Determination of Dynamical Heterogeneity from Dynamic Neutron Scattering of Proteins

Abstract

Motional displacements of hydrogen (H) in proteins can be measured using incoherent neutron-scattering methods. These displacements can also be calculated numerically using data from molecular dynamics simulations. An enormous amount of data on the average mean-square motional displacement (MSD) of H as a function of protein temperature, hydration, and other conditions has been collected. H resides in a wide spectrum of sites in a protein. Some H are tightly bound to molecular chains, and the H motion is dictated by that of the chain. Other H are quite independent. As a result, there is a distribution of motions and MSDs of H within a protein that is denoted dynamical heterogeneity. The goal of this paper is to incorporate a distribution of MSDs into models of the H incoherent intermediate scattering function, I(Q,t), that is calculated and observed. The aim is to contribute information on the distribution as well as on the average MSD from comparison of the models with simulations and experiment. For example, we find that simulations of I(Q,t) in lysozyme are well reproduced if the distribution of MSDs is bimodal with two broad peaks rather than a single broad peak.

Figure 1

MSD in lysozyme obtained in (34) from fitting a model $I\left(Q,t\right)$, which neglects dynamical heterogeneity to $I\left(Q,t\right)$ calculated from Eq. 1 using molecular dynamics simulation.

Figure 2

The histogram of hydrogen (H) mean-square displacements (MSDs) in lysozyme calculated from MD simulation, ${\langle r^2\rangle }_{MD}=\langle {\Delta }^2\left(t\right)\rangle /2$ at $t=1$ ns, compared with the model $\rho \left(s\right)$ given by Eq. 23 (red solid line). The parameters in $\rho \left(s\right)$ are chosen to get the best fit to the histogram: $s_1=0.25$ Ų, $s_L=0.76$ Ų, $\Delta s=1.4$ Ų, $s_2=s_1+\Delta s$, ${\sigma }_1^2=0.03$ Å⁴, and ${\sigma }_2^2=0.08$ Å⁴. To see this figure in color, go online.

Figure 3

(Top) The ISF $I\left(Q,t\right)$ calculated using Eq. 1 from an MD simulation of lysozyme (open circles) and a fit of the model $I_M\left(Q,t\right)$ given by Eqs. 4, 7, and 23 (solid line). Shown is $I\left(Q,t\right)$ at 20 Q values from 0.2 to 4 Å⁻¹ in steps of 0.2 Å⁻¹, top to bottom. (Middle) The ${\rho }_N\left(s\right)$ vs. s for the Q values is shown: 0.2, 1.0, 2.0, 3.0, and 4.0 Å⁻¹. (Bottom) The best-fit parameter $3s_1$ vs. Q is shown. To see this figure in color, go online.

Figure 4

(Top) The average MSD obtained from fitting a model that includes DH (Eqs. 4, 7, and 23) to simulations of I(Q,t) for lysozyme (triangles) and the MSD obtained from fitting a similar model (34) without DH (squares). (Bottom) The parameter λ obtained for the model including DH for the fits is shown in Fig. 3 (top). To see this figure in color, go online.

Figure 5

(Top) The $S_R\left(Q,\omega =0\right)$ observed by Peters et al. (20) in hAChE (open circles) and the fit of our model (Eqs. 4, 7, 18, and 23) (solid line), which includes DH to the observed $S_R\left(Q,\omega =0\right)$. From top to bottom, the temperatures are 220, 240, 260, 280, and 300 K. (Middle and bottom) The parameter $3s_1$ and the relaxation parameter λ obtained from fits of the model $S_R\left(Q,\omega =0\right)$ to the experimental data are shown. To see this figure in color, go online.

Figure 6

(LHS) The normalized distribution of MSDs, ${\rho }_N\left(s\right)$, obtained by fitting Eq. 18 to the experimental $S_R\left(Q,\omega =0\right)$ for hAChE shown in Fig. 5 for five different temperatures: 220, 240, 260, 280, and 300 K. The height of the second peak in ${\rho }_N\left(s\right)$ is largest at 300 K. (RHS) The MSD obtained using Eq. 8 including DH (solid circles) and the MSD without DH obtained by fitting Eq. 22 to the observed $S_R\left(Q,\omega =0\right)$ in the Q-range are shown: 0.2–0.8 Å⁻¹ (solid triangles). To see this figure in color, go online.

Figure 7

(Top) The best-fit Gaussian ${\rho }_G^N\left(s\right)$ at eight Q-values, from 0.2 to 1.6 Å⁻¹ in steps of 0.2 Å⁻¹, with σ set at ${\sigma }^2$ = 0.2 Å⁴. (Bottom) The best-fit $s_0$ in Eq. 24 is shown at eight Q values with ${\sigma }^2$ set at the values shown (in Å⁴). To see this figure in color, go online.

Figure 8

(LHS) $I\left(Q,t\right)$ of lysozyme calculated using MD simulation (open circles) and a fit of the model $I_M\left(Q,t\right)$, Eq. 4, obtained using Eq. 7 and ${\rho }_{GT}\left(s\right)$ in Eq. 26 to the simulated $I\left(Q,t\right)$ for eight Q values, 0.2 to 1.6 Å⁻¹ from top to bottom. (RHS) The corresponding best-fit ${\rho }_{GT}\left(s\right)$ is shown. The broadest ${\rho }_{GT}\left(s\right)$ fits Q = 0.2 Å⁻¹. To see this figure in color, go online.

Figure 9

The best-fit parameters $s_T$, ${\sigma }^2$, and λ obtained from fits shown in Fig. 8. To see this figure in color, go online.