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Interaction network, protein structure, scale-free network
A protein interaction network is a graph whose vertices are the protein’s amino acids and whose edges are the interactions between them. Using a graph theory approach, we study the properties of these networks. In a ﬁrst time, we lead a topological description of structural families to observe how proteins from the same family have homogeneous topological properties. Second, we compare the studied graphs to the general model of scale-free networks. In particular, we are interested in the degree distribution and the mean degree of vertices. The results show a correlation between these two measures.
degree distributions to deduce that those distributions are speciﬁc and conﬁrm relative works. 1.1 Amino Acid Interaction Networks The 3D structure of a protein is represented by the coordinates of its atoms. We consider the residues of proteins to represent them. From ﬁles recorded in Protein Data Bank (PDB) , we compute the distances between pair of amino acids by considering that the Cα atom is their centre. We consider a contact map matrix which is a N × N 0–1 matrix whose element (i, j) is one if there is a contact between amino acids i and j and zero otherwise. A contact is deﬁned according to the distance between two residues, when this distance is inferior to 7 ˚ , a contact exists A between these residues. We construct a graph with N vertices (each vertex corresponds to an amino acid) and the contact map matrix as incidence matrix. It is called contact map graph. The contact map graph is an abstract description of the protein structure taking into account only the interactions between the amino acids. In this paper, we consider the subgraph induced by the set of amino acids participating in secondary structures. We call this graph secondary structure interaction network (SSE-IN). Thus, the structure determining interactions are those between amino acids belonging to the same SSE on local level and between diﬀerent SSEs on global level. Figure 1 gives an example of a protein and its SSE-IN. To generate a SSE-IN graph, we start from a PDB ﬁle from which we extract speciﬁc data to build the graph. For this purpose, we have developed a parser that is able to build the set of nodes representing the protein amino acids and the set of edges considered as the node interactions. Once the graph is generated, it may be displayed in two-or three-dimensional space. Towards this goal, we exploit the GraphStream library  which allows the manipulation of graphs. 2. A Topological Description Through the results presented in this section, we want to oﬀer a graph theory interpretation of the protein structural 95