Hierarchical structure of the energy landscape of proteins revisited by time series analysis. II. Investigation of explicit solvent effects

Time series analysis tools are employed on the principal modes obtained from the C(alpha) trajectories from two independent molecular-dynamics simulations of alpha-amylase inhibitor (tendamistat). Fluctuations inside an energy minimum (intraminimum motions), transitions between minima (interminimum motions), and relaxations in different hierarchical energy levels are investigated and compared with those encountered in vacuum by using different sampling window sizes and intervals. The low-frequency low-indexed mode relationship, established in vacuum, is also encountered in water, which shows the reliability of the important dynamics information offered by principal components analysis in water. It has been shown that examining a short data collection period (100 ps) may result in a high population of overdamped modes, while some of the low-frequency oscillations (<10 cm(-1)) can be captured in water by using a longer data collection period (1200 ps). Simultaneous analysis of short and long sampling window sizes gives the following picture of the effect of water on protein dynamics. Water makes the protein lose its memory: future conformations are less dependent on previous conformations due to the lowering of energy barriers in hierarchical levels of the energy landscape. In short-time dynamics (<10 ps), damping factors extracted from time series model parameters are lowered. For tendamistat, the friction coefficient in the Langevin equation is found to be around 40-60 cm(-1) for the low-indexed modes, compatible with literature. The fact that water has increased the friction and that on the other hand has lubrication effect at first sight contradicts. However, this comes about because water enhances the transitions between minima and forces the protein to reduce its already inherent inability to maintain oscillations observed in vacuum. Some of the frequencies lower than 10 cm(-1) are found to be overdamped, while those higher than 20 cm(-1) are slightly increased. As for the long-time dynamics in water, it is found that random-walk motion is maintained for approximately 200 ps (about five times of that in vacuum) in the low-indexed modes, showing the lowering of energy barriers between the higher-level minima.     

Application of time series analysis on molecular dynamics simulations of proteins: a study of different conformational spaces by principal component analysis

Time series analysis is applied on the collective coordinates obtained from principal component analysis of independent molecular dynamics simulations of alpha-amylase inhibitor tendamistat and immunity protein of colicin E7 based on the Calpha coordinates history. Even though the principal component directions obtained for each run are considerably different, the dynamics information obtained from these runs are surprisingly similar in terms of time series models and parameters. There are two main differences in the dynamics of the two proteins: the higher density of low frequencies and the larger step sizes for the interminima motions of colicin E7 than those of alpha-amylase inhibitor, which may be attributed to the higher number of residues of colicin E7 and/or the structural differences of the two proteins. The cumulative density function of the low frequencies in each run conforms to the expectations from the normal mode analysis. When different runs of alpha-amylase inhibitor are projected on the same set of eigenvectors, it is found that principal components obtained from a certain conformational region of a protein has a moderate explanation power in other conformational regions and the local minima are similar to a certain extent, while the height of the energy barriers in between the minima significantly change. As a final remark, time series analysis tools are further exploited in this study with the motive of explaining the equilibrium fluctuations of proteins.     

Time series analysis of collective motions in proteins

The dynamics of alpha-amylase inhibitor tendamistat around its native state is investigated using time series analysis of the principal components of the C(alpha) atomic displacements obtained from molecular dynamics trajectories. Collective motion along a principal component is modeled as a homogeneous nonstationary process, which is the result of the damped oscillations in local minima superimposed on a random walk. The motion in local minima is described by a stationary autoregressive moving average model, consisting of the frequency, damping factor, moving average parameters and random shock terms. Frequencies for the first 50 principal components are found to be in the 3-25 cm(-1) range, which are well correlated with the principal component indices and also with atomistic normal mode analysis results. Damping factors, though their correlation is less pronounced, decrease as principal component indices increase, indicating that low frequency motions are less affected by friction. The existence of a positive moving average parameter indicates that the stochastic force term is likely to disturb the mode in opposite directions for two successive sampling times, showing the modes tendency to stay close to minimum. All these four parameters affect the mean square fluctuations of a principal mode within a single minimum. The inter-minima transitions are described by a random walk model, which is driven by a random shock term considerably smaller than that for the intra-minimum motion. The principal modes are classified into three subspaces based on their dynamics: essential, semiconstrained, and constrained, at least in partial consistency with previous studies. The Gaussian-type distributions of the intermediate modes, called "semiconstrained" modes, are explained by asserting that this random walk behavior is not completely free but between energy barriers.     

