Structural basis of hierarchical multiple substates of a protein. I: Introduction

A computer experiment of protein dynamics is carried out, which consists of two steps: (1) A Monte Carlo simulation of thermal fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor; and (2) a simulation of the quick freezing of fluctuating conformations into energy minima by minimization of the energy of a number of conformations sampled in the Monte Carlo simulation. From the analysis of results of the computer experiment is obtained the following picture of protein dynamics: multiple energy minima exist in the native state, and they are distributed in clusters in the conformational space. The dynamics has a hierarchical structure which has at least two levels. In the first level, dynamics is restricted within one of the clusters of minima. In the second, transitions occur among the clusters. Local parts of a protein molecule, side chains and local main chain segments, can take multiple locally stable conformations in the native state. Many minima result from combinations of these multiple local conformations. The hierarchical structure in the dynamics comes from interactions among the local parts. Protein molecules have two types of flexibility, each associated with elastic and plastic deformations, respectively.     

Single Pair Förster Resonance Energy Transfer: A Versatile Tool To Investigate Protein Conformational Dynamics

Conformational changes of proteins and other biomolecules play a fundamental role in their functional mechanism. Single pair Förster resonance energy transfer (spFRET) offers the possibility to detect these conformational changes and dynamics, and to characterize their underlying kinetics. Using spFRET on microscopes with different modes of detection, dynamic timescales ranging from nanoseconds to seconds can be quantified. Confocal microscopy can be used as a means to analyze dynamics in the range of nanoseconds to milliseconds, while total internal reflection fluorescence (TIRF) microscopy offers information about conformational changes on timescales of milliseconds to seconds. While the existence of dynamics can be directly inferred from the FRET efficiency time trace or the correlation of FRET efficiency and fluorescence lifetime, additional computational approaches are required to extract the kinetic rates of these dynamics, a short overview of which is given in this review.

     

Protein conformational landscapes: Energy minimization and clustering of a long molecular dynamics trajectory

Using energy minimization and cluster analysis, we have analyzed a 1020 ps molecular dynamics trajectory of solvated bovine pancreatic trypsin inhibitor. Elucidation of conformational sub states in this way both illustrates the degree of conformational convergence in the simulation and reduces the structural data to a tractable subset. The relative movement of structures upon energy minimization was used to estimate the sizes of features on the protein potential energy surface. The structures were analyzed using their pairwise root-mean-square C_α deviations, which gave a global measure of conformational changes that would not be apparent by monitoring single degrees of freedom. At time scales of 0.1 ps, energy minimization detected sharp transitions between energy minima separated by 0.1 Å rms deviation. Larger conformational clusters containing these smaller minima and separated by 0.25 Å were seen at 1 ps time scales. Both of these small features of the conformational landscape were characterized by movements in loop regions associated with small, correlated backbone dihedral angle shifts. On a nanosecond time scale, the main features of the protein energy landscape were clusters separated by over 0.7 Å rms deviation, with only seven of these sub states visited over the 1 ns trajectory. These substates, discernible both before and after energy minimization, differ mainly in a monotonic pivot of the loop residues 11–18 over the course of the simulation. This loop contains lysine 17, which specifically binds to trypsin in the active site. The trajectory did not return to previously visited clusters, indicating that this trajectory has not been shown to have completely sampled the conformational substates available to it. Because the apparent convergence to a single region of conformation space depends on both the time scale of observation and the size of the conformational features examined, convergence must be operationally defined within the context of the simulation. © 1995 Wiley-Liss, Inc.     

Molecular dynamics simulation of Lewis blood groups and related oligosaccharides.

Molecular dynamics simulations without explicit inclusion of solvent molecules have been performed to study the motions of Lewisa and Lewisb blood group oligosaccharides, and two blood group A tetrasaccharides having type I and type II core chains. The blood group H trisaccharide has also been studied and compared with the blood group A type II core chain. The potential energy surface developed by Rasmussen and co-workers was used with the molecular mechanics code CHARMM. The lowest energy minima of the component disaccharide fragments were obtained from conformational energy mapping. The lowest energy minima of these disaccharide fragments were used to build the tri- and tetrasaccharides that were further minimized before the actual heating/equilibration and dynamics simulations. The trajectories of the disaccharide fragments, e.g., Fuc alpha- (1----4)GlcNAc, Gal beta-(1----4)GlcNAc, etc., show transitions among various minima. However, the oligosaccharides were found to be dynamically stable and no transitions to other minimum energy conformations were observed in the time series of the glycosidic dihedral angles even during trajectories as long as 300 ps. The stable conformations of the glycosidic linkages in the oligosaccharides are not necessarily the same as the minimum energy conformation of the corresponding isolated disaccharides. The average fluctuations of the glycosidic angles in the oligosaccharides were well within the range of +/- 15 degrees. The results of these trajectory calculations were consistent with the relatively rigid single-conformation models derived for these oligosaccharides from 1H-nmr data.     

