FIXATION PROBABILITIES AND FIXATION TIMES IN A SUBDIVIDED POPULATION.

Abstract

The rate of genetic evolution depends in part on the probabilities of fixation and times to fixation of alleles of different types. Using a diffusion approximation to the Wright-Fisher model of selection and genetic drift in a single panmictic population, Kimura (1962) found a general formula for the probability of fixation at a diallelic locus with arbitrary selection coefficients. That greatly extended the results obtained for haploid models or for diploid models with no dominance which were solved using branching processes (Fisher, 1922) and by the direct analysis of a Markov chain (Moran, 1962). Kimura and Ohta (1969) used the same type of diffusion approximation to find the expected time to fixation in a diallelic system. These results are all for models of a single population. Since many natural populations are widely distributed in space, the assumption of panmixis may not be valid. It is important, then, to investigate the effects of geographic subdivision of the probabilites of fixation and times to fixation in a subdivided population. Pollak (1966) and Maruyama (1970a, 1972, 1974) have obtained several results for the case of a diallelic locus with no dominance. Pollak used the theory of multi-type branching processes to find the fixation probabilities of mutants subject to different selection pressures in different local populations. Maruyama showed that if the same selection pressure is acting in each local population and if the migration process has the property that the average allele frequencies do not change during the migration stage, then the fixation probabilities are the same as in a single panmictic population of the same total size. Maruyama's results were obtained using both a diffusion approximation and a direct analysis of a Markov chain. The same result could be obtained by using one of Pollak's formulae but he did not note that fact. None of these results are applicable to the case with other degrees of dominance and no information is obtained on fixation times. More recently Lande (1979) found the fixation probability of an underdominant mutant with equal homozygote fitnesses in the limit of low migration rates. Maruyama (1980) has independently obtained the same results as presented below for the fixation probabilities of mutants with different degrees of dominance. The only attempt at determining fixation times in a subdivided population that I am aware of is my heuristic analysis of a linear stepping stone model (Slatkin, 1976). The approach taken in this paper to the problem of finding the probabilities of fixation and times to fixation in a subdivided population is necessarily indirect. There is no way to solve the complete multi-dimensional diffusion equation and obtain the required quantities. Instead, I will obtain bounds on these quantities by considering certain limiting cases for which approximate analytic results can be found. This indirect approach will be complemented by computer simulations.