Hydration dynamics and time scales of coupled water-protein fluctuations

We report experimental and theoretical studies on water and protein dynamics following photoexcitation of apomyoglobin. Using site-directed mutation and with femtosecond resolution, we experimentally observed relaxation dynamics with a biphasic distribution of time scales, 5 and 87 ps, around the site Trp7. Theoretical studies using both linear response and direct nonequilibrium molecular dynamics (MD) calculations reproduced the biphasic behavior. Further constrained MD simulations with either frozen protein or frozen water revealed the molecular mechanism of slow hydration processes and elucidated the role of protein fluctuations. Observation of slow water dynamics in MD simulations requires protein flexibility, regardless of whether the slow Stokes shift component results from the water or protein contribution. The initial dynamics in a few picoseconds represents fast local motions such as reorientations and translations of hydrating water molecules, followed by slow relaxation involving strongly coupled water-protein motions. We observed a transition from one isomeric protein configuration to another after 10 ns during our 30 ns ground-state simulation. For one isomer, the surface hydration energy dominates the slow component of the total relaxation energy. For the other isomer, the slow component is dominated by protein interactions with the chromophore. In both cases, coupled water-protein motion is shown to be necessary for observation of the slow dynamics. Such biologically important water-protein motions occur on tens of picoseconds. One significant discrepancy exists between theory and experiment, the large inertial relaxation predicted by simulations but clearly absent in experiment. Further improvements required in the theoretical model are discussed.     

Energetics and dynamics in MbCN: CN--vibrational relaxation from molecular dynamics simulations

The dynamics of the cyanide anion bound to sperm-whale myoglobin is investigated using atomistic simulations. With density-functional theory, a 2D potential energy surface for the cyanide-heme complex is calculated. Two deep minima with a stabilization energy of approximately 50 kcal/mol corresponding to two different binding orientations (Fe-CN and Fe-NC) of the ligand are found. The Fe-CN conformation is favored over Fe-NC by several kcal/mol. Mixed quantum mechanics/molecular mechanics calculations show that the binding orientation affects the bond strength of the ligand, with a significantly different bond length and a 25 cm-1 shift in the fundamental CN-frequency. For the molecular dynamics (MD) simulations, a 3-center fluctuating charge model for the Fe-CN unit is developed that captures polarization and ligand-metal charge transfer. Stability arguments based on the energetics around the active site and the CN- frequency shifts suggest that the Fe-CN conformation with epsilon-protonation of His epsilon 64 are most likely, which is in agreement with experiment. Both equilibrium and nonequilibrium MD simulations are carried out to investigate the relaxation time scale and possible relaxation pathways in bound MbCN. The nonequilibrium MD simulations with a vibrationally excited ligand reveal that vibrational relaxation takes place on a time scale of hundreds of picoseconds within the active site. This finding supports the hypothesis that the experimentally observed relaxation rate (3.6 ps) reflects the repopulation of the electronic ground state.     

