Under stabilizing selection in a finite population, X, the rate of aflelic substitutions at a locus is approximately A ,I(1 + S)-'½, where iA is the mutation rate. S, the stringency of selection upon new mutants, is defined by SNVm/Vs, where N is the population size and Vm/Vs is a measure of the average fitness decrease experienced by new mutants. The approximation holds both for a "lonely" locus, which is the sole provider of genetic variation for the character under selection, and for an "embedded" locus, which is not. In both cases I use the Crow and Kimura model of a continuum of alleles with Gaussian selection and mutation. Monte Carlo simulations corroborate the substitution rate formula. Some molecular evolution data suggest the potential utility and limitations of the formula for estimating population size, mutation, and selection parameters. This work agrees with the rest of nearly neutral theory in emphasizing the important role of population size for substitution rates.Under a strictly neutral theory of molecular evolution, allelic substitutions would occur at an expected rate of X = 2Nguu(1/2N) = g, [1] where X is the substitution rate, ,u is the mutation rate, N is the population size, and u(1/2N) is the substitution probability. Under a less strict, effectively neutral theory, one could expect [2] where p is the rate of appearance of neutral mutants, and p > p (1). But this model, which corresponds to Kimura's original proposal, depends on the deleterious mutants being so bad that they have no chance of becoming fixed. Meanwhile, new neutral alleles must have substitution probability 1/2N for Eq. 2 to work exactly. Thus, Ohta (2), Kimura (3), and others (4-7) have developed more sophisticated nearly neutral models in which new mutants can assume any of a continuum of selection coefficients. The effectively neutral mutation rate might then be defined, as in ref. 3, by the rate of mutations to alleles with a selection coefficient less than 1/2N. Kimura (3), for example, showed that Eq. 2 holds approximately when selection coefficients are distributed according to a Gamma density.Most students of natural selection agree that at the phenotypic level, stabilizing selection rules. In this paper, I obtain an approximation for X, when the locus of interest is at equilibrium under stabilizing selection. The present work builds on the work of Kimura (8) in which the nearly neutral theory was first extended to this important special case.I show that the approximation is fairly robust to deviations from the assumptions of the model that engenders it. Monte Carlo simulations using a variant of the pseudosamplingvariable method of Kimura and Takahata (9) also corroborate the analytic results. The substitution rate formula can be used, with some discretion, to investigate DNA sequencedivergence data.
Two Models with a Continuum of Possible AllelesCrow and Kimura (10) introduced a model for the study of quantitative characters under stabilizing selection that has since been put to considerable use (11,12)....