Quasielastic neutron scattering and relaxation processes in proteins: analytical and simulation-based models

Fractional Extended Diffusion Theory to capture anomalous relaxation from biased/accelerated molecular simulations

Biased and accelerated molecular simulations (BAMS) are widely used tools to observe relevant molecular phenomena occurring on time scales inaccessible to standard molecular dynamics, but evaluation of the physical time scales involved in the processes is not directly possible from them. For this reason, the problem of recovering dynamics from such kinds of simulations is the object of very active research due to the relevant theoretical and practical implications of dynamics on the properties of both natural and synthetic molecular systems. In a recent paper [A. Rapallo et al., J. Comput. Chem. 42, 586-599 (2021)], it has been shown how the coupling of BAMS (which destroys the dynamics but allows to calculate average properties) with Extended Diffusion Theory (EDT) (which requires input appropriate equilibrium averages calculated over the BAMS trajectories) allows to effectively use the Smoluchowski equation to calculate the orientational time correlation function of the head-tail unit vector defined over a peptide in water solution. Orientational relaxation of this vector is the result of the coupling of internal molecular motions with overall molecular rotation, and it was very well described by correlation functions expressed in terms of weighted sums of suitable time-exponentially decaying functions, in agreement with a Brownian diffusive regime. However, situations occur where exponentially decaying functions are no longer appropriate to capture the actual dynamical behavior, which exhibits persistent long time correlations, compatible with the so called subdiffusive regimes. In this paper, a generalization of EDT will be given, exploiting a fractional Smoluchowski equation (FEDT) to capture the non-exponential character observed in the relaxation of intramolecular distances and molecular radius of gyration, whose dynamics depend on internal molecular motions only. The calculation methods, proper to EDT, are adapted to implement the generalization of the theory, and the resulting algorithm confirms FEDT as a tool of practical value in recovering dynamics from BAMS, to be used in general situations, involving both regular and anomalous diffusion regimes.

Energy renormalization for temperature transferable coarse-graining of silicone polymer

The bottom-up prediction of thermodynamic and mechanical behaviors of polymeric materials based on molecular dynamics (MD) simulation is of critical importance in polymer physics. Although the atomistically informed coarse-grained (CG) model can access greater spatiotemporal scales and retain essential chemical specificity, the temperature-transferable CG model is still a big challenge and hinders widespread application of this technique. Herein, we use a silicone polymer, i.e., polydimethylsiloxane (PDMS), having an incredibly low chain rigidity as a model system, combined with an energy-renormalization (ER) approach, to systematically develop a temperature-transferable CG model. Specifically, by introducing temperature-dependent ER factors to renormalize the effective distance and cohesive energy parameters, the developed CG model faithfully preserved the dynamics, mechanical and conformational behaviors compared with the target all-atomistic (AA) model from glassy to melt regimes, which was further validated by experimental data. With the developed CG model featuring tremendously improved computational efficiency, we systematically explored the influences of cohesive interaction strength and temperature on the dynamical heterogeneity and mechanical response of polymers, where we observed consistent trends with other linear polymers with varying chain rigidity and monomeric structures. This study serves as an extension of our proposed ER approach of developing temperature transferable CG models with diverse segmental structures, highlighting the critical role of cohesive interaction strength on CG modeling of polymer dynamics and thermomechanical behaviors.

Free Energy Surface and Molecular Mechanism of Slow Structural Transitions in Lipid Bilayers

Lipid membrane remodeling, crucial for many cellular processes, is governed by the coupling of membrane structure and shape fluctuations. Given the importance of the ∼ nm length scale, details of the transition intermediates for conformational change are not fully captured by a continuum-mechanical description. Slow dynamics and the lack of knowledge of reaction coordinates (RCs) for biasing methods pose a challenge for all-atom (AA) simulations. Here, we map system dynamics on Langevin dynamics in a normal mode space determined from an elastic network model representation for the lipid-water Hamiltonian. AA molecular dynamics (MD) simulations are used to determine model parameters, and Langevin dynamics predictions for bilayer structural, mechanical, and dynamic properties are validated against MD simulations and experiments. Transferability to describe the dynamics of a larger lipid bilayer and a heterogeneous membrane-protein system is assessed. A set of generic RCs for pore formation in two tensionless bilayers is obtained by coupling Langevin dynamics to the underlying energy landscape for membrane deformations. Structure evolution is carried out by AA MD, wherein the generic RCs are used in a path metadynamics or an umbrella sampling simulation to determine the thermodynamics of pore formation and its molecular determinants, such as the role of distinct bilayer motions, lipid solvation, and lipid packing.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

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) = E_α(-(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 β-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.

