RANDOM GENETIC DRIFT IN MULTI-ALLELIC LOCUS

Abstract

In the theory that evolution is anl irregularly shifting state of balance, especially that of a subtle balance between the evolutionary pressures of mutation, immigration and selection as a group and random or stochastic processes which give rise to processes of trial and error (Wright, 1950), the process of random genetic drift is one of the problems first to be clarified. Actually the problem of the random drift has been discussed by various authors. As early as 1921, and pointed out that the group of organisms destined to become the parents of the next generation is usually considerably smaller than the number of individuals of their species so that some genes will be lost by chance. According to their opinion, reduction of potential variability is automatic, being independent of any sort of selection. They considered that this is the most important gain in knowledge that we owe to Mendel's work and to the biomechanical interpretation of it. The first mathematical treatment of this so-called Hagedoorn Effect was carried out by Fisher (1922). It was rather unfortunate that an abbreviation of a term in his differential equation led him to the erroneous result that the rate of decrease of the variance in the population is 1/4n per generation, where n is the number of individuals in the population. Independently, \Vright worked on this problem and obtained the correct answer of 1/2N for the first time, where N is the effective population number (see Wright, 1931). Using an integral equation he also arrived at the flat distribution of unfixed classes for the state of steady decay. This stimulated Fisher to re-examine his results and he found himself in entire agreement with Wright's results. Furthermore, Fisher elaborated the terminal part of the distribution for the case of steady decay by his method of a functional equation (Fisher, 1930a, b). This method was followed by Haldane (1939) to treat the more general case where the number of descendants does not necessarily follow the Poisson distribution. Until now, Fisher's method remains as an unique and powerful tool to find the exact distribution at the terminal part. In 1945, Wright introduced the Fokker-Planck equation to solve the problem of the gene frequency distribution and applied the equation to the case of steady decay. The problem of random drift was also treated as a problem of a finite Markov chain by Malecot (1944), who got the asymptotic formula for a very large number of generations which is essentially the same as the previously known formula for steady decay. His result on the characteristic roots was later corrected by Feller (1951), who succeeded in getting all characteristic values of the matrix of transition probabilities. In spite of all these works, the theoretical results have been largely at the level of asymptotic formulae, and no complete solution has been published until the present