Quantitative Interpretation of Solvent Paramagnetic Relaxation for Probing Protein-Cosolute Interactions

Protein-small cosolute molecule interactions are ubiquitous and known to modulate the solubility, stability, and function of many proteins. Characterization of such transient weak interactions at atomic resolution remains challenging. In this work, we develop a simple and practical NMR method for extracting both energetic and dynamic information on protein-cosolute interactions from solvent paramagnetic relaxation enhancement (sPRE) measurements. Our procedure is based on an approximate (non-Lorentzian) spectral density that behaves exactly at both high and low frequencies. This spectral density contains two parameters, one global related to the translational diffusion coefficient of the paramagnetic cosolute, and the other residue specific. These parameters can be readily determined from sPRE data, and then used to calculate analytically a concentration normalized equilibrium average of the interspin distance, ⟨r^-6⟩_norm, and an effective correlation time, τ_C, that provide measures of the energetics and dynamics of the interaction at atomic resolution. We compare our approach with existing ones, and demonstrate the utility of our method using experimental ¹H longitudinal and transverse sPRE data recorded on the protein ubiquitin in the presence of two different nitroxide radical cosolutes, at multiple static magnetic fields. The approach for analyzing sPRE data outlined here provides a powerful tool for deepening our understanding of extremely weak protein-cosolute interactions.