Renan Dos Reis

Luciano Da

F Costa

Enzyme Similarity Networks

Enzymes, which play an essential role in life, have their functionality defined by their shape and physical properties. In the present work, we describe how the coincidence methodology can be employed for obtaining similarity networks from physico-chemical features of seventy well-studied enzymes of the Glycoside Hydrolase Family 13, which can be classified into four categories based on the chemical reactions they catalyze. More specifically, each of these enzymes is mapped into a network node, while the links between pairs of enzymes are determined by the coincidence similarity values between their standardized physico-chemical feature values. The obtained networks have their modularity and number of isolated nodes optimized respectively to two parameters involved in the coincidence methodology, resulting in highly modular networks, providing important information not only about the enzymatic relationships between the considered enzymes, but also for choosing and designing proteins for specific biotechnological applications. In order to investigate the effect of the considered physico-chemical features on the enzymes relationships, a coincidencebased methodology is applied to create a meta-network, in which the enzymes similarity networks obtained by the combination of every possible feature are mapped into nodes of a feature combination network, while the coincidence similarity between those networks is taken into account to define the respective links. The obtained feature combination network systematically and comprehensively indicate the impact of the features combinations on the protein similarity. To find an optimal feature combination, a combined parameter optimization function is adopted promoting the maximization of the modularity simultaneously with the minimization of isolated nodes. Several interesting results are reported and discussed, including the identification of subgroups of enzymes with similar physico-chemical features among EC classes, which could not have been identified if were not for the possibility to vary the alpha coincidence parameter.

Introduction

Several of the most complex real-world systems can be found in biology, extending from micro to macroscopic time and spatial scales. Examples at micro and mesoscopic level include the immune system, the basal metabolism, and the nervous system. At more macroscopic levels, we have food webs and ecology.

A great deal of the biological complexity stems not only from the vast number of elements typically involved, but also and specially from the diversity of types of these entities. Of special relevance are biological molecules such as proteins, which appear in an impressive diversity of sizes and types, with specific properties regarding their functionalities.

Proteins are polymers of amino acids, each chosen among 22 possible types. Each of these amino acids have specific electrical, chemical, as well as other physical properties that will dictate how they behave and fold in the 3D space. The protein and all its resulting features on the 3D spaceincluding but not limited to its topology, volume, stability to environmental changes, mean hydropathy -will depend on its amino acid composition and will enable a biochemical function [START_REF] Creighton | The biophysical chemistry of nucleic acids and proteins[END_REF]. Thus, the great number of possibilities to assemble amino acid chains is directly connected to the numerous possible activities that proteins can perform [START_REF] Creighton | The biophysical chemistry of nucleic acids and proteins[END_REF].

A particularly important activity of proteins is the capacity to act as catalysts. This is crucial to the existence of life, as almost all biochemical reactions depends on that kind of polymer to occur in the rates needed to maintain biological systems [START_REF] Creighton | The biophysical chemistry of nucleic acids and proteins[END_REF]. This role is so important that proteins that acts as catalysts are specially called enzymes.

There are four perspectives from which the catalytic activity of enzymes can be better appreciated. The first is that enzymes increases the rates of very specific chemical reactions by selectively interacting with substrates, stabilizing transitions states and releasing chemical products [START_REF] Creighton | The biophysical chemistry of nucleic acids and proteins[END_REF]. The second concerns the fact that the same enzyme can have more than one distinct catalytic activity. For instance, the neopullulanase enzyme of Geobacillus stearothermophilus can participate in both the converting of pullulan in panose and the converting of cyclomaltodextrin in linear maltodextrin [START_REF] Hondoh | Threedimensional structure and substrate binding of bacillus stearothermophilus neopullulanase[END_REF]. The third perspective is that the same enzyme type can exist in several distinct organisms, extending along both related and unrelated phylogenetic levels. For example, pullulanase enzymes can be found in Bacillus subtilisand and Thermococcus kodakarensis, two bacteria species, as well as in Oryza sativa and Spinacia oleracea, two plant species [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. The forth perspective is that enzymes can be applied outside of its source organism to catalyze reactions of biotechnological interest [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF].

The catalytic activity of an enzyme stems from two principal aspects: (i) its 3D geometric shape; and (ii) the physical and chemical properties along its chain of amino acids [START_REF] Creighton | The biophysical chemistry of nucleic acids and proteins[END_REF]. The interaction between the enzyme and its substrates is favored by complementary shape and physical interactions, as well as access and proximity between them. For instance, a portion of an enzyme that assume a stable concave and positive charge surface in a reaction environment will have high probability of interacting to a convex and negatively charged substrate. However, although there are multiple computational and experimental techniques to study enzyme structure, this still is an arduous task.

Therefore, a complete understanding of the activity and application of an enzyme in biotechnological systems begins with the characterization and classification of its physical and chemical properties [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. A logical subsequent analysis involves organizing a set of enzymes of particular interest according to their mutual similarity. This is of particular interest because enzymes with similar properties can be synthesized, purified and applied in similar conditions [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. For instance, if two enzymes have the same catalytic activity and similar physico-chemical properties, they can be interchangeable in their biotechnological application, thus being two options that can be applied by industries according to economic criteria -such as cost of synthesis and production rates.

In the last few decades, multiple efforts been carried out with the intend of classifying and organizing enzymes based on their functional characteristics. The main one is the creation of the Enzyme Commission (EC) number system by the International Union of Biochemistry and Molecular Biology (IUBMB) [START_REF] Iubmb | Nomenclature Committee of the International Union of Biochemistry and Molecular Biology[END_REF]. This commission labels enzymes with a four-part EC number based on their catalytic activities. For example, the neopullulanase enzyme of Geobacillus stearothermophilus mentioned above receives the EC numbers 3.2.1.135 for its neopullulanase activity and 3.2.1.54 for its cyclomaltodextrinase [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. These four sequential numbers have different meanings: the first indicates the main catalytic class of the activity (e.g, 3 is for hydrolases); the second indicates the subclass based on the target chemical structure (e.g, 3.2 is for glycosylases -targets glycosyl groups); the third is for further information on the chemical reaction (e.g, 3.2.1 is for glycosydasestargets O-and S-glycosyl groups); and the fourth is a serial number for the specific reaction [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF][START_REF] Mcdonald | ExplorEnz: the primary source of the IUBMB enzyme list[END_REF]. This method of classification is useful for biotechnological purposes, as enzymes are promptly grouped to other proteins with the same biochemical applications, irrespective of its original species. The EC numbers system can be easily accessed on ExplorEns database, available at https://www.enzyme-database.org/ [START_REF] Mcdonald | ExplorEnz: the primary source of the IUBMB enzyme list[END_REF].

