Determining Protein Structures from NOESY Distance Constraints by Semidefinite Programming

International audienceContemporary practical methods for protein NMR structure determination use molecular dynamics coupled with a simulated annealing schedule. The objective of these methods is to minimize the error of deviating from the NOE distance constraints. However, the corresponding objective function is highly nonconvex and, consequently, difficult to optimize. Euclidean distance matrix (EDM) methods based on semidefinite programming (SDP) provide a natural framework for these problems. However, the high complexity of SDP solvers and the often noisy distance constraints provide major challenges to this approach. The main contribution of this paper is a new SDP formulation for the EDM approach that overcomes these two difficulties. We model the protein as a set of intersecting two- and three- dimensional cliques. Then, we adapt and extend a technique called semidefinite facial reduction to reduce the SDP problem size to approximately one quarter of the size of the original problem. The reduced SDP problem can be solved approximately 100 times faster, and it is also more resistant to numerical problems from erroneous and inexact distance bounds