# Faithful Interpretation of Protein Structures through Weighted Persistent Homology Improves Evolutionary Distance Estimation

Léa Bou Dagher, Dominique Madern, Philippe Malbos, Céline Brochier-Armanet

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Léa Bou Dagher, Dominique Madern, Philippe Malbos, Céline Brochier-Armanet. Faithful Interpretation of Protein Structures through Weighted Persistent Homology Improves Evolutionary Distance Estimation. Molecular Biology and Evolution, 2025, 42 (2), 10.1093/molbev/msae271. hal-04981247

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# Faithful Interpretation of Protein Structures through Weighted Persistent Homology Improves Evolutionary Distance Estimation

Léa Bou Dagher ID,<fnref n="1" />,<fnref n="2" />,<fnref n="3" /> Dominique Madern ID,<fnref n="4" /> Philippe Malbos ID,<fnref n="2" />,*

Céline Brochier-Armanet ID<fnref n="1" />,<fnref n="5" />,*

## Abstract

Phylogenetic inference is mainly based on sequence analysis and requires reliable alignments. This can be challenging, especially when sequences are highly divergent. In this context, the use of three-dimensional protein structures is a promising alternative. In a recent study, we introduced an original topological data analysis method based on persistent homology to estimate the evolutionary distances from structures. The method was successfully tested on 518 protein families representing 22,940 predicted structures. However, as anticipated, the reliability of the estimated evolutionary distances was impacted by the quality of the predicted structures and the presence of indels in the proteins. This paper introduces a new topological descriptor, called bio-topological marker (BTM), which provides a more faithful description of the structures, a topological analysis for estimating evolutionary distances from BTMs, and a new weight-filtering method adapted to protein structures. These new developments significantly improve the estimation of evolutionary distances and phylogenies inferred from structures.

**Keywords:** protein 3D structure, bio-topological marker, 3D structures vectorization, structural phylogeny, indels

## Introduction

Pioneering work in the second half of the 20th century showed that proteins and genetic sequences contain historical information (i.e. a phylogenetic signal) that can be used to trace their evolutionary history and that of the organisms (Zuckerkandl and Pauling 1965). For proteins, the phylogenetic signal is mainly analyzed at the sequence level, although three-dimensional structures (hereafter called structures) have been proposed as an interesting alternative because they are thought to evolve more slowly (Illergard et al. 2009). However, the number of structures available in databases has been a limiting factor. The situation is changing with the development of efficient prediction programs based on artificial intelligence, which provide almost unlimited access to structures (Du et al. 2021; Jumper et al. 2021; Lin et al. 2023; Abramson et al. 2024). It is therefore expected that the use of structures to study protein evolution will increase in the coming years, hence the need for efficient structure analysis methods.

We have recently proposed an original approach based on persistent homology (PH) to extract the phylogenetic signal contained in structures represented by point clouds, where points correspond to the spatial coordinates of amino acid atoms (Bou Dagher et al. 2024b) (supplementary Text S1 and

supplementary figs. S1 to S3, Supplementary Material online). PH is a theory from algebraic topology and topological data analysis that emerged in the late 1990s (Barannikov 1994; Frosini and Landi 1999; Robins 1999; Edelsbrunner et al. 2002). It provides robust and efficient algorithms to describe the multidimensional geometric features (e.g. connected components, cycles, and cavities) of data, represented by finite point clouds, at different spatial scales, by the mean of a filtration process (Fig. 1) (Edelsbrunner et al. 2002; Carlsson 2009). For proteins, cycles and cavities, corresponding to structural features of homological dimensions 1 and 2 ($k=1$ and $k=2$, respectively), were found to be the most informative (Xia and Wei 2014; Bou Dagher et al. 2024b). The filtration process is designed to record the appearance (birth), persistence, and disappearance (death) of these features along the filtration in the form of PH-markers (supplementary Text S2 and supplementary figs. S1 to S5, Supplementary Material online). Barcodes (BCs) are the most common PH-markers. They represent the persistence of geometric features in the form of intervals drawn in the plane. The left and right ends of an interval indicate the birth and death of a feature, and the length of the interval corresponds to its lifespan (Fig. 1) (Barannikov 1994; Carlsson 2009). BCs can also be represented in the form of persistence landscapes (PLs), which

