The structured ancestral selection graph and the many-demes limit

Abstract

We show that the unstructured ancestral selection graph applies to part of the history of a sample from a population structured by restricted migration among subpopulations, or demes. The result holds in the limit as the number of demes tends to infinity with proportionately weak selection, and we have also made the assumptions of island-type migration and that demes are equivalent in size. After an instantaneous sample-size adjustment, this structured ancestral selection graph converges to an unstructured ancestral selection graph with a mutation parameter that depends inversely on the migration rate. In contrast, the selection parameter for the population is independent of the migration rate and is identical to the selection parameter in an unstructured population. We show analytically that estimators of the migration rate, based on pairwise sequence differences, derived under the assumption of neutrality should perform equally well in the presence of weak selection. We also modify an algorithm for simulating genealogies conditional on the frequencies of two selected alleles in a sample. This permits efficient simulation of stronger selection than was previously possible. Using this new algorithm, we simulate gene genealogies under the many-demes ancestral selection graph and identify some situations in which migration has a strong effect on the time to the most recent common ancestor of the sample. We find that a similar effect also increases the sensitivity of the genealogy to selection.

Figure 1.—

Graphical depiction of the structured selection process for the case of D = 3 demes.

Figure 2.—

PDFs for the T_MRCA of a sample of size 48 spread evenly among four demes and with sample allele counts (1 A₁, 11 A₂). A–C depict results for three different mutation rates as indicated. The migration parameter was ℳ = 0.01, and the solid curve shows the PDF for σ = 0; the long-dashed curves, for σ = 5.0; and the short-dashed curves, for σ = 7.5. The case of θ = 0.01 and σ = 7.5 could not be completed due to prohibitive computational requirements.

Figure 3.—

PDFs for the T_MRCA of a sample of size 48 spread evenly among four demes, but comparing two sample distributions of alleles A₁ and A₂: (1, 11) as in Figure 2 and (11, 1). In all cases, ℳ = 10 and θ = 0.1. As in Figure 2, the solid curve shows the PDF for σ = 0; the long-dashed curves, for σ = 5.0; and the short-dashed curves, for σ = 7.5.