Heritable variation and heterozygosity under a balance between mutations and stabilizing selection

Summary

A model of the balance between mutations and stabilizing selection affecting a quantitative character is developed and analysed. This model is essentially a discretized version of the continuum-of-alleles models analysed previously by Kimura, Lande, Turelli and others, and is formally similar to the stepwise mutation models used to interpret electrophoretic data. The complete model cannot be solved even for a haploid species, but there are useful approximations for most parameter values of interest. The ‘house-of-cards’ approximation can be used when selection is strong relative to mutation, and a normal approximation can be used when selection is relatively weak. For intermediate levels of selection a new ‘five-allele’ approximation provides accurate results over a wide range of parameter values. The house-of-cards and five-allele approximations applied to recessive alleles in a diploid population show that, for a given mutation rate, a somewhat larger genetic variance is maintained at equilibrium than in a comparable model of additive alleles. Under directional selection, the increase in genetic variance is largest for alleles of large effect and is much smaller for alleles of intermediate or small effect. At an equilibrium under stabilizing selection, homozygotes would tend to have a higher average fitness than heterozygotes when each mutation has a relatively large effect (the house-of-cards approximation), with the reverse if each mutation has a small effect (the normal approximation).