A Continuous-State Coalescent and the Impact of Weak Selection on the Structure of Gene Genealogies

Coalescent theory provides an elegant and powerful method for understanding the shape of gene genealogies and resulting patterns of genetic diversity. However, the coalescent does not naturally accommodate the effects of heritable variation in fitness. Although some methods are available for studying the effects of strong selection (Ns >> 1), few tools beyond forward simulation are available for quantifying the impact of weak selection at many sites. Here, we introduce a continuous-state coalescent capable of accurately describing the distortions to genealogies caused by moderate to weak natural selection affecting many linked sites. We calculate approximately the full distribution of pairwise coalescent times, the lengths of coalescent intervals, and the time to the most recent common ancestor of a sample. Weak selection (Ns approximate to 1) is found to substantially decrease the tree depth, primarily through a shortening of the lengths of the basal coalescent intervals. Additionally, we demonstrate that only two parameters, population size and the variance of the distribution describing fitness heritability, are sufficient to describe most changes.