Mode-Coupling SRLS versus Mode-Decoupled Model-Free N-H Bond Dynamics: Mode-Mixing and Renormalization

The common approach to N-H motion in proteins is model-free (MF), where the global (R^C) and local (R^L) motions are assumed decoupled. We have recently applied to N-H bond dynamics the slowly relaxing local structure (SRLS) model, which accounts rigorously for mode-coupling. The original and extended MF formulas are perturbational expansions of SRLS with respect to the local ordering, (S₀²)², when R^L ≫ R^C. Their functional form, number of terms equal to the number of dynamic modes, is implied by mode-decoupling, and the free diffusion eigenvalue, 1/τ = 6R^L, by the absence of strong-potential-induced renormalization. However, for N-H motion, (S ₀²)² is high and in the extended MF regime R_⊥^L ≈ R^C. Although the functional form of the original MF formula is largely valid for R^C/R^L ≤ 0.01 and (S₀²)² ≥ 0.8, τ _e MF represents the significantly reduced potential-dependent renormalized value of τ. Hence, the application of this formula to calculate NMR variables is appropriate in this parameter range, but associating τ_e with the local motion correlation time is inappropriate. Means to derive τ from τ_e are provided. For a cosine squared potential, the cone-model-based MF formula that relates τ_e to τ can also be used, Deriving τ from τ_e is important for proper characterization of the site-specific local motion and in the context of τ-dependent MF functionalities. Mode-coupling dominates the extended MF regime where SRLS must be invariably used. Eigenmode and spectral density analysis is provided in this study for the two parameter ranges associated with N-H bond motion.