Slowly relaxing local structure (SRLS) analysis of ¹⁵N-H relaxation from the prototypical small proteins GB1 and GB3

¹⁵N-H relaxation parameters from the first (GB1) and third (GB3) immunoglobulin-binding domains of streptococcal protein G were analyzed previously with the traditional model-free (MF) method. These proteins comprise an α-helix and a four-stranded β-sheet. An extensive study of GB1 (GB3) used combined three-field (five-field) data acquired in the 278-323 K range (at 297 K). For successful analysis of the GB3 data, it was necessary to allow for variations in the ¹⁵N chemical shift anisotropy (CSA) tensor and virtually eliminate the local motion. In the case of GB1, the spectral density was parametrized. Here, we analyze these data with the slowly relaxing local structure (SRLS) approach, which is the generalization of MF in allowing for general tensorial properties, and accounting for mode-coupling. A standard (featuring constant magnetic tensors) SRLS fitting scheme is used. This analysis accounts for the important asymmetry of the local spatial restrictions; it provides physical order parameters, local diffusion rates, related activation energies, and key features of local geometry. Using data from GB3 we show that the main local ordering axis is C_i-1 ^α - C_i^α, and the average axial (rhombic) order parameter is -0.457 ± 0.017 (1.156 ± 0.015) for the α-helix and -0.484 ± 0.002 (1.10 ± 0.04) for the rest of the polypeptide chain. The N-H bonds within (outside of) the α-helix reorient locally with an average correlation time, 〈τ〉, of 310 (130) ps, as compared to 3.33 ns for the global tumbling. Several N-H bonds in the loops β₁/β₂, β₂/α- helix, and α-helix/β₃ have 〈τ〉 of 380, 320, and 750 ps, respectively. The distinctive experimental data of the α-helix are due to relatively weak and substantially rhombic local ordering and slow local motion. For GB1, we derive activation energies from local diffusion rates. They are 43.3 ± 7.1 kJ/mol for the β-strands, 24.7 ± 3.9 kJ/mol for the α-helix (and approximately for the loop β₃/β₄), and 18.9 ± 1.8 kJ/mol for the other loops. The physical SRLS description provides new insights into the backbone dynamics of GB1 and GB3 in particular, and proteins in general.