Modelling the unusual temperature dependence of atomic displacements in proteins by local nonharmonic potentials

Recent x-ray studies of metmyoglobin at different temperatures [Frauenfelder, H. Petsko, G. A. & Tsernoglou, D. (1979) Nature (London) 280, 558-563] show that the dependence of the mean square displacements of individual atoms on the absolute temperature T can be described analytically by T mu, in which the parameter mu undergoes local variations in the range -2 to +20. This strong deviation from harmonic behavior mu = 1 is modelled here, in the classical limit, by local nonharmonic potentials, in which a particle spends part of its time in one or more weakly bound states (traps) and the rest of it as an almost free particle in a restricted volume (cavity). Analytic calculations of the mean square atomic displacement in such cavity--traps potentials for a simplified three-dimensional geometry yield the following results: A parametrically described T mu behavior characterizes a transition region in temperature below which the particle is "trapped" (solid-like) and above which the particle is "free" (liquid-like). The magnitude of mu increases with an increase in the ratio of cavity volume to trap volume. Mu greater than 0 type of behavior can be obtained by a single trap and mu less than 0 by two symmetric traps. A comparison is made between the predictions of the model and the experimental results, and the elucidation of the model parameters from x-ray data is discussed.