Clifford algebra and discretizable distance geometry

Abstract

Protein structure calculations using nuclear magnetic resonance (NMR) experiments are one of the most important applications of distance geometry. The chemistry of proteins and the NMR data allow us to define an atomic order, such that the distances related to the pairs of atoms {i−3,i},{i−2,i},{i−1,i} are available, and solve the problem iteratively using a combinatorial method, called branch‐and‐prune. The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i−3,i−2,i, with radius given by the atomic distances di−3,i,di−2,i,di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i may not be precise or even not be available. Using conformal Clifford algebra, in addition to take care of NMR uncertainties, which implies that we have to calculate sphere intersections considering that their centers and radius may not be fixed anymore, we consider a more flexible atomic order, where distances di−3,i are replaced by dj,i, where j⩽i−3. Copyright © 2017 John Wiley & Sons, Ltd.

Minimal NMR distance information for rigidity of protein graphs

“…Very recent results [2,3] using Clifford algebra propose an alternative that avoids the sampling process in the branching phase of i BP. However, in order to apply these results to protein structure calculations, a new re-order must be defined that avoids the situation (3).…”

Boolean Combination of Circular Arcs using Orthogonal Spheres

“…In the context of the distance geometry puzzle, we need to be able to combine arcs on the same circle: determining whether or not they overlap, and if they do, find a smaller arc representing their intersection. That is rather involved to execute in the direct point pair representation; this led the authors of [9] to represent the arc by sampled points, and later in [2,6] to introduce rotors whose motion is the arc. In the present paper, we show that arc combination becomes easily tractable when we factorize the point pair in terms of spheres; and subsequent computations are easier when those spheres are orthogonal to the circle, and to each other.…”

“…Furthermore, in the Discretizable Molecular Distance Geometry Problem (DMDGP) [8], it is assumed that a few more distances between neighboring atoms along the chain are known as intervals [7]. For a set of This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECC-UNICAMP, Campinas, Brazil, edited by Sebastià Xambó-Descamps and Carlile Lavor. vertices of the molecular graph, one can then provide a distance matrix such as this one from [2] and [6]:…”

“…Because of the prevalence of spheres and their intersections in this problem, [1,2] and [6] have addressed the problem by conformal geometric algebra (CGA). This is the geometric algebra of a vector space whose vectors can be interpreted as spheres, whose 2D subspaces can be interpreted as circles, and whose 3D subspaces are point pairs (the intersection of three spheres).…”

“…In an example from [6], an interval distance of X 4 to X 1 constrains the possibilities for X 4 to two arcs Starting from the first few vertices with interval distances, the DMDGP problem is one of combining circular arcs in space. However, conformal geometric algebra has no natural element to represent such arcs, and so the authors of [2,6] used the motion representation provided by the spinors to generate discretely sampled points (in [2]) or continuously parametrized orbits (in [6]) as their arc representation.…”