On a 3D-matrix representation of the tertiary structure of a protein

The main object of the present paper is a preliminary discussion of a way to describe any three-dimensional structure of a folded protein by means of a suitable three-dimensional matrix, as well as of its possible use for the forecast of the natural spatial structure of a protein of which only the amino acid sequence is known. The contents of this paper are confined to the introduction of the language and to its application to some particularly simple (and short) proteins as samples of this kind of description and as a start-point to draw the mathematical tools to be added in this context to express the geometric information in such a way as to apply the methods of inference of computational statistics. As a matter of fact, we hope to be able to show in future papers that such representation, when suitably completed by the introduction of a suitable discrete version of the notion of “deformation field” and of other geometric notions, allows the reconstruction of the folded configuration of a protein by a suitable “abstract” version of some recent and powerful statistical methods usually applied to image recognition, in turn suitably extended from the two-dimensional to the three-dimensional case (cf. [D. Geman, S. Geman, Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell. 6 (1984)]).