The Distribution of Mutant Alleles

The results are presented from a simulation study of the spatial distribution of mutant alleles in a subdivided population. Statistical measures of the spatial pattern are defined in such a way that the same quantities could be measured in a geographic survey of allele frequencies in natural populations. Two types of quantities are discussed in this paper: (I) the occupancy distribution provides information on the presence or absence of the mutant in different numbers of demes; and (2) the conditional frequency distribution provides information about the extent of local differentiation when the mutant is present in different numbers of demes. Properties of these distributions are found for different types of natural selection acting on the mutant. Some results are presented for the same statistical measures based on samples of individuals from a fraction of the total number of demes. The simulation results for intermediate levels of the migration rates are compared with analytic results obtained on the limits of high and low migration rates. The main conclusion is that these measures of the spatial distribution of mutants in a subdivided population have simple properties that could provide a new perspective on data from natural populations. T H E spatial distribution of alleles in a subdivided population may reveal some aspects of the breeding structure of the population and some indication of the selection acting on those alleles. CAVALLI-SFORZA ( 1966) has emphasized that, even when the breeding structure the migration rates and effective local population sizes is unknown, the breeding structure must be the same for all alleles and loci. Using this fact, LEWONTIN and KRAKAUER (1973) attempted to look for consistency of allele distributions to determine whether selective neutrality could, under any breeding structure, account for the observed distributions. While there were statistical defects with their analysis that weaken their conclusions (NEI and MARUYAMA 1975; ROBERTSON 1975), their basic approach remains valid. To use that approach, however, new theoretical results are needed that provide useful measures of allele distributions in subdivided populations. In this paper, I will show that there are some simple properties of the spatial distribution of mutant alleles in subdivided populations. In another paper (SLATKIN, in preparation) , I will discuss the application of these results to data. Geretics 95: 503-524 June, 1980