Accurate and efficient description of protein vibrational dynamics: comparing molecular dynamics and Gaussian models

Current all-atom potential based molecular dynamics (MD) allows the identification of a protein's functional motions on a wide-range of timescales, up to few tens of nanoseconds. However, functional, large-scale motions of proteins may occur on a timescale currently not accessible by all-atom potential based MD. To avoid the massive computational effort required by this approach, several simplified schemes have been introduced. One of the most satisfactory is the Gaussian network approach based on the energy expansion in terms of the deviation of the protein backbone from its native configuration. Here, we consider an extension of this model that captures in a more realistic way the distribution of native interactions due to the introduction of effective side-chain centroids. Since their location is entirely determined by the protein backbone, the model is amenable to the same exact and computationally efficient treatment as previous simpler models. The ability of the model to describe the correlated motion of protein residues in thermodynamic equilibrium is established through a series of successful comparisons with an extensive (14 ns) MD simulation based on the AMBER potential of HIV-1 protease in complex with a peptide substrate. Thus, the model presented here emerges as a powerful tool to provide preliminary, fast yet accurate characterizations of protein near-native motion.     

A Physical Picture of Protein Dynamics and Conformational Changes

A physical model is reviewed which explains different aspects of protein dynamics consistently. At low temperatures, the molecules are frozen in conformational substates. Their average energy is 3/2RT. Solid-state vibrations occur on a time scale of femtoseconds to nanoseconds. Above a characteristic temperature, often called the dynamical transition temperature, slow modes of motions can be observed occurring on a time scale between about 140 and 1 ns. These motions are overdamped, quasidiffusive, and involve collective motions of segments of the size of an α-helix. Molecules performing these types of motion are in the “flexible state”. This state is reached by thermal activation. It is shown that these motions are essential for conformational relaxation. Based on this picture, a new approach is proposed to understand conformational changes. It connects structural fluctuations and conformational transitions.     

Relaxation kinetics and the glassiness of native proteins: coupling of timescales

We provide evidence that the onset of functional dynamics of folded proteins with elevated temperatures is associated with the effective sampling of its energy landscape under physiological conditions. The analysis is based on data describing the relaxation phenomena governing the backbone dynamics of bovine pancreatic trypsin inhibitor derived from molecular dynamics simulations, previously reported by us. By representing the backbone dynamics of the folded protein by three distinct regimes, it is possible to decompose its seemingly complex dynamics, described by a stretch exponential decay of the backbone motions. Of these three regimes, one is associated with the slow timescales due to the activity along the envelope of the energy surface defining the folded protein. Another, with fast timescales, is due to the activity along the pockets decorating the folded-state envelope. The intermediate regime emerges at temperatures where jumps between the pockets become possible. It is at the temperature window where motions corresponding to all three timescales become operative that the protein becomes active.     

Conformational dynamics of cytochrome c: correlation to hydrogen exchange.

We study the dynamical fluctuations of horse heart cytochrome c by molecular dynamics (MD) simulations in aqueous solution, at four temperatures: 300 K, 360 K, 430 K, and 550 K. Each simulation covers a production time of at least 1.5 nanoseconds (ns). The conformational dynamics of the system is analyzed in terms of collective motions that involve the whole protein, and local motions that involve the formation and breaking of intramolecular hydrogen bonds. The character of the MD trajectories can be described within the framework of rugged energy landscape dynamics. The MD trajectories sample multiple conformational minima, with basins in protein conformational space being sampled for a few hundred picoseconds. The trajectories of the system in configurational space can be described in terms of diffusion of a particle in real space with a waiting time distribution due to partial trapping in shallow minima. As a consequence of the hierarchical nature of the dynamics, the mean square displacement autocorrelation function, <|x(t) - x(0)|2>, exhibits a power law dependence on time, with an exponent of around 0.5 for times shorter than 100 ps, and an exponent of 1.75 for longer times. This power law behavior indicates that the system exhibits suppressed diffusion (sub-diffusion) in sampling of configurational space at time scales shorter than 100 ps, and enhanced (super-diffusion) at longer time scales. The multi-basin feature of the trajectories is present at all temperatures simulated. Structural changes associated with inter-basin displacements correspond to collective motions of the Omega loops and coiled regions and relative motions of the alpha-helices as rigid bodies. Similar motions may be involved in experimentally observed amide hydrogen exchange. However, some groups showing large correlated motions do not expose the amino hydrogens to the solvent. We show that large fluctuations are not necessarily correlated to hydrogen exchange. For example, regions of the proteins forming alpha helices and turns show significant fluctuations, but as rigid bodies, and the hydrogen bonds involved in the formation of these structures do not break in proportion to these fluctuations. Proteins 1999;36:175-191. Published 1999 Wiley-Liss, Inc.     