Other important efforts on curating and classifying proteins are: the BRENDA database (https://www.brenda-enzymes.org), with nomenclature, functional, kinetic, organism-related, structural and experimental information on enzymes [START_REF] Chang | BRENDA, the ELIXIR core data resource in 2021: new developments and updates[END_REF]; the BioCatNet database system (https://biocatnet.de/), focused on catalytic and amino acid sequence information of enzymes for engineering purposes [START_REF] Patrick | BioCatNet: A database system for the integration of enzyme sequences and biocatalytic experiments[END_REF]; the carbohydrate-active enzymes database (CAZy) (http://www.cazy.org/), specialized on the organization of genomic, structural and biochemical features of carbohydrate-active enzymes (CAZymes), as well as their classification into protein families [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. These resources can assist biotechnology researchers to both prospect and engineer enzymes with better features for their intended application [START_REF] Fasim | Large-scale production of enzymes for biotechnology uses[END_REF].

The utilization of physical and chemical knowledge of enzymes available on those databases could be further improved by identifying communities of proteins with similar properties within the groups assembled through phylogenetic, structural and catalytic analysis. For this purpose, in the present work, we aim at applying the coincidence methodology [START_REF] Da | Coincidence complex networks[END_REF] to obtain enzyme similarity networks based on physico-chemical data that are particularly detailed and modular. The coincidence similarity was developed as an extension of the Jaccard similarity index capable of being applied to real values [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF] and able to incorporate information about the relative interiority (or overlap, e.g..g. [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF]) between two datasets. When compared to more traditional similarity indices as the cosine similarity and correlation, the coincidence index has been found to allow more strict quantification of the pairwise similarity between two datasets [START_REF] Da | On similarity[END_REF][START_REF] Da | Comparing cross correlation-based similarities[END_REF].

The coincidence methodology has been proposed [START_REF] Da | Coincidence complex networks[END_REF] as a means to translate datasets of entities characterized by respective features into respective networks presenting enhanced connectivity details and modularity. It has been applied with encouraging success to several types of data [START_REF] Da | A caleidoscope of datasets represented as networks by using the coincidence methodology[END_REF][START_REF] Da | Elementary particles networks as revealed by their spin, charge and mass[END_REF] and interesting problems including city characterization [START_REF] Luciano Da | A similarity approach to cities and features[END_REF] and motifs identification [START_REF] Domingues | City Motifs as Revealed by Similarity Between Hierarchical Features[END_REF].

In the present work, we apply the coincidence methodology to translate seventy well-studied enzymes of biotechnological interest into similarity networks, where each node corresponds to an enzyme, and the interconnection values between pairs of nodes are specified by the pairwise coincidence similarity between their respective physico-chemical features -namely number of amino acids, molecular weight, theoretical isoelectric point (pI), instability index, aliphatic index, and grand average of hydropathicity (GRAVY) [START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF].

The translation procedure involves two parameters, namely α and T , controlling respectively the relative contribution of negative and positive feature values and the threshold for removing weaker connections [START_REF] Da | Coincidence complex networks[END_REF]. Thus, similarity networks with optimal modularity can be obtained by optimizing the modularity value while systematically varying α and T [START_REF] Da | Coincidence complex networks[END_REF][START_REF] Domingues | City Motifs as Revealed by Similarity Between Hierarchical Features[END_REF]. However, it often occurs that the parameter configuration leading to the maximum modularity is also characterized by a large number of isolated nodes. In order to identify parameter configurations leading to high (but not necessarily optimal) modularity while the number of isolated nodes is simultaneously minimized, we adopted a supervised optimization index reflecting both these sought properties.

The enzymes selected for this work are CAZymes of the glycoside hydrolase family 13 (GH13) [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. They span four types of catalytic activities [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF], each with particular biotechnological interests: hexosyltransferases (EC 2.4.1.-), applied in the production of sugars with novel properties, both in material and food sciences [START_REF] Franceus | Sucrose phosphorylase and related enzymes in glycoside hydrolase family 13: Discovery, application and engineering[END_REF][START_REF] Tian | Amylosucrase as a transglucosylation tool: From molecular features to bioengineering applications[END_REF][START_REF] Qi | Cyclodextrin glucanotransferase: from gene to applications[END_REF]; phosphate α-maltosyltransferases (EC 2.4.99.16), enzymes of potential pharmaceutical interest [START_REF] Puchart | Glycoside phosphorylases: structure, catalytic properties and biotechnological potential[END_REF][START_REF] Leiba | Mycobacterium tuberculosis maltosyltransferase GlgE, a genetically validated antituberculosis target, is negatively regulated by Ser/Thr phosphorylation[END_REF][START_REF] Syson | Structure of streptomyces maltosyltransferase GlgE, a homologue of a genetically validated antituberculosis target[END_REF]; glycosidases, enzymes that hydrolyse O-and S-glycosyl compounds (EC 3.2.1.-), applied in starch and baking industries [START_REF] Nisha | Recombinant bacterial amylopullulanases: developments and perspectives[END_REF]; and intramolecular transferases of maltodextrin, sucrose or maltose (EC 5.4.99.-), used for the production of valuable sugars, such as trehalose [START_REF] Cai | Biotechnical production of trehalose through the trehalose synthase pathway: current status and future prospects[END_REF]. In the translation of these enzymes into similarity networks, we calculate the modularity based on the EC number of each enzyme, which was tuned via the supervised optimization index. The proposed method led to enzyme similarity networks not only with a small number of isolated nodes, but also with high levels of catalytic similarity within the obtained clusters.

