A steady-state approach for analysis of high-resolution relaxometry

High-resolution relaxometry measures nuclear spin longitudinal relaxation rate constants at low static magnetic field, either in the fringe field of a high-field NMR magnet or in an external electromagnetic coil, while polarizing and detecting nuclear magnetization at high field to optimize resolution and sensitivity for biological macromolecules. Detected magnetization depends on relaxation in the low magnetic field and on relaxation during transfer to and from the high magnetic field. Relaxation for backbone amide ¹⁵N magnetization in proteins is inherently multiexponential because of dipole-dipole and chemical shift anisotropy interactions with the amide ¹H spin and dipole-dipole interactions between the amide ¹H spin and ¹H remote spins. Nevertheless, relaxation decay profiles for backbone amide ¹⁵N spins in proteins are empirically observed to be essentially monoexponential with a single effective relaxation rate constant at magnetic fields as low as 1 T. The present work derives an expression for the effective relaxation rate constant under that assumption that relaxation in the network of dipole-dipole coupled ¹H spins is sufficiently rapid. This result enables efficient analysis of relaxometry data without explicit integration of the stochastic Liouville equation for relaxation of the amide N-H moiety and remote amide ¹H spins. The approach is validated by relaxometry measurements for ¹⁵N-labeled human ubiquitin and E. coli ribonuclease HI. The results obtained with the proposed approach agree well with results obtained using the MINOTAUR program (N. Bolik-Coulon et al., 2023), which integrates the full stochastic Liouville equation.