequilibrium secondary-contact simulations produced equivalent distributions of F(ST) when anchored by the global mean F(ST) observed in oysters, 0.0917. The 99th quantile of simulated neutral F(ST) encompassed most of the variation among oyster loci. Three AFLP loci exhibited F(ST) values higher than this threshold. Although no locus was significant after correcting for multiple tests, our results show in geographically clinal organisms: AFLPs can efficiently characterize the genomic distribution of F(ST); equilibrium models can be used to evaluate outliers; these procedures help focus research on genomic regions of interest.     

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

Effects of polymorphism for locally adapted genes on rates of neutral introgression in structured populations

Adaptation to local conditions within demes balanced by migration can maintain polymorphisms for variants that reduce fitness in certain ecological contexts. Here, we address the effects of such polymorphisms on the rate of introgression of neutral marker genes, possibly genetically linked to targets of selection. Barriers to neutral gene flow are expected to increase with linkage to targets of local selection and with differences between demes in the frequencies of locally adapted alleles. This expectation is borne out under purifying and disruptive selection, regimes that promote monomorphism within demes. In contrast, overdominance within demes induces minimal barriers to neutral introgression even in the face of very large differences between demes in the frequencies of locally adapted alleles. Further, segregation distortion, a phenomenon observed in a number of interspecific hybrids, can in fact promote transmission by migrants to future generations at rates exceeding those of residents.     

The genomic and ecological context of hybridization affects the probability that symmetrical incompatibilities drive hybrid speciation

Despite examples of homoploid hybrid species, theoretical work describing when, where, and how we expect homoploid hybrid speciation to occur remains relatively rare. Here, I explore the probability of homoploid hybrid speciation due to “symmetrical incompatibilities” under different selective and genetic scenarios. Through simulation, I test how genetic architecture and selection acting on traits that do not themselves generate incompatibilities interact to affect the probability that hybrids evolve symmetrical incompatibilities with their parent species. Unsurprisingly, selection against admixture at “adaptive” loci that are linked to loci that generate incompatibilities tends to reduce the probability of evolving symmetrical incompatibilities. By contrast, selection that favors admixed genotypes at adaptive loci can promote the evolution of symmetrical incompatibilities. The magnitude of these outcomes is affected by the strength of selection, aspects of genetic architecture such as linkage relationships and the linear arrangement of loci along a chromosome, and the amount of hybridization following the formation of a hybrid zone. These results highlight how understanding the nature of selection, aspects of the genetics of traits affecting fitness, and the strength of reproductive isolation between hybridizing taxa can all be used to inform when we expect to observe homoploid hybrid speciation due to symmetrical incompatibilities.     

Blessed are the meek for they shall inherit the earth: quietness component of introversion is associated with single nucleotide polymorphisms in two circadian clock genes

Individual differences studied by chronobiologists and personality psychologists are usually shaped by polygenic selection occurring by small allele frequency shifts spreading across many loci. Therefore, the candidate gene association studies suffer from increased likelihood of false positive findings. We previously associated a PER3 single nucleotide polymorphism (SNP, rs228697) with self-ratings on personality-relevant nouns exemplifying personality dimension of Extraversion/Introversion in a sample of 88 female students. To replicate and extend this finding, we genotyped three more SNPs in three circadian clock genes. The results indicated that the minor alleles of PER3 rs228697 and PER2 rs934945 were rather similar in terms of their association with a personality type nicknamed “demure persona” (i.e. described by such nouns as “quietness”, “restraint”, “taciturnity”, “bashfulness”, “timidity”, “constraint”, and “reticence”). Analysis of data from populations of the 1000 Genomes Project suggested that, like frequencies of the minor alleles of many SNPs in circadian clock genes, the frequencies of these two SNPs were higher in populations of out-of-African ancestry compared to populations of African ancestry. We suggested that genetic candidates for Extraversion/Introversion can be prioritized in future association studies by means of identification of genetic signatures of polygenic selection imposed by out-of-Africa expansion of ancestral populations.     

Approximate Bayesian computation for modular inference problems with many parameters: the example of migration rates

We propose a two‐step procedure for estimating multiple migration rates in an approximate Bayesian computation (ABC) framework, accounting for global nuisance parameters. The approach is not limited to migration, but generally of interest for inference problems with multiple parameters and a modular structure (e.g. independent sets of demes or loci). We condition on a known, but complex demographic model of a spatially subdivided population, motivated by the reintroduction of Alpine ibex (Capra ibex) into Switzerland. In the first step, the global parameters ancestral mutation rate and male mating skew have been estimated for the whole population in Aeschbacher et al. (Genetics 2012; 192 : 1027). In the second step, we estimate in this study the migration rates independently for clusters of demes putatively connected by migration. For large clusters (many migration rates), ABC faces the problem of too many summary statistics. We therefore assess by simulation if estimation per pair of demes is a valid alternative. We find that the trade‐off between reduced dimensionality for the pairwise estimation on the one hand and lower accuracy due to the assumption of pairwise independence on the other depends on the number of migration rates to be inferred: the accuracy of the pairwise approach increases with the number of parameters, relative to the joint estimation approach. To distinguish between low and zero migration, we perform ABC‐type model comparison between a model with migration and one without. Applying the approach to microsatellite data from Alpine ibex, we find no evidence for substantial gene flow via migration, except for one pair of demes in one direction.     

The interpretation of selection coefficients

Evolutionary biologists have an array of powerful theoretical techniques that can accurately predict changes in the genetic composition of populations. Changes in gene frequencies and genetic associations between loci can be tracked as they respond to a wide variety of evolutionary forces. However, it is often less clear how to decompose these various forces into components that accurately reflect the underlying biology. Here, we present several issues that arise in the definition and interpretation of selection and selection coefficients, focusing on insights gained through the examination of selection coefficients in multilocus notation. Using this notation, we discuss how its flexibility—which allows different biological units to be identified as targets of selection—is reflected in the interpretation of the coefficients that the notation generates. In many situations, it can be difficult to agree on whether loci can be considered to be under “direct” versus “indirect” selection, or to quantify this selection. We present arguments for what the terms direct and indirect selection might best encompass, considering a range of issues, from viability and sexual selection to kin selection. We show how multilocus notation can discriminate between direct and indirect selection, and describe when it can do so.     

Effects of Interference Between Selected Loci on the Mutation Load,Inbreeding Depression,and Heterosis

A classical prediction from single-locus models is that inbreeding increases the efficiency of selection against partially recessive deleterious alleles (purging), thereby decreasing the mutation load and level of inbreeding depression. However, previous multilocus simulation studies found that increasing the rate of self-fertilization of individuals may not lead to purging and argued that selective interference among loci causes this effect. In this article, I derive simple analytical approximations for the mutation load and inbreeding depression, taking into account the effects of interference between pairs of loci. I consider two classical scenarios of nonrandomly mating populations: a single population undergoing partial selfing and a subdivided population with limited dispersal. In the first case, correlations in homozygosity between loci tend to reduce mean fitness and increase inbreeding depression. These effects are stronger when deleterious alleles are more recessive, but only weakly depend on the strength of selection against deleterious alleles and on recombination rates. In subdivided populations, interference increases inbreeding depression within demes, but decreases heterosis between demes. Comparisons with multilocus, individual-based simulations show that these analytical approximations are accurate as long as the effects of interference stay moderate, but fail for high deleterious mutation rates and low dominance coefficients of deleterious alleles.