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

Abstract

We revisit the 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, often the NOE distance restraints are too imprecise and 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. Determining the rotation of each rigid unit gives the full protein structure. We propose solving the non-convex optimization problems using the sum-of-squares (SOS) hierarchy, a hierarchy of convex relaxations with increasing complexity and approximation power. Unlike classical global optimization approaches, SOS optimization returns a certificate of optimality if the global optimum is found. Based on the SOS method, we proposed two algorithms-RDC-SOS and RDC-NOE-SOS, that have polynomial time complexity in the number of amino-acid residues and run efficiently on a standard desktop. In many instances, the proposed methods exactly recover the solution to the original non-convex optimization problem. To the best of our knowledge this is the first time SOS relaxation is introduced to solve non-convex optimization problems in structural biology. 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 using any unbiased estimator. Our simulation results show that when the RDC measurements are corrupted by Gaussian noise of realistic variance, both SOS based algorithms attain the CRB. We successfully apply our method in a divide-and-conquer fashion to determine the structure of ubiquitin from experimental NOE and RDC measurements obtained in two alignment media, achieving more accurate and faster reconstructions compared to the current state of the art.

Systematic Exploration of Protein Conformational Space Using a Distance Geometry Approach

“…This is in contrast to classical methods for structure determination, 1 which might at best produce different protein conformations. The approach is different with respect to the more recently proposed methods aiming at determining the global minimum configuration of the system 4, [55][56][57][58][59] or at determining all relative positions of monomers within a protein homo-oligomer. 60 On the contrary, the exhaustive list of conformations generated by TAiBP provides solutions for a larger range of problems.…”

Double-Distance Frameworks and Mixed Sparsity Graphs

“…It seems to us, moreover, that multiple constraint rigidity theory is of potential significance in a range of applications. We note, for example, that in three dimensions, the measure of residual dipolar coupling (RDC) between rigid units of a protein may be interpreted as a secondary nonmetric constraint [5]. Also, in the area of 3D sensor networks it is natural to consider the augmentation of Euclidean distances by (possibly partially available) data such as altitudes to a reference surface [9].…”