The balance between pleiotropic mutation and selection, when alleles have discrete effects

The theory of pleiotropic mutation and selection is investigated and developed for a large population of asexual organisms. Members of the population are subject to stabilising selection on Ω phenotypic characters, which each independently affect fitness. Pleiotropy is incorporated into the model by allowing each mutation to simultaneously affect all characters. To expose differences with continuous-allele models, the characters are taken to originate from discrete-effect alleles and thus have discrete genotypic effects. Each character can take the values n×Δ where n=0,±1,±2,…, and the splitting in character effects, Δ, is a parameter of the model. When the distribution of mutant effects is normally distributed around the parental value, and Δ is large, a “stepwise” model of mutation arises, where only adjacent trait effects are accessible from a single mutation. The present work is primarily concerned with the opposite limit, where Δ is small and many different trait effects are accessible from a single mutation. In contrast to what has been established for continuous-effect models, discrete-effect models do not yield a singular equilibrium distribution of genotypic effects for any value of Ω. Instead, for different values of Ω, the equilibrium frequencies of trait values have very different dependencies on Δ. For Ω=1 and 2, decreasing Δ broadens the width of the frequency distribution and hence increases the equilibrium level of polymorphism. For all sufficiently large values of Ω, however, decreasing Δ decreases the width of the frequency distribution and the equilibrium level of polymorphism. The connection with continuous trait models follows when the limit Δ→0 is considered, and a singular probability density of trait values is obtained for all sufficiently large Ω.