Rigidity of protein structure revealed by incoherent neutron scattering

The rigidity and flexibility of a protein is reflected in its structural dynamics. Studies on protein dynamics often focus on flexibility and softness; this review focuses on protein structural rigidity. The extent of rigidity can be assessed experimentally with incoherent neutron scattering; a method that is complementary to molecular dynamics simulation. This experimental technique can provide information about protein dynamics in timescales of pico- to nanoseconds and at spatial scales of nanometers; these dynamics can help quantify the rigidity of a protein by indices such as force constant, Boson peak, dynamical transition, and dynamical heterogeneity. These indicators also reflect the rigidity of a protein's secondary and tertiary structures. In addition, the indices reveal how rigidity is influenced by different environmental parameters, such as hydration, temperature, pressure, and protein-protein interactions. Hydration affects both rigidity and softness more than other environmental factors. Interestingly, hydration affects harmonic and anharmonic motions in opposite ways. This difference is probably due to the protein's dynamic coupling with water molecules via hydrogen bonding.     

Neutron frequency windows and the protein dynamical transition

Proteins undergo an apparent dynamical transition on temperature variation that has been correlated with the onset of function. The transition in the mean-square displacement, , that is observed using a spectrometer or computer simulation, depends on the relationship between the timescales of the relaxation processes activated and the timescale accessible to the instrument or simulation. Models are described of two extreme situations---an "equilibrium" model, in which the long-time dynamics changes with temperature and all motions are resolved by the instrument used; and a "frequency window" model, in which there is no change in the long-time dynamics but as the temperature increases, the relaxation frequencies move into the instrumental range. Here we demonstrate that the latter, frequency-window model can describe the temperature and timescale dependences of both the intermediate neutron scattering function and derived from molecular dynamics simulations of a small protein in a cryosolution. The frequency-window model also describes the energy-resolution and temperature-dependences of obtained from experimental neutron scattering on glutamate dehydrogenase in the same solvent. Although equilibrium effects should also contribute to dynamical transitions in proteins, the present results suggests that frequency-window effects can play a role in the simulations and experiments examined. Finally, misquotations of previous findings are discussed in the context of solvent activation of protein dynamics and the possible relationship of this to activity.     

How does it really move? Recent progress in the investigation of protein nanosecond dynamics by NMR and simulation

Nuclear magnetic resonance (NMR) spin relaxation experiments currently probe molecular motions on timescales from picoseconds to nanoseconds. The detailed interpretation of these motions in atomic detail benefits from complementarity with the results from molecular dynamics (MD) simulations. In this mini-review, we describe the recent developments in experimental techniques to study the backbone dynamics from ¹⁵N relaxation and side-chain dynamics from ¹³C relaxation, discuss the different analysis approaches from model-free to dynamics detectors, and highlight the many ways that NMR relaxation experiments and MD simulations can be used together to improve the interpretation and gain insights into protein dynamics.     

Biophysical study of thermal denaturation of apo-calmodulin: dynamics of native and unfolded states

Apo-calmodulin, a small, mainly α, soluble protein is a calcium-dependent protein activator. This article presents a study of internal dynamics of native and thermal unfolded apo-calmodulin, using quasi-elastic neutron scattering. This technique can probe protein internal dynamics in the picosecond timescale and in the nanometer length-scale. It appears that a dynamical transition is associated with thermal denaturation of apo-calmodulin. This dynamical transition goes together with a decrease of the confinement of hydrogen atoms, a decrease of immobile protons proportion and an increase of dynamical heterogeneity. The comparison of native and unfolded states dynamics suggests that the dynamics of protein atoms is more influenced by their distance to the backbone than by their solvent exposure.     