Harmonic analysis of large systems. III. Comparison with molecular dynamics

Atomic motions in bovine pancreatic trypsin inhibitor (BPTI), derived from molecular dynamics, harmonic analysis, and quasiharmonic analysis, are compared when a single protein model, energy parameters, and environment are employed. Molecular dynamics (MD) was carried out for 2 nanoseconds. An average structure was determined from the last nanosecond of the MD simulation, when no major structural changes were observed. This structure was used for several harmonic analysis calculations as well as for a reference structure for the quasiharmonic analysis, for both full basis and reduced basis sets. In contrast to the harmonic analysis results, the quasiharmonic reduced basis calculation using a spherical harmonics reduced basis provided good agreement with the full basis calculation, suggesting that when anharmonic effects are considered, BPTI can behave as a homogeneous object. An extensive analysis of the normal modes from a diverse set of 201 minimized MD simulation frames was performed. On only the sub-picosecond time scale were energy minima revisited after a transition to another state. This analysis shows that the dynamics average structure is not representative of the simulation frames in terms of energy and vibrational frequencies. For this model of BPTI, 42% of the motion (mean-squared fluctuation) can be attributed to harmonic limit behavior. A spectral analysis of the correlation function of deformation for a particular normal mode or quasiharmonic mode can be used to determine the time scales of motions which correspond to harmonic vibration, large-scale drift, or sharp transitions between local substrates. © 1995 John Wiley & Sons, Inc.     

Molecular dynamics simulations and instantaneous normal-mode analysis of the vibrational relaxation of the C-H stretching modes of N-methylacetamide-d in liquid deuterated water

Nonequilibrium molecular dynamics (MD) simulations and instantaneous normal mode (INMs) analyses are used to study the vibrational relaxation of the C-H stretching modes (ν(s)(CH?)) of deuterated N-methylacetamide (NMAD) in aqueous (D2O) solution. The INMs are identified unequivocally in terms of the equilibrium normal modes (ENMs), or groups of them, using a restricted version of the recently proposed Min-Cost assignment method. After excitation of the parent ν(s)(CH?) modes with one vibrational quantum, the vibrational energy is shown to dissipate through both intramolecular vibrational redistribution (IVR) and intermolecular vibrational energy transfer (VET). The decay of the vibrational energy of the ν(s)(CH?) modes is well fitted to a triple exponential function, with each characterizing a well-defined stage of the entire relaxation process. The first, and major, relaxation stage corresponds to a coherent ultrashort (τ(rel) = 0.07 ps) energy transfer from the parent ν(s)(CH?) modes to the methyl bending modes δ(CH?), so that the initially excited state rapidly evolves into a mixed stretch-bend state. In the second stage, characterized by a time of 0.92 ps, the vibrational energy flows through IVR to a number of mid-range-energy vibrations of the solute. In the third stage, the vibrational energy accumulated in the excited modes dissipates into the bath through an indirect VET process mediated by lower-energy modes, on a time scale of 10.6 ps. All the specific relaxation channels participating in the whole relaxation process are properly identified. The results from the simulations are finally compared with the recent experimental measurements of the ν(s)(CH?) vibrational energy relaxation in NMAD/D?O(l) reported by Dlott et al. (J. Phys. Chem. A 2009, 113, 75.) using ultrafast infrared-Raman spectroscopy.     

General framework for studying the dynamics of folded and nonfolded proteins by NMR relaxation spectroscopy and MD simulation

A general framework is presented for the interpretation of NMR relaxation data of proteins. The method, termed isotropic reorientational eigenmode dynamics (iRED), relies on a principal component analysis of the isotropically averaged covariance matrix of the lattice functions of the spin interactions responsible for spin relaxation. The covariance matrix, which is evaluated using a molecular dynamics (MD) simulation, is diagonalized yielding reorientational eigenmodes and amplitudes that reveal detailed information about correlated protein dynamics. The eigenvalue distribution allows one to quantitatively assess whether overall and internal motions are statistically separable. To each eigenmode belongs a correlation time that can be adjusted to optimally reproduce experimental relaxation parameters. A key feature of the method is that it does not require separability of overall tumbling and internal motions, which makes it applicable to a wide range of systems, such as folded, partially folded, and unfolded biomolecular systems and other macromolecules in solution. The approach was applied to NMR relaxation data of ubiquitin collected at multiple magnetic fields in the native form and in the partially folded A-state using MD trajectories with lengths of 6 and 70 ns. The relaxation data of native ubiquitin are well reproduced after adjustment of the correlation times of the 10 largest eigenmodes. For this state, a high degree of separability between internal and overall motions is present as is reflected in large amplitude and collectivity gaps between internal and overall reorientational modes. In contrast, no such separability exists for the A-state. Residual overall tumbling motion involving the N-terminal beta-sheet and the central helix is observed for two of the largest modes only. By adjusting the correlation times of the 10 largest modes, a high degree of consistency between the experimental relaxation data and the iRED model is reached for this highly flexible biomolecule.     