Revisiting the modeling of quasielastic neutron scattering from bulk water

Quasi-elastic neutron scattering (QENS) from bulk-water at 300 K, measured on the IRIS backscattering neutron spectrometer (ISIS, UK), is interpreted using the jump diffusion model (JDM), a “minimalistic” multi-timescale relaxation model (MRM) and molecular dynamics simulations (MD). In the case of MRM data analysis is performed in the time domain, where the relaxation of the intermediate scattering function is described by a stretched Mittag-Leffler function, Eα(−(|t|/τ)α). This function displays an asymptotic power law decay and contains the exponential relaxation function as a special case (α = 1). To further compare the two approaches, MD simulations of bulk water were performed using the SPCE force field and the resulting MD trajectories analysed using the nMoldyn software. We show that both JDM and MRM accurately describe the diffusion of bulk water observed by QENS at all length scales, and confirm that MD simulations do not fully describe the quantum effects of jump diffusion.

Notes on Fitting and Analysis Frameworks for QENS Spectra of (Soft) Colloid Suspensions

With continuously improving signal-to-noise ratios, a statistically sound analysis of quasi-elastic neutron scattering (QENS) spectra requires to fit increasingly complex models which poses several challenges. Simultaneous fits of the spectra for all recorded values of the momentum transfer become a standard approach. Spectrometers at spallation sources can have a complicated non-Gaussian resolution function which has to be described most accurately. At the same time, to speed up the fitting, an analytical convolution with this resolution function is of interest. Here, we discuss basic concepts to efficient approaches for fits of QENS spectra based on standard MATLAB and Python fit algorithms. We illustrate the fits with example data from IN16B, BASIS, and BATS.

Origin of subdiffusions in proteins: Insight from peptide systems

Subdiffusive kinetics are popular in proteins and peptides as observed in experiments and simulations. For protein systems with diverse interactions, are there multiple mechanisms to produce the common subdiffusion behavior? To approach this problem, long trajectories of two model peptides are simulated to study the mechanism of subdiffusion and the relations with their interactions. The free-energy profiles and the subdiffusive kinetics are observed for these two peptides. A hierarchical plateau analysis is employed to extract the features of the landscape from the mean square of displacement. The mechanism of subdiffusions can be postulated by comparing the exponents by simulations with those based on various models. The results indicate that the mechanisms of these two peptides are different and are related to the characteristics of their energy landscapes. The subdiffusion of the flexible peptide is mainly caused by depth distribution of traps on the energy landscape, while the subdiffusion of the helical peptide is attributed to the fractal topology of local minima on the landscape. The emergence of these different mechanisms reflects different kinetic scenarios in peptide systems though the peptides behave in a similar way of diffusion. To confirm these ideas, the transition networks between various conformations of these peptides are generated. Based on the network description, the controlled kinetics based only on the topology of the networks are calculated and compared with the results based on simulations. For the flexible peptide, the feature of controlled diffusion is distinct from that of simulation, and for the helical peptide, two kinds of kinetics have a similar exponent of subdiffusion. These results further exemplify the importance of the landscape topology in the kinetics of structural proteins and the effect of depth distribution of traps for the subdiffusion of disordered peptides.

