Linkage and the maintenance of variation for quantitative traits by mutation-selection balance: an infinitesimal model

The infinitesimal model is extended to cover linkage in finite populations. General equations to predict the dynamics of the genetic variation under the joint effects of mutation, selection and drift are derived. Under truncation and stabilizing selection, the quadratic equations for the asymptotic genetic variance (V-G) are respectively V-G(2) (1 + kS) + V-G(V-e - 2N(e)V(m)) - 2N(e)V(m)V(e) = 0 and V-G(2) (1 + S) + V-G(V-e + gamma - 2 NeVm) - 2N(e)V(m)(V-e + gamma) = 0, where N-e is the effective population size, V-m is the mutational variance, V-e is the environmental variance, gamma is the parameter that measures the spread of fitness around the optimum under stabilizing selection, k is equal to i(i-x) where i is the selection intensity and x is the cut-off point under truncation selection. The term S is a function of the number of chromosomes (v) and the average chromosome length (l): S approximate to v-1/v + 1/vl ln (N(e)l+1). These predictions are accurate when compared with results of simulations of small populations unless the number of genes is small. The infinitesimal model reduces to the continuum of alleles model if there is no recombination between homologous chromosomes.