Two-deme Model Research Articles

Locally adaptive inversions in structured populations.

Inversions have been proposed to facilitate local adaptation, by linking together locally coadapted alleles at different loci. Prior work addressing this question theoretically has considered the spread of inversions in "continent-island" scenarios in which there is a unidirectional flow of maladapted migrants into the island population. In this setting, inversions capturing locally adaptive haplotypes are most likely to invade when selection is weak, because stronger local selection (i) more effectively purges maladaptive alleles and (ii) generates linkage disequilibrium between adaptive alleles, thus lessening the advantage of inversions. We show this finding only holds under limited conditions by studying the establishment of inversions in a more general two-deme model, which explicitly considers the dynamics of allele frequencies in both populations linked by bidirectional migration. In this model, the level of symmetry between demes can be varied from complete asymmetry (continent-island) to complete symmetry. For symmetric selection and migration, strong selection increases the allele frequency divergence between demes thereby increasing the frequency of maladaptive alleles in migrants, favoring inversions-the opposite of the pattern seen in the asymmetric continent-island scenario. We also account for the likelihood that a new inversion captures an adaptive haplotype in the first instance. When considering the combined process of capture and invasion in "continent island" and symmetric scenarios, relatively strong selection increases inversion establishment probability. Migration must also be low enough that the inversion is likely to capture an adaptive allele combination, but not so low as to eliminate the inversion's advantage. Overall, our analysis suggests that inversions are likely to harbor larger effect alleles that experience relatively strong selection.

When does gene flow facilitate evolutionary rescue?

Experimental and theoretical studies have highlighted the impact of gene flow on the probability of evolutionary rescue in structured habitats. Mathematical modeling and simulations of evolutionary rescue in spatially or otherwise structured populations showed that intermediate migration rates can often maximize the probability of rescue in gradually or abruptly deteriorating habitats. These theoretical results corroborate the positive effect of gene flow on evolutionary rescue that has been identified in experimental yeast populations. The observations that gene flow can facilitate adaptation are in seeming conflict with traditional population genetics results that show that gene flow usually hampers (local) adaptation. Identifying conditions for when gene flow facilitates survival chances of populations rather than reducing them remains a key unresolved theoretical question. We here present a simple analytically tractable model for evolutionary rescue in a two-deme model with gene flow. Our main result is a simple condition for when migration facilitates evolutionary rescue, as opposed as no migration. We further investigate the roles of asymmetries in gene flow and/or carrying capacities, and the effects of density regulation and local growth rates on evolutionaryrescue.

Establishment of Locally Adapted Mutations Under Divergent Selection.

We study the establishment probabilities of locally adapted mutations using a multi-type branching process framework. We find a surprisingly simple and intuitive analytical approximation for the establishment probabilities in a symmetric two-deme model under the assumption of weak (positive) selection. This is the first analytical closed-form approximation for arbitrary migration rate to appear in the literature. We find that the establishment probability lies between the weak and the strong migration limits if we condition the origin of the mutation to the deme where it is advantageous. This is not the case when we condition the mutation to first occur in a deme where it is disadvantageous. In this case we find that an intermediate migration rate maximizes the probability of establishment. We extend our results to the cases of multiple demes, two demes with asymmetric rates of gene flow, and asymmetric carrying capacities. The latter case allows us to illustrate how density regulation can affect establishment probabilities. Finally, we use our results to investigate the role of gene flow on the rate of local adaptation and identify cases in which intermediate amounts of gene flow facilitate the rate of local adaptation as compared to two populations without gene flow.

Divergence and evolution of assortative mating in a polygenic trait model of speciation with gene flow.

Assortative mating is an important driver of speciation in populations with gene flow and is predicted to evolve under certain conditions in few-locus models. However, the evolution of assortment is less understood for mating based on quantitative traits, which are often characterized by high genetic variability and extensive linkage disequilibrium between trait loci. We explore this scenario for a two-deme model with migration, by considering a single polygenic trait subject to divergent viability selection across demes, as well as assortative mating and sexual selection within demes, and investigate how trait divergence is shaped by various evolutionary forces. Our analysis reveals the existence of sharp thresholds of assortment strength, at which divergence increases dramatically. We also study the evolution of assortment via invasion of modifiers of mate discrimination and show that the ES assortment strength has an intermediate value under a range of migration-selection parameters, even in diverged populations, due to subtle effects which depend sensitively on the extent of phenotypic variation within these populations. The evolutionary dynamics of the polygenic trait is studied using the hypergeometric and infinitesimal models. We further investigate the sensitivity of our results to the assumptions of the hypergeometric model, using individual-based simulations.

Local adaptation and the evolution of chromosome fusions.

We use forward and coalescent models of population genetics to study chromosome fusions that reduce the recombination between two locally adapted loci. Under a continent-island model, a fusion spreads and reaches a polymorphic equilibrium when it causes recombination between locally adapted alleles to be less than their selective advantage. In contrast, fusions in a two-deme model always spread; whether it reaches a polymorphic equilibrium or becomes fixed depends on the relative recombination rates of fused homozygotes and heterozygotes. Neutral divergence around fusion polymorphisms is markedly increased, showing peaks at the point of fusion and at the locally adapted loci. Local adaptation could explain the evolution of many of chromosome fusions, which are some of the most common chromosome rearrangements in nature.

