The effect of a barrier to gene flow on patterns of geographic variation

Abstract

SummaryExplicit formulae are given for the effects of a barrier to gene flow on random fluctuations in allele frequency; these formulae can also be seen as generating functions for the distribution of coalescence times. The formulae are derived using a continuous diffusion approximation, which is accurate over all but very small spatial scales. The continuous approximation is confirmed by comparison with the exact solution to the stepping stone model. In both one and two spatial dimensions, the variance of fluctuations in allele frequencies increases near the barrier; when the barrier is very strong, the variance doubles. However, the effect on fluctuations close to the barrier is much greater when the population is spread over two spatial dimensions than when it occupies a linear, one-dimensional habitat : barriers of strength comparable with the dispersal range (B s) can have an appreciable effect in two dimensions, whereas only barriers with strength comparable with the characteristic scale (B L=s= ffiffiffiffiffi ffi 2m p ) are significant in one dimension (m is the rate of mutation or long-range dispersal). Thus, in a two-dimensional population, barriers to gene flow can be detected through their effect on the spatial pattern of genetic marker alleles.

Genetic consequences of habitat fragmentation during a range expansion

“…LDD does not influence the F ST distribution within a patch (Figure 6), where there are no barriers to dispersal and the sampled demes will likely be part of the same sectors (being close to each other). Our findings are consistent with the simulation study of Wegmann et al (2006) and the theoretical results of Barton (2008), where an increase in the variance of F ST was observed in the presence of spatial heterogeneity or dispersal barriers.…”

Estimating Barriers to Gene Flow from Distorted Isolation-by-Distance Patterns

“…For a population occupying a linear habitat, Nagylaki (1988) derived continuous equations for identity by state by taking the limit of a model of linear demes that exchange migrants (the so called stepping stone model). Barton (2008) expanded this by solving an analogous equation for two-dimensional populations. His formulas are given as a numerical Fourier transform that diverges for nearby individuals.…”

“…His formulas are given as a numerical Fourier transform that diverges for nearby individuals. These equations for two-dimensional populations are formally problematic, as they were not obtained by rescaling (which is impossible in two spatial dimensions), but Barton (2008) demonstrated that the solution is in close agreement with the solutions from a discrete stepping stone model for all but very close distances.…”

“…This approximation ignores that lineages do not move apart again once they have coalesced. An equivalent simplification is made by Barton (2008), and it is accurate as long as coalescence is sufficiently rare (Wilkins 2004). Since the dispersal process is symmetrical in time in our model of constant population density, the chance that two lineages at current position p 1 and p 2 are close at time t back equals the probability density that a single lineage moves from p 1 to p 2 in time 2t.…”

“…This signal therefore holds big potential for demographic inference. The derivation of Barton (2008) also shows that the effect of a barrier depends primarily on short-lived, localized fluctuations. In general, isolation by distance patterns equilibrate relatively quickly and depend mostly on recent demography (Barton et al 2013;Aguillon et al 2017).…”