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

Abstract

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.

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

“…For example, Nakagawa et al [40] find a bimodal distribution of MSDs is consistent with data. A somewhat bimodal distribution of MSDs is found in simulations of SNase [25] and a clearly bimodal distribution here in Fig. 9 for lysozyme.…”

“…Tokuhisa and coworkers [25] reached an equivalent conclusion for SNase. This opens the way to using the Q dependence of r 2 to test models of distribution of the (r j ) 2 without concern that the Q dependence could arise from other factors.…”

“…Tokuhisa and coworkers [25] also find that the isotropic approximation, in which [Q · r j (t)] 2 is replaced by Q 2 r j 2 /3, is valid in SNase and not responsible for any deviation of the EISF from a Gaussian. Although some 10 % of H in SNase show an anisotropic r j 2 the 10 % have small r 2 values and contribute little to the EISF.…”

“…As noted in the Introduction, Tokuhisa and coworkers [25] evaluated the cumulant expansion of the EISF for individual atoms in Staphylococcal Nuclase (SNase) keeping terms up to 4th and 6th order. They found that the 4th and 6th order terms were negligible compared to the Gaussian when the average over all the H in the protein was taken.…”

“…Tokuhisa et al [25] evaluated (1) two higher-order terms (fourth and sixth order) beyond the Gaussian in a cumulant expansion of I i (Q, t) for individual atoms, (2) the degree of anisotropy in the motion of individual atoms and (3) the degree of dynamical heterogeneity among the atoms in the sum over j in Eq. (6), all from an MD simulation of staphylococal nuclase (SNase).…”