From Power-Law to Correlation-Time Distributions: A Unified Framework for the Analysis of Nuclear Magnetic Relaxation...

A recurring and significant finding across diverse biological and macromolecular systems is that the frequency dependence of the spin-lattice relaxation rate often cannot be well fitted with a single correlation time but rather follows a power-law function. This power-law dependence is attributed to the dynamics of rare, strongly bound water molecules trapped on rugged macromolecular surfaces, with a Pareto distribution of correlation times. Here, we show that power-law dependences naturally emerge from a broad distribution of correlation times with weighting factors proportional to 1/τ^(1-α). We derive analytical expressions for limiting cases and perform numerical simulations demonstrating that this distribution of correlation times generates power-law exponents closely matching α over wide frequency windows. We validate this framework by fitting Nuclear Magnetic Relaxation Dispersion (NMRD) profiles of sedimented proteins, biological tissues, cross-linked hydrogels, and protein solutions. This approach establishes a physical interpretation of power-law relaxation, enabling the extraction of dynamic information otherwise inaccessible.