The Average Number of Generations until Fixation of a Mutant Gene in a Finite Population.

A mutant gene which appeared in a finite population will eventually either be lost from the population or fixed (established) in it. The mean time until either of these alternative events takes place was studied by WATTERSON (1962) and EWENS (1963). They made use of a method previously announced by DARLING and SIEGERT (1953), and, independently by FELLER (1954). Actually, DARLING and SIEGERT refer to its application to genetics. From the standpoint of population genetics, however, it is much more desirable to determine separately the mean time until fixation and that until loss. Since the gene substitution in a population plays a key role in the evolution of the species, it may be of particular interest to know the mean time for a rare mutant gene to become fixed in a finite population, excluding the cases in which such a gene is lost from the population. In the present paper, a solution to this problem will be presented together with Monte Carlo experiments to test some of the theoretical results. Throughout this paper, the senior author (M. K.) is responsible for the mathematical treatments, while the junior author (T. 0.) is responsible for the numerical treatments based on computers.