Solution Structure and Conformational Flexibility in the Active State of the Orange Carotenoid Protein. Part II: Quasielastic Neutron Scattering

Orange carotenoid proteins (OCPs), which are protecting cyanobacterial light-harvesting antennae from photodamage, undergo a pronounced structural change upon light absorption. In addition, the active state is anticipated to boost a significantly higher molecular flexibility similar to a "molten globule" state. Here, we used quasielastic neutron scattering to directly characterize the vibrational and conformational molecular dynamics of OCP in its ground and active states, respectively, on the picosecond time scale. At a temperature of 100 K, we observe mainly (vibronic) inelastic features with peak energies at 5 and 6 meV (40 and 48 cm^-1, respectively). At physiological temperatures, however, two (Lorentzian) quasielastic components represent localized protein motions, that is, stochastic structural fluctuations of protein side chains between various conformational substates of the protein. Global diffusion of OCP is not observed on the given time scale. The slower Lorentzian component is affected by illumination and can be well-characterized by a jump-diffusion model. While the jump diffusion constant D is (2.82 ± 0.01) × 10^-5 cm²/s at 300 K in the ground state, it is increased by ∼20% to (3.48 ± 0.01) × 10^-5 cm²/s in the active state, revealing a strong enhancement of molecular mobility. The increased mobility is also reflected in the average atomic mean square displacement ⟨u²⟩; we determine a ⟨u²⟩ of 1.47 ± 0.05 Å in the ground state, but 1.86 ± 0.05 Å in the active state (at 300 K). This effect is assigned to two factors: (i) the elongated structure of the active state with two widely separated protein domains is characterized by a larger number of surface residues with a concomitantly higher degree of motional freedom and (ii) a larger number of hydration water molecules bound at the surface of the protein. We thus conclude that the active state of the orange carotenoid protein displays an enhanced conformational dynamics. The higher degree of flexibility may provide additional channels for nonradiative decay so that harmful excess energy can be more efficiently converted to heat.

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.

Dynamics of proteins in solution

The dynamics of proteins in solution includes a variety of processes, such as backbone and side-chain fluctuations, interdomain motions, as well as global rotational and translational (i.e. center of mass) diffusion. Since protein dynamics is related to protein function and essential transport processes, a detailed mechanistic understanding and monitoring of protein dynamics in solution is highly desirable. The hierarchical character of protein dynamics requires experimental tools addressing a broad range of time- and length scales. We discuss how different techniques contribute to a comprehensive picture of protein dynamics, and focus in particular on results from neutron spectroscopy. We outline the underlying principles and review available instrumentation as well as related analysis frameworks.

Model-free description of polymer-coated gold nanoparticle dynamics in aqueous solutions obtained by Bayesian analysis of neutron spin echo data

We present a neutron spin echo study of the nanosecond dynamics of polyethylene glycol (PEG) functionalized nanosized gold particles dissolved in D_{2}O at two temperatures and two different PEG molecular weights (400D and 2000D). The analysis of the neutron spin echo data was performed by applying a Bayesian approach to the description of time correlation function decays in terms of exponential terms, recently proved to be theoretically rigorous. This approach, which addresses in a direct way the fundamental issue of model choice in any dynamical analysis, provides here a guide to the most statistically supported way to follow the decay of the intermediate scattering functions I(Q,t) by basing on statistical grounds the choice of the number of terms required for the description of the nanosecond dynamics of the studied systems. Then, the presented analysis avoids from the start resorting to a preselected framework and can be considered as model free. By comparing the results of PEG-coated nanoparticles with those obtained in PEG2000 solutions, we were able to disentangle the translational diffusion of the nanoparticles from the internal dynamics of the polymer grafted to them, and to show that the polymer corona relaxation follows a pure exponential decay in agreement with the behavior predicted by coarse grained molecular dynamics simulations and theoretical models. This methodology has one further advantage: in the presence of a complex dynamical scenario, I(Q,t) is often described in terms of the Kohlrausch-Williams-Watts function that can implicitly represent a distribution of relaxation times. By choosing to describe the I(Q,t) as a sum of exponential functions and with the support of the Bayesian approach, we can explicitly determine when a finer-structure analysis of the dynamical complexity of the system exists according to the available data without the risk of overparametrization. The approach presented here is an effective tool that can be used in general to provide an unbiased interpretation of neutron spin echo data or whenever spectroscopy techniques yield time relaxation data curves.

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.

