On a characterization of the folding of proteins

Abstract

We consider a characterization of the folding of 3-D model proteins. Using as the input the geometry of molecules, we first construct a distance/distance matrix in which element i, j is given by the quotient of the Euclidean and the graph theoretical distance between the two vertices. The leading eigenvalue of the D/D matrix gives a measure of the folding of the protein structure. Using higher powers of the elements of D/D matrices, we generate the corresponding leading eigenvalue λ(k) for different exponents (k=1, 2, 3,…). So, derived invariants represent a characterization of the folding of a structure—here, model proteins. The derived invariants are analogous to the characterization of proteins based on the average distance, referred to as a protein profile. The folding index ϕ=λ/n, that is, the leading eigenvalue, is normalized to the number of “points” representing the structure. Structures that are more folded have a smaller folding index. We illustrate the use of the folding indices to measure the degree of similarity of molecules. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 1017–1026, 1999