Modelling of solid polymers in chemical detail - gases in amorphous polymers

The diffusion of gases in dense polymers, above and below the glass-transition temperature, is described with a new Transition State Theory model that is based on the concept that the dynamics of small molecules dissolved in dense polymers is separated from the structural relaxation of the dense polymers. The model is used to study the dynamics of rare gases dissolved in atomistic micro-structures of four polymers at 300 K: poly(dimethylsiloxane), poly(isobutylene), atactic poly(vinylchloride) and the polycarbonate of 4,4′-isopropylidenediphenol (bisphenol-A). Short-time-scale MD runs (5 ps) are used to characterize the elastic thermal motion of the host matrix; this information on mobility is then used for a stochastic simulation of solute dynamics up to ca. 1ms. All dissolved molecules show similar behavior by displaying three time regimes: a short-time, high-mobility domain, an intermediate time domain of anomalous diffusion, and a diffusive regime at long times. From the long-time data diffusion coefficients are estimated; comparison with experimental data results in good agreement.     

Multidimensional infrared spectroscopy of the N-H bond motions in formamide

The heterodyned two-dimensional (2D) IR spectra and equilibrium dynamics of the N-H stretching motion of DCONHD in deuterated formamide, DCOND(2), were studied with 80 fs pulses at 3 microm. The time evolution of the heterodyned 2D IR spectra, pump-probe spectra, and photon echo peak shift demonstrate that interstate dynamics is occurring by relaxation of the original N-H excitation. The N-H vibrational frequency correlation function can be expressed as a sum of three exponentials with correlation times 0.24 ps, 0.8 ps, and 11 ps. The intermediate component is attributed to motions of the N-Hcdots, three dots, centeredO unit involving only slight angular variations of the N-H bond. The slow component is attributed to the structure breaking and making. The anisotropy decay confirmed that the significant angular N-H bond motion occurs on the 11 ps time scale. The fast component, which is the least well determined, might correspond to the modulation of the H-bond distance without angular motion. The correlation coefficient between the pumped and relaxed state distributions was +0.51, implying that the excited state phase memory is only slightly diminished by the relaxation of the N-H excitation. The relaxed modes are concluded to be local to the driven N-H mode.     

Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: the case of domain motions

Protein dynamics on a long time scale was investigated using all-atom molecular dynamics (MD) simulation and time-structure based independent component analysis (tICA). We selected the lysine-, arginine-, ornithine-binding protein (LAO) as a target protein and focused on its domain motions in the open state. A MD simulation of the LAO in explicit water was performed for 600 ns, in which slow and large-amplitude domain motions of the LAO were observed. After extracting domain motions by rigid-body domain analysis, the tICA was applied to the obtained rigid-body trajectory, yielding slow modes of the LAO's domain motions in order of decreasing time scale. The slowest mode detected by the tICA represented not a closure motion described by a largest-amplitude mode determined by the principal component analysis but a twist motion with a time scale of tens of nanoseconds. The slow dynamics of the LAO were well described by only the slowest mode and were characterized by transitions between two basins. The results show that tICA is promising for describing and analyzing slow dynamics of proteins.     

Temperature dependence of solvation dynamics and anisotropy decay in a protein: ANS in bovine serum albumin

Temperature dependence of solvation dynamics and fluorescence anisotropy decay of 8-anilino-1-naphthalenesulfonate (ANS) bound to a protein, bovine serum albumin (BSA), are studied. Solvation dynamics of ANS bound to BSA displays a component (300 ps) which is independent of temperature in the range of 278-318 K and a long component which decreases from 5800 ps at 278 K to 3600 ps at 318 K. The temperature independent part is ascribed to a dynamic exchange of bound to free water with a low barrier. The temperature variation of the long component of solvation dynamics corresponds to an activation energy of 2.1 kcal mol(-1). The activation energy is ascribed to local segmental motion of the protein along with the associated water molecules and polar residues. The time scale of solvation dynamics is found to be very different from the time scale of anisotropy decay. The anisotropy decays are analyzed in terms of the wobbling motion of the probe (ANS) and the overall tumbling of the protein.     