A two-locus model of spatially varying stabilizing or directional selection on a quantitative trait

The consequences of spatially varying, stabilizing or directional selection on a quantitative trait in a subdivided population are studied. A deterministic two-locus two-deme model is employed to explore the effects of migration, the degree of divergent selection, and the genetic architecture, i.e., the recombination rate and ratio of locus effects, on the maintenance of genetic variation. The possible equilibrium configurations are determined as functions of the migration rate. They depend crucially on the strength of divergent selection and the genetic architecture. The maximum migration rates are investigated below which a stable fully polymorphic equilibrium or a stable single-locus polymorphism can exist. Under stabilizing selection, but with different optima in the demes, strong recombination may facilitate the maintenance of polymorphism. However usually, and in particular with directional selection in opposite direction, the critical migration rates are maximized by a concentrated genetic architecture, i.e., by a major locus and a tightly linked minor one. Thus, complementing previous work on the evolution of genetic architectures in subdivided populations subject to diversifying selection, it is shown that concentrated architectures may aid the maintenance of polymorphism. Conditions are obtained when this is the case. Finally, the dependence of the phenotypic variance, linkage disequilibrium, and various measures of local adaptation and differentiation on the parameters is elaborated.

The consequences of gene flow for local adaptation and differentiation: a two-locus two-deme model

We consider a population subdivided into two demes connected by migration in which selection acts in opposite direction. We explore the effects of recombination and migration on the maintenance of multilocus polymorphism, on local adaptation, and on differentiation by employing a deterministic model with genic selection on two linked diallelic loci (i.e., no dominance or epistasis). For the following cases, we characterize explicitly the possible equilibrium configurations: weak, strong, highly asymmetric, and super-symmetric migration, no or weak recombination, and independent or strongly recombining loci. For independent loci (linkage equilibrium) and for completely linked loci, we derive the possible bifurcation patterns as functions of the total migration rate, assuming all other parameters are fixed but arbitrary. For these and other cases, we determine analytically the maximum migration rate below which a stable fully polymorphic equilibrium exists. In this case, differentiation and local adaptation are maintained. Their degree is quantified by a new multilocus version of F_mathrm{ST} and by the migration load, respectively. In addition, we investigate the invasion conditions of locally beneficial mutants and show that linkage to a locus that is already in migration-selection balance facilitates invasion. Hence, loci of much smaller effect can invade than predicted by one-locus theory if linkage is sufficiently tight. We study how this minimum amount of linkage admitting invasion depends on the migration pattern. This suggests the emergence of clusters of locally beneficial mutations, which may form ‘genomic islands of divergence’. Finally, the influence of linkage and two-way migration on the effective migration rate at a linked neutral locus is explored. Numerical work complements our analytical results.

Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ 0 (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ 0 is an increasing function of the panmictic rate and to deduce the effect on λ 0 of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ 0 ; bounds on λ 0 are established; and λ 0 is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.

Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes

The maintenance of genetic variation in a spatially heterogeneous environment has been one of the main research themes in theoretical population genetics. Despite considerable progress in understanding the consequences of spatially structured environments on genetic variation, many problems remain unsolved. One of them concerns the relationship between the number of demes, the degree of dominance, and the maximum number of alleles that can be maintained by selection in a subdivided population. In this work, we study the potential of maintaining genetic variation in a two-deme model with deme-independent degree of intermediate dominance, which includes absence of G×E interaction as a special case. We present a thorough numerical analysis of a two-deme three-allele model, which allows us to identify dominance and selection patterns that harbor the potential for stable triallelic equilibria. The information gained by this approach is then used to construct an example in which existence and asymptotic stability of a fully polymorphic equilibrium can be proved analytically. Noteworthy, in this example the parameter range in which three alleles can coexist is maximized for intermediate migration rates. Our results can be interpreted in a specialist–generalist context and (among others) show when two specialists can coexist with a generalist in two demes if the degree of dominance is deme independent and intermediate. The dominance relation between the generalist allele and the specialist alleles play a decisive role. We also discuss linear selection on a quantitative trait and show that G×E interaction is not necessary for the maintenance of more than two alleles in two demes.

Single-Locus Polymorphism in a Heterogeneous Two-Deme Model

Environmental heterogeneity has long been considered a likely explanation for the high levels of genetic variation found in most natural populations: selection in a spatially heterogeneous environment can maintain more variation. While this theoretical result has been extensively studied in models with limited parameters (e.g., two alleles, fixed gene flow, and particular selection schemes), the effect of spatial heterogeneity is poorly understood for models with a wider range of parameters (e.g., multiple alleles, different levels of gene flow, and more general selection schemes). We have compared the volume of fitness space that maintains variation in a single-deme model to the volume in a two-deme model for multiple alleles, random selection schemes, and various levels of migration. Furthermore, equilibrium allele-frequency vectors were examined to see if particular patterns of variation are more prevalent than first expected. The two-deme model maintains variation for substantially larger volumes of fitness space with lower heterozygote fitness than the single-deme model. This result implies that selection schemes in the two-deme model can have a wider range of fitness patterns while still maintaining variation. The equilibrium allele-frequency patterns emerging from the two-deme model are more variable and strongly influenced by gene flow.