<footnote n="1">¹Université Claude Bernard Lyon 1, LBBE, UMR 5558, CNRS, VAS, Villeurbanne F-69622, France</footnote>

<footnote n="2">²Université Claude Bernard Lyon 1, CNRS, Institut Camille Jordan, UMR5208, Villeurbanne F-69622, France</footnote>

<footnote n="3">³Laboratoire de mathématiques, École Doctorale en Science et Technologie, Université Libanaise, Post Box 5, Hadath, Liban</footnote>

<footnote n="4">⁴Univ. Grenoble Alpes, CEA, CNRS, IBS, Grenoble 38000, France</footnote>

<footnote n="5">⁵Institut Universitaire de France</footnote>

<footnote>*Corresponding authors: E-mails: celine.brochier-armanet@univ-lyon1.fr; malbos@math.univ-lyon1.fr.</footnote>

<footnote>Associate editor: Julian Echave</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Examples of the filtration process and PH-marker computation. a) Schematic representation of geometric features captured by PH in homological dimensions $k=0$, 1, and 2. For $k=0$, they correspond to connected components, for $k=1$ to cycles, and for $k=2$ to cavities. Cycles and cavities must contain an empty space to be considered as geometric features. b) Illustration of the filtration process and barcode (BC) calculation of a point cloud in $\mathbb{R}^2$. The filtration is a continuous process, in which balls centered on points of the point cloud have a progressively increasing radius. Step 0 shows the initial point cloud, with no connection between points. Accordingly, the BC on the right contains only 12 intervals corresponding to these 12 connected components in homological dimension $k=0$, as no higher-dimensional features are observed. Step 1, the radius of the balls increases. Step 2, seven balls make contacts. Six persistent components remain alive, while six die. Balls continue to grow. At step 3, all the balls make contacts, reducing the number of connected components to one. In addition, a cycle is formed. It is represented by the birth of an interval in the BC in homological dimension $k=1$. This cycle will persist until its death at step 4. Note that in this example, no cavity is observed in homological dimension $k=2$, since the point cloud is in $\mathbb{R}^2$. However, when dealing with point clouds representing protein structures, the presence of cavities is common. c) The BC obtained at the end of the filtration process records the birth and death of geometric features observed during the filtration process. Each interval represents a geometric feature, with the length corresponding to its persistence or lifespan (i.e. the difference between its birth and death times). The persistence landscape (PL) is a smoother, more stable representation of the BC, where each interval is transformed into a piecewise linear function. The peaks of these functions correspond to the midpoint of each geometric feature's lifespan. The Betti curves (BT) plot the number of geometric features in each homological dimension as a function of the filtration parameter.</figcaption></figure>
allows the most persistent geometric features to be given greater importance than those with a shorter lifespan (Bubenik 2015). Finally, Betti curves (BT) provide another visualization of PH, plotting the number of features in each dimension along the filtration (Edelsbrunner et al. 2002).

PH's corpus of methods allows the calculation of distances, called PH-distances, between PH-markers and thus between structures. One of the main strengths of the PH is that it does not require the alignment of structures or the direct comparison of their structural characteristics (e.g. intramolecular distances between Ca, secondary structures). This is in contrast to most structural comparison methods (see for example Holm and Sander 1995; Choi et al. 2004; Krissinel and Henrick 2004; Zhang and Skolnick 2005; Zotenko et al. 2006; Akdel et al. 2020; Mirzaei et al. 2021; Xia et al. 2022; Zhang et al. 2022; Bale et al. 2023; van Kempen et al. 2024). In essence, the PH is an algebraic formalism for measuring the geometric features of structures (e.g. the size and number of cycles and cavities) through a combinatorial representation of their multiscale and multidimensional organization. The resulting measures are intrinsic to the structure and do not depend on its representation in any coordinate system. In addition, the filtering method is specifically designed to eliminate uncertainties in the point coordinates of point clouds, making the approach particularly relevant to the analysis of numerically and experimentally resolved structures.