Local segmental dynamics of cis-1,4-polybutadiene,polypropylene and polyethylene terephthalate via molecular dynamics simulations

The local segmental dynamics of cis-1,4-polybutadiene, polypropylene and polyethylene terephthalate have been investigated via isothermal-isobaric molecular dynamics simulations. The simulation pressure was 1 atm for all systems, with all simulation temperatures being at least 150 K above the polymer's glass transition temperature. The trajectories have been analysed via autocorrelation functions (ACFs) of chord vectors spanning different number of chain backbone bonds. Inverse Laplace transformations of these ACFs using the CONTIN algorithm afforded the corresponding distribution of relaxation times (DRTs) for the simulated dynamics. All DRTs illustrated a peak on fast timescales corresponding to short length scale segmental motion and a peak at longer timescales corresponding to longer length scale relaxations. A third peak, intermediate between the fast and slow processes, appears as the relaxation of chord vectors spanning increasing number of backbone bonds is considered. The temperature dependence of the relaxation dynamics is also investigated.     

Long-timescale simulation methods

The outstanding challenges in computer simulations of biological macromolecules are related to their complexity. Part of the complexity of biological systems concerns their physical size. Enumerating atoms ranging from a few in small signal molecules to the millions of particles in biological complexes is an obvious example of biological hierarchy. Another aspect is the extremely broad range of timescales of life science processes (many orders of magnitude); this adds another dimension of complexity. This extended range of timescales may even be observed for a single biomolecular process. Consider, for example, the R to T transition in hemoglobin. The complete conformational change occurs in tens of microseconds. However, the system has more than one timescale. Considerable activity occurs on a range of timescales before the final event (heme relaxation, picoseconds; tertiary relaxation, nanoseconds; ligand escape from the protein matrix and rebinding, hundreds of nanoseconds and so on). Whereas the basic time-step of atomically detailed simulations is about a femtosecond, it is not difficult to find molecular processes in biology that span more than ten orders of magnitude of relevant times, making the straightforward simulation of these events very difficult. Several techniques have been developed in recent years to address these problems.     

Analysis of Sub-τc and Supra-τc Motions in Protein Gβ1 Using Molecular Dynamics Simulations

The functions of proteins depend on the dynamical behavior of their native states on a wide range of timescales. To investigate these dynamics in the case of the small protein Gβ1, we analyzed molecular dynamics simulations with the model-free approach of nuclear magnetic relaxation. We found amplitudes of fast timescale motions (sub-τ_c, where τ_c is the rotational correlation time) consistent with S² obtained from spin relaxation measurements as well as amplitudes of slow timescale motions (supra-τ_c) in quantitative agreement with S² order parameters derived from residual dipolar coupling measurements. The slow timescale motions are associated with the large variations of the ³J couplings that follow transitions between different conformational substates. These results provide further characterization of the large structural fluctuations in the native states of proteins that occur on timescales longer than the rotational correlation time.     

Radially softening diffusive motions in a globular protein

Molecular dynamics simulation, quasielastic neutron scattering and analytical theory are combined to characterize diffusive motions in a hydrated protein, C-phycocyanin. The simulation-derived scattering function is in approximate agreement with experiment and is decomposed to determine the essential contributions. It is found that the geometry of the atomic motions can be modeled as diffusion in spheres with a distribution of radii. The time dependence of the dynamics follows stretched exponential behavior, reflecting a distribution of relaxation times. The average side chain and backbone dynamics are quantified and compared. The dynamical parameters are shown to present a smooth variation with distance from the core of the protein. Moving outward from the center of the protein there is a progressive increase of the mean sphere size, accompanied by a narrowing and shifting to shorter times of the relaxation time distribution. This smooth, "radially softening" dynamics may have important consequences for protein function. It also raises the possibility that the dynamical or "glass" transition with temperature observed experimentally in proteins might be depth dependent, involving, as the temperature decreases, progressive freezing out of the anharmonic dynamics with increasing distance from the center of the protein.     