The results typically obtained in supervised classification are known to depend critically on the set of features adopted to characterize the data elements of interest (i.e. individual enzymes). In the present work, we employ the feature analysis methodology described in [START_REF] Da | Elementary particles networks as revealed by their spin, charge and mass[END_REF] as the means for studying, in a systematic and complete manner, the contribution of each of the considered physico-chemical features on the obtained enzyme similarity networks. More specifically, each of the possible combinations of physico-chemical features are considered for obtaining optimal enzyme similarity networks, and the similarity between each of these networks is then gauged by using the coincidence index, resulting in a new network expressing in a comprehensive and systematic manner the effects of considering different sets of features on the construction of enzyme similarity networks. Therefore, one of the contributions of the approach reported in this work relates to a verification of how much the adopted biophysical properties, that emerges from the chemical properties of each amino acid in the enzyme structure, are congruent with the four original types of catalytic activities, which are defined in terms of other properties. In this sense, a good correspondence between the four original types with the groups obtained by using the reported approach would indicate, among other implications, that these measurements can be used as subsidies for identifying those groups.

Several relevant contributions and results are reported and discussed in this work. First, we have the optimal index favoring simultaneous maximization of the modularity as well as minimization of the number of isolated nodes. Combined with the potential of the coincidence methodology for providing more strict quantification of the similarity between two mathematical structures, the optimization procedure yielded markedly uniform connected groups of enzymes (respectively to their EC number) while minimizing the number of isolated nodes. This is mostly due to the possibility provided by the α parameter to control the contribution of the features with same or opposite signs on the resulting similarity. Remarkably, the optimal configuration could not have been achieved by using the standard coincidence formula without resourcing to the α parameter. Interestingly, most of the four original types of enzyme were split into two or more connected groups, indicating that those types can be subcategorized in terms of the adopted physico-chemical properties. These results provide an important subsidy not only for selecting and designing new enzymes for biotechnological applications, but also for better understanding the physico-chemical space of each enzyme classes.

The current work starts by presenting the adopted enzyme data and basic concepts and methods to be applied, including the coincidence similarity index, network modularity and its optimization, as well as the methodology for systematic analysis of the effect of the adopted features on the obtained protein similarity networks.

Materials and Methods

In this section are presented the model enzymes in study, their respective physico-chemical features, the coincidence similarity and its application to translated datasets into respective networks, the concept of network mod-ularity and its optimization, as well as the description of the systematic methodology for studying the effect of the adopted features on the obtained enzyme similarity networks.

The Model Enzymes

The seventy enzymes considered for this work were selected from the CAZy database, which have been used as a tool for the study of CAZymes since 1998 [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. This database categorizes CAZymes into five classes, each class with several families of structurally-related proteins [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. One family of great biotechnological interest and with extensive experimental data available is the GH13, of the glycoside hydrolases class, composed of an evolutionary diversified set of enzymes that hydrolyze glycosidic bonds [START_REF] Mark | Dividing the large glycoside hydrolase family 13 into subfamilies: towards improved functional annotations of alpha-amylase-related proteins[END_REF].

The CAZy database has 129333 assigned enzymes under the GH13 family, from which only 147 are both characterized and have their structure available [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. Thus, in this work we will refer to these 147 enzymes as "wellstudied".

Furthermore, the GH13 enzymes have 30 different assigned EC numbers [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. To simplify this collection of catalytic activities, we shorten the EC number to its third digit (e.g., the EC number 3.2.1.1 becomes 3.2.1), thus keeping enzymes with similar catalytic activities on the same group while increasing the size of those groups. In this manner, all of the GH13 enzymes result into four enzyme types: 2.4.1, 2.4.99, 3.4.1, 5.4.99. The description of the enzyme types is summarized in Table 1.

Among the 147 well-studied enzymes, we selected 70 model enzymes based on their evolutionary diversity, reported cases of biotechnological interest, and quality of the data available on CAZy and UniProt [START_REF]UniProt: the universal protein knowledgebase in 2021[END_REF] databases. Additional information on the adopted model enzymes is in Supplementary Material 1. The distribution of the model enzymes into the four enzyme types, the description of their catalytic activity and their potential biotechnological uses are presented in Table 1.

Table 1: Distribution of the 70 model enzymes in the four types of enzymes under study. The model enzymes were selected from a pool of 147 well-studied CAZymes from the GH13 family. Each enzyme type corresponds to a simplified EC number relative to a set of similar catalytic activities. The asterisk indicates that all well-studied enzymes of that specific type have been used in this work. We collected the amino acid sequences of all chosen model enzymes in Uniprot [START_REF]UniProt: the universal protein knowledgebase in 2021[END_REF] by using the accession numbers provided by CAZy [START_REF] Drula | The carbohydrate-active enzyme database: functions and literature[END_REF]. The amino acid sequences were used to calculate six physico-chemical features important for the synthesis, purification and application of enzymes, namely:

1. Number of amino acids -It is one of the most fundamental features of proteins. Since it is directly proportional to the length of its respective genes, this enzyme feature play a role in assembling the right expression system, which can have restrictions based on gene length [START_REF] Bajpai | High capacity vectors[END_REF]. Furthermore, the number of amino acids is important for size exclusion chromatography, a technique that separates proteins based on their capability to infiltrate and be retained in a porous gel medium, which depends on protein size and weight [START_REF] Wilson | Principles and techniques of biochemistry and molecular biology[END_REF][START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. The target proteins can be purified by selecting the gel with compatible pore size. Therefore, purification of enzymes with similar sizes via molecular exclusion chromatography can be performed using the same chromatography equipment.

2. Molecular weight -Just like the previous, this is a important feature on size-exclusion chromatography [START_REF] Wilson | Principles and techniques of biochemistry and molecular biology[END_REF][START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. Furthermore, molecular weight can be exploited in ultrafiltration, a size-exclusion technique that consists in passing the solution with the target protein through a semipermeable membrane that retains molecules with a minimal molecular weight [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. Similarly, enzymes can be retained in semipermeable membrane to be used in membrane reactors [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. Therefore, enzymes with similar molecular weights could be purified or used on membrane reactors using semipermeable membranes with the same cutoff.