By analyzing a large dataset, we have shown strong correlations between PH-distances computed on structures and phylogenetic distances on sequences (Bou Dagher et al. 2024b). The strength of this correlation is influenced by several factors, including the filtration method (i.e. Vietoris-Rips or alpha complex), the type of atoms represented in the point clouds (e.g. Ca, C, N, O, all atoms), the length of the proteins, the confidence index (CI) of the amino acid position in the structures, and the presence of indels (Bou Dagher et al. 2024b). In particular, we observed that omitting amino acids involved in indels or with poor CI improved distance estimation on structures. However, these omissions are questionable because they disrupt and distort the representation of structures, which can lead to inaccurate descriptions of their geometry, with, for instance, the artificial disappearance or appearance of cycles and cavities. Furthermore, since indels are part of the evolution, neglecting them may lead to an underestimation of evolutionary distances. Finally, although the best estimate of evolutionary distances was obtained using Wasserstein's distance (Cohen-Steiner et al. 2010) on point clouds filtered with the alpha complex filtration, this and other distances documented in the literature (e.g. the bottleneck distance (Carlsson et al. 2004; Collins et al. 2004), the landscape distance (Bubenik 2015)) are "off-the-shelf" methods, i.e. they were not specifically designed to analyze structures.

This follow-up study introduces new topological descriptors, called bio-topological markers (BTMs), and a new topological analysis for estimating evolutionary distances from BTMs. A BTM combines different types of PH-markers within a single vectorization to capture the geometric characteristics of structures in a more detailed and informative way. These vectorizations take full account of the combinatorial and statistical properties of PH-markers when calculating PH-distances. We show that combining BTMs with a weighted filtration, which integrates atom coordinate CIs, as well as the global and local density of point clouds, significantly improves the estimation of structural distances and phylogenetic reconstructions.

# Materials and Methods

## Dataset Assembly

The dataset analyzed comes from a study we published recently (Bou Dagher et al. 2024b). This dataset was composed of 518 monomeric protein families from 10 taxonomic groups of prokaryotes: 63 families are from *Bacillales* (Firmicutes), 31 from *Bacteroidales* (CFB group bacteria), 56 from *Corynebacterales* (Actinomycetota), 102 from *Enterobacterales* (*Gamma proteobacteria*), 39 from *Pseudomonadales* (*Gammaproteo bacteria*), 21 from *Hyphomicrobiales* (*Alphaproteobacteria*), 85 from *Escherichia* (*Enterobacterales*), 27 from *Methano coccales* (*Euryarchaeotes*), 65 from *Thermococcales* (*Euryarchaeotes*), and 29 from *Sulfolobales* (*Crenarchaeotes*) (Bou Dagher et al. 2024a). The complete dataset contained 22,940 proteins representing 763,648 pairs of homologous proteins. For each family, at least one experimentally resolved structure with high quality (<1.5 Å for bacteria and <2 Å for archaea) was available in the RCSDB PDB. The structure of each protein was predicted using AlphFold2 (see Bou Dagher et al. 2024b for more details).

## Weighted Point Cloud Filtration and PH-Marker Computation

Structures were represented by point clouds, whose points correspond to the spatial coordinates of the amino acid Ca (Fig. 2). Each point cloud, then denoted by PC(Cα, fω), was filtered using the weighted alpha complex with respect to a weight function fω on its points, which determines the order of their entry into the filtration process (Fig. 2) (supplementary Text S3 and supplementary figs. S3 to S15, Supplementary Material online). The higher the Cα weight, the earlier the Cα point enters the filtration process (Fig. 2). As a result, the geometric features to which they contribute have a longer lifetime. Note that when the weights are identically distributed (fω ≡ 0), all the Cα start to be filtered at the same time (supplementary fig. S7, Supplementary Material online). Here, the function fω included the CI of the Cα (CI Ca), the global density of the point clouds, and the local density of Cα within the point clouds. We tested seven functions combining these parameters (supplementary Text S3 and supplementary figs. S3 to S15, Supplementary Material online). Six PH-markers corresponding to barcodes (BC), persistent landscapes (PL), and Betti curves (BT), in homological dimensions 1 and 2, were computed from the filtered point clouds (Fig. 2). The BCs represent the chronology of the birth and death of geometric features (i.e. cycles and cavities) within the structure throughout the filtration process. The PLs represent their mid-lives and the BTs their number along the filtration (supplementary Text S2 and supplementary figs. S1 to S5, Supplementary Material online). For a given protein, the filtration parameter ranges from 0 to the diameter of its structure.

