## Abstract

Epistasis is defined as the influence of the genotype at one locus on the effect of a mutation at another locus. As such it plays a crucial role in a variety of evolutionary phenomena such as speciation, population bottle necks, and the evolution of genetic architecture (i.e., the evolution of dominance, canalization, and genetic correlations). In mathematical population genetics, however, epistasis is often represented as a mere noise term in an additive model of gene effects. In this paper it is argued that epistasis needs to be scaled in a way that is more directly related to the mechanisms of evolutionary change. A review of general measurement theory shows that the scaling of a quantitative concept has to reflect the empirical relationships among the objects. To apply these ideas to epistatic mutation effects, it is proposed to scale A x A epistatic effects as the change in the magnitude of the additive effect of a mutation at one locus due to a mutation at a second locus. It is shown that the absolute change in the additive effect at locus A due to a substitution at locus B is always identical to the absolute change in B due to the substitution at A. The absolute A x A epistatic effects of A on B and of B on A are identical, even if the relative effects can be different. The proposed scaling of A x A epistasis leads to particularly simple equations for the decomposition of genotypic variance. The Kacser Burns model of metabolic flux is analyzed for the presence of epistatic effects on flux. It is shown that the non-linearity of the Kacser Burns model is not sufficient to cause A x A epistasis among the genes coding for enzymes. It is concluded that non-linearity of the genotype-phenotype map is not sufficient to cause epistasis. Finally, it is shown that there exist correlations among the additive and epistatic effects among pairs of loci, caused by the inherent symmetries of Mendelian genetic systems. For instance, it is shown that a mutation that has a larger than average additive effect will tend to decrease the additive effect of a second mutation, i.e., it will tend to have a negative (canalizing) interaction with a subsequent gene substitution. This is confirmed in a preliminary analysis of QTL-data for adult body weight in mice.

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