3. Theoretical pI -The pI is the medium pH when the net charge of the protein is 0. When the pH is above the pI, the protein will have a negative net charge, while when the pH is below the pI, the protein will have a positive net charge. This property is important for setting the experimental apparatus of ion-exchange chromatography. This type of chromatography consists in passing the solution with the target protein in a matrix with selected ion-exchange groups [START_REF] Wilson | Principles and techniques of biochemistry and molecular biology[END_REF][START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. If the target protein is in a solution with a pH above its pI, it will have a negative surface charge and will be trapped in a matrix with positively charged ion-exchange group (anion-exchange chromatography), while if the target protein is in a solution with a pH below its pI, it will have a positive surface charge and will be trapped in a matrix with negatively charged ion-exchange group (cation-exchange chromatography) [START_REF] Wilson | Principles and techniques of biochemistry and molecular biology[END_REF][START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF]. Therefore, proteins with similar pI that will be produced in similar conditions could be purified from a solution using the same chromatography apparatus. Such principle is also used in bifunctional or mixed-mode chromatography [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF] 4. Instability index -Is a fundamental property used to predict the stability of the enzyme in vitro [START_REF] Kunchur Guruprasad | Correlation between stability of a protein and its dipeptide composition: a novel approach for predicting in vivo stability of a protein from its primary sequence[END_REF][START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF]. Enzymes with an instability index lower than 40 are considered stable, and the lower this value, the more likely the enzyme is to be stable. Therefore, enzymes with similar instability index should demand similar experimental and industrial treatments during its purification and application -e.g., suitable temperature to perform the enzyme purification, the possibility to perform denaturation fractionation, or the need to use protease inhibitors in solution to avoid the destruction of the desired enzyme on the synthesis and purification processes [START_REF] Burgess | Use of bioinformatics in planning a protein purification[END_REF].

5. Aliphatic index -The feature corresponds to the relative volume occupied by aliphatic side chains, and it indirectly indicates the thermostability of the enzyme [START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF][START_REF] Ikai | Thermostability and aliphatic index of globular proteins[END_REF]. Similar to the previous index, enzymes with the same aliphatic index show similar stability responses to temperature, being a very important characteristic to evaluate the suitability of an enzyme for industrial applications.

6. Grand average of hydropathicity (GRAVY) -It indicates the average hydropathicity of the amino acids that composes the enzyme, i.e. the mean hydrophilic or hydrophobic behaviour of the whole enzyme [START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF][START_REF] Kyte | A simple method for displaying the hydropathic character of a protein[END_REF]. A positive GRAVY value indicates that the protein is hydrophobic and a negative value that it is hydrophilic [START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF][START_REF] Kyte | A simple method for displaying the hydropathic character of a protein[END_REF]. Enzymes with similar GRAVY values will have a similar behaviour towards water, thus enabling the use of the same technical apparatus for purification through hydrophobic/hydrophilic interaction chromatography [START_REF] Buchholz | Biocatalysts and Enzyme Technology[END_REF][START_REF] Wilson | Principles and techniques of biochemistry and molecular biology[END_REF][START_REF] Chiu | Evaluation of peptide fractionation strategies used in proteome analysis[END_REF]. Additionally, enzymes with similar GRAVY will have similar solubility to different solvents, therefore increasing the possibility to apply either of them in the same solution.

The calculations were performed with the ProtParam tool, available on the ExPASy server (https://web.expasy.org/protparam/) [START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF]. Although several experimental methods are available to estimate such physico-chemical properties, this procedure was used in present work to standardize the data acquisition to this large dataset.

The Coincidence Similarity

Quantifying the pairwise similarity between two mathematical structures represents one of the most essential operations in science and technology. While the Jaccard [START_REF] Jaccard | Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines[END_REF] and other related indices (e.g. [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF][START_REF] Cem Emre Akbas | L1 norm based multiplication-free cosine similarity measures for big data analysis[END_REF][START_REF] Mirkin | Mathematical classification and clustering. Nonconvex optimization and its applications[END_REF]) have been often adopted for quantifying the similarity between two sets (categorical or binary data), concepts such as the Pearson correlation and cosine similarity have been widely employed for measuring the similarity between two realvalued feature vectors (e.g. [START_REF] Guilherme F De Arruda | A complex networks approach for data clustering[END_REF][START_REF] Wira | Evaluating text coherence based on semantic similarity graph[END_REF][START_REF] Brusco | A comparison of 71 binary similarity coefficients: The effect of base rates[END_REF][START_REF] Hamers | Similarity measures in scientometric research: The jaccard index versus salton's cosine formula[END_REF]).

The coincidence similarity index has been proposed as an extension of the Jaccard index in the sense of being applicable to real-valued data and being able to reflect the relative interiority between the datasets [START_REF] Da | Further generalizations of the Jaccard index[END_REF]. The extension of the Jaccard index to real-valued data has been accomplished through multiset theory, suitably adapted to take into account negative values [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | Multisets[END_REF][START_REF] Da | Multiset neurons[END_REF]. The Jaccard index has been shown [START_REF] Da | Further generalizations of the Jaccard index[END_REF] to be unable to take into account the relative interiority, or overlap (e.g. [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF]) between two sets. This can be addressed by multiplying the Jaccard index with the interiority index, therefore yielding the coincidence similarity index.

The potential of the coincidence index for more strict quantification of the similarity between two generic mathematical structures has been verified respectively to several comparative approaches (e.g. [START_REF] Da | Comparing cross correlation-based similarities[END_REF][START_REF] Da | Multiset neurons[END_REF]) and applications (e.g. [START_REF] Domingues | City Motifs as Revealed by Similarity Between Hierarchical Features[END_REF][START_REF] Luciano Da | A similarity approach to cities and features[END_REF]).

In addition, the real-valued coincidence similarity index can be adapted, with the introduction of a parameter α, so as to allow the contributions of pairs of features with the same or opposite signs to be taken with different weights respectively to the resulting coincidence values [START_REF] Da | On similarity[END_REF][START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | Multiset neurons[END_REF].

A method for translating datasets, with data elements characterized by respective features, into respective detailed networks has been reported [START_REF] Da | Coincidence complex networks[END_REF] which is based on assigning a node to each of the data elements, while the pairwise links between nodes correspond to the coincidence similarity between the respective standardized (or not) features. This method has been verified to allow networks characterized by a marked level of interconnection detail as well as by enhanced modularity [START_REF] Da | Coincidence complex networks[END_REF]. As discussed in [START_REF] Da | Multiset neurons[END_REF], the coincidence index naturally performs the comparison relatively to the magnitudes of the vectors, being little susceptible to perturbations or noise on single features.

Only two parameters are involved in the coincidence methodology, namely the above mentioned α as well as an overall threshold T which is eventually adopted in case binary connections are required. By varying α and T , the coincidence approach to translating dataset into networks can be optimized respectively to sought properties such as network modularity, number of components, among many alternative possibilities.