Neutron scattering in the biological sciences: progress and prospects

The scattering of neutrons can be used to provide information on the structure and dynamics of biological systems on multiple length and time scales. Pursuant to a National Science Foundation-funded workshop in February 2018, recent developments in this field are reviewed here, as well as future prospects that can be expected given recent advances in sources, instrumentation and computational power and methods. Crystallography, solution scattering, dynamics, membranes, labeling and imaging are examined. For the extraction of maximum information, the incorporation of judicious specific deuterium labeling, the integration of several types of experiment, and interpretation using high-performance computer simulation models are often found to be particularly powerful.

Dynamic Neutron Scattering by Biological Systems

Dynamic neutron scattering directly probes motions in biological systems on femtosecond to microsecond timescales. When combined with molecular dynamics simulation and normal mode analysis, detailed descriptions of the forms and frequencies of motions can be derived. We examine vibrations in proteins, the temperature dependence of protein motions, and concepts describing the rich variety of motions detectable using neutrons in biological systems at physiological temperatures. New techniques for deriving information on collective motions using coherent scattering are also reviewed.

The effects of pressure on the energy landscape of proteins

Protein dynamics is characterized by fluctuations among different conformational substates, i.e. the different minima of their energy landscape. At temperatures above ~200 K, these fluctuations lead to a steep increase in the thermal dependence of all dynamical properties, phenomenon known as Protein Dynamical Transition. In spite of the intense studies, little is known about the effects of pressure on these processes, investigated mostly near room temperature. We studied by neutron scattering the dynamics of myoglobin in a wide temperature and pressure range. Our results show that high pressure reduces protein motions, but does not affect the onset temperature for the Protein Dynamical Transition, indicating that the energy differences and barriers among conformational substates do not change with pressure. Instead, high pressure values strongly reduce the average structural differences between the accessible conformational substates, thus increasing the roughness of the free energy landscape of the system.

Homogeneous and heterogeneous dynamics in native and denatured bovine serum albumin

Quasielastic incoherent neutron spectroscopy experiments reveal that chemical denaturation significantly modifies the internal dynamics of bovine serum albumin.

Thermodynamics of lipid multi-lamellar vesicles in presence of sterols at high hydrostatic pressure

We compared the effect of cholesterol at different concentration on the phase behaviour of DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) multilamellar vesicles. We used pressure perturbation differential scanning calorimetry (PPC) that studies a system on the whole by giving access to relevant thermodynamic quantities, and elastic incoherent neutron scattering (EINS) that probes local motions of a system at the atomic level by allowing extraction of dynamical parameters. PPC revealed that the volume expansion coefficient of DMPC and DMPC/Cholesterol samples with 13 and 25 mol% cholesterol is a linear function of the heat capacity measured by differential scanning calorimetry. Neutron backscattering spectroscopy showed that the mean square displacements of H atoms do exhibit an increase with temperature and a decrease under increasing pressure. Cholesterol added at concentrations of 25 and 50 mol% led to suppression of the main phase transition. Taking advantage of these results, the present study aims (i) to show that calorimetry and EINS using the Bicout and Zaccai model equally permit to get access to thermodynamic quantities characterizing pure DMPC and DMPC/cholesterol mixtures, thus directly confirming the theoretical method, and (ii) to validate our approach as function of temperature and of pressure, as both are equally important and complementary thermodynamic variables.

Investigating Structure and Dynamics of Proteins in Amorphous Phases Using Neutron Scattering

In order to increase shelf life and minimize aggregation during storage, many biotherapeutic drugs are formulated and stored as either frozen solutions or lyophilized powders. However, characterizing amorphous solids can be challenging with the commonly available set of biophysical measurements used for proteins in liquid solutions. Therefore, some questions remain regarding the structure of the active pharmaceutical ingredient during freezing and drying of the drug product and the molecular role of excipients. Neutron scattering is a powerful technique to study structure and dynamics of a variety of systems in both solid and liquid phases. Moreover, neutron scattering experiments can generally be correlated with theory and molecular simulations to analyze experimental data. In this article, we focus on the use of neutron techniques to address problems of biotechnological interest. We describe the use of small-angle neutron scattering to study the solution structure of biological molecules and the packing arrangement in amorphous phases, that is, frozen glasses and freeze-dried protein powders. In addition, we discuss the use of neutron spectroscopy to measure the dynamics of glassy systems at different time and length scales. Overall, we expect that the present article will guide and prompt the use of neutron scattering to provide unique insights on many of the outstanding questions in biotechnology.

