# From Power‐Law to Correlation‐Time Distributions: A Unified Framework for the Analysis of Nuclear Magnetic Relaxation Dispersion (NMRD) Profiles of Complex Biological Systems

## TL;DR

This paper introduces a new framework to interpret nuclear magnetic relaxation data in biological systems using a distribution of correlation times.

## Contribution

The paper provides a unified physical model explaining power-law relaxation in NMRD profiles through a distribution of correlation times.

## Key findings

- Power-law relaxation in NMRD profiles arises from a broad distribution of correlation times.
- The framework was validated across various biological systems including proteins and tissues.
- The model enables extraction of dynamic information from NMRD profiles.

## Abstract

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.

1H NMRD profiles of biological systems often follow a power‐law dependence rather than a Lorentzian dispersion. This behavior can arise from a broad distribution of correlation times, weighted by 1/τ 1−α
. Validated across diverse systems such as sedimented proteins and biological tissues, this framework provides a physical interpretation of power‐law relaxation and enables direct access to dynamic information.