The effects of linkage and gene flow on local adaptation: A two-locus continent–island model

Abstract

Population subdivision and migration are generally considered to be important causes of linkage disequilibrium (LD). We explore the combined effects of recombination and gene flow on the amount of LD, the maintenance of polymorphism, and the degree of local adaptation in a subdivided population by analyzing a diploid, deterministic continent–island model with genic selection on two linked loci (i.e., no dominance or epistasis). For this simple model, we characterize explicitly all possible equilibrium configurations. Simple and intuitive approximations for many quantities of interest are obtained in limiting cases, such as weak migration, weak selection, weak or strong recombination. For instance, we derive explicit expressions for the measures Dnbsp(=pAB−pApB) and r2 (the squared correlation in allelic state) of LD. They depend in qualitatively different ways on the migration rate. Remarkably high values of r2 are maintained between weakly linked loci, especially if gene flow is low. We determine how the maximum amount of gene flow that admits preservation of the locally adapted haplotype, hence of polymorphism at both loci, depends on recombination rate and selection coefficients. We also investigate the evolution of differentiation by examining the invasion of beneficial mutants of small effect that are linked to an already present, locally adapted allele. Mutants of much smaller effect can invade successfully than predicted by naive single-locus theory provided they are at least weakly linked. Finally, the influence of linkage on the degree of local adaptation, the migration load, and the effective migration rate at a neutral locus is explored. We discuss possible consequences for the evolution of genetic architecture, in particular, for the emergence of clusters of tightly linked, slightly beneficial mutations and the evolution of recombination and chromosome inversions.

Genomic rearrangements and the evolution of clusters of locally adaptive loci

“…The establishment probabilities, EP clustered and EP unlinked , can be estimated by using the continent-island model of Bürger and Ackerman (24) to calculate the relative fitness of linked and unlinked mutations, and splicing this into Kimura's (51) equation for fixation probability, as shown by Yeaman and Otto (52). This approach assumes that a genomic island begins as a single differentiated allele (at a "focal locus") and expands as other linked locally adapted mutations establish (details are given in Methods).…”

“…If many such focal loci exist throughout the genome, then the maximum strength of selection at each focal locus will typically be lower, thereby reducing both the potential size of the cluster [due to s vs. r (19,53)] and the divergence hitchhiking advantage for any linked mutations. Thus, divergence hitchhiking is unlikely to be a broadly important explanation for the existence of islands of divergence without some genomic mechanism that dramatically reduces the rate of recombination around the initially divergent locus, such as a large inversion (24,32) or modifiers creating a recombination coldspot. Alternatively, if genomic rearrangements bring together the loci that contribute to local adaptation (thereby creating a genomic island), this would effectively increase the ratio of μ clustered /μ unlinked , and could facilitate divergence hitchhiking in subsequent bouts of local adaptation (if the alleles forming the genomic island happened to be lost owing to genetic homogenization among populations).…”

“…If adaptation to a given environment is based on changes in allele frequency at multiple loci, then recombination between the chromosomes of a locally adapted parent and a maladapted immigrant will tend to break up combinations of alleles with high fitness that have been established by selection (21)(22)(23)(24). As long as migration rates are low enough that the descendents of a locally adapted allele spend a large proportion of their time in the patch where they are fittest, then it is beneficial to be tightly linked to other alleles that are also locally adapted.…”

“…For all four of these mechanisms, the fitness benefits due to linkage tend to decrease with increasing rates of recombination between locally adapted alleles (21,24,25,28,29). As a result, genomic islands are more likely to evolve in areas of the genome with low rates of recombination, such as coldspots or regions with segregating inversions (17,(30)(31)(32)(33).…”

Population-genomic inference of the strength and timing of selection against gene flow

“…The recent explosion of genome-wide DNA sequencing data allows us to directly observe this mosaic and has spurred theoretical and empirical studies aiming to better understand the mechanisms underlying local adaptation and speciation (e.g., refs. [35][36][37][38]. However, an explicit, model-based framework linking observed genome-wide patterns of divergence with the underlying mechanism has hitherto been missing.…”