Long-Time Mean Square Displacements in Proteins

We propose a method for obtaining the intrinsic, long time mean square displacement (MSD) of atoms and molecules in proteins from finite time molecular dynamics (MD) simulations. Typical data from simulations are limited to times of 1 to 10 ns and over this time period the calculated MSD continues to increase without a clear limiting value. The proposed method consists of fitting a model to MD simulation-derived values of the incoherent intermediate neutron scattering function, $I_{inc}(Q,t)$, for finite times. The infinite time MSD, $<r^2>$, appears as a parameter in the model and is determined by fits of the model to the finite time $I_{inc}(Q,t)$. Specifically, the $<r^2>$ is defined in the usual way in terms of the Debye-Waller factor as $I(Q,t = \infty) = \exp(- Q^2 <r^2 > /3)$. The method is illustrated by obtaining the intrinsic MSD $<r^2>$ of hydrated lysozyme powder (h = 0.4 g water/g protein) over a wide temperature range. The intrinsic $<r^2>$ obtained from data out to 1 ns and to 10 ns is found to be the same. The intrinsic $<r^2>$ is approximately twice the value of the MSD that is reached in simulations after times of 1 ns which correspond to those observed using neutron instruments that have an energy resolution width of 1 \mu eV.