Derivation of Mean-Square Displacements for Protein Dynamics from Elastic Incoherent Neutron Scattering

Abstract

The derivation of mean-square displacements from elastic incoherent neutron scattering (EINS) of proteins is examined, with the aid of experiments on camphor-bound cytochrome P450cam and complementary molecular dynamics simulations. It is shown that a q(4) correction to the elastic incoherent structure factor (where q is the scattering vector) can be simply used to reliably estimate from the experiment both the average mean-square atomic displacement, <Δr(2)> of the nonexchanged hydrogen atoms in the protein and its variance, σ(2). The molecular dynamics simulation results are in broad agreement with the experimentally derived <Δr(2)> and σ(2) derived from EINS on instruments at two different energy resolutions, corresponding to dynamics on the ∼100 ps and ∼1 ns time scales. Significant dynamical heterogeneity is found to arise from methyl-group rotations. The easy-to-apply q(4) correction extends the information extracted from elastic incoherent neutron scattering experiments and should be of wide applicability.

Elastic and Conformational Softness of a Globular Protein

“…This finding leads to a simple means of experimentally separating the conformational from the elastic atomic displacements. 87.15.ap, 87.64.Bx, 87.14.E-+ Both authors contributed equally 2 *Author to whom correspondence should be addressed: smithjc@ornl.gov Protein molecules in their native states, although highly structured, have a degree of flexibility (sometimes called 'softness' [1]) required for their biological function [2,3].On short time scales (≤ 1 ns) this flexibility has often been characterized by the thermal fluctuations of atomic positions as determined using simulation and scattering techniques [1,[4][5][6][7]. These fluctuations are partly conformational, i.e., involving transitions between energy wells [8][9][10], and partly "elastic", i.e., dynamics confined within single energy wells [9,10].…”

Long-time mean-square displacements in proteins

“…S(Q, ω = 0)) at low Q (0 < Q < 0.4Å −1 ) are usually selected. Secondly and most importantly, Zheng et al [34] have shown that I(Q, t) departs from a Gaussian approximation because of dynamical heterogeneity. The heterogeneity introduces a Q 4 term in Eq.…”

“…The heterogeneity introduces a correction to fourth order in Q that can be used to extract the variance of the distribution of mean-square displacements. [33,34] Similarly, the global MSD obtained from neutron scattering measurements depends on the energy resolution employed. The MSD is obtained from the elastic (ω = 0) component of the resolution broadened dynamic structure factor (DSF), S R (Q, ω = 0), as…”

Intrinsic mean-square displacements in proteins

“…In a real protein, the H occupies a variety of positions that have different values of r 2 . If a distribution of r 2 values is included [26][27][28], there are terms in I ∞ beyond a simple Gaussian represented by a single parameter r 2 . In future applications a generalization to include a distribution of r 2 values based on Eq.…”

“…The neutrons scatter from the H at points r i (t) in the protein. The incoherent DSF is proportional to an average over the N self correlation functions of the r i (t) of each individual H. The H is distributed in very different environments throughout the protein and these correlation functions may be quite different with different time scales [26][27][28]. Our first approximation is to represent this average over all H by a single representative H so that the I inc (Q, t) reduces to I(Q, t) = exp(−iQ · r(t)) exp(iQ · r(0)) .…”