Power-law and logarithmic relaxations of hydrated proteins: a molecular dynamics simulations study

We use molecular dynamics simulations to study anomalous internal protein dynamics observed for the backbone atoms of hydrated elastin and hydrated myoglobin in the picoseconds and nanoseconds regimes. The anomalous dynamics manifests itself in a sublinear increase of the atomic mean square displacements and in a power-law or logarithmic-like decay of correlation functions. We find that several, but not all, observations can be described in the frameworks of rugged potential-energy landscape and fractional Fokker-Planck approaches, in particular, a fractional Ornstein-Uhlenbeck process. Furthermore, mode-coupling theory allows us to rationalize findings at ambient temperatures, but there are deviations between theoretical predictions and simulation results related to the anomalous dynamics at cryogenic temperatures. We argue that the observations are consistent with a scenario where a broad β-relaxation peak shifts through the picoseconds and nanoseconds regimes when cooling from 300 to 200 K, say. Inspection of trajectories of consecutive nitrogen atoms along the protein backbone reveals that correlated forward-backward jumps, which exhibit a substantial degree of cooperativity, are a key feature of the anomalous dynamics.