# Identifying coevolving residues by factoring out the evolutionary distance covariance matrix

Pablo Espana Gutierrez, François Coste

► To cite this version:

Pablo Espana Gutierrez, François Coste. Identifying coevolving residues by factoring out the evolutionary distance covariance matrix. LEGEND 2025 - Machine Learning for Evolutionary Genomics Data, Dec 2025, Aussois, France. 2025. hal-05424193

HAL Id: hal-05424193

https://inria.hal.science/hal-05424193v1

Submitted on 18 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Identifying coevolving residues by factoring out the evolutionary distance covariance matrix

<figure><img src="image_3.png" /></figure>

Université
de Rennes

IRISA

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

DyLiSS

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

École
normale
supérieure

Pablo Espana Gutierrez, François Coste

# Towards modeling proteins with distant dependencies

Models integrating explicit dependencies between pairs of residues contain precious structural information to model a protein given its sequence. [1] [2] [3]

<figure><img src="image_8.png" /></figure>

DNA sequencer

<figure><img src="image_9.png" /></figure>

Covarying residues
Coevolution ?
3D contact point ?

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

Sequence

Multiple sequences
alignment (MSA)

Model with
explicit residue
dependencies

<figure><img src="image_15.png" /></figure>

Couplings
parameters

Positional
parameters

Model
inference

## Covariance-based methods

Data matrix X

<figure><img src="image_16.png" /></figure>

n One-hot
encoding

<figure><img src="image_17.png" /></figure>

n Column
centering

<figure><img src="image_18.png" /></figure>

n

<figure><img src="image_19.png" /></figure>

Covariance

$$C = \frac{1}{n} X^T X$$

<figure><img src="image_20.png" /></figure>

$$f_{i,j}(a, b) - f_i(a) f_j(b)$$

+ Pseudo-counts

Inversion

+ Frobenius norm
+ APC

<figure><img src="image_21.png" /></figure>

2008
mpDCA

2011

mfDCA [4]

2012

PSICOV [5]

2013

GREMLIN

2014

CCMPred

2014

GaussDCA [6]

2018

bmDCA

2024

eaDCA

Direct Coupling Analysis (DCA)

# A theoretical framework to factor out phylogeny from couplings

<figure><img src="image_22.png" /></figure>

We modelize here that the data matrix X follows a **matrix normal law**, with **two** covariance matrices.

An **evolutionary distance covariance matrix**, representing dependencies between sequences (row dependencies of the MSA).

A **coupling covariance matrix**, capturing coevolution among residue positions driven by structural or functional constraints (column dependencies of the MSA).

$$X \sim NM(\mu, C^s, C^{\mathrm{c}})$$

Estimator

Pseudo-inverse

$$\hat{C}^c = \frac{1}{n} X^T (\hat{C}^s)^\dagger X$$

Given $C^s$, we can factor this signal out to only keep the signal of coevolution among residues

An estimator for $C^s$ is to compute the sequence distance covariance matrix:

$$C^s = \frac{1}{L} XX^T$$

$\frac{1}{n}$

$n$

<figure><img src="image_23.png" /></figure>

$$C^s$$

<figure><img src="image_24.png" /></figure>

(Frobenius norm)

<figure><img src="image_25.png" /></figure>

But doing so corresponds to a spectral flattening:

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

How to precisely infer the **evolutionary history signal** to better isolate the **couplings signal** ?

# Better estimation of the phylogenetic signal

## 1 Averaging among various alignments with shared evolutionary history

<figure><img src="image_28.png" /></figure>

$C^s_1$

<figure><img src="image_29.png" /></figure>

$C^s_2$

<figure><img src="image_30.png" /></figure>

$C^s_3$

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

Average

## 2 Projecting to a hierarchical block matrix

<figure><img src="image_33.png" /></figure>

<footnote>[1] Wilburn and Eddy. "Remote homology search with hidden Potts models". (2020)</footnote>

<footnote>[2] Muntoni et al. "Aligning biological sequences by exploiting residue conservation and coevolution" (2020)</footnote>

<footnote>[3] Talibart. "Comparison of homologous protein sequences using direct coupling information by pairwise Potts model alignments". Thesis (2021)</footnote>

<footnote>[4] Morcos et al. "Direct-coupling analysis of residue coevolution captures native contacts across many protein families". (2011)</footnote>

<footnote>[5] Jones et al. "PSICOV: precise structural contact prediction using sparse inverse covariance estimation on large MSA". (2012)</footnote>

<footnote>[6] Baldassi et al. "Fast and accurate multivariate Gaussian modeling of protein families: Predicting residue contacts and protein-interaction partners". (2014)</footnote>