The context-dependence of mutations: a linkage of formalisms

TL;DR

This paper unifies various definitions of epistasis under the mathematical formalism of the weighted Walsh-Hadamard transform, highlighting the importance of background-averaged epistasis for understanding biological systems.

Contribution

It reveals that different epistasis definitions are formal variants of the Walsh-Hadamard transform, providing a unified framework for analyzing complex genetic interactions.

Findings

Different definitions of epistasis are formal variants of the Walsh-Hadamard transform.

Background-averaged epistasis is most informative for understanding general epistatic structure.

Proposes approaches for efficiently learning epistatic interactions in complex systems.

Abstract

Defining the extent of epistasis - the non-independence 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 obscure in general. Here, we show that different definitions of epistasis are versions of a single mathematical formalism - the weighted Walsh-Hadamard transform. We discuss that one of the definitions, the backgound-averaged epistasis, is the most informative when the goal is to uncover the general epistatic structure of a biological system, a description that can be rather different from the local epistatic structure of specific model systems. 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 possible approaches for optimally learning the epistatic structure of biological systems.