## Vectorization of PH-Markers

The vectorization of the PH-markers was designed to reflect and accommodate their main characteristics (Fig. 2). For BCs, the aim was to extract statistics rather than simply seeing them as a collection of intervals. We therefore considered 20 statistical measurements on the intervals: minimum birth time, number of features, mean lifespan, standard deviation, skewness, kurtoisis, first quartile, median, third quartile, and interquartile range of lifespan and mid-lifespan; BC persistence entropy and BC norm. A detailed explanation of these statistics and their
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Pipeline describing the computation of BTMs from protein structures. We started with structures predicted by Alphafold2. Each structure is represented by a point cloud corresponding to the three-dimensional coordinates of the amino acid Cα. The simplicial complexes were built by applying a weighted alpha complex filtration, which allows points with higher weights to enter at earlier steps of the filtration. Using linear algebra, the evolution of geometric features in the point cloud filtered with the weight function fω is encoded by an algebraic structure called the persistence module. This persistence module has an algebraic signature that can be featured to PH-markers (i.e. barcodes [BCs], Persistent landscapes [PLs], and Betti curves [BTs]) in different homological dimensions. For proteins, only homological dimension 1 (k=1) and 2 (k=2) are considered. In order to provide richer descriptions of the geometric features of the point cloud, we introduce BTMs that correspond to vectorizations of PH-markers. More precisely, BTs, PLs, and BTs are vectorized into 40-dimensional vectors (called V[BC], V[PL], and V[BT]). This allows capturing the combinatorial and statistical properties of the corresponding PH-markers. The combination of V(BC), V(PL), and V(BT) allows the construction of a 120-dimensional BTM.</figcaption></figure>

meaning can be found in supplementary Text S3, Supplementary Material online. Vectorization was also used to discretize PLs and BTs. To give them the same importance as the BC, we discretized them into 20 values for each homological dimension. Each PH-marker was thus converted into a 20-dimensional vector (supplementary text S4, supplementary figs. S5 to S9 and supplementary table S1, Supplementary Material online). For a given structure, the vectorization allowed PH-markers of different types and homological dimensions to be combined into a single vector. We defined a bio-topological marker (BTM) of a structure as the vectorization of a given PH-marker or a combination of PH-markers (Fig. 2). For example, for a given structure, the vectorization of the BC in homological dimension 1 is a BTM of dimension 20, called V(BC). Here, we constructed BTMs corresponding to V(BC), V(PL), and V(BT) from BCs, PLs, and BTs in homological dimensions 1 and 2 and we denoted by V(BC, PL, BT) the BTM of dimension 120 combining the six vectors (supplementary Text S4 and supplementary figs. S8 to S12, Supplementary Material online).

## PH-Distances Computation

PH-distances on structures allow them to be compared on the basis of their geometric features. For BTM based on BCs (i.e. V[BC]), we used the Wasserstein distance, which measures the distances between two BCs as a minimal cost of a matching between them. For BTMs constructed from PLs (i.e. V[PL]) and BT (i.e. V[BT]), we used the L² distance. Now, let us define these distances in detail.

To define Wasserstein's distance, let us recall the formal definition of a BC. A BC is a multiset of intervals (or horizontal bars) [b, d], where b, d ∈ ℝ ∪ (+∞) correspond to the birth value b of a geometric feature and to its death value d. We define the p-norm on intervals as follows. Given two intervals [b₁, d₁] and [b₂, d₂] we set:

$$ \| [b_1, d_1] - [b_2, d_2] \|_p = (|b_1 - b_2|^p + |d_1 - d_2|^p)^{\frac{1}{p}}. $$

For q ∈ [0, +∞], the Wasserstein q-distance between two BCs B and B' is defined to be:

$$ W_s(B, B') = \inf_{\varphi: B \to B'} \left( \sum_{a \in B} \|a - \varphi(a)\|_p^q \right)^{\frac{1}{q}}, $$

where the infimum is taken over all bijections φ between multisets B and B', and a is an interval in B. If the BCs B and B' do not contain the same number of intervals, a nonpaired interval [b, d] is paired with the interval $[\frac{b+d}{2}, \frac{b+d}{2}]$. In this study, the Wasserstein distance was computed using GUDHI library with p = q = 2 (Maria et al. 2014).

