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We revisit the problem of protein structure determination from geometricalrestraints from NMR, using convex optimization. It is well-known that theNP-hard distance geometry problem of determining atomic positions from pairwisedistance restraints can be relaxed into a convex semidefinite program. Oftenthe NOE distance restraints are too imprecise and sparse for accurate structuredetermination. Residual dipolar coupling (RDC) measurements provide additionalgeometric information on the angles between atom-pair directions and axes ofthe principal-axis-frame. The optimization problem involving RDC is highlynon-convex and requires a good initialization even within the simulatedannealing framework. In this paper, we model the protein backbone as anarticulated structure composed of rigid units. Determining the rotation of eachrigid unit gives the full protein structure. We propose solving the non-convexoptimization problems using the sum-of-squares (SOS) hierarchy. The twoalgorithms - RDC-SOS and RDC-NOE-SOS, have polynomial time complexity in thenumber of amino-acid residues and run efficiently on a standard desktop. Inmany instances, the proposed methods exactly recover the solution to theoriginal non-convex optimization problem. We introduce a statistical tool, theCramer-Rao bound (CRB), to provide an information theoretic bound on thehighest resolution one can hope to achieve when determining protein structurefrom noisy measurements using any methodology. Our simulation results show thatwhen the RDC measurements are corrupted by Gaussian noise of realisticvariance, both SOS based algorithms attain the CRB. We successfully apply ourmethod in a divide-and-conquer fashion to determine the structure of ubiquitinfrom experimental NOE and RDC measurements, achieving more accurate and fasterreconstructions compared to the current state of the art.

Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination