Protein dynamics in solution and powder measured by incoherent elastic neutron scattering: the influence of Q-range and energy resolution

Incoherent elastic neutron scattering (IENS) has been widely used to measure intramolecular atomic mean square displacements (MSDs) of proteins in powder and in solution. The instrumental energy resolution and the wave vector transfer (Q-range) determine, respectively, the time and length scales of observable motions. In order to investigate contributions of diffusive motions to MSDs measured by this method, we calculated the elastic intensity for several simple scattering functions. We showed that continuous translational diffusion contributes to MSDs in a Q-range where the energy width of the scattering function is of the order of the instrumental energy resolution. We discuss the choice of instruments adapted to focus on intramolecular motions in the presence of solvent or global macromolecular diffusion. The concepts developed are applied to interpret experimental data from H₂O- and D₂O-hydrated proteins. Finally, analogies between the Gaussian approximation in IENS and the Guinier approximation in small-angle scattering are discussed.