The Euclidean distance on the vector V(BC) is defined as follows. Let V<sub>B</sub>(BC) = (p<sub>1</sub>, ..., p<sub>40</sub>) and V<sub>B</sub>'(BC) = (q<sub>1</sub>, ..., q<sub>40</sub>) be vectorizations of BCs B and B'. The Euclidean distance between V<sub>B</sub>(BC) and V<sub>B</sub>'(BC) is defined to be:

$$ \text{Euc}(V_B(\text{BC}), V_{B'}(\text{BC})) = \left( \sum_{i=1}^{40} (p_i - q_i)^2 \right)^{\frac{1}{2}}. $$

The L² distance on the vector V(PL) is defined as follows. Let V<sub>B</sub>(PL) = (p<sub>1</sub>, ..., p<sub>40</sub>) and V<sub>B</sub>'(PL) = (q<sub>1</sub>, ..., q<sub>40</sub>) be vectorizations of PL B and B'. The L²-distance between V<sub>B</sub>(PL) and V<sub>B</sub>'(PL) is defined to be:

$$ L^2(V_B(\text{PL}), V_{B'}(\text{PL})) = \left( \sum_{i=1}^{40} |p_i(t) - q_i(t)|^2 dt \right)^{\frac{1}{2}}. $$

The L² distance on the vector V(BT) is defined in the same way.

Finally, the distance between two BTMs of type V(BC, PL, BT) is defined as the sum of the Euclidian distance on the V(BC) part of the vector, the L² distance on V(PL), and the L² distance on V(BT). According to Bou Dagher et al (2024b), all PH-distances were normalized by the average number of Cα in point clouds.
## Other Structural Distance Computation

We used DALI and TM-align (Zhang and Skolnick 2005; Holm et al. 2023) (supplementary Text S5, Supplementary Material online) to compute distances between pairs of homologous protein structures (Tkaczuk et al. 2007; Matsui and Iwasaki 2020; Lomoio et al. 2023). The TM-align distance is computed from the structural alignment of the two structures to be compared, while the DALI distance is computed by comparing their intra-molecular distances. We have also considered a distance based on PC_ali, a recently introduced approach that is based on both sequence and structure comparisons (Bastolla et al. 2023). Details of these distances can be found in supplementary Text S5, Supplementary Material online.

## Phylogenetic Inference and Comparison of Tree Topology

For each of the 518 protein families studied, we inferred: (i) a maximum likelihood (ML) phylogeny based on the sequences, (ii) a neighbor-joining (NJ) tree based on the patristic distances of the ML tree, (iii) a NJ tree based on the PH-distances calculated on BTMs, (iv) a NJ tree based on the DALI distances, (v) a NJ tree based on the TM-align distances, (vi) a NJ tree based on the PC_ali distances.

The ML tree was inferred as follows: (i) protein sequences were aligned using MAFFT v7.453 with the accurate L-INSI option (Katoh and Standley 2013), (ii) the resulting multiple alignment was trimmed with BMGE v2 with the BLOSUM45 substitution matrix (Criscuolo and Gribaldo 2010), (iii) the ML phylogenetic tree was inferred with IQ-TREE v2.1.2 (Nguyen et al. 2015) and the best fit model according to ModelFinder (see Bou Dagher et al. 2024b for more details). Patristic distances of the ML tree were computed with SEAVIEW 5.0.5 (Gouy et al. 2010). NJ trees were inferred using the Bio.Phylo TreeConstruction biopython library (Han and Zmasek 2009).

