Variations on a theme by Debye and Waller: from simple crystals to proteins

Debye and Waller showed how to adjust scattering intensities in diffraction experiments for harmonic motions of atoms about an average, static reference configuration. However, many motions, particularly in biological molecules as compared to simple crystals, are far from harmonic. We show how, using a variety of simple anharmonic, multiconformational models, it is possible to construct a variety of Generalized Debye-Waller Factors, and understand their meaning. A central result for these cases is that, in principle, intensity factors cannot be obtained from true total mean square displacements of the atoms. We make the distinction between the intensity factors for unimodal quasiharmonic motions and those for the anharmonic, multimodal (valley hopping) motions. Only the former affect the conventional B factors.