Genetic variability under step-wise discrete mutation and stabilizing selection

Abstract

To explain the nature of genetic variability for quantitative traits in infinitely large natural populations, a model involving step-wise mutation with discrete allelic effects and stabilizing selection of optimal type is considered. When the number of alleles at a locus is taken as finite instead of an infinitely large number, the properties of the equilibrium seem to change. In particular, cases of up to fifteen alleles at the locus are discussed in detail. The results obtained are more general and encompass on the one hand Turelli's [22] findings based on the 'house of cards' approximation for strong selection and on the other the results of the normal approximation for weak selection. The results of Slatkin [21] based on a five alleles approximation for intermediate selection are also made more exact by solving the set of recurrence equations without assumtng that the outermost alleles are negligible in frequency. The results obtained bring out clearly the behaviour of the genetic variability and heterozygosity at equilibrium as the ratio of mutation and selection parameters change from very low values to very high values. It seems the number of alleles considered at each locus could be a crucial factor In mutation-selection balance equilibria In large natural populations unless selection forces are sufficiently large that no more than two alleles can segregate at the locus.