Protein structure determination using Riemannian approach

The protein nuclear magnetic resonance (NMR) structure determination is one of the most extensively studied problems due to its increasing importance in biological function analysis. We adopt a novel method, based on one of the matrix completion (MC) techniques-the Riemannian approach, to solve the protein structure determination problem. We formulate the protein structure in terms of low-rank matrix which can be solved by an optimization problem in the Riemannian spectrahedron manifold whose objective function has been delimited with the derived boundary condition. Two efficient algorithms in Riemannian approach-the trust-region (Tr) algorithm and the conjugate gradient (Cg) algorithm are used to reconstruct protein structures. We first use the two algorithms in a toy model and show that the Tr algorithm is more robust. Afterwards, we rebuild the protein structure from the NOE distance information deposited in NMR Restraints Grid (http://restraintsgrid.bmrb.wisc.edu/NRG/MRGridServlet). A dataset with both X-ray crystallographic structure and NMR structure deposited in Protein Data Bank (PDB) is used to statistically evaluate the performance of our method. By comparing both our rebuilt structures and NMR counterparts with the \"standard\" X-ray structures, we conclude that our rebuilt structures have similar (sometimes even smaller) RMSDs relative to \"standard\" X-ray structures in contrast with the reference NMR structures. Besides, we also validate our method by comparing the Z-scores between our rebuilt structures with reference structures using Protein Structure Validation Software suit. All the validation scores indicate that the Riemannian approach in MC techniques is valid in reconstructing the protein structures from NOE distance information. The software based on Riemannian approach is freely available at https://github.com/xubiaopeng/Protein_Recon_MCRiemman.\n\nAuthor summaryMatrix Completion is a technique widely used in many aspects, such as the global positioning in sensor networks, collaborative filtering in recommendation system for many companies and face recognition, etc. In biology, distance geometry used to be a popular method for reconstructing protein structures related to NMR experiment. However, due to the low quality of the reconstructed results, those methods were replaced by other dynamic methods such as ARIA, CYANA and UNIO. Recently, a new MC technique named Riemannian approach is introduced and proved mathematically, which promotes us to apply it in protein structure determination from NMR measurements. In this paper, by combining the Riemannian approach and some post-processing procedures together, we reconstruct the protein structures from the incomplete distance information measured by NMR. By evaluating our results and comparing with the corresponding PDB NMR deposits, we show that the current Riemannian approach method is valid and at least comparable with (if not better than) the state-of-art methods in NMR structure determination.