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How pairwise coevolutionary models capture the collective residue variability in proteins
Matteo Figliuzzi, Pierre Barrat-Charlaix, Martin Weigt
TL;DR
The paper asks whether pairwise coevolutionary models capture the collective sequence variability of homologous proteins, including higher-order features. It analyzes inferred pairwise models and finds that they reproduce several such features, while empirical alignments also contain phylogenetic correlations that remain difficult to disentangle.
Problem
It remained unclear whether pairwise couplings capture higher-order covariation in homologous protein families, despite the practical success of global coevolutionary models.
Source: §1 · p. 2Method
The paper infers pairwise models from empirical one- and two-residue statistics and examines correlations mediated by coupling paths.
Source: Direct coupling analysis – methodology… · p. 3, Correlations are mediated collectively… · p. 6Results
Pairwise models reproduce non-fitted higher-order correlations and sequence-space features, although small distances between closely related natural sequences are not reproduced.
Source: §1 · p. 3, Discussion · p. 11Takeaways & Limitations
The findings support pairwise models as sufficient to describe residue variability across homologous proteins within the examined scope.
Source: §1 · p. 3, Discussion · p. 11Empirical alignments mix functional and phylogenetic correlations, and how to disentangle them remains an open question.
Source: Discussion · p. 12
Abstract
from arXiv · showhide
Global coevolutionary models of homologous protein families, as constructed by direct coupling analysis (DCA), have recently gained popularity in particular due to their capacity to accurately predict residue-residue contacts from sequence information alone, and thereby to facilitate tertiary and quaternary protein structure prediction. More recently, they have also been used to predict fitness effects of amino-acid substitutions in proteins, and to predict evolutionary conserved protein-protein interactions. These models are based on two currently unjustified hypotheses: (a) correlations in the amino-acid usage of different positions are resulting collectively from networks of direct couplings; and (b) pairwise couplings are sufficient to capture the amino-acid variability. Here we propose a highly precise inference scheme based on Boltzmann-machine learning, which allows us to systematically address these hypotheses. We show how correlations are built up in a highly collective way by a large number of coupling paths, which are based on the protein's three-dimensional structure. We further find that pairwise coevolutionary models capture the collective residue variability across homologous proteins even for quantities which are not imposed by the inference procedure, like three-residue correlations, the clustered structure of protein families in sequence space or the sequence distances between homologs. These findings strongly suggest that pairwise coevolutionary models are actually sufficient to accurately capture the residue variability in homologous protein families.
1 Introduction
Homologous proteins can retain structure and function despite extensive sequence change, and their sequence variability carries information about evolutionary pressures. The paper investigates whether pairwise coevolutionary models capture this variability and reports that they reproduce several higher-order features not fitted by the models.
1 Introduction
- Homologous proteins may retain structure and biological function despite pairwise sequence identities of 20–30% or less.Source: §1 · p. 1
- Residue conservation can identify important sites, while profile models capture individual-column amino-acid statistics and support alignment and homology detection.Source: §1 · p. 2
- Pairwise residue covariation motivates global models such as DCA, which use sequence information for contact prediction and protein-structure modeling.Source: §1 · p. 2
- It remained unclear whether observed covariation reflects direct contacts or indirect paths, and whether pairwise couplings capture higher-order variation.Source: §1 · p. 2
- Exact pairwise inference is computationally infeasible for typical alignment lengths, while approximate methods do not reproduce pairwise statistics accurately enough to analyze higher-order correlations and collective effects.The paper introduces a highly precise inference approach and applies it to large protein families.Source: §1 · p. 2
- Indirect pair correlations arise collectively from networks of direct couplings based on structural vicinity between residues.Source: §1 · p. 3
- Pairwise DCA models reproduce non-fitted higher-order features, including three-residue correlations and sequence-space organization, supporting their sufficiency for capturing homologous-protein variability.The reported features also include sequence-distance distributions.Source: §1 · p. 3
Results
The authors use precise Boltzmann-machine learning to fit pairwise DCA models and examine how their coupling networks generate correlations and capture sequence variability beyond fitted statistics. They find that correlations are distributed across many coupling paths, while pairwise models reproduce several non-fitted properties of homologous protein families.
