Using known QTLs to detect directional epistatic interactions.

Epistasis plays important roles in evolution, for example in the evolution of recombination, but each of the current methods to study epistasis has limitations. Here, we propose a new strategy. If a quantitative trait locus (QTL) affecting a quantitative character has been identified, individuals who have the same genotype at that QTL can be regarded as comprising a subpopulation whose response to selection depends in part on interactions with other loci affecting the character. We define the marginal differences to be the differences in the average phenotypes of individuals with different genotypes of that QTL. We show that the response of the marginal differences to directional selection on the quantitative character depends on epistatic gene interactions. For a model with no interactions, the marginal differences do not differ on average from their starting values once linkage equilibrium has been re-established. If there is directional epistasis, meaning that interactions between the QTL and other loci tend to increase or decrease the character more than under an additive model, then the marginal differences will tend to increase or decrease accordingly when larger values of the character are selected for. We develop a likelihood ratio test for significant changes in the marginal differences and show that it has some power to detect directional epistasis for realistic sample sizes. We also show that epistatic interactions which affect the evolution of the marginal differences do not necessarily result in a substantial epistatic component of the genetic variance.