Clifford Algebra and the Discretizable Molecular Distance Geometry Problem

Abstract

Nuclear Magnetic Resonance experiments can provide distances between pairs of atoms of a protein that are close enough and the problem is how to determine the 3D protein structure based on this partial distance information, called Molecular Distance Geometry Problem. It is possible to define an atomic order 1, ..., n and solve the problem iteratively using an exact method, called Branch-and-Prune (BP). The main step of BP algorithm is to solve a quadratic system to get the two possible positions for i, i > 3, in terms of the positions of i−3, i−2, i−1 and the distances di−1, i, di−2, i, di−3, i. Because of uncertainty in NMR data, some of the distances di−3, i may not be precise and the main problem to apply BP is related to the difficulty of obtaining an analytical expression of the position of atom i in terms of the positions of the three previous ones and the corresponding distances. We present such expression and although it is similar to one already existing in the literature, based on polyspherical coordinates, a new proof is given, based on Clifford algebra, and we also explain how such expression can be useful in BP using a parameterization which depends on di−3, i. The results suggest that a master equation might exist, what is generally not believed by many researchers.