On statistically meaningful geometric properties of digital three-dimensional structures of proteins

In the framework of the language introduced in a previous paper [B. Carbonaro, F. Vitale, C. Giordano, On a 3D-matrix representation of the tertiary structure of a protein, Math. Comput. Modelling 43 (2006) 1434–1464] to represent the natural three-dimensional structure of a protein as a three-dimensional numerical matrix, which could be viewed as the “digital structure” of the protein, a number of notions, which will be shown to play an effective rôle in the reconstruction of unknown natural configurations of newly discovered proteins, are introduced. These notions are of two kinds: first, some classical local geometric properties of curves (represented by curvature and torsion parameters), suitably re-defined in a discrete framework, that are needed to describe–at least in statistical terms–the “trend” of deformation of secondary structures depending on amino acids surrounding a given amino acid A ; second, the notion of the background of a four-tuple of amino acids within a protein chain to which it belongs, whose influence on curvature and torsion associated with the four-tuple is the object of statistical study. The dependence of the values of curvature and torsion, calculated for triples and four-tuples of amino acids sampled on the whole family of Myoglobins, on their background, is shown and discussed, just as a preliminary application, for the occurrences of the following particular four-tuple of amino acids: (Lysine, Glutamic Acid, Valine, Alanine).