NMR Protein Structure Calculation and Sphere Intersections

Carlile Lavor,Michael Souza,Luis Aragón José,Rafael Alves

doi:10.1515/cmb-2020-0103

Abstract

Abstract Nuclear Magnetic Resonance (NMR) experiments can be used to calculate 3D protein structures and geometric properties of protein molecules allow us to solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i − 3, i − 2, i − 1, with radii 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 should be represented as interval distances [ d _ i - 3 , i , d ¯ i - 3 , i {\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}} ], where d _ i - 3 , i ≤ d i - 3 , i ≤ d ¯ i - 3 , i {\underline{d}_{i - 3,i}} \le {d_{i - 3,i}} \le {\bar d_{i - 3,i}} . In the literature, an extension of the BP algorithm was proposed to deal with interval distances, where the idea is to sample values from [ d _ i - 3 , i , d ¯ i - 3 , i {\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}} ]. We present a new method, based on conformal geometric algebra, to reduce the size of [ d _ i - 3 , i , d ¯ i - 3 , i {\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}} ], before the sampling process. We also compare it with another approach proposed in the literature.

Highlights

Nuclear Magnetic Resonance (NMR) experiments provide short distance values between atoms of a protein molecule
Because of uncertainty in NMR data, some of the distances di−,i should be represented as interval distances [di−,i, di−,i], where di−,i ≤ di−,i ≤ di−,i
The MDGP is called the Discretizable Molecular Distance Geometry Problem (DMDGP) [19, 20], which can be stated as follows, where we use xi instead of xvi and di,j in place of dvi,vj : (DMDGP) Given a simple undirected graph G = (V, E, d) in which the vertex set V is ordered as V = {v, ..., vn}, whose edges are weighted by d : E → (, ∞), subject to the following three constraints: 1. For the initial three vertices v, v, v ∈ V, there exist points x, x, x ∈ R satisfying equations (1); 2

Summary

Introduction

Nuclear Magnetic Resonance (NMR) experiments provide short distance values between atoms of a protein molecule. The requirements (3) and (4) imply that, at each iteration of the BP algorithm, we intersect three spheres centered at the positions for vertices vi− , vi− , vi− with radius di− ,i , di− ,i , di− ,i, respectively, resulting in two possible positions for vi, i > .

Results

Conclusion