[Paper Review] The relation between alignment covariance and background-averaged epistasis

Frank J. Poelwijk, Rama Ranganathan|arXiv (Cornell University)|Mar 31, 2017

Genomics and Phylogenetic Studies3 references3 citations

TL;DR

This paper establishes a mathematical link between alignment-based covariance and background-averaged epistasis, showing that covariance in multiple sequence alignments implies epistatic effects but not vice versa. By reparametrizing sequence data and applying an inverse Walsh-Hadamard transform to alignment statistics, the authors demonstrate that functional predictions for combinatorial mutants can be accurately reconstructed from natural alignments—even with limited depth—achieving high Spearman correlation (R² = 0.86) with experimental data.

ABSTRACT

Epistasis, or the context-dependence of the effects of mutations, limits our ability to predict the functional impact of combinations of mutations, and ultimately our ability to predict evolutionary trajectories. Information about the context-dependence of mutations can essentially be obtained in two ways: First, by experimental measurement the functional effects of combinations of mutations and calculating the epistatic contributions directly, and second, by statistical analysis of the frequencies and co-occurrences of protein residues in a multiple sequence alignment of protein homologs. In this manuscript, we derive the mathematical relationship between epistasis calculated on the basis of functional measurements, and the covariance calculated from a multiple sequence alignment. There is no one-to-one mapping between covariance and epistatic terms: covariance implies epistasis, but epistasis does not necessarily lead to covariance, indicating that covariance in itself is not the directly relevant quantity for functional prediction. Having calculated epistatic contributions from the alignment, we can directly obtain a functional prediction from the alignment statistics by applying a Walsh-Hadamard transform, fully analogous to the transformation that reconstructs functional data from measured epistatic contributions. This embedding into the Hadamard framework is directly relevant for solidifying our theoretical understanding of statistical methods that predict function and three-dimensional structure from natural alignments.

Motivation & Objective

• To establish a theoretical framework linking statistical patterns in multiple sequence alignments to functional epistasis.
• To resolve the conceptual gap between covariance-based inference and experimentally measured epistasis in protein sequences.
• To enable functional prediction of combinatorial mutants using only alignment statistics, without experimental assays.
• To validate that alignment-derived epistatic terms can reconstruct phenotypes with high accuracy using inverse Hadamard transforms.

Proposed method

• Assumes a binary phenotype (functional/non-functional) to map alignment frequencies to epistatic contributions.
• Reparametrizes genotypes from {0,1} to {-1,1} to align with the background-averaged epistasis formalism.
• Defines alignment-based epistatic terms using the mean of element-wise products of columns in the alignment, scaled by sequence space size.
• Applies the inverse Walsh-Hadamard transform to reconstruct phenotypic values from alignment-derived epistatic terms.
• Uses the inverse transform $\boldsymbol{\hat{y}}^{\mathrm{aln}} = \boldsymbol{H}^{-1}\boldsymbol{V}^{-1}\boldsymbol{\bar{\omega}}^{\mathrm{aln}}$ to predict functional outcomes.
• Compares predictions from alignment statistics to experimental epistasis data from a 2^13 mutant library in fluorescent protein.

Experimental results

Research questions

• RQ1 How is alignment covariance mathematically related to background-averaged epistasis in protein sequences?
• RQ2 Can functional predictions for combinatorial mutants be accurately derived from natural sequence alignments alone?
• RQ3 Does the absence of a one-to-one mapping between covariance and epistasis limit the predictive power of alignment-based methods?
• RQ4 To what extent do limited-depth alignments still yield reliable epistatic estimates for functional prediction?
• RQ5 Can the Hadamard framework be extended to natural protein sequence spaces with 20 amino acid choices?

Key findings

• Covariance in multiple sequence alignments implies epistasis, but epistasis does not necessarily lead to detectable covariance, indicating that covariance is not a direct proxy for functional context-dependence.
• Alignment-derived epistatic terms, computed as column means of reparametrized sequences, can be used to reconstruct phenotypic values via the inverse Walsh-Hadamard transform.
• Even with only 200–300 sequences from a total of 8,192 possible mutants, the method achieves high Spearman correlation (ρ ≈ 0.9) between predicted and measured phenotypes.
• For the full functional mutant dataset, the R² between alignment-based and experimentally measured epistasis was 0.86 for first- and second-order terms combined.
• The method remains robust when alignment depth is limited, suggesting that representative subsets of functional sequences suffice for accurate prediction.
• The theoretical framework provides a solid foundation for using natural sequence alignments to predict function and 3D structure, though extension to 20 amino acid states remains an open challenge.