A Mendelian Markov process with binomial transition probabilities

Abstract

In any finite genetic population, the effect of chance involved in the sampling by which one generation replaces another causes gene frequency change as long as fixation or extinction is not achieved. In genetic terminology this fluctuation in gene frequency due to the finiteness of a population is called 'random genetic drift'. Fisher (1922), using an equation of the diffusion type, was the first to make an effort to describe quantitatively the phenomenon of random genetic drift. In the corrected version of this work (1930), he found the general result that the rate of decrease of the genetic variance in the population due to drift was 1/(2N) per generation, where N is the number of individuals of mating potential. This result was confirmed by various authors using different methods (e.g. Wright, 1931, 1937; Malecot, 1944; Feller, 1950). The particular form of the diffusion model which has had the widest use was introduced into genetical literature by Wright (1945). This equation which plays a fundamental role in the theory of gene frequencies is the Kolmogorov forward equation or-as it is commonly known among physicists-the Fokker-Planck equation. Earlier Kolmogorov (1935) had introduced the steady-state form of the forward equation in genetics. The complete solutions to the problem, however, were obtained much later and are due to Kimura (1955, 1957). The problem of random genetic drift as a problem in finite Markov chains was first considered by Malecot (1944). Based on the largest non-unit characteristic root of the transition probability matrix Malecot obtained the asymptotic rate of decrease of heterozygosity. This is essentially the same as the previously known results due to Fisher and Wright for steady decay. Feller (1951) succeeded in finding a general expression for all the characteristic roots of the transition probability matrix. The use of diffusion approximations requires the assumption of an infinite population: while in fact the state space is discrete, it is treated as though it were continuous. The tacit assumption in such an approach is that for large population size the approximation of the discrete time, discrete space process by the continuous diffusion process does not result in serious error. It was only recently that a rigorous justification of the diffusion approach was given by Watterson (1962) with respect to the processes considered by Wright (1945) and Kimura (1954, 1955). The validity of the diffusion approximation as providing adequate numerical solution to the problem concerning the probability that a particular gene is lost or fixed by a certain time was one of the questions considered by Ewens (1963). Since exact expressions were not available his method of getting exact results consisted in simply powering the transition matrix using a high-speed computer and collecting the appropriate terms.