Quantitative Interpretation of Transverse Spin Relaxation by Translational Diffusion in Liquids

Intermolecular spin relaxation by translational diffusion of spin pairs has been widely used to study the properties of biomolecules in liquids. Notably, solvent paramagnetic relaxation enhancement (sPRE) arising from paramagnetic cosolutes has gained significant attention for various applications in structural biology, including the structural refinement of intrinsically disordered proteins, the elucidation of the molecular mechanisms driving cosolute-induced protein denaturation, and the characterization of residue-specific effective near-surface electrostatic potentials (ENS). Furthermore, sPRE has been extensively applied in magnetic resonance imaging (MRI), where paramagnetic ions, such as Gd(III)-based ions, are used as contrast agents. Among these applications, the transverse sPRE rate (Γ₂) has predominantly been interpreted empirically as being proportional to the average interspin distance ⟨r^-6⟩_norm. In this study, we present a rigorous theoretical interpretation of Γ₂ for spherically symmetric intermolecular potentials, demonstrating that it is proportional to ⟨r^-4⟩_norm. We provide an explicit formula for calculating ⟨r^-4⟩_norm without any adjustable parameters, offering valuable insights into the interaction potential independent of the type or strength of interactions. It has broad applicability, including the precise interpretation of the relaxation properties of the MRI contrast agents and calculation of the ENS.