Non-Gaussian behavior of elastic incoherent neutron scattering profiles of proteins studied by molecular dynamics simulation

Elastic incoherent neutron scattering (EINS) data can be approximated with a Gaussian function of $q$ in a low $q$ region. However, in a higher $q$ region the deviation from a Gaussian function becomes non-negligible. Protein dynamic properties can be derived from the analyses of the non-Gaussian behavior, which has been experimentally investigated. To evaluate the origins of the non-Gaussian behavior of protein dynamics, we conducted a molecular dynamics (MD) simulation of staphylococcal nuclease. Instead of the ordinary cumulant expansion, we decomposed the non-Gaussian terms into three components: (i) the component originating from the heterogeneity of the mean-square fluctuation, (ii) that from the anisotropy, and (iii) that from higher-order terms such as anharmonicity. The MD simulation revealed various dynamics for each atom. The atomic motions are classified into three types: (i) “harmonic,” (ii) “anisotropic,” and (iii) “anharmonic.” However, each atom has a different degree of anisotropy. The contribution of the anisotropy to the total scattering function averages out due to these differences. Anharmonic motion is described as the jump among multiple minima. The jump distance and the probability of the residence at one site vary from atom to atom. Each anharmonic component oscillates between positive and negative values. Thus, the contribution of the anharmonicity to the total scattering is canceled due to the variations in the anharmonicity. Consequently, the non-Gaussian behavior of the total EINS from a protein can be analyzed by the dynamical heterogeneity.

Figure 1

(Color) (a) EISF as a function of $q$ calculated using the result of 1-ns MD simulation at $300\mathrm{K}$. Total $S(q,0)$ (black line) and its components $S^{\mathrm{gauss}}(q,0)$ (yellow line), $S^{2\text{−}\mathrm{hetero}}(q,0)$ (blue line), $S^{2\text{−}\mathrm{aniso}}(q,0)$ (green line), and $S^{\mathrm{higher}}(q,0)$ (red line) are shown. (b) Total $S(q,0)$ (solid line) and the sum of $S^{\mathrm{gauss}}(q,0)$ and $S^{2\text{−}\mathrm{hetero}}(q,0)$ (broken line) are shown.

Figure 2

(Color) Relationship between the second-order heterogeneity and the mean-square fluctuations of the hydrogen atoms. (a) ${\varsigma }_k$ given by Eq. (13) as a function of $\{⟨\mathrm{\Delta }{r}_k^2⟩{\}}^{-1∕2}$. (b) Magnitudes of ${\varsigma }_k$ are shown by colors mapped onto the hydrogen atoms along with the ribbon model of SNase using PYMOL [30]. The ${\varsigma }_k$ values decrease in the order of red, orange, yellow, green, light blue, and blue. The atoms selected for Fig. 3 are highlighted with large spheres. (c) Distribution of $⟨\mathrm{\Delta }{r}_k^2⟩$.

Figure 3

(Color) Probability distributions of the atomic fluctuations (a–d) and the atomic scattering function with its components (e–h) for typical hydrogen atoms. Selected atoms are as follows: for (a) and (e) the hydrogen atom bound to Ca of Met 98 as an example of a “harmonic” atom; for (b) and (f) the hydrogen atom bound to Ca of Asn 68 as an example of an “anisotropic” atom; for (c) and (g) the hydrogen atom bound to Cg of Glu67 as one example of an “anharmonic” atom; and for (d) and (h) the hydrogen atom bound to Cg of Ile18 as another example of an “anharmonic” atom. From (a) to (d), solid lines indicate the distributions of atomic fluctuations and broken lines indicate the approximated Gaussian function with the variances $⟨\mathrm{\Delta }{x}_k^2⟩$, $⟨\mathrm{\Delta }{y}_k^2⟩$, and $⟨\mathrm{\Delta }{z}_k^2⟩$. Blue, black, and red indicate the $x$, $y$, and $z$ direction, respectively. From (e) to (h), $s_k(q,0)$, $S^{gauss}(q,0)$, $s_k^{2\text{−}\mathrm{hetero}}(q,0)$, $s_k^{2\text{−}\mathrm{aniso}}(q,0)$, and $s_k^{\mathrm{higher}}(q,0)$ are depicted with black, yellow, blue, green, and red lines, respectively.

Figure 4

Relation of the maximum value of $\mid s_k^{\mathrm{higher}}(q,0)\mid$ $(\mid s_k^{\mathrm{higher}}(q,0\left){\mid }_{\max }\right)$ with the maximum value of $s_k^{2\text{−}\mathrm{aniso}}(q,0)$ $\left[s_k^{2\text{−}\mathrm{aniso}}\right(q,0{)}_{\max }]$. A circle (drawn with a broken line) indicates the densely distributed region. Atoms shown in Figs. 3a, 3b, 3c, 3d are marked with $\circ$.

Figure 5

Histograms of $s_k^{2\text{−}\mathrm{aniso}}(q,0)$ (grey bars) and $s_k^{\mathrm{higher}}(q,0)$ (black bars) values at $q=1$, 2, 3, 4, and $5{\mathrm{\AA }}^{-1}$. Open arrows and solid arrows indicate the average values of $s_k^{2\text{−}\mathrm{aniso}}(q,0)$ and $s_k^{\mathrm{higher}}(q,0)$ at each $q$ value, which equal $S^{2\text{−}\mathrm{aniso}}(q,0)$ and $S^{\mathrm{higher}}(q,0)$ at given $q$.