Generalized Model-Free Analysis of Nuclear Spin Relaxation Experiments

NMR spectroscopy is one of the most widely used experimental tools in chemistry and physics. Compared with methods such as X-ray crystallography, both structural and dynamical information is obtained. The analytic formulations in NMR spectroscopy are complementary to numerical molecular dynamics (MD) simulations in terms of theoretical force fields based on experimental data. Generalized model-free (GMF) analysis bridges theory and experiment by introducing an irreducible representation of the nuclear spin interactions (dipolar and quadrupolar coupling and chemical shift), which transforms under rotations by the Wigner rotation matrix. Solid-state NMR experiments characterize the dynamical variables by including the amplitudes and rates of motions within the alignment frame, e.g., crystal axes system, or director frame for liquid crystals or biomembranes. According to time-dependent perturbation theory, the NMR relaxation rates depend on the spectral densities of motion due to the irreducible components of the coupling Hamiltonian. The mean-squared amplitudes and correlation times together with the activation barriers characterize the structural dynamics. Application to rhodopsin gives an example where the methyl groups of the retinal cofactor have different motional rates and activation barriers that change with light activation. The simple framework of GMF analysis can be applied to relaxation experiments for various biomolecular systems, including membrane proteins, amyloid fibrils, and aligned biopolymers.