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.

Fast antibody fragment motion: flexible linkers act as entropic spring

A flexible linker region between three fragments allows antibodies to adjust their binding sites to an antigen or receptor. Using Neutron Spin Echo Spectroscopy we observed fragment motion on a timescale of 7 ns with motional amplitudes of about 1 nm relative to each other. The mechanistic complexity of the linker region can be described by a spring model with Brownian motion of the fragments in a harmonic potential. Displacements, timescale, friction and force constant of the underlying dynamics are accessed. The force constant exhibits a similar strength to an entropic spring, with friction of the fragment matching the unbound state. The observed fast motions are fluctuations in pre-existing equilibrium configurations. The Brownian motion of domains in a harmonic potential is the appropriate model to examine functional hinge motions dependent on the structural topology and highlights the role of internal forces and friction to function.

Effects of pressure on the dynamics of an oligomeric protein from deep-sea hyperthermophile

Inorganic pyrophosphatase (IPPase) from Thermococcus thioreducens is a large oligomeric protein derived from a hyperthermophilic microorganism that is found near hydrothermal vents deep under the sea, where the pressure is up to 100 MPa (1 kbar). It has attracted great interest in biophysical research because of its high activity under extreme conditions in the seabed. In this study, we use the quasielastic neutron scattering (QENS) technique to investigate the effects of pressure on the conformational flexibility and relaxation dynamics of IPPase over a wide temperature range. The β-relaxation dynamics of proteins was studied in the time ranges from 2 to 25 ps, and from 100 ps to 2 ns, using two spectrometers. Our results indicate that, under a pressure of 100 MPa, close to that of the native environment deep under the sea, IPPase displays much faster relaxation dynamics than a mesophilic model protein, hen egg white lysozyme (HEWL), at all measured temperatures, opposite to what we observed previously under ambient pressure. This contradictory observation provides evidence that the protein energy landscape is distorted by high pressure, which is significantly different for hyperthermophilic (IPPase) and mesophilic (HEWL) proteins. We further derive from our observations a schematic denaturation phase diagram together with energy landscapes for the two very different proteins, which can be used as a general picture to understand the dynamical properties of thermophilic proteins under pressure.

Adaptation of Extremophilic Proteins with Temperature and Pressure: Evidence from Initiation Factor 6

In this work, we study dynamical properties of an extremophilic protein, Initiation Factor 6 (IF6), produced by the archeabacterium Methanocaldococcus jannascii, which thrives close to deep-sea hydrothermal vents where temperatures reach 80 °C and the pressure is up to 750 bar. Molecular dynamics simulations (MD) and quasi-elastic neutron scattering (QENS) measurements give new insights into the dynamical properties of this protein with respect to its eukaryotic and mesophilic homologue. Results obtained by MD are supported by QENS data and are interpreted within the framework of a fractional Brownian dynamics model for the characterization of protein relaxation dynamics. IF6 from M. jannaschii at high temperature and pressure shares similar flexibility with its eukaryotic homologue from S. cerevisieae under ambient conditions. This work shows for the first time, to our knowledge, that the very common pattern of corresponding states for thermophilic protein adaptation can be extended to thermo-barophilic proteins. A detailed analysis of dynamic properties and of local structural fluctuations reveals a complex pattern for "corresponding" structural flexibilities. In particular, in the case of IF6, the latter seems to be strongly related to the entropic contribution given by an additional, C-terminal, 20 amino-acid tail which is evolutionary conserved in all mesophilic IF6s.