Increasing temperature accelerates protein unfolding without changing the pathway of unfolding

We have traditionally relied on extremely elevated temperatures (498K, 225 degrees C) to investigate the unfolding process of proteins within the timescale available to molecular dynamics simulations with explicit solvent. However, recent advances in computer hardware have allowed us to extend our thermal denaturation studies to much lower temperatures. Here we describe the results of simulations of chymotrypsin inhibitor 2 at seven temperatures, ranging from 298K to 498K. The simulation lengths vary from 94ns to 20ns, for a total simulation time of 344ns, or 0.34 micros. At 298K, the protein is very stable over the full 50ns simulation. At 348K, corresponding to the experimentally observed melting temperature of CI2, the protein unfolds over the first 25ns, explores partially unfolded conformations for 20ns, and then refolds over the last 35ns. Above its melting temperature, complete thermal denaturation occurs in an activated process. Early unfolding is characterized by sliding or breathing motions in the protein core, leading to an unfolding transition state with a weakened core and some loss of secondary structure. After the unfolding transition, the core contacts are rapidly lost as the protein passes on to the fully denatured ensemble. While the overall character and order of events in the unfolding process are well conserved across temperatures, there are substantial differences in the timescales over which these events take place. We conclude that 498K simulations are suitable for elucidating the details of protein unfolding at a minimum of computational expense.     

Studying backbone torsional dynamics of intrinsically disordered proteins using fluorescence depolarization kinetics

Intrinsically disordered proteins (IDPs) do not autonomously adopt a stable unique 3D structure and exist as an ensemble of rapidly interconverting structures. They are characterized by significant conformational plasticity and are associated with several biological functions and dysfunctions. The rapid conformational fluctuation is governed by the backbone segmental dynamics arising due to the dihedral angle fluctuation on the Ramachandran ?–ψ conformational space. We discovered that the intrinsic backbone torsional mobility can be monitored by a sensitive fluorescence readout, namely fluorescence depolarization kinetics, of tryptophan in an archetypal IDP such as α-synuclein. This methodology allows us to map the site-specific torsional mobility in the dihedral space within picosecond-nanosecond time range at a low protein concentration under the native condition. The characteristic timescale of ~?1.4 ns, independent of residue position, represents collective torsional dynamics of dihedral angles (? and ψ) of several residues from tryptophan and is independent of overall global tumbling of the protein. We believe that fluorescence depolarization kinetics methodology will find broad application to study both short-range and long-range correlated motions, internal friction, binding-induced folding, disorder-to-order transition, misfolding and aggregation of IDPs.     

Molecular dynamics decomposition of temperature-dependent elastic neutron scattering by a protein solution

Molecular dynamics simulations are performed of bovine pancreatic trypsin inhibitor in a cryosolution over a range of temperatures from 80 to 300 K and the origins identified of elastic dynamic neutron scattering from the solution. The elastic scattering and mean-square displacement calculated from the molecular dynamics trajectories are in reasonable agreement with experiments on a larger protein in the same solvent. The solvent and protein contributions to the scattering from the simulation model are determined. At lower temperatures (< approximately 200 K) or on shorter timescales ( approximately 10 ps) the scattering contributions are proportional to the isotopic nuclear scattering cross-sections of each component. However, for T > 200 K marked deviations from these cross-sections are seen due to differences in the dynamics of the components of the solution. Rapid activation of solvent diffusion leads to the variation with temperature of the total elastic intensity being determined largely by that of the solvent. At higher temperatures (>240 K) and longer times ( approximately 100 ps) the protein makes the only significant contribution to the scattering, the solvent scattering having moved out of the accessible time-space window. Decomposition of the protein mean-square displacement shows that the observed dynamical transition in the solution at 200-220 K involves activation of both internal motions and external whole-molecule rotational and translational diffusion. The proportion that the external dynamics contributes to the protein mean-square displacement increases to approximately 30 and 60% at 300 K on the 10- and 100-ps timescales, respectively.     

