Sphere intersection algorithms for Molecular Distance Geometry Problem

Abstract

The problem of estimating the full three-dimensional structure of a molecule, determining the position in space of all the atoms that compose it, is called Molecular Distance Geometry Problem (MDGP). To do this from an incomplete set of distances is NP-hard computational problem, where to get a feasible solution in a reasonable execution time presenting interesting mathematical and computational challenges. In this work, continuous and discrete mathematical approaches to solve MDGP is revised, based on the analysis of two types of calculating of sphere intersection: solving nonlinear systems from interatomic Euclidean distance equations, or solving internal coordinate systems using matrix multiplication techniques. We adapted the Branch-and-Prune (BP) method considering four spheres intersection. Computational experiments using instances from PDB benchmark are performed, determining the 3D structure based on our theoretical assumptions in a competitive computational processing time.