THE MAINTENANCE OF POLYGENIC VARIATION IN FINITE POPULATIONS.

Models of the maintenance of genetic variance in a polygenic trait have usually assumed that population size is infinite and that selection is weak. Consequently, they will overestimate the amount of variation maintained in finite populations. I derive approximations for the equilibrium genetic variance, V^G in finite populations under weak stabilizing selection for triallelic loci and for an infinite "rare alleles" model. These are compared to results for neutral characters, to the "Gaussian allelic" model, and to Wright's approximation for a biallelic locus under arbitrary selection pressures. For a variety of parameter values, the three-allele, Gaussian, and Wrightian approximations all converge on the neutral model when population size is small. As expected, far less equilibrium genetic variance can be maintained if effective population size, N, is on the order of a few hundred than if N is infinite. All of the models predict that comparisons among populations with N less than about 10 should show substantial differences in V^G. While it is easier to maintain absolute V^G when alleles interact to yield dominance or overdominance for fitness, less additivity also makes V^G more susceptible to differences in N. I argue that experimental data do not seem to reflect the predicted degree of relationship between N and V^G. This calls into question the ability of mutation-selection balance or simple balancing selection to explain observed V^G. The dependence of V^G on N could be used to test the adequacy of mutation-selection balance models.