The topology of NJ trees was compared with the topology of the ML tree, used as a reference. To compare tree topologies, we used the Robinson-Foulds (RF) distance (Robinson and Foulds 1981). Since the RF distance depends on the number of leaves in the compared trees, we normalized the RF distance between two topologies by the maximum expected distance, i.e. $2\ell - 6$, where $\ell$ is the number of leaves. The RF distances were computed using IQ-TREE v2.1.2 (Nguyen et al. 2015).

## Results and Discussion

### Weighted PH Improves the Accuracy of Structure-Based Evolutionary Distance Estimation

In a previous study, we estimated structural evolutionary distances with the Wasserstein distance on BCs (Bou Dagher et al. 2024b). We showed that omitting $Ca$, whose position is unreliable (CI <70%), improves the estimate. However, the omission of some $Ca$ means that the distance computation is based on a discontinuous representation of the structures. Here, we proposed to refine the approach by considering the CI $Ca$ during filtration, i.e. by weighting $Ca$ according to their CI (Fig. 2). In this way, a $Ca$ with an unreliable position is not omitted, *de facto* ensuring the preservation of the geometric integrity of the structures, but it is given a lower weight, which means that it enters into the filtration later. In addition, we also considered the influence of local $Ca$ density, since $Ca$ with low CI are generally located in less dense areas

(supplementary figs. S10 to S11, Supplementary Material online, see also Radivojac et al. 2004; Jumper et al. 2021). We tested seven weighting functions. Regardless of the weighting system, we observed stronger correlations between the Wasserstein and patristic ML-distances ($r=0.84$ to $0.90$ [$k=1$], $r=0.81$ to $0.87$ [$k=2$]) than when $Ca$ were unweighted ($r=0.72$ [$k=1$] and $r=0.74$ [$k=2$]) or when $Ca$ with low CI were omitted ($r=0.74$ [$k=1$] and $z=0.75$ [$k=2$]) (supplementary table S2, Supplementary Material online). The best results were obtained when, in addition to the CI $Ca$, both the global density of the point clouds and the local density of the $Ca$ within point clouds were considered ($r=0.90$ [$k=1$] and $0.87$ [$k=2$]). Not surprisingly, close results were obtained when considering the squared CI $Ca$ or the squared local density of $Ca$ (supplementary table S2, Supplementary Material online). These results showed the importance of considering all $Ca$ but to relativize their contribution in geometric models. Indeed, even if the position of $Ca$ with low CI is uncertain, they are never far from their true neighbors. By delaying their entry into the filtration, they will only contribute to the largest geometric features.

### Weighted PH Greatly Reduces the Impact of Indels on Distance Estimates

Structural distances are sensitive to indel events that occur in protein sequences (Holm 2020; Malik et al. 2020). This is also the case for the PH-distances, as omitting $Ca$ corresponding to indels led to better estimates of evolutionary distances (Table 1; Bou Dagher et al. 2024b). Note that this was not the case when $Ca$ were weighted according to their CI (Table 1). This may be due to the fact that $Ca$ corresponding to indels mainly correspond to disordered regions, which on average have weaker CIs (Wrabl and Grishin 2004; Jumper et al. 2021; Ruff and Pappu 2021).

### Vectorization of PH-Markers Improves the Accuracy Structural Evolutionary Distance Estimation

While Wasserstein distances on BCs focus solely on the information provided by the persistence of geometric features along the filtration process, distances on BTMs integrate more information (Fig. 2). The relevance of BTMs was tested by comparing Wasserstein distances computed on BC and unweighted PH-distances on four BTMs (i.e. the 40-dimensionnal BTMs corresponding to V[BC], V[PL], and V[BT], and the 120-dimentional BTM corresponding to V[BC, PL, BT]), with the patristic ML-distances used as reference (supplementary tables S1 and S3, Supplementary Material online). We observed stronger correlations with PH-distances computed on BTMs than with the Wasserstein distances on BCs (Table 1); the best being obtained with the 120-dimensional BTM corresponding to V(BC, PL, BT) ($r=0.83$). However, we obtained even better results using the weighted PH on BTMs ($r=0.85$ to $0.97$, Table 1), showing that BTMs provide more accurate topological descriptions of structures. Once again, taking indels into account has almost no effect ($r=0.86$ to $0.97$, Table 1).

