The average mean-square displacement, $\langle r^2\rangle$, of $H$ atoms in a protein is frequently determined using incoherent neutron-scattering experiments. $\langle r^2\rangle$ is obtained from the observed elastic incoherent dynamic structure factor, $S_i(Q,\omega =0)$, assuming the form $S_i(Q,\omega =0) =exp(-Q^2\langle r^2\rangle /3)$. This is often referred to as the Gaussian approximation (GA) to $S_i(Q,\omega =0)$. $\langle r^2\rangle$ obtained in this way depends on the value of the wave vector, $Q$ considered. Equivalently, the observed $S_i(Q,\omega =0)$ deviates from the GA. We investigate the origin of the $Q$ dependence of $\langle r^2\rangle$ 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 $S(Q,\omega )$ is an accurate approximation and is not the origin of the $Q$ dependence of $\langle r^2\rangle$. 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 $\langle r^2\rangle$. The observed $Q$ dependence of $\langle r^2\rangle$ can be used to provide information on the dynamical heterogeneity in proteins.

Close

PrevNext

View figure in article

• 
• 
• 
• 
• 
• 
• 
• 
•