Diffusion crossover of protein molecules: Two-step coarse-graining and oscillating memory

A consistent treatment of anomalous diffusion requires a microcosmic framework that captures the underlying couplings between the relevant degrees of freedom. To this end, we employ a two-step coarse-graining procedure, in which protein molecules are modeled as generalized Brownian particles interacting harmonically with their neighbors, while the latter are coupled to a thermal bath. Under this construction, the power-law memory kernel in the generalized Langevin equation can be approximated as a sum of several response functions of damped oscillators, revealing, in particular, oscillatory behavior on sub-picosecond timescales. When applied to diffusion of proteins, the model predicts that the protein molecules exhibit ballistic diffusion up to ∼0.3 ps and subdiffusion up to 100 ps. Owing to the limited measurement window and the initial velocity preparation, we find that the time-averaged mean-square displacement along the reaction coordinate displays an upward-tail behavior. Finally, we extend the Markovianized dynamics to the super-diffusive regime by introducing velocity-dependent coupling.