Quantitative trait loci and gene interaction: the quantitative genetics of metapopulations

Genetic population differentiation is typically viewed as differentiation of population means. However, several theories of evolution and speciation postulate that populations differentiate not only with respect to the population means, but also with respect to the effects of alleles within these populations. I develop herein a measure of population differentiation for the 'local average effects' of alleles, where local average effect is defined as the average effect of an allele in a deme measured as a deviation from the metapopulation mean. The differentiation for local average effects has two components, a component attributable to the population mean and a residual component that is attributable to changes in the local average effects independent of the population mean. The variance in local average effects attributable to the population mean is measured as the variance in the mean local average effect of all alleles. The variance in the residual local average effects is measured as the difference between the variance local average effects of individual alleles and the variance in the mean local average effects of all alleles. Differentiation for population means and differentiation for residual local average effects need not be related. I show that when there is only additive gene action, populations can differentiate for population means, but not for residual local average effects. However, if there is gene interaction then populations can also differentiate for local average effects of alleles. The consequence of this differentiation is that the local average effects of alleles change relative to each other such that an allele that is favoured by selection in one population may be removed by selection in other populations. I discuss the evolutionary consequences of differentiation for local average effects, and the interpretation of QTL data in light of this model.