### Accuracy and Reliability of Weighted PH-Distances for Phylogenetic Inference

We compared the weighted PH-distances computed on the 120-dimentional BTMs with those computed with DALI and
<table><caption>**Table 1** Table showing the correlation coefficients and *P*-values between structural distances and patristic ML-distances</caption><thead><tr><th>Structural distances</th><th>PC(Cα), with all Cα</th><th>PC(Cα), with Cα of amino acids involved in indels being omitted</th><th>PC(Cα, f<sub>CI,Δα</sub>), with all Cα</th><th>PC(Cα, f<sub>CI,Δα</sub>), with Cα of amino acids involved in indels being omitted</th></tr></thead><tbody><tr><td>Ws1-distances on BC</td><td>0.72/5.1 × 10<sup>-4</sup></td><td>0.74/3.1 × 10<sup>-5</sup></td><td>0.90/2.5 × 10<sup>-4</sup></td><td>0.90/1.5 × 10<sup>-5</sup></td></tr><tr><td>Ws2-distances on BC</td><td>0.74/1.9 × 10<sup>-4</sup></td><td>0.75/7.4 × 10<sup>-4</sup></td><td>0.87/7.3 × 10<sup>-4</sup></td><td>0.87/2.8 × 10<sup>-4</sup></td></tr><tr><td>Euc-distances on V(BC)</td><td>0.79/2.1 × 10<sup>-3</sup></td><td>0.83/4.0 × 10<sup>-5</sup></td><td>0.91/1.1 × 10<sup>-4</sup></td><td>0.91/2.2 × 10<sup>-4</sup></td></tr><tr><td>L<sup>2</sup>-distances on V(PL)</td><td>0.73/4.9 × 10<sup>-4</sup></td><td>0.77/9.1 × 10<sup>-3</sup></td><td>0.85/1.7 × 10<sup>-5</sup></td><td>0.86/2.3 × 10<sup>-5</sup></td></tr><tr><td>L<sup>2</sup>-distances on V(BT)</td><td>0.72/1.7 × 10<sup>-5</sup></td><td>0.75/1.7 × 10<sup>-5</sup></td><td>0.87/0.6 × 10<sup>-5</sup></td><td>0.89/1.3 × 10<sup>-5</sup></td></tr><tr><td>PH-distances on V(BC, PL, BT)</td><td>0.83/3.8 × 10<sup>-4</sup></td><td>0.87/3.1 × 10<sup>-4</sup></td><td>0.97/3.2 × 10<sup>-4</sup></td><td>0.97/1.2 × 10<sup>-4</sup></td></tr><tr><td>DALI distances</td><td>0.85/1.3 × 10<sup>-3</sup></td><td>...</td><td>...</td><td>...</td></tr><tr><td>TM-align distances</td><td>0.87/5.2 × 10<sup>-3</sup></td><td>...</td><td>...</td><td>...</td></tr><tr><td>PC_ali distance</td><td>0.92/4.1 × 10<sup>-4</sup></td><td>...</td><td>...</td><td>...</td></tr></tbody></table>

