Signature of functional enzyme dynamics in quasielastic neutron scattering spectra: The case of phosphoglycerate kinase.

We present an analysis of high-resolution quasi-elastic neutron scattering spectra of phosphoglycerate kinase which elucidates the influence of the enzymatic activity on the dynamics of the protein. We show that in the active state the inter-domain motions are amplified and the intra-domain asymptotic power-law relaxation ∝t-α is accelerated, with a reduced coefficient α. Employing an energy landscape picture of protein dynamics, this observation can be translated into a widening of the distribution of energy barriers separating conformational substates of the protein.

Quasi-analytical resolution-correction of elastic neutron scattering from proteins.

Elastic neutron scattering from proteins reflects the motional amplitudes resulting from their internal collective and single-atom dynamics and is observable if the global diffusion of whole molecules is either blocked or cannot be resolved by the spectrometer under consideration. Due to finite instrumental resolution, the measured elastic scattering amplitude always contains contaminations from quasielastic neutron scattering and some model must be assumed to extract the resolution-corrected counterpart from corresponding experimental spectra. Here, we derive a quasi-analytical method for that purpose, assuming that the intermediate scattering function relaxes with a "stretched" Mittag-Leffler function, Eα(-(t/τ)α) (0 < α < 1), toward the elastic amplitude and that the instrumental resolution function has Gaussian form. The corresponding function can be integrated into a fitting procedure and allows for eliminating the elastic intensity as a fit parameter. We illustrate the method for the analysis of two proteins in solution, the intrinsically disordered Myelin Basic Protein, confirming recently published results [Hassani et al., J. Chem. Phys. 156, 025102 (2022)], and the well-folded globular protein myoglobin. We also briefly discuss the consequences of our findings for the extraction of mean square position fluctuations from elastic scans.

Multiscale relaxation dynamics and diffusion of myelin basic protein in solution studied by quasielastic neutron scattering.

We report an analysis of high-resolution quasielastic neutron scattering spectra from Myelin Basic Protein (MBP) in solution, comparing the spectra at three different temperatures (283, 303, and 323 K) for a pure D2O buffer and a mixture of D2O buffer with 30% of deuterated trifluoroethanol (TFE). Accompanying experiments with dynamic light scattering and Circular Dichroism (CD) spectroscopy have been performed to obtain, respectively, the global diffusion constant and the secondary structure content of the molecule for both buffers as a function of temperature. Modeling the decay of the neutron intermediate scattering function by the Mittag-Leffler relaxation function, ϕ(t) = Eα(-(t/τ)α) (0 < α < 1), we find that trifluoroethanol slows down the relaxation dynamics of the protein at 283 K and leads to a broader relaxation rate spectrum. This effect vanishes with increasing temperature, and at 323 K, its relaxation dynamics is identical in both solvents. These results are coherent with the data from dynamic light scattering, which show that the hydrodynamic radius of MBP in TFE-enriched solutions does not depend on temperature and is only slightly smaller compared to the pure D2O buffer, except for 283 K, where it is much reduced. In accordance with these observations, the CD spectra reveal that TFE induces essentially a partial transition from ß-strands to α-helices, but only a weak increase in the total secondary structure content, leaving about 50% of the protein unfolded. The results show that MBP is for all temperatures and in both buffers an intrinsically disordered protein and that TFE essentially induces a reduction in its hydrodynamic radius and its relaxation dynamics at low temperatures.

Franck-Condon picture of incoherent neutron scattering.

A spectroscopic interpretation of incoherent neutron scattering experiments is presented which is based on Franck-Condon-type probabilities for scattering-induced transitions between quantum states of the target. The resulting expressions for the scattering functions enable an energy landscape-oriented analysis of neutron scattering spectra as well as a physical interpretation of Van Hove's space-time correlation functions in the quantum regime that accounts for the scattering kinematics. They suggest moreover a combined analysis of quasielastic and elastic scattering that become inseparable for complex systems with slow power-law relaxation.

Memory effects in a random walk description of protein structure ensembles.

In this paper, we show that ensembles of well-structured and unstructured proteins can be distinguished by borrowing concepts from non-equilibrium statistical mechanics. For this purpose, we represent proteins by two different polymer models and interpret the resulting polymer configurations as random walks of a diffusing particle in space. The first model is the trace of the Cα-atoms along the protein main chain, and the second is their projections onto the protein axis. The resulting trajectories are subsequently analyzed using the theory of the generalized Langevin equation. Velocities are replaced by displacements relating consecutive points on the discrete protein axes and equilibrium ensemble averages by averages over appropriate protein structure ensembles. The resulting displacement autocorrelation functions resemble those of the velocity autocorrelation functions of simple liquids and display a minimum, which can be related to the lengths of secondary structure elements. This minimum is clearly more pronounced for well-structured proteins than for unstructured ones, and the corresponding memory function displays a slower decay, indicating a stronger "folding memory."

Frequency domain modeling of quasielastic neutron scattering from hydrated protein powders: Application to free and inhibited human acetylcholinesterase.

