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)).