A note on the Cayley-Menger determinant and the Molecular Distance Geometry Problem

Abstract

Using information from protein geometry and distance data provided by Nuclear Magnetic Resonance (NMR) experiments, the Molecular Distance Geometry Problem (MDGP) can be solved by a combinatorial approach, called Branch-and-Prune (BP). The primal version of BP algorithm seeks MDGP graph realizations, while the dual BP looks for completions of associated partial distance matrices. These two algorithms are very similar when distance values are precise. In the literature, there are some proposals for extending the primal BP to take care of NMR uncertainties. Using Cayley-Menger determinant, we present a global optimization approach that also allows the dual BP to deal with NMR uncertainties.