Modeling the evolution of genetic architecture: A continuum of alleles model with pairwise AxA epistasis.

A continuum of alleles model with pair-wise AxA epistasis is proposed and its transmission genetic, and variational properties are analysed. The basic idea is that genes control the values of underlying variables, which affect the genotypic value of phenotypic characters proportional to a "scaling factor". Epistasis is the influence of one gene on the average effect of another gene. In this model, epistasis is introduced as a mutational effect of one gene on the scaling factors of another gene. In accordance with empirical results, the model assumes that the average direct effect of mutations is zero, as is the average epistatic effect. The model predicts that, on average, a mutation at one locus increases the expected mutational variance of mutations at another interacting locus. The increase in mutational variance is predicted to be equal to the variance of the pair-wise epistatic effects. This result is consistent with the observation that mutant phenotypes tend to be more variable than the wildtype phenotype. Another generic result of this model is that the frequency of canalizing mutations can at most be equal to the frequency of de-canalizing mutations. Furthermore, it is predicted that the mutational variance of a character increases at least linearly with the size of the character; hence this model is scale variant. In the case of two characters it is shown that the dimensionality of the locus-specific mutational effect distribution is invariant, i.e. the rank of the mutational covariance matrix M is invariant. While in additive models the mutational covariance matrix is always and entirely invariant, the invariance in the case of epistatic models is unexpected. Epistatic interactions can change the magnitude of the mutational (co)variances at a locus and can thus influence the structure of the mutational covariance matrix. However, in the present model the dimensionality of the mutational effect distribution remains the same. A consequence of this result is that, in this model, the genetic architecture of a set of characters is always evolvable i.e. no hard constraints can evolve.