Motional displacements in proteins: The origin of wave-vector-dependent values

The average mean-square displacement, 〈r(2)〉, of H atoms in a protein is frequently determined using incoherent neutron-scattering experiments. 〈r(2)〉 is obtained from the observed elastic incoherent dynamic structure factor, S(i)(Q,ω=0), assuming the form S(i)(Q,ω=0) =exp(-Q(2)〈r(2)〉/3). This is often referred to as the Gaussian approximation (GA) to S(i)(Q,ω=0). 〈r(2)〉 obtained in this way depends on the value of the wave vector, Q considered. Equivalently, the observed S(i)(Q,ω=0) deviates from the GA. We investigate the origin of the Q dependence of 〈r(2)〉 by evaluating the scattering functions in different approximations using molecular dynamics (MD) simulation of the protein lysozyme. We find that keeping only the Gaussian term in a cumulant expansion of S(Q,ω) is an accurate approximation and is not the origin of the Q dependence of 〈r(2)〉. This is demonstrated by showing that the term beyond the Gaussian is negligible and that the GA is valid for an individual atom in the protein. Rather, the Q dependence (deviation from the GA) arises from the dynamical heterogeneity of the H in the protein. Specifically it arises from representing, in the analysis of data, this diverse dynamics by a single average scattering center that has a single, average 〈r(2)〉. The observed Q dependence of 〈r(2)〉 can be used to provide information on the dynamical heterogeneity in proteins.