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, $I_{\mathrm{inc}}(\mathbf{Q},t)$, for finite times. The infinite-time MSD, $\langle r^2\rangle$, appears as a parameter in the model and is determined by fits of the model to the finite-time $I_{\mathrm{inc}}(\mathbf{Q},t)$. Specifically, the $\langle r^2\rangle$ is defined in the usual way in terms of the Debye-Waller factor as $I(\mathbf{Q},t=\infty )=\exp (-Q^2\langle r^2\rangle /3)$. The method is illustrated by obtaining the intrinsic MSD $\langle r^2\rangle$ of hydrated lysozyme powder ($h=0.4$ g water/g protein) over a wide temperature range. The intrinsic $\langle r^2\rangle$ obtained from data out to 1 and to 10 ns is found to be the same. The intrinsic $\langle r^2\rangle$ 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.

• Received 28 July 2013

DOI:https://doi.org/10.1103/PhysRevE.88.052706