Analysis of a systematic search-based algorithm for determining protein backbone structure from a minimum number of residual dipolar couplings

We have developed an ab initio algorithm for determining a protein backbone structure using global orientational restraints on internuclear vectors derived from residual dipolar couplings (RDCs) measured in one or two different aligning media by solution nuclear magnetic resonance (NMR) spectroscopy. Specifically, the conformation and global orientations of individual secondary structure elements are computed, independently, by an exact solution, systematic search-based minimization algorithm using only 2 RDCs per residue. The systematic search is built upon a quartic equation for computing, exactly and in constant time, the directions of an internuclear vector from RDCs, and linear or quadratic equations for computing the sines and cosines of backbone dihedral (φ, ψ) angles from two vectors in consecutive peptide planes. In contrast to heuristic search such as simulated annealing (SA) or Monte-Carlo (MC) used by other NMR structure determination algorithms, our minimization algorithm can be analyzed rigorously in terms of expected algorithmic complexity and the coordinate precision of the protein structure as a function of error in the input data. The algorithm has been successfully applied to compute the backbone structures of three proteins using real NMR data.