Secondary structures of peptides and proteins via NMR chemical-shielding anisotropy (CSA) parameters

Complete nuclear magnetic resonance (NMR) chemical-shielding tensors, sigma, have been computed at different levels of density-functional theory (DFT), within the gauge-including atomic orbital (GIAO) formalism, for the atoms of the peptide model For-L-Ala-NH2 as a function of the backbone dihedral angles phi and psi by employing a dense grid of 10 degrees. A complete set of rigorously orthogonal symmetric tensor invariants, {sigma iso, rho, tau}, is introduced, where sigma iso is the usual isotropic chemical shielding, while the newly introduced rho and tau parameters describe the magnitude and the orientation/shape of the chemical-shielding anisotropy (CSA), respectively. The set {sigma iso, rho, tau} is unaffected by unitary transformations of the symmetric part of the shielding tensor. The mathematically and physically motivated {rho, tau} anisotropy pair is easily connected to more traditional shielding anisotropy measures, like span (Omega) and skew (kappa). The effectiveness of the different partitions of the CSA information in predicting conformations of peptides and proteins has been tested throughout the Ramachandran space by generating theoretical NMR anisotropy surfaces for our For-L-Ala-NH2 model. The CSA surfaces, including Omega(phi, psi), kappa(phi, psi), rho(phi, psi), and tau(phi, psi) are highly structured. Individually, none of these surfaces is able to distinguish unequivocally between the alpha-helix and beta-strand secondary structural types of proteins. However, two- and three-dimensional correlated plots, including Omega versus kappa, rho versus tau, and sigma iso versus rho versus tau, especially for 13Calpha, have considerable promise in distinguishing among all four of the major secondary structural elements.