The distribution of rare alleles
Population geneticists have long been interested in the behavior of rare variants. The definition of a rare variant has been the subject of some debate, centered mainly on whether alleles with small relative frequency should be considered rare, or whether alleles with small numbers should be. We study the behavior of the counts of rare alleles in samples taken from a population genetics model that allows for selection and infinitely-many-alleles mutation structure. We show that in large samples the counts of rare alleles — those represented once, twice, ... — are approximately distributed as a Poisson process, with a parameter that depends on the total mutation rate, but not on the selection parameters. This result is applied to the problem of estimating the fraction of neutral mutations.