Direct coupling analysis – methodology and approximate solutions
- DCA fits fields and pairwise couplings to reproduce single-position and position-pair amino-acid frequencies, thereby also reproducing connected pairwise correlations.Exact marginal calculations require exponential sums, motivating approximate inference; those approximations can predict contacts well but do not reproduce empirical frequencies.Source: Direct coupling analysis – methodology… · pp. 3–4
Accurate fitting is needed to reproduce the empirical residue covariation in homologous protein families
- bmDCA reproduces empirical frequencies and connected two-point correlations more accurately than plmDCA, yet their contact predictions are close to identical.The comparison uses the strongest 2L = 224 couplings for PF00072; the figure caption reports similar results across studied families.Source: Accurate fitting is needed to… · p. 4, Figure 1 · p. 5
Correlations are mediated collectively by distributed networks of coevolutionary couplings
- The strongest coupling path typically contributes only about 12.5% of total mutual information, with many partially overlapping paths supplying the remaining indirect correlation.The strongest path is usually direct; the 12.5% average is dominated by the shortest protein families.Source: Correlations are mediated collectively… · pp. 6–7
Indirect correlations result collectively from networks of direct couplings
- Length-two information captures more mutual information than direct information, but longer coupling chains contribute substantially, especially in larger proteins.Length-two information includes the direct path and all paths with one intermediate residue, but depends on only 2L −3 pair couplings.Source: On the structural basis of… · p. 7
On the structural basis of coevolutionary coupling networks
- Among the top residue pairs, direct information tracks structural contacts, while length-two information reaches 85% true positives at distances of 11–12 Å, consistent with second structural neighbours.Direct-information predictions reach about 85% true positives at 8 Å; mutual information shows no exponential distance behavior or characteristic scale.Source: On the structural basis of… · p. 7, Figure 2 · p. 8
Pairwise coevolutionary models accurately reproduce the residue variability beyond the fitted two-residue statistics
- Pairwise bmDCA reproduces non-fitted three-residue correlations, protein-family clustering in sequence space, and most pairwise Hamming distances, unlike profile models.The small-distance tail differs: natural sequences can be nearly identical through close phylogenetic relationships, whereas equilibrium bmDCA samples do not show such small distances.Source: Pairwise coevolutionary models… · p. 9, Figure 3 · p. 10
Discussion
Pairwise models reproduce higher-order and global features of homologous protein sequence variability beyond the fitted one- and two-residue statistics. The discussion qualifies this finding with regularization bias and unresolved phylogenetic correlations.
Discussion
- Residue correlations arise collectively from many residues, limiting local statistical measures and motivating global modeling approaches.The paper says correlations are not the result of a few strong coupling chains.Source: Discussion · p. 11
- Pairwise models reproduce three-point correlations, pairwise Hamming-distance histograms, and clustered sequence-space structure, but not the smallest distances between close relatives.These features are not explicitly fitted by the model.Source: Discussion · p. 11
- This statistical agreement does not rule out higher-order evolutionary effects, whose signatures were not detectable as deviations from pairwise-model behavior.Random samples from DCA models based on empirical one- and two-residue statistics appear statistically indistinguishable from natural sequences.Source: Discussion · p. 11
- WW-domain scrambling studies found that preserving pairwise correlations yielded some folding, functional sequences, which had the highest probabilities in pairwise coevolutionary models.The authors connect these findings to evolution-guided protein design.Source: Discussion · p. 11
- Finite MSA size requires regularization, which slightly overestimates small pair frequencies; stronger regularization gradually erases sampled sequence clusters.Larger MSAs permit weaker regularization, limiting overfitting and inference bias.Source: Discussion · p. 11
- Sequence distributions mix functional and phylogenetic correlations from finite divergence times and species-selection bias; disentangling them remains open.An ideal model would retain functional correlations while discarding phylogenetic ones.Source: Discussion · pp. 11–12
Methods
The methods use precise Boltzmann-machine learning to fit pairwise residue statistics, then quantify correlations carried by direct couplings and paths through intermediate residues. They also combine direct and one-intermediary paths to measure their joint effect.
Protein families
- 10 large Pfam families with known structures and aligned lengths below 200 residues were selected to estimate three-point correlations accurately.Sequences with more than 50 alignment gaps were removed.Source: Protein families · p. 12
Boltzmann machine learning
- Boltzmann-machine learning iteratively adjusts Potts-model parameters until sampled one- and two-point frequencies reproduce the empirical MSA frequencies.The likelihood is convex, and ℓ2-regularisation is used to reduce finite-sample effects.Source: Boltzmann machine learning · pp. 12–13
Quantifying the strength of a coupling chain
- Path probability generalizes direct probability to a chain, multiplying direct probabilities across links and summing over intermediate-site configurations.Dynamic programming performs the sum, and the definition preserves empirical marginals at sites along the path.Source: Quantifying the strength of a coupling… · p. 13
The joint effect of paths of length 2
- Combining the direct path with all paths through one intermediary yields a joint distribution whose mutual information is called L2I.The combined path probabilities can be multiplied because each intermediate residue appears only once and is conditionally independent given the endpoints.Source: The joint effect of paths of length 2 · p. 14