Evolution and maintenance of quantitative genetic variation by mutations.

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Abstract
The genotypic variance within, .sigma.w2 and between, .sigma.b2, random mating populations and rates and times for convergence to equilibrium values from different founder populations are formulated for an additive genetic model with an arbitary number of alleles .kappa., number of loci m, population size N, and mutation rate u, with unequal mutation rates for alleles. As a base of reference, the additive variance .sigma.a2 in an infinite equilibrium population is used. .sigma.a2 increases as k increases and decreases with variation in the mutation rates. Both transitional and equilibrium values of the variance within populations could be expressed as .sigma.w2 = (1 - .theta.).sigma.a2, where .theta. is the coancestry with mutations of individuals within populations. Thus, rates of convergence and evolutionary times are a function of those for .theta., which involves both N and u. When the founder population is fixed, very long times are required to obtain a perceptible increase of .sigma.w2 and equilibrium values of .sigma.w2 are very small when 4Nu .ltoreq. 10-1. The variance between populations can be expressed as .sigma.b2 = 2.theta..sigma.a2 when the founder population is an infinite equilibrium population, and as .sigma.b2 = 2(.theta. - .phi.) .sigma.a2 when the founder population is fixed, where .phi. is a function only of u. In this latter case, rates of divergence, while affected by both N and u, are dominated by u and asymptotically a function of u only. With u = 10-5, very long times (103 generations) are required for any perceptible divergence, even for N = 1-10. At equilibrium, most of the variance is between small populations and within very large populations. Migration increases the variance within populations and decreases the variance between populations.