The Context-Dependence of Mutations: A Linkage of Formalisms

There has been much recent interest in the prevalence of epistasis in the relationships between genotype, phenotype, and fitness in biological systems [1–7]. Epistasis here is defined as the nonindependence (or context-dependence) of the effect of a mutation, which is a generalization of Bateson’s original definition of epistasis as a genetic interaction in which a mutation “masks” the effect of variation at another locus [8]. It is also in line with Fisher’s broader definition of “epistacy” [9]. Epistasis limits our ability to predict the function of a system that harbors several mutations, given knowledge of the effects of those mutations taken independently [10–13], and makes these relationships increasingly more complex [14–19]. From an evolutionary perspective, the presence of epistatic interactions may limit or entirely preclude trajectories of single-mutation steps towards peaks in the fitness landscape [20–29]. With regard to human health, epistasis complicates our understanding of the origin and progression of disease [30–37]. Thus, interest in the extent of epistatic interactions in biological systems has originated from the fields of protein biochemistry, protein engineering, medicine, systems biology, and evolutionary biology alike. Originally, epistasis was considered in the context of two genes, but we can define it more broadly as the nonindependence of mutational effects in the genome, whether the effects are within, between, or even outside protein coding regions (e.g., in regulatory regions). The perturbations may go beyond point mutagenesis, but we limit the discussion here for clarity of presentation. Importantly, the definition of epistasis can be extended beyond pairwise effects to comprise a hierarchy of three-way, four-way, and higher-order terms that represent the complete theoretical description of epistasis between the parts that make up a biological system. How can we quantitatively assign an epistatic interaction given experimentally determined effects of mutations? Because epistasis is deviation from independence, it is crucial to first explicitly state the null hypothesis—asserting what exactly it means to have independent contributions of mutations. This by itself is typically nontrivial. In some cases the phenotype is directly related to a thermodynamic state variable, and the issue is straightforward: independence implies additivity in the state variable. For example, for equilibrium binding reactions between two proteins, independence means additivity in the free energy of binding ΔGbind, such that the energetic effect of a double mutation is the sum of the energetic effects of each single mutation taken independently. However, in general, many phenotypes cannot be so directly linked to a thermodynamic state variable, and quantification of epistasis needs to be accompanied by a proper rationale for the choice of null hypothesis. In what follows, we will assume this step has already been carried out and we will equate independence with additivity of mutational effects. Epistasis between two mutations is then defined as the degree to which the effect of both mutations together differs from the sum of the effects of the single mutations. In this paper, we describe three theoretical frameworks that have been proposed for characterizing the epistasis between components of biological systems; these frameworks originate in different fields and use seemingly different calculations to describe the nonindependence of mutations [2,14,24,33,38–46]. We extend previous observations [47–50] to show that these formalisms are different manifestations of a common mathematical principle, which explains their conceptual similarities and distinctions. Each of these formalisms has its value depending on depth of coverage and nature of sampling in the experimental data and the objective of the analysis. In the end, the fundamental issue is to develop practical approaches for optimally learning the epistatic structure of biological systems in the face of the explosive combinatorial complexity of possible epistatic interactions between mutations. Understanding the mathematical relationships between the different frameworks for analyzing epistasis is a key step in this process.