In its parameterless version, the coincidence similarity index between two vectors X and Y can be defined as corresponding to the product between the respective Jaccard and interiority indices, namely:

C(X, Y ) = J (X, Y )I(X, Y ) (1)
The real-valued coincidence can be immediately obtained as:

C R (X, Y ) = J R (X, Y )I R (X, Y ) (2)
The interiority (or overlap, e.g. [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF]) index, in its real-valued version [START_REF] Da | On similarity[END_REF], can be expressed as:

I R (X, Y ) = i∈S min{|x i |, |y i |} min{S X , S Y } ( 3
)
where S is the support of the vectors, and

S X = i∈S |x i |; S Y = i∈S |x i | (4) with 0 ≤ I R (X, Y ) ≤ 1.
The real-valued Jaccard index can be defined [START_REF] Da | On similarity[END_REF][START_REF] Da | Further generalizations of the Jaccard index[END_REF] as:

J R (X, Y ) = sign(x i y i ) min{|x i |, |y i |} i∈S max{|x i |, |y i |} (5) with 0 ≤ J R (X, Y ) ≤ 1.
Now, the parametrized real-valued coincidence index can be written as:

J R (X, Y, α) = [α]s + (X, Y ) -[1 -α]s -(X, Y ) i∈S max{|x i |, |y i |} (6)
where:

s + (X, Y ) = i∈S |sign(x i ) + sign(y i )| min{|x i |, |y i |} (7) s -(X, Y ) = i∈S |sign(x i ) -sign(y i )| min{|x i |, |y i |} (8)
The parameter α, with 0 ≤ α ≤ 1 provides an interesting manner to control the relative contributions of the pairwise features combinations with the same or opposite signs onto the resulting similarity values. In particular, for α > 0.5, the pairwise combinations of features with the same sign will have larger relative contribution to the resulting index than those with opposite signs. By varying α, it becomes possible to optimize the obtained coincidence similarity values with respect to several indices, such as the modularity of the resulting networks [START_REF] Da | Coincidence complex networks[END_REF].

It can be verified that when α = 0.5, the parameterized real valued coincidence becomes identical to its parameterless version.

Translating Datasets into Networks by Using the Coincidence Methodology

As proposed in [START_REF] Da | Coincidence complex networks[END_REF], the coincidence index can be effectively employed in order to translate datasets into complex networks. Each of the N data elements x i is characterized by M respective features j, 1 ≤ j ≤ M , organized into respective feature vectors, which are then standardized respectively to each of the features so as yield null means and unit variance. The standardization is performed respectively to each feature j of each data element i as:

xi,j = x i,j -µ i σ i (9)
where µ i and σ i are the means and standard deviation along the values of the feature j.

The coincidence similarity is then calculated respectively to each pair of data elements, yielding a respective adjacency matrix which can be thresholded by T (the coincidence values of the entries larger or equal to T are kept). The parameter α can be set so as to optimize some property of interest, such as the network modularity or the number of isolated nodes.

Provided the parameters α and T are set in a suitable manner, this simple methodology has been found [START_REF] Da | Coincidence complex networks[END_REF][START_REF] Da | Comparing cross correlation-based similarities[END_REF][START_REF] Da | A caleidoscope of datasets represented as networks by using the coincidence methodology[END_REF][START_REF] Luciano Da | A similarity approach to cities and features[END_REF][START_REF] Domingues | City Motifs as Revealed by Similarity Between Hierarchical Features[END_REF] to allow networks to be obtained that are characterized by enhanced level of details and modularity. These enhanced features are a direct consequence not only of the more strict ability of coincidence to quantify similarity [START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF], but also of the possibility to control, by setting the parameter α, the relative contribution of features with the same or opposite sign on the resulting similarity value.

Network Optimization

Conceptually, a community or module of a network is a set of nodes that are more intensely interconnected one another than with the remainder of the network [START_REF] Newman | Finding and evaluating community structure in networks[END_REF][START_REF] Clauset | Finding community structure in very large networks[END_REF]. Networks that have communities, which are said to be modular, appear frequently in real-world systems, including social networks, scientific collaboration, as well as several biological networks, including protein interaction and, very probably, also protein similarity [START_REF] Girvan | Community structure in social and biological networks[END_REF][START_REF] Anna | The function of communities in protein interaction networks at multiple scales[END_REF].

In order to quantify how modular a network is, given a proposed division criterion, a possibility is to calculate the modularity of the network [START_REF] Clauset | Finding community structure in very large networks[END_REF][START_REF] Newman | Finding and evaluating community structure in networks[END_REF][START_REF] Brandes | On modularity clustering[END_REF]. The modularity quantifies the difference between the number of edges that connect nodes within and among each candidate modules. There are several ways in which the modularity of a network can be defined, but in this work we adopt the equation [START_REF] Brandes | On modularity clustering[END_REF]:

⨿(C) = c∈C   |E(c)| m - v∈c deg(v) 2m 2   (10)
Where -1/2 ≤ ⨿(C) ≤ 1, C is a proposed community structure, c is a module, m is the number of nodes in the network, E(c) is the number of edges within the c module, and deg(v) is the total degree of the vertex v. This simple equation can be used to optimize the modularity of a network: the first term should be maximized with modules that have many edges among its nodes, while the second term should be minimized with the module having nodes with low degree. An equivalent and computationally efficient approach to calculate this equation [START_REF] Clauset | Finding community structure in very large networks[END_REF] was used in this work.

Although it is possible to propose modules in multiple ways considering different network properties [START_REF] Newman | Finding and evaluating community structure in networks[END_REF][START_REF] Fortunato | Community detection in graphs[END_REF], the proposed community structures for biological networks usually have a biological meaning transcending the network topology. In our case, the enzyme similarity networks will be divided on four modules respective to the four enzyme types in study. Therefore, the proposed community structure will preserved. Thus, here we aim to determine suitable interconnections between the network nodes, which lead to optimal agreement with the original types, by using the coincidence method while varying the α and T parameters. Thus, in our analysis, the modularity ⨿ can be seen not as a function of C but as ⨿(α, T ).

An unfortunate consequence of the optimization of the equation 10 is that networks with low quantity of edges and with multiple isolated nodes may have high values of modularity. Although this is not necessarily a problem, the proposed coincidence method to create enzyme similarity networks are intended to be used to create modular networks -thus with biologically relevant information -with a reduced number of isolated nodes -thus integrating the most part of the enzymes in analysis.