Discrimination of near-native structures in protein-protein docking by testing the stability of local minima

Fast Fourier transform (FFT) correlation methods of protein-protein docking, combined with the clustering of low energy conformations, can find a number of local minima on the energy surface. For most complexes, the locations of the near-native structures can be constrained to the 30 largest clusters, each surrounding a local minimum. However, no reliable further discrimination can be obtained by energy measures because the differences in the energy levels between the minima are comparable with the errors in the energy evaluation. In fact, no current scoring function accounts for the entropic contributions that relate to the width rather than the depth of the minima. Since structures at narrow minima loose more entropy, some of the nonnative states can be detected by determining whether or not a local minimum is surrounded by a broad region of attraction on the energy surface. The analysis is based on starting Monte Carlo Minimization (MCM) runs from random points around each minimum, and observing whether a certain fraction of trajectories converge to a small region within the cluster. The cluster is considered stable if such a strong attractor exists, has at least 10 convergent trajectories, is relatively close to the original cluster center, and contains a low energy structure. We studied the stability of clusters for enzyme-inhibitor and antibody-antigen complexes in the Protein Docking Benchmark. The analysis yields three main results. First, all clusters that are close to the native structure are stable. Second, restricting considerations to stable clusters eliminates around half of the false positives, that is, solutions that are low in energy but far from the native structure of the complex. Third, dividing the conformational space into clusters and determining the stability of each cluster, the combined approach is less dependent on a priori information than exploring the potential conformational space by Monte Carlo minimizations.     

Microfolding: conformational probability map for the alanine dipeptide in water from molecular dynamics simulations

A direct attack on the protein-folding problem has been initiated with the free energy perturbation methods of molecular dynamics. The complete conformational probability map for the alanine dipeptide is presented. This work uses the SPC model for the explicit hydration of the dipeptide. Free energy differences for the four observed minima (beta, alpha R, alpha L, C7ax) are given, and the free energy barriers between minima are outlined.     

Conformation spaces of proteins

We report a simple method for measuring the accessible conformational space explored by an ensemble of protein structures. The method is useful for diverse ensembles derived from molecular dynamics trajectories, molecular modeling, and molecular structure determinations. It can be used to examine a wide range of time scales. The central tactic we use, which has been previously employed, is to replace the true mechanical degrees of freedom of a molecular system with the conformationally effective degrees of freedom as measured by the root-mean squared cartesian distances among all pairs of conformations. Each protein conformation is treated as a point in a high dimensional euclidean space. In this article, we model this space in a novel way by representing it as an N-dimensional hypercube, describable with only two parameters: the number of dimensions and the edge length. To validate this approach, we provide a number of elementary test cases and then use the N-cube method for measuring the size and shape of conformational space covered by molecular dynamics trajectories spanning 10 orders of magnitude in time. These calculations were performed on a small protein, the villin headpiece subdomain, exploring both the native state and the misfolded/folding regime. Distinct features include single, vibrationally averaged, substate minima on the 0.1-1-ps time scale, thermally averaged conformational states that persist for 1-100 ps and transitions between these local minima on nanosecond time scales. Large-scale refolding modes appear to become uncorrelated on the microsecond time scale. Associated length scales for these events are 0.2 A for the vibrational minima; 0.5 A for the conformational minima; and 1-2 A for the nanosecond events. We find that the conformational space that is dynamically accessible during folding of villin has enough volume for approximately 10(9) minima of the variety that persist for picoseconds. Molecular dynamics trajectories of the native protein and experimentally derived solution ensembles suggest the native state to be composed of approximately 10(2) of these thermally accessible minima. Thus, based on random exploration of accessible folding space alone, protein folding for a small protein is predicted to be a milliseconds time scale event. This time can be compared with the experimental folding time for villin of 10-100 micros. One possible explanation for the 10-100-fold discrepancy is that the slope of the "folding funnel" increases the rate 1-2 orders of magnitude above random exploration of substates.     