Polymer dynamics in torsion space

The large scale motion of proteins, or covalently bonded polymers in general, is governed by the dynamics of the torsion angles, with bond lengths and bond angles kept approximately constant. In the present work, the Lagrangian equations of torsion motion are derived for a general macromolecule. The dynamics is implemented numerically for a test protein, using the velocity Verlet method as the integrator. The results indicate time steps of up to about 30 fs can be used for short time (up to at least 20 ps) simulations, before the dynamics and energy start to differ significantly from results obtained with smaller time steps. For longer time simulations, up to 1000 ps, a time step of 10 fs is relatively safe. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Dynamics and thermodynamics of water in PAMAM dendrimers at subnanosecond time scales

Atomistic molecular dynamics simulations are used to study generation 5 polyamidoamine (PAMAM) dendrimers immersed in a bath of water. We interpret the results in terms of three classes of water: buried water well inside of the dendrimer surface, surface water associated with the dendrimer-water interface, and bulk water well outside of the dendrimer. We studied the dynamic and thermodynamic properties of the water at three pH values: high pH with none of the primary or tertiary amines protonated, intermediate pH with only the primary amines protonated, and low pH with all amines protonated. For all pH values we find that both buried and surface water exhibit two relaxation times: a fast relaxation ( approximately 1 ps) corresponding to the libration motion of the water and a slow ( approximately 20 ps) diffusional component related to the escaping of water from one domain to another. In contrast for bulk water the fast relaxation is approximately 0.4 ps while the slow relaxation is approximately 14 ps. These results are similar to those found in biological systems, where the fast relaxation is found to be approximately 1 ps while the slow relaxation ranges from 20 to 1000 ps. We used the 2PT MD method to extract the vibrational (power) spectrum and found substantial differences for the three classes of water. The translational diffusion coefficient for buried water is 11-33% (depending on pH) of the bulk value while the surface water is about 80%. The change in rotational diffusion is quite similar: 21-45% of the bulk value for buried water and 80% for surface water. This shows that translational and rotational dynamics of water are affected by the PAMAM-water interactions as well as due to the confinement in the interior of the dendrimer. We find that the reduction of translational or rotational diffusion is accompanied by a blue shift of the corresponding libration motions ( approximately 10 cm(-1) for translation, approximately 35 cm(-1) for rotation), indicating higher local force constants for these motions. These effects are most pronounced for the lowest pH, probably because of the increased rigidity caused by the internal charges. From the vibrational density of states we also calculate the enthalpies and entropies of the various waters. We find that water molecules are enthalpically favored near the PAMAM dendrimer: energy for surface water is approximately 0.1 kcal/mol lower to that in the bulk, and approximately 0.5-0.9 kcal/mol lower for buried water. In contrast, we find that both the buried and surface water are entropically unfavored: buried water is 0.9-2.2 kcal/mol lower than the bulk while the surface water is 0.1-0.2 kcal/mol lower. The net result is a thermodynamically unfavored state of the water surrounding the PAMAM dendrimer: 0.4-1.3 kcal/mol higher for buried water and 0.1-0.2 kcal/mol for surface water. This excess free energy of the surface and buried waters is released when the PAMAM dendrimer binds to DNA or metal ions, providing an extra driving force.     

