Memory function for protein diffusion

Standard algorithms to calculate the diffusion constant from computer simulations are based on either the mean-squared displacement or the velocity autocorrelation function of the tagged particle. They register displacements/velocities caused by random forces, but do not address their physical nature. This deficiency is resolved in the force route to the diffusion constant leading to Kirkwood equation for massive diffusive particles (Brownian motion). Approximate Kirkwood equation becomes exact when the force relaxation time is replaced with the memory time. To formulate the force route to the diffusion constant, memory functions were calculated here from molecular dynamics simulations of six charge mutants of the green fluorescent protein and the plastocyanin protein in a wide range of temperatures. The memory time falls between the velocity and force relaxation times, with the Kirkwood equation overestimating diffusion constants of proteins by a factor of ∼4. Diffusion constants from the velocity/displacement route strongly increase with increasing system size. Standard protocols accounting for finite-size effects show serious flaws when applied to protein diffusion by producing system-size corrections far exceeding both the finite-size diffusion constants and their infinite-size extrapolations. Diffusion constants from the force route show much less system-size dependence, and corrected values are mostly independent of the system size.