Extracting Structural Information from Residual Chemical Shift Anisotropy: Analytic Solutions for Peptide Plane Orientations and Applications to Determine Protein Structure

Residual dipolar coupling (RDC) and residual chemical shift anisotropy (RCSA) provide orientational restraints on internuclear vectors and the principal axes of chemical shift anisotropy (CSA) tensors, respectively. Mathematically, while an RDC represents a single sphero-conic, an RCSA can be interpreted as a linear combination of two sphero-conics. Since RDCs and RCSAs are described by a molecular alignment tensor, they contain inherent structural ambiguity due to the symmetry of the alignment tensor and the symmetry of the molecular fragment, which often leads to more than one orientation and conformation for the fragment consistent with the measured RDCs and RCSAs. While the orientational multiplicities have been long studied for RDCs, structural ambiguities arising from RCSAs have not been investigated. In this paper, we give exact and tight bounds on the number of peptide plane orientations consistent with multiple RDCs and/or RCSAs measured in one alignment medium. We prove that at most 16 orientations are possible for a peptide plane, which can be computed in closed form by solving a merely quadratic equation, and applying symmetry operations. Furthermore, we show that RCSAs can be used in the initial stages of structure determination to obtain highly accurate protein backbone global folds. We exploit the mathematical interplay between sphero-conics derived from RCSA and RDC, and protein kinematics, to derive quartic equations, which can be solved in closed-form to compute the protein backbone dihedral angles (φ, ψ). Building upon this, we designed a novel, sparse-data, polynomial-time divide-and-conquer algorithm to compute protein backbone conformations. Results on experimental NMR data for the protein human ubiquitin demonstrate that our algorithm computes backbone conformations with high accuracy from 13C′ -RCSA or 15N-RCSA, and N-H N RDC data. We show that the structural information present in 13C′ -RCSA and 15N-RCSA can be extracted analytically, and used in a rigorous algorithmic framework to compute a high-quality protein backbone global fold, from a limited amount of NMR data. This will benefit automated NOE assignment and high-resolution protein backbone structure determination from sparse NMR data.