Structure and dynamics of ionic micelles: MD simulation and neutron scattering study

Fully atomistic molecular dynamics (MD) simulations have been carried out on sodium dodecyl sulfate (SDS), an anionic micelle, and three cationic (CnTAB; n = 12, 14, 16) micelles, investigating the effects of size, the form of the headgroup, and chain length. They have been used to analyze neutron scattering data. MD simulations confirm the dynamical model of global motion of the whole micelle, segmental motion (headgroup and alkyl chain), and fast torsional motion associated with the surfactants that is used to analyze the experimental data. It is found that the solvent surrounding the headgroups results in their significant mobility, which exceeds that of the tails on the nanosecond time scale. The middle of the chain is found to be least mobile, consolidating the micellar configuration. This dynamical feature is similar for all the ionic micelles investigated and therefore independent of headgroup form and charge and chain length. Diffusion constants for global and segmental motion of the different micelles are consistent with experimentally obtained values as well as known structural features. This work provides a more realistic model of micelle dynamics and offers new insight into the strongly fluctuating surface of micelles which is important in understanding micelle dispersion and related functionality, like drug delivery.

Shake, rattle, and roll: Impact of the dynamics of flavivirus particles on their interactions with the host

Remarkable progress in structural biology has equipped virologists with insight into structures of viral proteins and virions at increasingly high resolution. Structural information has been used extensively to address fundamental questions about virtually all aspects of how viruses replicate in cells, interact with the host, and in the design of antiviral compounds. However, many critical aspects of virology exist outside the snapshots captured by traditional methods used to generate high-resolution structures. Like all proteins, viral proteins are not static structures. The conformational flexibility and dynamics of proteins play a significant role in protein-protein interactions, and in the structure and biology of virus particles. This review will discuss the implications of the dynamics of viral proteins on the biology, antigenicity, and immunogenicity of flaviviruses.

Correlation of the dynamics of native human acetylcholinesterase and its inhibited huperzine A counterpart from sub-picoseconds to nanoseconds

It is a long debated question whether catalytic activities of enzymes, which lie on the millisecond timescale, are possibly already reflected in variations in atomic thermal fluctuations on the pico- to nanosecond timescale. To shed light on this puzzle, the enzyme human acetylcholinesterase in its wild-type form and complexed with the inhibitor huperzine A were investigated by various neutron scattering techniques and molecular dynamics simulations. Previous results on elastic neutron scattering at various timescales and simulations suggest that dynamical processes are not affected on average by the presence of the ligand within the considered time ranges between 10 ps and 1 ns. In the work presented here, the focus was laid on quasi-elastic (QENS) and inelastic neutron scattering (INS). These techniques give access to different kinds of individual diffusive motions and to the density of states of collective motions at the sub-picoseconds timescale. Hence, they permit going beyond the first approach of looking at mean square displacements. For both samples, the autocorrelation function was well described by a stretched-exponential function indicating a linkage between the timescales of fast and slow functional relaxation dynamics. The findings of the QENS and INS investigation are discussed in relation to the results of our earlier elastic incoherent neutron scattering and molecular dynamics simulations.

On the reliability of NMR relaxation data analyses: a Markov Chain Monte Carlo approach

The analysis of NMR relaxation data is revisited along the lines of a Bayesian approach. Using a Markov Chain Monte Carlo strategy of data fitting, we investigate conditions under which relaxation data can be effectively interpreted in terms of internal dynamics. The limitations to the extraction of kinetic parameters that characterize internal dynamics are analyzed, and we show that extracting characteristic time scales shorter than a few tens of ps is very unlikely. However, using MCMC methods, reliable estimates of the marginal probability distributions and estimators (average, standard deviations, etc.) can still be obtained for subsets of the model parameters. Thus, unlike more conventional strategies of data analysis, the method avoids a model selection process. In addition, it indicates what information may be extracted from the data, but also what cannot.

High hydrostatic pressure effects investigated by neutron scattering on lipid multilamellar vesicles

The effects of high hydrostatic pressure on the structure and dynamics of model membrane systems were investigated using neutron scattering. Diffraction experiments show shifts of the pre- and main-phase transitions of multilamellar vesicles of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) to higher temperatures with increased pressure which are close to results observed previously by other techniques, namely (10.4 ± 1.0) K kbar(-1) and (20.0 ± 0.5) K kbar(-1) for the two transitions. Backscattering spectroscopy reveals that the mean square displacements in the liquid phase are about 10% smaller at 300 bar and about 20% smaller at 600 bar compared to atmospheric pressure, whereas in the gel phase below the main phase transition the mean square displacements show a smaller difference in the dynamics of the three pressure values within the studied pressure range.

