Genealogical distances under low levels of selection

Abstract

For a panmictic population of constant size evolving under neutrality, Kingman's coalescent describes the genealogy of a population sample in equilibrium. However, for genealogical trees under selection, not even expectations for most basic quantities like height and length of the resulting random tree are known. Here, we give an analytic expression for the distribution of the total tree length of a sample of size n under low levels of selection in a two-alleles model. We can prove that trees are shorter than under neutrality under genic selection and if the beneficial mutant has dominance h < 1/2, but longer for h > 1/2. The difference to neutrality is O(α 2 ) for genic selection with selection intensity α and O(α) for other modes of dominance. * Genic selectionConsider a Moran model of size N, where every individual has type either • or •, selection is genic, type • is advantageous with selection coefficient α, and mutation is bi-directional. In other words, consider a population of N (haploid) individuals with the following transitions:1. Every pair of individuals resamples at rate 1; upon such a resampling event, one of the two involved individuals dies, the other one reproduces.