The dynamics of water at DNA interfaces: computational studies of Hoechst 33258 bound to DNA

Together, spectroscopy combined with computational studies that relate directly to the experimental measurements have the potential to provide unprecedented insight into the dynamics of important biological processes. Recent time-resolved fluorescence experiments have shown that the time scales for collective reorganization at the interface of proteins and DNA with water are more than an order of magnitude slower than in bulk aqueous solution. The molecular interpretation of this change in the collective response is somewhat controversial some attribute the slower reorganization to dramatically retarded water motion, while others describe rapid water dynamics combined with a slower biomolecular response. To connect directly to solvation dynamics experiments of the fluorescent probe Hoechst 33258 (H33258) bound to DNA, we have generated 770 ns of molecular dynamics (MD) simulations and calculated the equilibrium and nonequilibrium solvation response to excitation of the probe. The calculated time scales for the solvation response of H33258 free in solution (0.17 and 1.4 ps) and bound to DNA (1.5 and 20 ps) are highly consistent with experiment (0.2 and 1.2 ps, 1.4 and 19 ps, respectively). Decomposition of the calculated response revealed that water solvating the probe bound to DNA was still relatively mobile, only slowing by a factor of 2-3, while DNA motion was responsible for the long-time component (approximately 20 ps).     

Vibrational energy relaxation dynamics of Si---H stretching modes on stepped H/Si(111) 1×1 surfaces

The vibrational energy relaxation rates of excited Si---H stretching modes on the monohydride steps of miscut H/Si(111) 1×1 surfaces are calculated using Bloch-Redfield theory combined with classical molecular dynamics (MD) simulation. The structure and vibrational frequencies of the surface are first investigated using the Car-Parrinello ab initio MD method. The calculated Si---Si---H bending frequencies and relaxed structures are then used to refine the empirical potential for the classical MD simulations. The lifetime of the excited Si---H stretching mode at the step is found to be shorter than the modes on the terrace. Both the magnitude and the trend of the calculated results agree well with the experimental measurement on the 9° monohydride stepped surface. The vibrational relaxation rate of the Si---H stretching modes on the 15° monohydride stepped surface are also calculated and predicted to have a slightly shorter lifetime than for the 9° surface.     

Ultrafast dynamics of myoglobin probed by time-resolved resonance Raman spectroscopy

Recent experimental work carried out in this laboratory on the ultrafast dynamics of myoglobin (Mb) is summarized with a stress on structural and vibrational energy relaxation. Studies on the structural relaxation of Mb following CO photolysis revealed that the structural change of heme itself, caused by CO photodissociation, is completed within the instrumental response time of the time-resolved resonance Raman apparatus used (approximately 2 ps). In contrast, changes in the intensity and frequency of the iron-histidine (Fe-His) stretching mode upon dissociation of the trans ligand were found to occur in the picosecond regime. The Fe-His band is absent for the CO-bound form, and its appearance upon photodissociation was not instantaneous, in contrast with that observed in the vibrational modes of heme, suggesting appreciable time evolution of the Fe displacement from the heme plane. The band position of the Fe-His stretching mode changed with a time constant of about 100 ps, indicating that tertiary structural changes of the protein occurred in a 100-ps range. Temporal changes of the anti-Stokes Raman intensity of the v4 and v7 bands demonstrated immediate generation of vibrationally excited heme upon the photodissociation and decay of the excited populations, whose time constants were 1.1 +/- 0.6 and 1.9 +/- 0.6 ps, respectively. In addition, the development of the time-resolved resonance Raman apparatus and prospects in this research field are described.     

Time scales and pathways of vibrational energy relaxation in liquid CHBr3 and CDBr3

Molecular dynamics simulations are used in conjunction with Landau-Teller, fluctuating Landau-Teller, and time-dependent perturbation theories to investigate energy flow out of various vibrational states of liquid CHBr3 and CDBr3. The CH stretch overtone is found to relax with a time scale of about 1 ps compared to the 50 ps rate for the fundamental. The relaxation pathways and rates for the CD stretch decay in CDBr3 are computed in order to understand the changes arising from deuteration. While the computed relaxation rate agrees well with experiments, the pathway is found to be more complex than anticipated. In addition to the above channels for CH(D) stretch relaxation that involve only the hindered translations and rotations of the solvent, routes involving off-resonant and resonant excitations of solvent vibrational modes are also examined. Finally, the decay of energy from low frequency states to near-lying solute states and solvent vibrations are studied.