Combinatorial Methods for Epistasis and Dominance.

We develop computational tools for the analysis of nonlinear genotype-phenotype relationships with epistasis among multiple loci or dominance interactions among multiple alleles within the same locus. Theory distinguishes between separable traits, with removable epistasis, and traits with essential epistasis. Separable traits can be transformed to a natural scale where additive methods apply. The methods we present solve for the natural scale, exactly when possible and approximately when not. Through graph methods, our methods allow for enumeration, counting, or sampling of distinct trait architectures satisfying constraints from the separability theory. A tool is provided for diagnosing which separability constraints are violated by a given nonseparable architecture. For genetic traits controlled by limited numbers of loci and alleles, our algorithm enumerates all possible trait structures and finds exact or error-minimizing linearizing transformations by formulating a constrained optimization program. We find that the fraction of possible distinct genetic traits satisfying simple criteria that can be fully or approximately linearized is high for small systems and falls as the number of alleles or loci increases.