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.     

beta-hairpin stability and folding: molecular dynamics studies of the first beta-hairpin of tendamistat

The stability and (un)folding of the 19-residue peptide, SCVTLYQSWRYSQADNGCA, corresponding to the first beta-hairpin (residues 10 to 28) of the alpha-amylase inhibitor tendamistat (PDB entry 3AIT) has been studied by molecular dynamics simulations in explicit water under periodic boundary conditions at several temperatures (300 K, 360 K and 400 K), starting from various conformations for simulation lengths, ranging from 10 to 30 ns. Comparison of trajectories of the reduced and oxidized native peptides reveals the importance of the disulphide bridge closing the beta-hairpin in maintaining a proper turn conformation, thereby insuring a proper side-chain arrangement of the conserved turn residues. This allows rationalization of the conservation of those cysteine residues among the family of alpha-amylase inhibitors. High temperature simulations starting from widely different initial configurations (native beta-hairpin, alpha and left-handed helical and extended conformations) begin sampling similar regions of the conformational space within tens of nanoseconds, and both native and non-native beta-hairpin conformations are recovered. Transitions between conformational clusters are accompanied by an increase in energy fluctuations, which is consistent with the increase in heat capacity measured experimentally upon protein folding. The folding events observed in the various simulations support a model for beta-hairpin formation in which the turn is formed first, followed by hydrogen bond formation closing the hairpin, and subsequent stabilization by side-chain hydrophobic interactions.     

Molecular dynamics simulation of oligosaccharides containing N-acetyl neuraminic acid

αD -N-acetyl neuraminic acid (Neu5Ac, sialic acid) is a commonly occurring carbohydrate residue in various cell surface glycolipids and glycoproteins. This residue is linked terminally or internally to Gal residues via an α(2 → 3) or α(2 → 6) linkage. In the cell surface receptor, sialyl-Lewis^X, a terminal α(2 → 3) linkage is present. Previous studies from our laboratory have shown that in solution Lewis^X adopts a relatively rigid structure. In order to model the Neu5Ac residue, vacuum molecular dynamics of this monosaccharide were compared with simulations that explicitly include solvent water. The dynamical average of the monosaccharide conformation obtained from the two simulations was similar. Vacuum calculations for the disaccharide Neu5Ac α(2 → 3) Gal β-O-methyl show that a number of low energy minima are accessible to this disaccharide. Molecular dynamics simulations starting from the low energy minima show conformational transitions with a time scale of 10–50 ps among several of the minima while large barriers between other minima prevent transitions on the time scale studied. Simulations of this disaccharide in the presence of solvent show fewer conformational transitions, illustrating a dampening effect of the solvent that has been observed in some other studies. Our results are most consistent with an equilibrium among multiple conformations for the Neu5Ac α(2 → 3) Gal β linkage. © 1994 John Wiley & Sons, Inc.     

Exploring the energy landscape of a beta hairpin in explicit solvent

We studied the energy landscape of the peptide Ace-GEWTYDDATKTFTVTE-Nme, taken from the C-terminal fragment (41-56) of protein G, in explicit aqueous solution by a highly parallel replica-exchange approach that combines molecular dynamics trajectories with a temperature exchange Monte Carlo process. The combined trajectories in T and configurational space allow a replica to overcome a free energy barrier present at one temperature by increasing T, changing configurations, and cooling in a self-regulated manner, thus allowing sampling of broad regions of configurational space in short (nanoseconds) time scales. The free energy landscape of this system over a wide range of temperatures shows that the system preferentially adopts a beta hairpin structure. However, the peptide also samples other stable ensembles where the peptide adopts helices and helix-turn-helix states, among others. The helical states become increasingly stable at low temperatures, but are slightly less stable than the beta turn ensemble. The energy landscape is rugged at low T, where substates are separated by large energy barriers. These barriers disappear at higher T (approximately 330 K), where the system preferentially adopts a "molten globule" state with structures similar to the beta hairpin.     

