Pleiotropic stabilizing selection limits the number of polymorphic loci to at most the number of characters

We demonstrate that, in a model incorporating weak Gaussian stabilizing selection on n additively determined characters, at most n loci are polymorphic at a stable equilibrium. The number of characters is defined to be the number of independent components in the Gaussian selection scheme. We also assume linkage equilibrium, and that either the number of loci is large enough that the phenotypic distribution in the population can be approximated as multivariate Gaussian or that selection is weak enough that the mean fitness of the population can be approximated using only the mean and the variance of the characters in the population. Our results appear to rule out antagonistic pleiotropy without epistasis as a major force in maintaining additive genetic variation in a uniform environment. However, they are consistent with the maintenance of variability by genotype-environment interaction if a trait in different environments corresponds to different characters and the number of different environments exceeds the number of polymorphic loci that affect the trait.