Coalescent Size Versus Coalescent Time with Strong Selection

Abstract

This work studies the coalescent (ancestral pedigree, genealogy) of the entire population. The coalescent structure (topology) is robust, but selection changes the rate of coalescence (the time between branching events). The change in the rate of coalescence is not uniform, rather the reduction in the time between branching events is greatest when the coalescent is small (immediately after the common ancestor is the only member of the coalescent) with little change when the coalescent is large (immediately preceding when that common ancestor becomes fixed and the size of the coalescent is N). This provides that the reduction in the coalescent time due to selection is much greater than the reduction in the cumulative size of the coalescent (total number of ancestors of the present population after and including the most recent common ancestor) due to selection. If Ns ≫ 1, the coalescent and fixation times are approximately equal to $\tfrac{{\ln 2N s}} {s}$ , which is much less than the value N which would result from neutral drift (N rather than the canonical haploid neutral fixation time 2N is the appropriate comparison for the model considered here), in particular, it is 70% less for Ns=10 and 95% less for Ns=100. However, for those values of Ns, and N ranging between 10³ and 10⁶, the reduction in the cumulative size of the coalescent of the entire population compared to the neutral case ranges from 17% to 65% (depending on the values of N and s). The coalescent time for two individuals for Ns of 10 and 100 is reduced by approximately 70% and 94%, respectively, compared with the neutral case. Because heterozygosity is proportional to the coalescent time for two individuals and the number of segregating alleles is proportional to the cumulative size of the coalescent, selection reduces heterozygosity more than it reduces the number of segregating alleles.