Oriented Conformal Geometric Algebra and the Molecular Distance Geometry Problem

Abstract

The problem of 3D protein structure determination using distance information from nuclear magnetic resonance (NMR) experiments is a classical problem in distance geometry. NMR data and the chemistry of proteins provide a way to define a protein backbone order such that the distances related to the pairs of atoms $$\{i-3,i\},\{i-2,i\},\{i-1,i\}$$ are available, implying a combinatorial method to solve the problem, called branch-and-prune (BP). There are two main steps in BP algorithm: the first one (the branching phase) is to intersect three spheres centered at the positions for atoms $$ i-3,i-2,i$$ , with radius given by the atomic distances $$ d_{i-3,i},d_{i-2,i},d_{i-1,i}$$ , respectively, to obtain two possible positions for atom i; and the second one (the pruning phase) is to check if additional spheres (related to distances $$d_{j,i}$$ , $$j<i-3$$ ) can be used to select one of the two possibilities for atom i. Differently from distances $$d_{i-2,i},d_{i-1,i}$$ (associated to bond lenghts and bond angles), distances $$d_{j,i}$$ , $$j\le i-3$$ , may not be precise. BP algorithm has difficulties to deal with uncertainties, and this paper proposes the oriented conformal geometric algebra to take care of intersection of spheres when their centers and radius are not precise.