Random Genetic Drift in a Tri-Allelic Locus; Exact Solution with a Continuous Model

Abstract

Random genetic drift is a stochastic process of change in gene frequency in finite populations due to random sampling of gametes in reproduction. Since the pioneerinig works by Fisher (1922, 1930) and Wright (1931), much attention has been paid to this phenomenon and many theoretical as well as experimental studies have been attempted. A brief review on this topic will be found in a previous paper (Kimura 1955b). The evolutionary significance of random drift is still in dispute (Fisher and Ford 1947, 1950; Wright 1948, 1951), and decisive evidence for any conclusion is still missing. Haldane (1954) has suggested an analysis of frequencies of antigenic characters among neighboring populations for this purpose. Recently, Glass (1954) reviewed some evidence for the operation of random drift in human populations. From the genetical point of view, it is highly probable that there exists a class of genes so nearly neutral in selective value that random genetic drift plays a prominent role in determining the local differentiation of the gene frequencies. The best examples are to be found in certain isoalleles in Drosophila and other organisms. From the standpoint of mathematical genetics, the problem of random drift provides an area where the theory of Markov processes finds important applications (Feller 1951, Crow and Kimura 1955). In a recent paper the present author reported a complete solution of the process for the case of a pair of alleles (Kimura 1955a). With multiple alleles the problem becomes more difficult, and the solution even for three alleles (Kimura 1955b) contains a function CQ (x, y) and only the first three terms in the expansion are given explicitly. The