Therefore, we propose to combine the increase of the modularity ⨿(α, T ) with the minimization of the number of isolated node I into a single optimization index F [START_REF] Audet | Multiobjective optimization through a series of single-objective formulations[END_REF]:

F(α, T ) = (I(α, T )/m -0) 2 + (⨿(α, T ) -1) 2 (11)
Through minimizing the index F (α, T ) by varying (α, T ) to construct the enzyme similarity network supervised by a community structure C, it becomes possible to both minimize the relative number of isolated nodes I(α, T )/m to 0 and optimize the modularity ⨿(α, T ) of the network to 1, considering the biologically relevant communities defined by C.

In brief, in the present work, given any set of the 6 physico-chemical features in study, we apply the coincidence methodology to create 121 networks between the 70 adopted model enzymes, each network corresponding to a pair of (α, T ), with α and T assuming values in 0.0, 0.1, 0.2, ..., 0.9, 1.0. Then, we estimate the optimization index F for each network considering the division of the nodes into four enzyme types (modules), and we define the optimal network as the one with (α, T ) that yielded the minimal calculated value of F.

Feature Analysis

Before real-world or abstract patterns can be recognized or classified, they need to have their most relevant properties translated into respective mathematical measurements, or features. While there is no definite method for selecting which are the best features relatively to each pattern recognition problem, it becomes of particular relevance to devise means for studying the effect of specific combinations of features into the obtained result(e.g. [START_REF] Luciano Da | Shape classification and analysis[END_REF]).

Of particular relevance is the potential impact of different feature choices on the performance of the therefore implemented approaches respectively to each particular dataset.

In the present work, each of the 70 adopted model enzymes are characterized by 6 physico-chemical features. In order to investigate how possible combinations of these features impact on the obtained enzyme similarity network, we adopt the coincidence-based approach reported in [START_REF] Da | Coincidence complex networks[END_REF]. This approach involves obtaining optimal similarity networks for each of the possible feature combinations and then obtaining a feature combination network, in which each node corresponds to a similarity network, while the links between any two of these nodes are determined by the coincidence between the adjacency matrix of these networks.

The feature combination network expresses in a quantitative and effective manner the effect of each feature in the obtained similarity networks. This type of network has been found to be strongly modular respective to specific sets of features [START_REF] Da | Coincidence complex networks[END_REF][START_REF] Luciano Da | A similarity approach to cities and features[END_REF][START_REF] Domingues | City Motifs as Revealed by Similarity Between Hierarchical Features[END_REF][START_REF] Da | Elementary particles networks as revealed by their spin, charge and mass[END_REF], so that each of the identified communities of features can be taken as a set of related models for describing the dataset. Communities that are strongly interconnected within itself indicate particularly homogeneous models, characterized by intense synergy between the features involved.

In the present work, we consider the optimal enzyme similarity network for each of the possible combinations of the six physico-chemical features in study. In order to provide a direct indication of the effect of the considered feature combinations on the obtained enzyme similarity networks, the respective optimization index is also shown in the resulting feature combination network.

Results and Discussion

In this section we derive the optimal enzyme similarity network from the adopted six physico-chemical features respectively to the adopted optimization index (which needs to be minimized), and investigate the impact of the combinations between the six features on the resulting networks.

We start by presenting, in Figure 1, the two-axes Linear Discriminant Analysis (LDA, e.g. [START_REF] Luciano Da | Shape classification and analysis[END_REF]) projection of the 70 model proteins considering the adopted six physico-chemical features. The LDA consists of a supervised approach in which the original data elements are projected into a lower dimensional space so as to maximize their separation based on the respective intra and intercluster distance matrices (e.g. [START_REF] Luciano Da | Shape classification and analysis[END_REF]).

It can be readily observed from this result that the LDA projection is unable to effectively cluster the four types of enzymes, being mostly devoid of clustering tendency. This result is partially due to the fact that LDA is a two-dimensional projection of six-dimensional data, data that represents physico-chemical and non-catalytic characteristics of the enzymes. herefore, the already expected weak interrelationship between different EC groups in a six-dimensional space may have been weakened by a substantial loss of information. In addition, LDA is based on correlation-related similarity, which has been found to be less strict than the coincidence approach (e.g. [START_REF] Da | Comparing cross correlation-based similarities[END_REF][START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF]).

Next, we construct 63 optimal enzyme similarity networks respectively to the adopted combined optimization index F(α, T ) respective to each of the possible combinations of the six physico-chemical features. This optimization took place over the two dimensional space defied by the parameters α and T . The 63 optimal networks were gathered in a feature combination network in Figure 2. A large connected component can be readily identified in Figure 2, corresponding to protein similarity networks that are similar to one another, as well as some smaller groups and isolated nodes. Each of these groups can be understood as corresponding to a possible model of the enzyme similarities, in the sense that the nodes within each of these groups will lead to enzyme similarity networks that are similar [START_REF] Da | Elementary particles networks as revealed by their spin, charge and mass[END_REF]. Distinct connected components therefore provide alternative models. Each of these models can be summarized in terms of the respective hub [START_REF] Luciano Da | A similarity approach to cities and features[END_REF].

The feature groups FG9 are worthy of special attention. This group corresponds to the largest obtained connected group, composed of combinations encompassing all the six adopted physico-chemical features, which also happens to include one of the largest modularity values. The minimum observed combined optimization index F is to be found within this connected group -respectively to the features 1, 3, 4, 5, and 6.

Figure 3 presents the number of occurrences of each of the six physicochemical features within each of the three largest feature groups, namely FG22, FG6 and FG9. These histograms indicate that all the six features are almost equally frequent in FG9. A similar situation, except for a substantially less frequent presence of the first feature, can be observed for FG22, while FG6 contains mostly the three first features as well as one occurrence of the fifth feature. This shows that these six physico-chemical characteristics are mostly interchangeable for creating enzyme networks with similar connections, at least when a minimum of three of them are taking into account in the calculations. Table 2 presents the five feature combinations (nodes in the network of Figure 2) leading to optimal enzyme similarity networks with the lowest observed values of the combined optimization index. It is presented the α and T values considered to achieve the optimal enzyme similarity networks with its respective features, as well as the respective resulting modularities, number of isolated nodes, number of components, combined index and feature group (FG). Except for the fifth entry in this table (features 1, 4 and 6), all other configurations involve at least four out of the six adopted physicochemical features.

