Accounting for the temperature dependence of 13C spin–lattice relaxation of methyl groups in the glycyl–alanyl-leucine model system under MAS with spin diffusion

The difficulties in quantitatively modeling the temperature dependence of spin–lattice relaxation in a model isotope-enriched peptide are explored as a prelude to obtaining dynamics parameters for motions in proteins from such measurements. The degree to which this can be handled by adding spin diffusion to a bath in standard rate matrix relaxation theory is studied using a small tri-peptide model system, glycyl–alanyl-leucine (GAL). We observe in this molecule that the relaxation of backbone carbons CO and Cα is not dominated by local fluctuations of the 13 C– 1 H dipolar couplings, but rather by 13 C– 13 C spin diffusion to nearby methyl relaxation sinks. A treatment of the methyl relaxation itself, which ignores 13 C– 13 C spin diffusion effects back to the otherwise slowly relaxing bath, provides poor agreement between theory and experimental data obtained for the temperature dependence of the methyl relaxation rates. Closed form approximate spectral densities and relaxation rates for a methyl group during magic angle spinning are obtained to compute the needed transition rates. These average computed rates, in conjunction with an extended form of the Solomon equations, are found to adequately model the temperature dependence of the methyl relaxation rates when spin diffusion is included. The barrier to rotation for the alanine methyl in GAL is determined to be 3.5 kcal mol −1 .