Residual Dipolar Coupling, Protein Backbone Conformation and Semidefinite Programming

We revisit the established problem of protein structure determination from geometrical restraints from NMR, using convex optimization. It is well-known that the NP-hard distance geometry problem of determining atomic positions from pairwise distance restraints can be relaxed into a convex semidefinite program (SDP). However, in practice the distance restraints are imprecise, and sometimes sparse, for accurate structure determination. Residual dipolar coupling (RDC) measurements provide additional geometric information on the angles between atom-pair directions and axes of the principal-axis-frame. The optimization problem involving RDC is highly non-convex and requires a good initialization even within the simulated annealing framework. In this paper, we model the protein backbone as an articulated structure composed of rigid units. We estimate the rotation of each rigid unit using SDP relaxation that incorporates chirality constraints. The two SDP based methods we propose RDCSDP and RDC-NOE-SDP have polynomial time complexity in the number of amino-acids and run efficiently on a regular PC. We further introduce a statistical tool, the Cramér-Rao bound (CRB) to provide an information theoretic bound on the highest resolution one can hope to achieve when determining protein structure from noisy measurements. Our simulation results show that when the RDC measurements are Y. Khoo Department of Physics, Princeton University, Princeton, NJ 08540, USA E-mail: [email protected] A. Singer Department of Mathematics and PACM, Princeton University, Princeton, NJ 08540, USA E-mail: [email protected] D. Cowburn Departments of Biochemistry and Physiology and Biophysics, Albert Einstein College of Medicine, Bronx, NY 10461, USA E-mail: [email protected] corrupted by Gaussian noise, for realistic noise magnitude our SDP algorithm attains the CRB. Through such comparison, the utility of CRB for benchmarking other procedures for structure determination in NMR is demonstrated. Finally, we apply our proposed method in a divide-andconquer fashion to determine the structure of ubiquitin from experimental distance restraints and RDC measurements obtained in two alignment media.