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=∞)=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 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 μeV.