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The context-dependence of mutations: a linkage of formalisms

Frank J. Poelwijk, Vinod Krishna, Rama Ranganathan

arXiv:1502.00726v2q-bio.QMq-bio.PE

TL;DR

Mutation effects can be evaluated in one reference genotype or averaged across many backgrounds, which distinguishes genotype-specific effects from broadly important ones. The paper links epistasis formalisms through a weighted Walsh–Hadamard transform and finds that background averaging adjusts mutation effects for higher-order interactions.

  • Problem

    Epistatic effects can be defined relative to one reference genotype or averaged over many backgrounds, with averaging distinguishing genotype-specific effects from broadly important ones.

    Source: The ensemble view of epistasis · p. 2

  • Method

    The paper expresses the different epistasis formalisms as versions of the weighted Walsh-Hadamard transform, selecting terms according to reference choice or limits on epistatic order.

    Source: Link between the formalisms · p. 4, Discussion · p. 6

  • Results

    1.67 kcal mol−1 is the significant three-way term under both mutant-cycle and background-averaged definitions in the PDZ example, while lower-order values differ.

    Source: Link between the formalisms · p. 4

  • Takeaways & Limitations

    Background averaging adjusts mutation effects for higher-order interactions and represents their general value across related systems.

    Source: Link between the formalisms · p. 4

    Regression cannot identify independent effects when mutations always co-occur, leaving infinitely many solutions.

    Source: The epistatic structure of real systems · p. 5

Abstract

from arXiv · show

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.

Results

The Results link local mutant-cycle, background-averaged, and regression definitions of epistasis to Walsh-Hadamard transforms, using a PDZ example to illustrate how averaging accounts for context-dependent interactions. They also address learning epistatic structure from sparse combinatorial data and choosing representative genotype ensembles, discussing regularization, sparsity-based learning, and ensemble-averaged sequence methods as possible approaches.

Basic definitions

  • Absence of pairwise epistasis does not rule out higher-order epistasis; without any epistatic interactions, multiple-mutant effects reduce to sums of single-mutation effects.A triple mutant’s phenotype can depend on the wild type, three single-mutant effects, three pairwise terms, and one three-way term.Source: The biochemical view of epistasis · p. 2

Estimating epistasis with linear regression

Link between the formalisms

  • All three epistasis formalisms are versions of the Walsh-Hadamard transform, differing in term selection for a reference genotype or limits on epistatic order.The full background-averaged transform uses weights to normalize for different numbers of terms across epistatic orders; lower-order regression is compressive and loses information.Source: Link between the formalisms · pp. 3–4

The ensemble view of epistasis

The epistatic structure of real systems

  • Complete analysis requires phenotypes for all 2^n genotypes, which is impractical for typical protein domains of about 150 mutable positions and can include nonfunctional variants.When data are complete only up to order p, partial background averaging offers an intermediate measure that omits unavailable higher-order mutant phenotypes.Source: The epistatic structure of real systems · pp. 4–5
  • When data are sparse or mutations always co-occur, regression may not uniquely identify epistatic terms; regularization such as kernel ridge regression or LASSO can help without requiring specific data structure or coverage depth.Regularization has caveats, and co-occurring mutations can make independent effects impossible to calculate.Source: The epistatic structure of real systems · p. 5
  • Choosing representative genotype ensembles for background averaging remains unresolved; an ℓ1-constrained approach could exploit a sparse core of strong couplings amid weaker interactions to infer background-averaged epistasis.Protein studies report a subset of amino acids with strong, distributed couplings amid more weakly and locally coupled amino acids, while sparse strong couplings have been argued to characterize evolvable systems.Source: The epistatic structure of real systems · p. 5
  • The proposed sparsity-based strategy resembles compressed sensing and could be explored through focused high-order mutational analyses in well-chosen model systems.The authors note that the required experimental technologies are becoming available to help define practical data-collection strategies.Source: The epistatic structure of real systems · p. 5
  • Protein sequence-alignment methods average over homologous sequences and have revealed cooperative interactions spanning local structural contacts to global functional modes.The authors suggest formally mapping their constrained optimization problem to the ensemble averaging used by coevolution methods.Source: The epistatic structure of real systems · p. 5

Discussion

Defining epistatic structure is central to understanding how phenotype arises from genotype. The discussion distinguishes background-averaged descriptions from analyses local to a chosen reference sequence.

Discussion

  • The weighted Walsh-Hadamard transform unifies different approaches to epistasis as versions of one mathematical formalism.The target is epistatic structure underlying how phenotype arises from genotype.Source: Discussion · pp. 5–6
  • In its general form, the transform averages mutant effects over all genetic backgrounds at every order, correcting effects for higher-order terms.Source: Discussion · p. 6
  • Background-averaged effects indicate how transferable mutations are between model systems, unlike thermodynamic mutant cycles tied to one reference genotype.Source: Discussion · p. 6
  • Regression captures epistatic terms only up to a chosen lower order and focuses on the local mutational environment of a reference sequence.Source: Discussion · p. 6

A. Expressing the biochemical epistasis operator G as a Hadamard transform:

The derivation expresses the biochemical epistasis operator G as a Hadamard transform by relating recursive matrix forms and using induction.

A. Expressing the biochemical epistasis operator G as a Hadamard transform:

B. Expressing the regression operator as a Hadamard transform:

The regression-operator derivation uses matrix identities and explains why zeroing selected rows can correspond to zeroing matching columns for the structured regression order.

B. Expressing the regression operator as a Hadamard transform:

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