According to the combined optimization index method, the best enzymatic similarity network achieved was constructed considering the features Interestingly, the region of largest modularity corresponds to a diagonal ridge extending roughly from the bottom-left to the upper-right portions of the graph in Figure 4(a). This indicates that larger modularity values tend to occur for mutually similar, though not identical, values of α and T . It is worth noticing that this modularity ridge coincides with an abrupt transition (a 'cliff') of the number of isolated nodes taking place along the same region Of particular relevance is the observation that the optimal configuration was obtained for α = 0.9, being therefore otherwise impossible to be identified by using the parameterless coincidence index equation (Eq. 5), which would keep α fixed ad 0.5. This result corroborates the importance of resourcing to the α parameter while translating datasets into respective coincidence networks.

Figure 5 shows the scatterplot of the number of isolated nodes in terms of modularity obtained for the enzyme networks of multiple α and T configurations using the features 1, 3, 4, 5, 6. The optimal combined index configuration is shown in purple, corresponding to α = 0.9 and T = 1.0. Other combinations of special interest are also highlighted in the figure. It is interesting to observe a relatively well-defined non-linear (V-like) relationship between the two optimization parameters.

The enzyme similarity network yielding the optimal (minimum) combined index is presented in Figure 6.

Several relatively large modules can be identified in Figure 6, most of which are highly uniform in the sense of involving mainly enzymes that catalyze the same type of reaction. Several of the obtained groups involve strongly interconnected enzymes, indicating respectively high uniformity of The enzyme similarity network with the optimal combined index, obtained with the features 1, 3, 4, 5, 6. The nodes are labeled with the respective Uniprot accession numbers of the enzymes. The components are labeled as Protein Groups (PG). Observe the several highly uniform connected components obtained for this configuration. Most of these components are strongly interconnected, except for PG5 and PG14, which correspond to chains implied by transitivity of the coincidence pairwise interactions. Of particular interest is the good homogeneity of enzyme types within each identified connected component, as well as the fact that those four types have yielded separated clusters, suggesting presence of respective subcategories.

physico-chemical properties. Interestingly, some of the obtained communities, such as PG5 and PG14, consists of a string of strongly connected nodes, which therefore indicates transitive similarity between the involved enzymes, with the enzymes corresponding to the two extremities being the least similar among the others enzymes in that group. Also of particular interest is the fact that all the four enzyme types yielded more than one related connected component, which indicates the presence of subgroups of physico-chemical characteristics within the original enzyme types.

From the biological perspective, the relatively good agreement between the obtained groups and the four original enzyme types with the formation of pure type groups -such as PG15, PG11, PG12 and PG7 -substantiate the potential of the physico-chemical features 1, 3, 4, 5, 6 in reflecting, as well as providing subsidies for predicting and enhancing, the catalytic properties adopted for the definition of the four original enzyme types. This correlation between physico-chemical properties and enzyme types is biochemically supported by the fact that this physico-chemical features are directly calculated from the amino acids that composes the protein sequences [START_REF] Gasteiger | Protein identification and analysis tools on the expasy server[END_REF], which will also dictate the folding of the enzyme in its final three-dimensional structure and, consequently, its catalytic activity [START_REF] Creighton | The biophysical chemistry of nucleic acids and proteins[END_REF].

Figure 7 illustrates the enzyme similarity networks obtained for several equally spaced values of the parameter α, more specifically α = 0.7, 0.8, 0.9, 1, with T fixed at 1.0. The protein similarity network obtained for α = 0.7 is composed almost exclusively of individual, disconnected nodes, except for few groups involving from two to four proteins. As α is progressively increased, the nodes become more and more interconnected, while the combined index reaches its maximum at α = 0.9, then decreasing for α = 1.0. At the same time, the number of connected components decreases monotonically with the increase of α. Even though the combined index values obtained for α ̸ = 0.9 are smaller than the peak value, the respectively obtained networks can still be observed to contain mostly uniform groups of nodes, that are progressively interconnected. It is interesting to keep in mind that, as α increases, with new links being incorporated into the network, the previous connections are maintained.

Figure 8 depicts the interval of variation for each of the five considered physico-chemical features characterizing the PGs with at least four enzymes: PG15, PG14, PG12, PG11 and PG5.

According to Figure 8, the enzymes of each group have quite similar number of amino acids, thus having similar protein and gene sizes. Therefore, the enzymes of a given group could be purified with the same size exclusion chromatography equipment, while different groups would require different chromatography gels. Because the enzymes of the groups PG5 and PG11 have pI values varying from neutral to slightly acidic conditions, these proteins could be purified through both cation-and anion-exchange chromatography, depending on altering the pH of the solutions to acid or basic, respectively. However, the enzymes of groups PG14, PG12 and, specially, PG11, because they have lower values of pI, should be more easily purified with anion-exchange chromatography.

Furthermore, by choosing enzymes of the group PG14 or PG15, the biotechnologist will have enzymes with low values instability index and high aliphatic index, thus having enzymes with highest predicted stability for uses in industrial processes. However, those enzymes might have just a slightly hydrophilic behaviour, thus they could present problems in solubility. Alternatively, enzymes of the PG12 group might have more stability problems, but a higher hydrophilic behaviour. Overall, the above analysis illustrates how the proposed methodology could help setting groups of technically related enzymes for biotechnological uses, while taking into account specific enzyme properties and requirements.

Concluding Remarks

Proteins, which include enzymes, have been largely acknowledged to constitute the building blocks of life. Their essential role lies in the fact that they implement different types and levels of functionality across species and biological systems, including many functionalities useful for biotechnological applications. As a consequence, the study of not only the catalytic activities of proteins but their physico-chemical properties and applicability outside its original species, as well as how the proteins can be grouped in classes are ubiquitous in biological, biophysical and biochemical sciences.

The present work has addressed the classification of enzymes based on their physico-chemical properties relying on the following three key aspects: (i) the coincidence method is employed to map from several combinations of enzyme features to respective similarity networks; (ii) this mapping is optimized with respect to an optimality index related to the modularity and number of isolated nodes of the resulting networks; and (iii) a coincidencebased approach is applied in order to characterize in a systematic and comprehensive manner the impact of the enzyme features on the resulting optimal similarity networks.

Several interesting results are reported and discussed, including the identification of optimized networks characterized by relatively high modularity levels and small number of isolated nodes obtained from the physico-chemical properties of the considered enzymes.

