Graph-structured populations and the Hill–Robertson effect

The Hill–Robertson effect describes how, in a finite panmictic diploid population, selection at one diallelic locus reduces the fixation probability of a selectively favoured allele at a second, linked diallelic locus. Here we investigate the influence of population structure on the Hill–Robertson effect in a population of size N . We model population structure as a network by assuming that individuals occupy nodes on a graph connected by edges that link members who can reproduce with each other. Three regular networks (fully connected, ring and torus), two forms of scale-free network and a star are examined. We find that (i) the effect of population structure on the probability of fixation of the favourable allele is invariant for regular structures, but on some scale-free networks and a star, this probability is greatly reduced; (ii) compared to a panmictic population, the mean time to fixation of the favoured allele is much greater on a ring, torus and linear scale-free network, but much less on power-2 scale-free and star networks; (iii) the likelihood with which each of the four possible haplotypes eventually fix is similar across regular networks, but scale-free populations and the star are consistently less likely and much faster to fix the optimal haplotype; (iv) increasing recombination increases the likelihood of fixing the favoured haplotype across all structures, whereas the time to fixation of that haplotype usually increased, and (v) star-like structures were overwhelmingly likely to fix the least fit haplotype and did so significantly more rapidly than other populations. Last, we find that small ( N < 64) panmictic populations do not exhibit the scaling property expected from Hill & Robertson (1966 Genet. Res. 8 , 269–294. ( doi:10.1017/S0016672300010156 )).

Further Properties of Gavrilets’ One-Locus Two-Allele Model of Maternal Selection

I derive several properties of the model proposed by Gavrilets for maternal selection at a single diallelic locus. Most notably, (i) stable oscillations of genotype frequencies (i.e., cycling) can occur and (ii) in the special case in which maternal effects and standard viability selection act multiplicatively, maternal selection effectively acts on maternally derived alleles only.

Genetic Diversity at a Single Locus Under Viability Selection and Facultative Apomixis: Equilibrium Structure and Deviations from Hardy-Weinberg Frequencies

We extensively analyze the maintenance of genetic variation and deviations from Hardy-Weinberg frequencies at a diallelic locus under mixed mating with apomixis and constant viability selection. Analytical proofs show that: (1) at most one polymorphic equilibrium exists, (2) polymorphism requires overdominant or underdominant selection, and (3) a simple, modified overdominance condition is sufficient to maintain genetic variation. In numerical analyses, only overdominant polymorphic equilibria are stable, and these are stable whenever they exist, which happens for ~78% of random fitness and mating parameters. The potential for maintaining both alleles increases with increasing apomixis or outcrossing and decreasing selfing. Simulations also indicate that equilibrium levels of heterozygosity will often be statistically indistinguishable from Hardy-Weinberg frequencies and that adults, not seeds, should usually be censused to maximize detecting deviations. Furthermore, although both censuses more often have an excess rather than a deficit of heterozygotes, analytical sign analyses of the fixation indices prove that, overall, adults are more likely to have an excess and seeds a deficit at equilibrium.

CLINES WITH ASYMMETRIC MIGRATION

ABSTRACT The consequences of asymmetric dispersion on the maintenance of an allele in a one-dimensional environmental pocket are examined. The diffusion model of migration and selection is restricted to a single diallelic locus in a monoecious population in the absence of mutation and random drift. It is further supposed that migration is homogeneous and independent of genotype, the population density is constant and uniform, and Hardy-Weinberg proportions obtain locally. If dispersion is preferentially out of an environmental pocket at the end of a very long habitat, the condition for maintaining the allele favored in the pocket becomes less stringent than for symmetric migration; dispersion preferentially into the pocket increases the severity of the condition for polymorphism. If an allele is harmful in large regions on both sides of an environmental pocket, the probability for polymorphism is decreased by asymmetric migration. The criterion for the existence of a cline is independent of the sense of the asymmetry; the cline itself is not. These phenomena are studied both analytically and numerically.—It is shown for symmetric migration and variable population density that the more densely populated parts of the habitat are more influential in determining gene frequency than the others. Thus, the higher the population density in an environmental pocket, the more easily an allele beneficial in the pocket will be maintained in the population.