Consortium, Rio de Janeiro, Brazil Stability of Ramachandran Plots under Small Perturbations of Bond and Dihedral Angles of a Backbone Protein Structure

Programming Approach [1]. The proposed cost function is the expression of a dihedral angle in terms of three variables: two bond angles, and the angle corresponding to alpha-carbon geometry. The average value of this dihedral angle is calculated to be 180 degrees by exhaustive statistical analysis of dipeptide structures [2] and this was the first indirect confirmation of Pauling’s insight on the existence of amide planes of Protein Structure [3]. In this work we show how the methods of Mathematical Programing can be applied to derive the same average value which we believe to be the first of a series of introductory results belonging to a new modelling process of biomolecular structure. We then consider the small perturbation of the three variables above and we introduce the expressions of the two remaining dihedral angles. From the observation of some regularities on the distances between atom sites along a protein backbone, we start our modelling by considering 3-dimensional curves through sequences of evenly spaced atom sites, according to an Euclidean definition of distance. This will lead to the introduction of elementary Ansätze for the coordinates of the position vectors of these evenly spaced atom sites and we make the assumption of local equilibria on the configuration of an atom site and its nearest neighbors. This assumption is enough for deriving the position of the central atom site from a generalized Fermat’s problem, since it also means the minimization of the potential energy function [4]. There are two interesting previous results comin