The stationary distribution of the infinitely-many neutral alleles diffusion model

An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate, and that the population is sufficiently large for the diffusion model to apply. Bounds on some moments are calculated from the density, and some applications are made to the problem of which allele is oldest in a population. A postulate made by Ewens (1972), concerning the distribution of allele numbers in a finite random sample from the neutral diffusion population, is shown to be correct. ALLELE FREOUENCIES; SELECTIVE NFUTRAI.ITY; I)IRICHLET DISTRIBUTION; ORDER STATISTICS. SAMPL.E DISTRIBUTION I. Assumptions and results Let X,, X,..., XK denote the allele relative frequencies in a population consisting of K alleles. For the present, we assume that each allele has the (constant) total mutation rate u per gene per generation, such that the probability of a mutation from an allele to any specific other allele is u/(K 1). We denote the effective population size by N,, and introduce the scaled mutation rates (1) = 4Nu, and e = 0/(K I). The diffusion approximation to the stationary density of X,, X, ,... XK is (see Wright (1969), (14.7)) (2) (X1,,X,2,XK)= IF(E)] K where x, + x, + + x = 1, and 0O x, _for each j. If we let K -,* , we are considering a limiting population in which all mutations produce new allelic types, there being infinitely many such types available. The best way to describe the limiting population seems to be by means of the order statistics Received 7 April 1976. 639 This content downloaded from 157.55.39.153 on Mon, 19 Sep 2016 04:49:18 UTC All use subject to http://about.jstor.org/terms