Stretching of single poly-ubiquitin molecules revisited: dynamic disorder in the non-exponential unfolding kinetics

A theoretical framework based on a generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and a power-law memory kernel is presented to describe the non-exponential kinetics of the unfolding of a single poly-ubiquitin molecule under a constant force [T.-L. Kuo, S. Garcia-Manyes, J. Li, I. Barel, H. Lu, B. J. Berne, M. Urbakh, J. Klafter, and J. M. Fernández, Proc. Natl. Acad. Sci. U.S.A. 107, 11336 (2010)]. Such a GLE-fGn strategy is made on the basis that the pulling coordinate variable x undergoes subdiffusion, usually resulting from conformational fluctuations, over a one-dimensional force-modified free-energy surface U(x, F). By using the Kramers' rate theory, we have obtained analytical formulae for the time-dependent rate coefficient k(t, F), the survival probability S(t, F) as well as the waiting time distribution function f(t, F) as functions of time t and force F. We find that our results can fit the experimental data of f(t, F) perfectly in the whole time range with a power-law exponent γ = 1/2, the characteristic of typical anomalous subdiffusion. In addition, the fitting of the survival probabilities for different forces facilitates us to reach rather reasonable estimations for intrinsic properties of the system, such as the free-energy barrier and the distance between the native conformation and the transition state conformation along the reaction coordinate, which are in good agreements with molecular dynamics simulations in the literatures. Although static disorder has been implicated in the original work of Kuo et al., our work suggests a sound and plausible alternative interpretation for the non-exponential kinetics in the stretching of poly-ubiquitin molecules, associated with dynamic disorder.

Continuous time random walk with linear force applied to hydrated proteins

An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations. Analytical expressions for transition probability density, mean square displacement, and intermediate scattering function are presented. The mean square displacement and intermediate scattering function can fit well the simulation data of the temperature-dependent translational dynamics of nitrogen atoms of elastin for a wide range of temperatures and various scattering vectors. Moreover, the numerical results are also compared with those of a fractional diffusion equation.

Kinetics of molecular transitions with dynamic disorder in single-molecule pulling experiments

Macromolecular transitions are subject to large fluctuations of rate constant, termed as dynamic disorder. The individual or intrinsic transition rates and activation free energies can be extracted from single-molecule pulling experiments. Here we present a theoretical framework based on a generalized Langevin equation with fractional Gaussian noise and power-law memory kernel to study the kinetics of macromolecular transitions to address the effects of dynamic disorder on barrier-crossing kinetics under external pulling force. By using the Kramers' rate theory, we have calculated the fluctuating rate constant of molecular transition, as well as the experimentally accessible quantities such as the force-dependent mean lifetime, the rupture force distribution, and the speed-dependent mean rupture force. Particular attention is paid to the discrepancies between the kinetics with and without dynamic disorder. We demonstrate that these discrepancies show strong and nontrivial dependence on the external force or the pulling speed, as well as the barrier height of the potential of mean force. Our results suggest that dynamic disorder is an important factor that should be taken into account properly in accurate interpretations of single-molecule pulling experiments.

Intrinsic mean-square displacements in proteins

The thermal mean-square displacement (MSD) of hydrogen in proteins and its associated hydration water is measured by neutron scattering experiments and used an indicator of protein function. The observed MSD as currently determined depends on the energy resolution width of the neutron scattering instrument employed. We propose a method for obtaining the intrinsic MSD of H in the proteins, one that is independent of the instrument resolution width. The intrinsic MSD is defined as the infinite time value of (r(2)) that appears in the Debye-Waller factor. The method consists of fitting a model to the resolution broadened elastic incoherent structure factor or to the resolution dependent MSD. The model contains the intrinsic MSD, the instrument resolution width, and a rate constant characterizing the motions of H in the protein. The method is illustrated by obtaining the intrinsic MSD (r(2)) of heparan sulphate (HS-0.4), ribonuclease A, and staphysloccal nuclase (SNase) from data in the literature.

