Multiplicative vs. arbitrary gene action in heterosis

In this article we investigate multiplicative effects between genes in relation to heterosis. The extensive literature on heterosis due to multiplicative effects between characters is reviewed, as is earlier work on the genetic description of heterosis. A two-locus diallelic model of arbitrary gene action is used to derive linear parameters for two multiplicative models. With multiplicative action between loci, epistatic effects are nonlinear functions of one-locus effects and the mean. With completely multiplicative action, the mean and additive effects form similar restrictions for all the rest of the effects. Extensions to more than two loci are indicated. The linear parameters of various models are then used to describe heterosis, which is taken as the difference between respective averages of a cross (F1) and its two parent populations (average of P). The difference (F2 - average of P) is also discussed. Two parts of heterosis are distinguished: part I arising from dominance, and part II due to additive X additive (a X a)-epistasis. Heterosis with multiplicative action between loci implies multiplicative accumulation of heterosis present at individual loci in part I, in addition to multiplicative (a X a)-interaction in part II. Heterosis with completely multiplicative action can only be negative (i.e., the F1 values must be less than the midparent), but the difference (F2 - average of P) can be positive under certain conditions. Heterosis without dominance can arise from multiplicative as well as any other nonadditive action between loci, as is exemplified by diminishing return interaction. The discussion enlarges the scope in various directions: the genetic significance of multiplicative models is considered. The description of heterosis is extended to three loci to show that multiplicative action between loci can make part I very large, but not part II. The genetic role of part II is explained. Finally, we compare multiplicative to arbitrary gene action in general, suggesting that the former may serve to measure nonadditivity of gene interactions in the latter.