An analysis of the influence of protein intrinsic dynamical properties on its thermal unfolding behavior

The influence of the protein topology-encoded dynamical properties on its thermal unfolding motions was studied in the present work. The intrinsic dynamics of protein topology was obtained by the anisotropic network model (ANM). The ANM has been largely used to investigate protein collective functional motions, but it is not well elucidated if this model can also reveal the preferred large-scale motions during protein unfolding. A small protein barnase is used as a typical case study to explore the relationship between protein topology-encoded dynamics and its unfolding motions. Three thermal unfolding simulations at 500 K were performed for barnase and the entire unfolding trajectories were sampled and partitioned into several windows. For each window, the preferred unfolding motions were investigated by essential dynamics analysis, and then associated with the intrinsic dynamical properties of the starting conformation in this window, which is detected by ANM. The results show that only a few slow normal modes imposed by protein structure are sufficient to give a significant overlap with the preferred unfolding motions. Especially, the large amplitude unfolding movements, which imply that the protein jumps out of a local energy basin, can be well described by a single or several ANM slow modes. Besides the global motions, it is also found that the local residual fluctuations encoded in protein structure are highly correlated with those in the protein unfolding process. Furthermore, we also investigated the relationship between protein intrinsic flexibility and its unfolding events. The results show that the intrinsic flexible regions tend to unfold early. Several early unfolding events can be predicted by analysis of protein structural flexibility. These results imply that protein structure-encoded dynamical properties have significant influences on protein unfolding motions.     

Protein Collective Motions Coupled to Ligand Migration in Myoglobin

Ligand migration processes inside myoglobin and protein dynamics coupled to the migration were theoretically investigated with molecular dynamics simulations. Based on a linear response theory, we identified protein motions coupled to the transient migration of ligand, carbon monoxide (CO), through channels. The result indicates that the coupled protein motions involve collective motions extended over the entire protein correlated with local gating motions at the channels. Protein motions, coupled to opening of a channel from the distal pocket to a neighboring xenon site, were found to share the collective motion with experimentally observed protein motions coupled to a doming motion of the heme Fe atom upon photodissociation of the ligand. Analysis based on generalized Langevin dynamics elucidated slow and diffusive features of the protein response motions. Remarkably small transmission coefficients for rates of the CO migrations through myoglobin were found, suggesting that the CO migration dynamics are characterized as motions governed by the protein dynamics involving the collective motions, rather than as thermally activated transitions across energy barriers of well-structured channels.     

Dynamics of Protein and its Hydration Water: Neutron Scattering Studies on Fully Deuterated GFP

We present a detailed analysis of the picosecond-to-nanosecond motions of green fluorescent protein (GFP) and its hydration water using neutron scattering spectroscopy and hydrogen/deuterium contrast. The analysis reveals that hydration water suppresses protein motions at lower temperatures (<∼200 K), and facilitates protein dynamics at high temperatures. Experimental data demonstrate that the hydration water is harmonic at temperatures <∼180–190 K and is not affected by the proteins’ methyl group rotations. The dynamics of the hydration water exhibits changes at ∼180–190 K that we ascribe to the glass transition in the hydrated protein. Our results confirm significant differences in the dynamics of protein and its hydration water at high temperatures: on the picosecond-to-nanosecond timescale, the hydration water exhibits diffusive dynamics, while the protein motions are localized to <∼3 Å. The diffusion of the GFP hydration water is similar to the behavior of hydration water previously observed for other proteins. Comparison with other globular proteins (e.g., lysozyme) reveals that on the timescale of 1 ns and at equivalent hydration level, GFP dynamics (mean-square displacements and quasielastic intensity) are of much smaller amplitude. Moreover, the suppression of the protein dynamics by the hydration water at low temperatures appears to be stronger in GFP than in other globular proteins. We ascribe this observation to the barrellike structure of GFP.