This article reports on a frequency domain analysis of quasielastic neutron scattering spectra from free and Huperzine-A-inhibited human acetylcholinesterase, extending a recent time domain analysis of the same experimental data [M. Saouessi et al., J. Chem. Phys. 150, 161104 (2019)]. An important technical point here is the construction of a semianalytical model for the resolution-broadened dynamic structure factor that can be fitted to the experimental spectra. We find comparable parameters as in our previous study and demonstrate that our model is sensitive to subpercent changes in the experimental data, which are caused by reversible binding of the inhibitor Huperzine A.

Asymptotic analysis of quasielastic neutron scattering data from human acetylcholinesterase reveals subtle dynamical changes upon ligand binding.

In this paper, we show that subtle changes in the internal dynamics of human acetylcholinesterase upon ligand binding can be extracted from quasielastic neutron scattering data by employing a nonexponential relaxation model for the intermediate scattering function. The relaxation is here described by a stretched Mittag-Leffler function, which exhibits slow power law decay for long times. Our analysis reveals that binding of a Huperzine A ligand increases the atomic motional amplitudes of the enzyme and slightly slows down its internal diffusive motions. This result is interpreted within an energy landscape picture for the motion of the hydrogen atoms.

Asymptotic neutron scattering laws for anomalously diffusing quantum particles.

The paper deals with a model-free approach to the analysis of quasielastic neutron scattering intensities from anomalously diffusing quantum particles. All quantities are inferred from the asymptotic form of their time-dependent mean square displacements which grow ∝t(α), with 0 ≤ α < 2. Confined diffusion (α = 0) is here explicitly included. We discuss in particular the intermediate scattering function for long times and the Fourier spectrum of the velocity autocorrelation function for small frequencies. Quantum effects enter in both cases through the general symmetry properties of quantum time correlation functions. It is shown that the fractional diffusion constant can be expressed by a Green-Kubo type relation involving the real part of the velocity autocorrelation function. The theory is exact in the diffusive regime and at moderate momentum transfers.

Communication: A multiscale Bayesian inference approach to analyzing subdiffusion in particle trajectories.

Anomalous diffusion is characterized by its asymptotic behavior for t → ∞. This makes it difficult to detect and describe in particle trajectories from experiments or computer simulations, which are necessarily of finite length. We propose a new approach using Bayesian inference applied directly to the observed trajectories sampled at different time scales. We illustrate the performance of this approach using random trajectories with known statistical properties and then use it for analyzing the motion of lipid molecules in the plane of a lipid bilayer.

Protein secondary-structure description with a coarse-grained model.

A coarse-grained geometrical model for protein secondary-structure description and analysis is presented which uses only the positions of the C(α) atoms. A space curve connecting these positions by piecewise polynomial interpolation is constructed and the folding of the protein backbone is described by a succession of screw motions linking the Frenet frames at consecutive C(α) positions. Using the ASTRAL subset of the SCOPe database of protein structures, thresholds are derived for the screw parameters of secondary-structure elements and demonstrate that the latter can be reliably assigned on the basis of a C(α) model. For this purpose, a comparative study with the widely used DSSP (Define Secondary Structure of Proteins) algorithm was performed and it was shown that the parameter distribution corresponding to the ensemble of all pure C(α) structures in the RCSB Protein Data Bank matches that of the ASTRAL database. It is expected that this approach will be useful in the development of structure-refinement techniques for low-resolution data.

Communication: Probing anomalous diffusion in frequency space.

Anomalous diffusion processes are usually detected by analyzing the time-dependent mean square displacement of the diffusing particles. The latter evolves asymptotically as W(t) ∼ 2Dαt(α), where Dα is the fractional diffusion constant and 0 < α < 2. In this article we show that both Dα and α can also be extracted from the low-frequency Fourier spectrum of the corresponding velocity autocorrelation function. This offers a simple method for the interpretation of quasielastic neutron scattering spectra from complex (bio)molecular systems, in which subdiffusive transport is frequently encountered. The approach is illustrated and validated by analyzing molecular dynamics simulations of molecular diffusion in a lipid POPC bilayer.

Communication: A scaling approach to anomalous diffusion.

The paper presents a rigorous derivation of the velocity autocorrelation function for an anomalously diffusing slow solute particle in a bath of fast solvent molecules. The result is obtained within the framework of the generalized Langevin equation and uses only scaling arguments and identities which are based on asymptotic analysis. It agrees with the velocity autocorrelation function of an anomalously diffusing Rayleigh particle whose dynamics is described by a fractional Ornstein-Uhlenbeck process in velocity space. A simple semi-analytical example illustrates under which conditions the latter model is appropriate.

Motional heterogeneity in human acetylcholinesterase revealed by a non-Gaussian model for elastic incoherent neutron scattering.