Temperature-dependent vibrational and conformational dynamics of photosystem II membrane fragments from spinach investigated by elastic and inelastic neutron scattering

Vibrational and conformational protein dynamics of photosystem II (PS II) membrane fragments from spinach were investigated by elastic and inelastic incoherent neutron scattering (EINS and IINS). As to the EINS experiments, the average atomic mean square displacement values of PS II membrane fragments hydrated at a relative humidity of 57% exhibit a dynamical transition at ~230K. In contrast, the dynamical transition was absent at a relative humidity of 44%. These findings are in agreement with previous studies which reported a "freezing" of protein mobility due to dehydration (Pieper et al. (2008) Eur. Biophys. J. 37: 657-663) and its correlation with an inhibition of electron transfer from Q(A)(-) to Q(B) (Kaminskaya et al. (2003) Biochemistry 42, 8119-8132). IINS spectra of a sample hydrated at a relative humidity of 57% show a distinct Boson peak at ~7.5meV at 20K, which shifts towards lower energy values upon temperature increase to 250K. This unexpected effect is interpreted in terms of a "softening" of the protein matrix along with the onset of conformational protein dynamics as revealed by the EINS experiments. Information on the density of vibrational states of pigment-protein complexes is important for a realistic calculation of excitation energy transfer kinetics and spectral lineshapes and is often routinely obtained by optical line-narrowing spectroscopy at liquid helium temperature. The data presented here demonstrate that IINS is a valuable experimental tool in determining the density of vibrational states not only at cryogenic, but also at nearly physiological temperatures up to 250K. This article is part of a Special Issue entitled: Photosynthesis Research for Sustainability: from Natural to Artificial.

Analytic treatment of nuclear spin-lattice relaxation for diffusion in a cone model

We consider nuclear spin-lattice relaxation rate resulted from a diffusion equation for rotational wobbling in a cone. We show that the widespread point of view that there are no analytical expressions for correlation functions for wobbling in a cone model is invalid and prove that nuclear spin-lattice relaxation in this model is exactly tractable and amenable to full analytical description. The mechanism of relaxation is assumed to be due to dipole-dipole interaction of nuclear spins and is treated within the framework of the standard Bloemberger, Purcell, Pound-Solomon scheme. We consider the general case of arbitrary orientation of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown to remain valid for systems with the distribution of the cone axes depending only on the tilt relative the magnetic field but otherwise being isotropic. We consider the case of random isotropic orientation of cone axes relative the magnetic field taking place in powders. Also we consider the cases of their predominant orientation along or opposite the magnetic field and that of their predominant orientation transverse to the magnetic field which may be relevant for, e.g., liquid crystals. Besides we treat in details the model case of the cone axis directed along the magnetic field. The latter provides direct comparison of the limiting case of our formulas with the textbook formulas for free isotropic rotational diffusion. The dependence of the spin-lattice relaxation rate on the cone half-width yields results similar to those predicted by the model-free approach.

From NMR relaxation to fractional Brownian dynamics in proteins: results from a virtual experiment

In a recent simulation study [J. Chem. Phys. 2010, 133, 145101], it has been shown that the time correlation functions probed by nuclear magnetic resonance (NMR) relaxation spectroscopy of proteins are well described by a fractional Brownian dynamics model, which accounts for the wide spectrum of relaxation rates characterizing their internal dynamics. Here, we perform numerical experiments to explore the possibility of using this model directly in the analysis of experimental NMR relaxation data. Starting from a molecular dynamics simulation of the 266 residue protein 6PGL in explicit water, we construct virtual (15)N R(1), R(2), and NOE relaxation rates at two different magnetic fields, including artificial noise, and test how far the parameters obtained from a fit of the model to the virtual experimental data coincide with those obtained from an analysis of the MD time correlation functions that have been used to construct these data. We show that in most cases, close agreement is found. Acceptance or rejection of parameter values obtained from relaxation rates are discussed on a physical basis, therefore avoiding overfitting.