The several combinations of these six features were systematically evaluated, and the optimal enzyme similarity network was obtained while considering features 1, 3, 4, 5 and 6, leaving out only the molecular weight. This result suggests that, among the adopted six biophysical features, the five selected measurements have more pronounced potential for characterizing the four types of considered enzymes.

Another particularly interesting result concerns the importance of using the parameters α and T while optimizing the properties of the networks obtained by the coincidence methodology. More specifically, it has been shown that the optimal enzyme network could not have been otherwise obtained with the parameterless (i.e. with α = 0.5) version of the coincidence similarity. At the same time, the mapping of the modularity and number of isolated nodes in terms of α and T provided interesting insights suggesting that the latter property defines a cliff that nearly coincides with the main diagonal characterized by α = T , while the maximum modularity configurations tending to occur along the respectively defined ridge. Therefore, the maximum modularity tends to take place at positions in the domain (α, T ) where the number of isolated nodes undergo an abrupt variation, which reminds of a phase transition.

The describe methodology and results pave the way to a number of interesting prospects for further related studies. For instance, the reported approach can be directly applied to other important enzyme and protein families, as well as to other types of biological molecules and interactions. It would also be interesting to consider additional biophysical properties, as well as measurements reflecting the geometrical structure of the considered molecules. Furthermore, this methodology could be incorporated as a complementary tool in the prospecting of molecules of biotechnological and pharmaceutical interest or in the rational design of proteins and ligands to improve its physico-chemical properties.

Figure 1 :

 1 Figure 1: The two-dimensional linear discriminant analysis (LDA) of the 70 model enzymes, derived from their six physico-chemical features. The dots are labeled with the respective Uniprot accession numbers of the enzymes. Little separation between the four enzyme types can be observed from this projection.

Figure 2 (

 2 a) depicts, in respective heatmap colors, the modularity values obtained for each of the 63 combinations of the six physico-chemical features. Figure 2(b) shows the same feature combination network, but with heatmap colors indicating the resulting number of isolated nodes for each feature combination, and Figure 2(c) presents the respective combined optimization index values in heatmap colors.

Figure 2 :

 2 Figure 2: Feature combination network, the network of coincidence similarities between the 63 possible combinations of the physico-chemical features. Each vertex corresponds to an optimal coincidence graph of enzymes created based on the labeled physico-chemical features, while the edges represents the coincidence index between pairs of graphs. The groups of connected vertex are labeled as Feature Groups (FG). Each vertex is colored according to the (a) modularity, (b) the number of isolated nodes, and (c) the combined index of their respective coincidence graph. The label numbers represents the following physico-chemical features: 1 -number of amino acids, 2 -molecular weight, 3 -theoretical pI, 4 -instability index, 5 -aliphatic index, 6 -GRAVY. This network was obtained for α = T = 0.5.

Figure 3 :

 3 Figure3: Histograms indicating the frequency of occurrence of each of the adopted six physico-chemical features within the main models FG22, FG6 and FG9. The optimal enzyme similarity network was identified within FG9.

1, 3 , 4 , 5 , 6 ,

 3456 with α = 0.9 and T = 1.0. Next, we proceeded to analyze this combination of resources now with multiple values of α and T , in order to study how the resulting enzymatic networks behave with the alteration of these two parameters. Figure4(a) depicts the modularity values, in heatmap colors, of the enzyme networks obtained for several values of α and T considering the features 1, 3, 4, 5, 6. The respective number of isolated nodes are shown in Figure 4(b), and the combined optimization index in (c).

Figure 4 :

 4 Figure 4: The modularity values (a) obtained for the features 1, 3, 4, 5, 6 in terms of several values of α and T . The respective numbers of isolated nodes are shown in (b), and the combined optimization index values in (c). This feature combination generates the enzyme similarity network with the lowest value of the combined index observed. The optimal parameter configuration is obtained for α = 0.9 and T = 1.0, indicated by the black box.

Figure 5 :

 5 Figure 5: The scatterplot obtained from the modularity and number of isolated nodes of the newtworks constructed for the features 1, 3, 4, 5, 6 in terms of several values of α and T . The optimal configuration is shown in purple, and those configurations with values of α and T equal to those of the optimal configuration are identified in red and blue, respectively. The values of these two parameters are shown for several of the combinations in the figure.

Figure 6 :

 6 Figure6: The enzyme similarity network with the optimal combined index, obtained with the features 1, 3, 4, 5, 6. The nodes are labeled with the respective Uniprot accession numbers of the enzymes. The components are labeled as Protein Groups (PG). Observe the several highly uniform connected components obtained for this configuration. Most of these components are strongly interconnected, except for PG5 and PG14, which correspond to chains implied by transitivity of the coincidence pairwise interactions. Of particular interest is the good homogeneity of enzyme types within each identified connected component, as well as the fact that those four types have yielded separated clusters, suggesting presence of respective subcategories.

Figure 7 :

 7 Figure 7: The enzyme similarity networks obtained with the features (1, 3, 4, 5, 6) respectively to four successively larger values of α, with T fixed at 1.0. The enzymes (nodes) become more and more interconnected as α increases, while the number of components decreases monotonically. The optimal combined index value considering this set of features is obtained for α = 0.9 and T = 1.0. The width of the links is proportional to the respective coincidence values.

Figure 8 :

 8 Figure 8: The ranges of the five considered physico-chemical features observed for the five largest enzymes groups obtained respectively to the optimal enzyme similarity network.

Table 2 :

 2 The five feature combinations leading to the most optimized combined indices, indicating the respective parameter configurations (α and T ), as well as other properties of the respective optimal enzyme similarity networks. Most of these configurations involve at least 4 of the six adopted physico-chemical features.

	α	T	Features	Modularity Isolated Nodes Components Combined index F FG
	0.9 1.0	1, 3, 4, 5, 6	0.4679	20	35	0.3648	9
	0.5 0.6	1, 3, 4, 6	0.5092	29	43	0.4125	21
	0.3 0.3	2, 3, 4, 5, 6	0.4659	25	36	0.4128	22
	0.3 0.3 1, 2, 3, 4, 5, 6	0.4157	19	32	0.4151	22
	0.5 0.6	1, 4, 6	0.3593	10	19	0.4309	17

Acknowledgements

L. da F. Costa thanks CNPq (307085/2018-0) and FAPESP (2015/22308-2) for support. R. dos Reis thanks CAPES (88887.637015/2021-00) for financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001