NMR Protein Structure Calculation and Sphere Intersections

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 [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], where {\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 [{\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 [{\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.