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The Context-Dependence of Mutations: A Linkage of Formalisms

Frank J. Poelwijk, Vinod Krishna, Rama Ranganathan

PLOS Computational Biology · 2016 · CC BY

doi:10.1371/journal.pcbi.1004771 (opens in a new tab)

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Abstract

OverviewDefining 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.

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Figures3

From PubMed Central, U.S. National Library of Medicine.

Fig 2
Examples of epistasis in a PDZ domain (A) and a K+ ion channel (B). (A) PDZ domains are small, mixed αβ proteins that bind target peptide ligand (in yellow stick bonds) in a groove formed between the β2 and α2 elements (PSD95pdz3 shown, Protein Data Bank (PDB) accession 1BE9). The study discussed in the main text and in Table 1 is focused on the epistatic interactions between three amino acid positions—two in the PDZ domain (H372 and G330) and one in the ligand (T-2) (red spheres). (B) a thermodynamic cube representing the energetics of mutations at the three positions; values are equilibrium dissociation constants (Kd) for the target ligand (CRIPT [58]) in μM for all eight possible combination of mutations; errors represent standard deviation. (C) structure of the homotetrameric KcsA K+ ion channel (PDB accession 1K4C), showing the four positions selected for mutation in Sadovsky and Yifrach (in red spheres, shown only for one subunit for clarity) [60]. Note that the experiments were carried out in the Shaker K+ ion channel, and the positions in Shaker numbering are given in parentheses. The positions form a network that roughly links the intracellular activation gate and the selectivity filter.
Full legend
Examples of epistasis in a PDZ domain (A) and a K+ ion channel (B). (A) PDZ domains are small, mixed αβ proteins that bind target peptide ligand (in yellow stick bonds) in a groove formed between the β2 and α2 elements (PSD95pdz3 shown, Protein Data Bank (PDB) accession 1BE9). The study discussed in the main text and in Table 1 is focused on the epistatic interactions between three amino acid positions—two in the PDZ domain (H372 and G330) and one in the ligand (T-2) (red spheres). (B) a thermodynamic cube representing the energetics of mutations at the three positions; values are equilibrium dissociation constants (Kd) for the target ligand (CRIPT [58]) in μM for all eight possible combination of mutations; errors represent standard deviation. (C) structure of the homotetrameric KcsA K+ ion channel (PDB accession 1K4C), showing the four positions selected for mutation in Sadovsky and Yifrach (in red spheres, shown only for one subunit for clarity) [60]. Note that the experiments were carried out in the Shaker K+ ion channel, and the positions in Shaker numbering are given in parentheses. The positions form a network that roughly links the intracellular activation gate and the selectivity filter.
Open Fig 2 full size (opens in a new tab)

Fig 1
Definitions of genotype, phenotype, and effects of mutations. Representation of (A) single mutant, (B) double mutant, and (C) triple mutant experiments. Phenotypes are denoted by yg, where g is the underlying genotype. g = {gN,…,g1} with gi ∈{0,1}; “0” or “1” indicates the state of the mutable site (e.g., amino acid position). The effect of a single, double, and triple mutation is given by the red arrows. Pairwise (or second-order) epistasis is defined as the differential effect of a mutation depending on the background in which it occurs; for example, in (B) it is the degree to which the effect of one mutation (e.g., y10−y00) deviates in the background of the second mutation (y11−y01). Thus, the expression for second-order epistasis is (y11−y10)−(y01−y00). The third order and higher cases are considered in the main text.
Full legend
Definitions of genotype, phenotype, and effects of mutations. Representation of (A) single mutant, (B) double mutant, and (C) triple mutant experiments. Phenotypes are denoted by yg, where g is the underlying genotype. g = {gN,…,g1} with gi ∈{0,1}; “0” or “1” indicates the state of the mutable site (e.g., amino acid position). The effect of a single, double, and triple mutation is given by the red arrows. Pairwise (or second-order) epistasis is defined as the differential effect of a mutation depending on the background in which it occurs; for example, in (B) it is the degree to which the effect of one mutation (e.g., y10−y00) deviates in the background of the second mutation (y11−y01). Thus, the expression for second-order epistasis is (y11−y10)−(y01−y00). The third order and higher cases are considered in the main text.
Open Fig 1 full size (opens in a new tab)

Fig 3
Examples of matrices Zp introduced to calculate the partial background-averaged epistasis for n = 3. (A) Z2 for when data for mutants up to second-order is available and (B) Z1 for when only first-order mutants are available. Both matrices are self-similar, which allows their generation for arbitrary order, and are related to the logic Sierpinski triangle. For example, Z2 = 1−AΣ, where A is the anti-diagonal identity matrix and Σ is the Sierpinski matrix (i.e., multigrade AND in Boolean logic) for three inputs.
Full legend
Examples of matrices Zp introduced to calculate the partial background-averaged epistasis for n = 3. (A) Z2 for when data for mutants up to second-order is available and (B) Z1 for when only first-order mutants are available. Both matrices are self-similar, which allows their generation for arbitrary order, and are related to the logic Sierpinski triangle. For example, Z2 = 1−AΣ, where A is the anti-diagonal identity matrix and Σ is the Sierpinski matrix (i.e., multigrade AND in Boolean logic) for three inputs.
Open Fig 3 full size (opens in a new tab)

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