Nonexponential decay of internal rotational correlation functions of native proteins and self-similar structural fluctuations

Structural fluctuations of a protein are essential for the function of native proteins and for protein folding. To understand how the main chain in the native state of a protein fluctuates on different time scales, we examined the rotational correlation functions (RCFs), C(t), of the backbone N-H bonds and of the dihedral angles γ built on four consecutive Cα atoms. Using molecular dynamics simulations of a model α/β protein (VA3) in its native state, we demonstrate that these RCFs decay as stretched exponentials, ln[C(t)] ≈ Dαtα with a constant Dα and an exponent α (0 < α < 0.35) varying with the free-energy profiles (FEPs) along the amino acid sequence. The probability distributions of the fluctuations of the main chain computed at short time scale (1 ps) were identical to those computed at large time scale (1 ns) if the time is rescaled by a factor depending on α < 1. This self-similar property and the nonexponential decays (α ≠ 1) of the RCFs are described by a rotational diffusion equation with a time-dependent diffusion coefficient D(t) = αDαtα-1. The present findings agree with observations of subdiffusion (α < 1) of fluorescent probes within a protein molecule. The subdiffusion of 15N-H bonds did not affect the value of the order parameter S2 extracted from the NMR relaxation data by assuming normal diffusion (α = 1) of 15N-H bonds on a nanosecond time scale. However, we found that the RCF does not converge to S2 on the nanosecond time scale for residues with multiple-minima FEPs.