The Context-Dependence of Mutations: A Linkage of Formalisms

Defining the extent of epistasis-the nonindependence of the effects of mutations-is essential for understanding the relationship of genotype, phenotype, and fitness in biological systems. The applications cover many areas of biological research, including biochemistry, genomics, protein and systems engineering, medicine, and evolutionary biology. However, the quantitative definitions of epistasis vary among fields, and its analysis beyond just pairwise effects remains problematic in general. Here, we bring together a number of previous results that show that different definitions of epistasis are versions of a single mathematical formalismthe weighted Walsh-Hadamard transform. We demonstrate that one of the definitions, the background-averaged epistasis, may be the most informative for describing the epistatic structure of a biological system. Key issues are the choice of effective ensembles for averaging and to practically contend with the vast combinatorial complexity of mutations. In this regard, we discuss strategies for optimally learning the epistatic structure of biological systems.