A Probability-Based Similarity Measure for Saupe Alignment Tensors with Applications to Residual Dipolar Couplings in NMR Structural Biology

High-throughput nuclear magnetic resonance (NMR) structural biology and NMR structural genomics pose a fascinating set of geometric challenges. A key bottleneck in NMR structural biology is the resonance assignment problem. We seek to accelerate protein NMR resonance assignment and structure determination by exploiting a priori structural information. In particular, a method known as nuclear vector replacement (NVR) has been proposed as a method for solving the assignment problem given a priori structural information. Among several different types of input data, NVR uses a particular type of NMR data known as residual dipolar couplings (RDCs). The basic physics of RDCs tells us that the data should be explainable by a structural model and set of parameters contained within the “Saupe alignment tensor”. In the NVR algorithm, one estimates the Saupe alignment tensors and then proceeds to refine those estimates. We would like to quantify the accuracy of such estimates, where we compare the estimated Saupe matrix to the correct Saupe matrix. In this work, we propose a way to quantify this comparison. Given a correct Saupe matrix and an estimated Saupe matrix, we compute an upper bound on the probability that a randomly rotated Saupe tensor would have an error smaller than the estimated Saupe matrix. This has the advantage of being a quantified upper bound, which also has a clear interpretation in terms of geometry and probability. While the specific application of our rotation probability results is given to NVR, our novel methods The International Journal of Robotics Research Vol. 24, No. 2–3, February–March 2005, pp. 165-182, DOI: 10.1177/0278364905050351 ©2005 Sage Publications can be used for any RDC-based algorithm to bound the accuracy of the estimated alignment tensors. Furthermore, they could also be used in X-ray crystallography or molecular docking to quantitate the accuracy of calculated rotations of proteins, protein domains, nucleic acids, or small molecules. KEY WORDS—SO(3), rotations, subgroup method, orthogonal image, alignment tensor, residual dipolar couplings, Saupe matrix, NMR structural biology ABBREVIATIONS USED—NMR, nuclear magnetic resonance; NVR, nuclear vector replacement; RDC, residual dipolar coupling; 3D, three-dimensional; MR, molecular replacement; SAR, structure activity relation; DOF, degrees of freedom; nt., nucleotides; SO(3), special orthogonal (rotation) group in 3D; S, the unit circle; S, the unit sphere; S, the unit 3-sphere; FAA, frame axis angle representation of rotations.