The structure of allelic diversity in the presence of purifying selection

Abstract: In the absence of selection, the structure of allelic diversity is described by the elegant sampling formula of Ewens. This formula has helped shape our expectations of empirical patterns of molecular variation. Along with coalescent theory, it provides statistical techniques for rejecting the null model of neutrality. However, we still do not fully understand the statistics of the allelic diversity we expect to see in the presence of natural selection. Earlier work has described the effects of strongly deleterious mutations linked to many neutral sites, and allelic variation in models where offspring fitness is unrelated to parental fitness, but it has proven difficult to understand allelic diversity in the presence of purifying selection at many linked sites. Here, we study the population genetics of infinitely many perfectly linked sites, some neutral and some deleterious. Our approach is based on studying the lineage structure within each class of individuals of similar fitness in the deleterious mutation-selection balance. Analogous to the Ewens sampling formula, we derive expressions for the likelihoods of any configuration of allelic types in a sample. We find that for moderate and weak selection pressures the patterns of allelic diversity cannot be described by a neutral model for any choice of the effective population size, indicating that there is power to detect selection from patterns of sampled allelic diversity.