Graph-structured populations and the Hill–Robertson effect

Abstract

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 )).

Martingales and the characteristic functions of absorption time on bipartite graphs

“…Determining when martingales provide accurate approximations of global statistics for such stochastic processes, or bounds on them, is an active research topic. But this issue is irrelevant for the Moran process [ 21 , 28 , 31 , 32 ] and related birth–death processes [ 7 , 15 ], because martingales yield exact results for these problems.…”

“…Martingales may also be applicable to other extensions of the Moran process constrained by graphs. Instead of haploid reproduction, we can consider diploid reproduction models, where two individuals can sexually reproduce only if they are connected by an edge on the graph [ 7 ]. We can consider birth–death processes with more than two competing species, each with different fitnesses [ 43 , 44 ].…”

“…For example, stochastic processes have modelled the spread of cancer cells in healthy tissue [3,4], disease in a population [5] and social trends [6]. Birthdeath processes are a subset of stochastic processes that model the spread of genetic mutations in a resident population [7][8][9]. One particularly popular birth-death process is the Moran process, which has generated significant research interest since its introduction over 60 years ago [10][11][12][13][14][15].…”