Structure and energy landscape of a photoswitchable peptide: a replica exchange molecular dynamics study

A replica exchange molecular dynamics (REMD) simulation of a bicyclic azobenzene peptide in explicit dimethyl sulfoxide solution is presented in order to characterize the conformational structures and energy landscape of a photoswitchable peptide. It is shown that an enhanced-sampling technique such as the REMD method is essential to obtain a converged conformational sampling of the peptide at room temperature. This is because conventional MD simulations of less than approximately 100-ns length are either trapped in local minima (at 295 K) or-if run at high temperature-do not resemble the room-temperature REMD results. Calculating various nuclear Overhauser effects (NOEs) and (3)J-couplings, a good overall agreement between the REMD simulations and the NMR experiments of Renner et al. (Biopolymers 2000;54:501-514) is found. In particular, the REMD study confirms the general picture drawn by Renner et al. that the trans-isomer of the azobenzene peptide exhibits a well-defined structure, while the cis-isomer is a conformational heterogeneous system; that is, the trans-isomer occurs in 2 well-defined conformers, while the cis-isomer represents an energetically frustrated system that leads to an ensemble of conformational structures. Employing a principal component analysis of the REMD data, the free energy landscape of the systems is studied at various temperatures. The implications for the folding and unfolding pathways of the system are discussed.     

Dynamics of water molecules buried in cavities of apolipoprotein E studied by molecular dynamics simulations and continuum electrostatic calculations

Molecular dynamics (MD) simulations of several nanoseconds each were used to monitor the dynamic behavior of the five crystal water molecules buried in the interior of the N-terminal domain of apolipoprotein E. These crystal water molecules are fairly well conserved in several apolipoprotein E structures, suggesting that they are not an artifact of the crystal and that they may have a structural and/or functional role for the protein. All five buried crystal water molecules leave the protein interior in the course of the longest simulations and exchange with water molecules from the bulk. The free energies of binding evaluated from the electrostatic binding free energy computed using a continuum model and estimates of the binding entropy changes represent shallow minima. The corresponding calculated residence times of the buried water molecules range from tens of picoseconds to hundreds of nanoseconds, which denote rather short times as for buried water molecules. Several water exchanges monitored in the simulations show that water molecules along the exit/entrance pathway use a relay of H bonds primarily formed with charged residues which helps either the exit or the entrance from or into the buried site. The exit/entrance of water molecules from/into the sites is permitted essentially by local motions of, at most, two side chains, indicating that, in these cases, complex correlated atomic motions are not needed to open the buried site toward the surface of the protein. This provides a possible explanation for the short residence times.     

Nanosecond Motions in Proteins Impose Bounds on the Timescale Distributions of Local Dynamics

We elucidate the physics of protein dynamical transition via 10-100-ns molecular dynamics simulations at temperatures spanning 160-300 K. By tracking the energy fluctuations, we show that the protein dynamical transition is marked by a crossover from nonstationary to stationary processes that underlie the dynamics of protein motions. A two-timescale function captures the nonexponential character of backbone structural relaxations. One timescale is attributed to the collective segmental motions and the other to local relaxations. The former is well defined by a single-exponential, nanosecond decay, operative at all temperatures. The latter is described by a set of processes that display a distribution of timescales. Although their average remains on the picosecond timescale, the distribution is markedly contracted at the onset of the transition. It is shown that the collective motions impose bounds on timescales spanned by local dynamical processes. The nonstationary character below the transition implicates the presence of a collection of substates whose interactions are restricted. At these temperatures, a wide distribution of local-motion timescales, extending beyond that of nanoseconds, is observed. At physiological temperatures, local motions are confined to timescales faster than nanoseconds. This relatively narrow window makes possible the appearance of multiple channels for the backbone dynamics to operate.     

Reconciling Mediating and Slaving Roles of Water in Protein Conformational Dynamics

Proteins accomplish their physiological functions with remarkably organized dynamic transitions among a hierarchical network of conformational substates. Despite the essential contribution of water molecules in shaping functionally important protein dynamics, their exact role is still controversial. Water molecules were reported either as mediators that facilitate or as masters that slave protein dynamics. Since dynamic behaviour of a given protein is ultimately determined by the underlying energy landscape, we systematically analysed protein self energies and protein-water interaction energies obtained from extensive molecular dynamics simulation trajectories of barstar. We found that protein-water interaction energy plays the dominant role when compared with protein self energy, and these two energy terms on average have negative correlation that increases with increasingly longer time scales ranging from 10 femtoseconds to 100 nanoseconds. Water molecules effectively roughen potential energy surface of proteins in the majority part of observed conformational space and smooth in the remaining part. These findings support a scenario wherein water on average slave protein conformational dynamics but facilitate a fraction of transitions among different conformational substates, and reconcile the controversy on the facilitating and slaving roles of water molecules in protein conformational dynamics.     