We study the dynamical transition of human acetylcholinesterase by analyzing elastic neutron scattering data with a simulation gauged analytical model that goes beyond the standard Gaussian approximation for the elastic incoherent structure factor [G. R. Kneller and K. Hinsen, J. Chem. Phys. 131, 045104 (2009)]. The model exploits the whole available momentum transfer range in the experimental data and yields not only a neutron-weighted average of the atomic mean square position fluctuations, but also an estimation for their distribution. Applied to the neutron scattering data from human acetylcholinesterase, it reveals a strong increase of the motional heterogeneity at the two transition temperatures T = 150 K and T = 220 K, respectively, which can be located with less ambiguity than with the Gaussian model. We find that the first transition is essentially characterized by a change in the form of the elastic scattering profile and the second by a homogeneous increase of all motional amplitudes. These results are in agreement with previous combined experimental and simulation studies of protein dynamics, which attribute the first transition to an onset of methyl rotations and the second to more unspecific diffusion processes involving large amplitude motions.

Model-free simulation approach to molecular diffusion tensors.

In the present work, we propose a simple model-free approach for the computation of molecular diffusion tensors from molecular dynamics trajectories. The method uses a rigid body trajectory of the molecule under consideration, which is constructed a posteriori by an accumulation of quaternion-based superposition fits of consecutive conformations. From the rigid body trajectory, we compute the translational and angular velocities of the molecule and by integration of the latter also the corresponding angular trajectory. All quantities can be referred to the laboratory frame and a molecule-fixed frame. The 6 × 6 diffusion tensor is computed from the asymptotic slope of the tensorial mean square displacement and, for comparison, also from the Kubo integral of the velocity correlation tensor. The method is illustrated for two simple model systems - a water molecule and a lysozyme molecule in bulk water. We give estimations of the statistical accuracy of the calculations.

A comparison of reduced coordinate sets for describing protein structure.

In all-atom molecular simulation studies of proteins, each atom in the protein is represented by a point mass and interactions are defined in terms of the atomic positions. In recent years, various simplified approaches have been proposed. These approaches aim to improve computational efficiency and to provide a better physical insight. The simplified models can differ widely in their description of the geometry and the interactions inside the protein. This study explores the most fundamental choice in the simplified protein models: the choice of a coordinate set defining the protein structure. A simplified model can use fewer point masses than the all-atom model and/or eliminate some of the internal coordinates of the molecule by setting them to an average or ideal value. We look at the implications of such choices for the overall protein structure. We find that care must be taken for angular coordinates, where even very small variations can lead to significant changes in the positions of far away atoms. In particular, we show that the φ/ψ torsion angles are not a sufficient coordinate set, whereas another coordinate set with two degrees of freedom per residue, virtual Cα backbone bond, and torsion angles performs satisfactorily.

ScrewFit: combining localization and description of protein secondary structure.

A new application of the ScrewFit algorithm [Kneller & Calligari (2006), Acta Cryst. D62, 302-311] is presented which adds the detection of protein secondary-structure elements to their detailed geometrical description in terms of a curve with intrinsic torsion. The extension is based on confidence and persistence criteria for the ScrewFit parameters which are established by analyzing the structural fluctuations of standard motifs in the SCOP fold classes. The agreement with the widely used DSSP method is comparable with the general consensus among other methods in the literature. This combination of secondary-structure detection and analysis is illustrated for the enzyme adenylate kinase.

nMoldyn 3: using task farming for a parallel spectroscopy-oriented analysis of molecular dynamics simulations.

We present a new version of the program package nMoldyn, which has been originally developed for a neutron-scattering oriented analysis of molecular dynamics simulations of macromolecular systems (Kneller et al., Comput. Phys. Commun. 1995, 91, 191) and was later rewritten to include in-depth time series analyses and a graphical user interface (Rog et al., J. Comput. Chem. 2003, 24, 657). The main improvement in this new version and the focus of this article are the parallelization of all the analysis algorithms for use on multicore desktop computers as well as distributed-memory computing clusters. The parallelization is based on a task farming approach which maintains a simple program structure permitting easy modification and extension of the code to integrate new analysis methods.

Impact of anisotropic atomic motions in proteins on powder-averaged incoherent neutron scattering intensities.

This paper addresses the question to which extent anisotropic atomic motions in proteins impact angular-averaged incoherent neutron scattering intensities, which are typically recorded for powder samples. For this purpose, the relevant correlation functions are represented as multipole series in which each term corresponds to a different degree of intrinsic motional anisotropy. The approach is illustrated by a simple analytical model and by a simulation-based example for lysozyme, considering in both cases the elastic incoherent structure factor. The second example shows that the motional anisotropy of the protein atoms is considerable and contributes significantly to the scattering intensity.

Communication: a minimal model for the diffusion-relaxation backbone dynamics of proteins.

We present a model for the local diffusion-relaxation dynamics of the C(α)-atoms in proteins describing both the diffusive short-time dynamics and the asymptotic long-time relaxation of the position autocorrelation functions. The relaxation rate spectra of the latter are represented by shifted gamma distributions, where the standard gamma distribution describes anomalous slow relaxation in macromolecular systems of infinite size and the shift accounts for a smallest local relaxation rate in macromolecules of finite size. The resulting autocorrelation functions are analytic for any time t ≥ 0. Using results from a molecular dynamics simulation of lysozyme, we demonstrate that the model fits the position autocorrelation functions of the C(α)-atoms exceptionally well and reveals moreover a strong correlation between the residue's solvent-accessible surface and the fitted model parameters.