Cross-relaxation between macromolecular and solvent spins: The role of long-range dipole couplings

Abstract

Nuclear spin relaxation by intermolecular dipole–dipole interactions between macromolecular and solvent nuclear moments forms the basis of a widely used method for investigating macromolecular solvation. In particular, intermolecular cross-relaxation [or nuclear Overhauser effect (NOE)] between protein and water protons has been used to probe the mobility of water molecules interacting with the protein surface. The method rests on the assumption that the intermolecular NOE is of short (4–5 Å) range and thus provides information about the mobility of individual water molecules in hydration sites near the monitored protein protons. Here, we present a theoretical analysis of the spectral density function (SDF) that governs the cross-relaxation rates in the laboratory-fixed and rotating frames. In contrast to the r−6 dependence of the intramolecular NOEs used for structure determination, the intermolecular NOE is shown to be long-ranged with important contributions from thousands of water molecules. For a consistent interpretation of such NOEs, it is necessary to use a model that explictly incorporates motionally retarded hydration water molecules as well as unperturbed bulk water molecules. We formulate a diffusion model with a nonuniform solvent mobility and solve it to obtain an analytical expression for the SDF. Calculations with this nonuniform diffusion model demonstrate that intermolecular NOEs with surface protons are dominated by long-range dipole couplings to bulk water and therefore provide little or no information about hydration dynamics. The physical basis of this unexpected phenomenon is that the characteristic time scale for relaxation-inducing fluctuations is longer for the more numerous remote water molecules, despite their higher mobility. The analytical results presented here are generally applicable to intermolecular dipolar relaxation of like or unlike (nuclear or electron) spins in a variety of experimental situations.