Structural basis of hierarchical multiple substates of a protein. IV: Rearrangements in atom packing and local deformations

Differences in atom packing are studied in the minimum energy conformations derived from the record of the Monte Carlo simulation of conformational fluctuation in the native state of a globular protein, bovine pancreatic trypsin inhibitor. It is found that local deformations observed among the minima which are found in the previous paper are accompanied by rearrangement of atom packing. Spatial locations of the local deformations in the three-dimensional folded structure are also studied. It is found that the local deformations are distributed in space in several clusters in the folded structure. The size and location of the clusters characterize the respective fluctuations of the first and the second levels observed in the simulation. In the fluctuations of the first level local deformations, each of which usually involves a few side chains and one main chain local segment, are thermally exited independently of each other near the surface of the molecule. The observed fluctuation of the second level involves a cooperative deformation involving many side chains and local main chain segments all in one cluster, which goes though the core of the molecule. The collective local deformations observed both in the first and second levels are plastic in the sense that they are accompanied with rearrangement of atom packing.     

On the convergence of the conformational coordinates basis set obtained by the essential dynamics analysis of proteins' molecular dynamics simulations.

In this article we present a quantitative evaluation of the convergence of the conformational coordinates of proteins, obtained by the Essential Dynamics method. Using a detailed analysis of long molecular dynamics trajectories in combination with a statistical assessment of the significance of the measured convergence, we obtained that simulations of a few hundreds of picoseconds are in general sufficient to provide a stable and statistically reliable definition of the essential and near constraints subspaces, at least within the nanoseconds time range. Proteins 1999;36:419-424.     

Limits of phase and amplitude sensitivity in the torus semicircularis ofEigenmannia

Eigenmannia can detect modulations in the time disparity of signals received by different regions of the body surface as small as several hundred nanoseconds. This study presents recordings of single units in the torus semicircularis that are sensitive to time disparities (differential-phase) between a sinusoidal signal received by the head region and a similar signal received by the body surface caudal to the fish's pectoral fins. The sensitivity of units to differential phase, measured by the change in spike rate per unit change in time disparity, was greatest when small phase modulations, rather than stationary phase differences, were presented. Thresholds of differential-phase coders ranged from 6.5 microseconds to several hundred microseconds, with approximately 20% of the units having thresholds in the 5-10 microseconds range. For most cells, sensitivity to small modulations of differential-phase was relatively unaffected by time disparity 'offsets' within a range of several hundred microseconds. A threshold of 5-10 microseconds is still an order of magnitude higher than that measured in the Jamming Avoidance Response (JAR). Neurons that were sensitive to amplitude modulations (AMs) had thresholds as low as 0.05%. This value is comparable to that observed at the behavioral level.     

Dynamics of the B-A transition of DNA double helices

Although the transition from the B-DNA double helix to the A-form is essential for biological function, as shown by the existence of the A-form in many protein–DNA complexes, the dynamics of this transition has not been resolved yet. According to molecular dynamics simulations the transition is expected in the time range of a few nanoseconds. The B–A transition induced by mixing of DNA samples with ethanol in stopped flow experiments is complete within the deadtime, showing that the reaction is faster than ~0.2 ms. The reaction was resolved by an electric field jump technique with induction of the transition by a dipole stretching force driving the A- to the B-form. Poly[d(A-T)] was established as a favourable model system, because of a particularly high cooperativity of the transition and because of a spectral signature allowing separation of potential side reactions. The time constants observed in the case of poly[d(A-T)] with ~1600 bp are in the range around 10 µs. An additional process with time constants of ~100 µs is probably due to nucleation. The same time constants (within experimental accuracy ±10%) were observed for a poly[d(A-T)] sample with ~70 bp. Under low salt conditions commonly used for studies of the B–A transition, the time constants are almost independent of the ionic strength. The experimental data show that a significant activation barrier exists in the B–A transition and that the helical states are clearly separated from each other, in contrast to predictions by molecular dynamics simulations.