Structural distances are: Ws1: Wasserstein distances in homological dimension 1 ($k=1$), Ws2: Wasserstein distance in homological dimension 2 ($k=2$), Euc: Euclidean distances, PH-distances defined on V(BC, PL, BT) as the sum of Euclidean distances on V(BC), L<sup>2</sup> distances on V(PL), and L<sup>2</sup> distances on V(BT), DALI distances (Holm et al. 2023), TM-align distances (Zhang and Skolnick 2005), and PC_ali distances (Bastolla et al. 2023). BC, barcode; PL, persistence landscape; BT, Betti curves.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Comparison between structure-based and sequence-based estimation of evolutionary distances and phylogenies. a) Comparison of ML-distances (x axis) and structural distances, corresponding, from left to right, to PH-distances computed on 120-dimensional BTMs, DALI distances, TM-align distances, and PC_ali distances. Each plot contains 763,648 points, corresponding to the 763,648 pairs of homologous proteins contained in the dataset. b) Boxplots showing the distribution of normalized RF distances computed between the ML-trees of the 518 protein families and the NJ trees inferred using structural distances, with, from left to right, PH-distances computed on 120-dimensional BTMs, DALI distances, TM-align distances, PC_ali distances, and patristic ML-distances. RF-distances computed between ML-trees and NJ-trees inferred with patristic ML-distances show the influence of the tree reconstruction method (ML vs. NJ) and are used as a reference. All *P*-values between pairs of boxplots are significant (<8 × 10<sup>-3</sup>). c) Comparison between the RF-distances computed between the 518 ML-trees and NJ-trees inferred using patristic ML-distances used as a reference (x axis), and, from left to right, the RF-distances computed between the 518 ML-trees and NJ-trees inferred using PH-distances on 120-dimensional BTMs, DALI distances, TM-align distances, and PC_ali distances.</figcaption></figure>
TM-align. As expected, we found strong correlations between the three distances ($r$=0.80 to 0.93, **supplementary fig. S16, Supplementary Material online**). However, the weighted PH-distances appeared to be more accurate as they correlated more strongly with the patristic ML-distances ($r$=0.97), than the DALI-distances ($r$=0.85) or the TM-align distances ($r$=0.88) (**Fig. 3a**). Moreover, PH-distances computed on BTMs outperformed also the recently proposed method PC_ali ($r$=0.97 and $r$=0.92, respectively).

To measure the reliability of structure-based distances, for each of the 518 protein families, we compared the ML-tree inferred on sequences with the NJ trees inferred with structural distances (**Fig. 3b and c**). We found that on average the topologies of NJ trees inferred using weighted PH-distances on BTMs were more similar to ML-trees (median RF distance = 0.22), than NJ trees inferred with DALI, TM-align, and PC_ali distances (median RF distance = 0.35, 0.28, and 0.25, respectively). This indicated that there is strong effect of the method used to calculate structural distances, and that distances based on PH represent a step forward. To go further, we wondered whether the differences in topologies reflected the inaccuracy of estimating evolutionary distances on structures or the influence of the method used to reconstruct trees (i.e. NJ for structures and ML for sequences). We inferred NJ trees using the patristic distances of the ML trees. The RF distances computed between these trees and ML trees were lower, but close to those computed between NJ trees inferred with weighted PH on BTMs and ML trees (median RF distance = 0.16 and 0.22, respectively, **Fig. 3b and c**). This indicates that there is a strong effect of the method used to reconstruct the trees (NJ vs. ML). This is not surprising, as ML and Bayesian methods are known to be more efficient than distance methods for phylogeny inference, and we believe that this is an important limitation for the use of protein structures for phylogenetic inference. Therefore, the next challenge will be to develop efficient tree reconstruction algorithms that can exploit the information extracted from structures.

## Conclusion

This work introduces two major improvements to protein studies by providing more accurate descriptions of structures and finer approaches to comparing structures. In particular, PH weighted by CIs made topological estimates more robust to resolution inaccuracies. Such an approach can be applied to predicted structures by using the pLDDT as weight, to experimentally resolved structures by using the B-factor, but also to structures for which no CI is available by using the squared local density as weight. It also provides an original solution to the problem of indels for structural distance estimation. Finally, the vectorized structure of BTMs will facilitate the application of machine learning algorithms to explore functional, biological, and evolutionary properties of structures. These advances open up new avenues in the field of structural phylogenetics. Our next goals are to refine both the vectorization of PH markers and their enrichment by integrating biological information into BTMs.

## Supplementary Material

Supplementary material is available at *Molecular Biology and Evolution* online.

## Acknowledgments

We would like to thank Jean-Pierre Flandrois and Joseph Nardin-Gennequin from the Claude Bernard University for

their valuable advice and fruitful discussions, as well as Fida El Chami from the Lebanese University for her advice and support.

## Funding

L.B.D. was supported by SAFAR scholarship funded by the Institut Français and Université Libanaise (129638R). This work has been supported by the French National Research Agency (grant: ANR-22-CE02-0027).

## Conflict of Interest

The authors declare that they have no conflicts of interest with the contents of this article.

## Data Availability

The data underlying this article are available in the permanent Mendeley data repository [https://data.mendeley.com/datasets/6h69hdh5bg/1] (DOI 10.17632/6h69hdh5bg.1).

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