Spectral properties of contact matrix: Application to proteins

A protein can be modelled by a set of points representing its amino acids. Topologically, this set of points is entirely defined by its contact matrix (adjacency matrix in graph theory). The contact matrix characterizing the relation between neighboring amino acids is deduced from Voronoi or Laguerre decomposition. This method allows contact matrices to be defined without any arbitrary cut-off that could induce arbitrary effects. Eigenvalues of these matrices are related with elementary excitations in proteins. We present some spectral properties of these matrices that reflect global properties of proteins. The eigenvectors indicate participation of each amino acids to the excitation modes of the proteins. It is interesting to compare the protein modelled as a close packing of amino acids, with a random close packing of spheres. The main features of the protein are those of a packing, a result that confirms the importance of the dense packing model for proteins. Nevertheless there are some properties, specific to the hierarchical organization of the protein: the primary chain order, the secondary structures and the domain structures.