Hierarchical structure of the energy landscape of proteins revisited by time series analysis. I. Mimicking protein dynamics in different time scales

Time series models, which are constructed from the projections of the molecular-dynamics (MD) runs on principal components (modes), are used to mimic the dynamics of two proteins: tendamistat and immunity protein of colicin E7 (ImmE7). Four independent MD runs of tendamistat and three independent runs of ImmE7 protein in vacuum are used to investigate the energy landscapes of these proteins. It is found that mean-square displacements of residues along the modes in different time scales can be mimicked by time series models, which are utilized in dividing protein dynamics into different regimes with respect to the dominating motion type. The first two regimes constitute the dominance of intraminimum motions during the first 5 ps and the random walk motion in a hierarchically higher-level energy minimum, which comprise the initial time period of the trajectories up to 20-40 ps for tendamistat and 80-120 ps for ImmE7. These are also the time ranges within which the linear nonstationary time series are completely satisfactory in explaining protein dynamics. Encountering energy barriers enclosing higher-level energy minima constrains the random walk motion of the proteins, and pseudorelaxation processes at different levels of minima are detected in tendamistat, depending on the sampling window size. Correlation (relaxation) times of 30-40 ps and 150-200 ps are detected for two energy envelopes of successive levels for tendamistat, which gives an overall idea about the hierarchical structure of the energy landscape. However, it should be stressed that correlation times of the modes are highly variable with respect to conformational subspaces and sampling window sizes, indicating the absence of an actual relaxation. The random-walk step sizes and the time length of the second regime are used to illuminate an important difference between the dynamics of the two proteins, which cannot be clarified by the investigation of relaxation times alone: ImmE7 has lower-energy barriers enclosing the higher-level energy minimum, preventing the protein to relax and letting it move in a random-walk fashion for a longer period of time.     

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.     

Dynamics of bulk polyethylene on a high coordination lattice

Monte Carlo simulations of bulk polyethylene were performed on a high-coordination lattice. In this study, we investigated the effect of coarse-grained short and long range interaction parameters on the dynamic properties of the bulk system. As a result of fine tuning of the simulation parameters, our coarse-grained simulations were successful in mimicing the local and large-scale dynamics of C₄₄H₉₀ and C₁₀₀H₂₀₂ melts.     

Fracture mechanics analysis of arc shaped specimens for pipe grade polymers

Instrumented three-point bend impact fracture tests are widely used to evaluate pipe grade polymers. Often specimens cut from small-diameter pipe are used, and these are necessarily arc-shaped. Because the orientation and thermal history may differ between extruded pipe and compression moulded plaque material, this additional difference in geometry must be properly accounted for, or it may mask any effects on material properties. This paper modifies a previously published solution for the geometry-dependent energy correction factor for arc-shaped specimens, and extends it to a wider range of standard pipe geometries. When the results are properly corrected, the effects of processing on a commercial PE100 appear to be minor.     

Effect of ligand binding on the intraminimum dynamics of proteins

Effects of ligand binding on protein dynamics are studied via molecular dynamics (MD) simulations on two different enzymes, dihydrofolate reductase (DHFR) and triosephosphate isomerase (TIM), in their unliganded (free) and liganded states. Domain motions in MD trajectories are analyzed by collectivities and rotation angles along the principal components (PCs). DHFR in the free state has well‐defined domain rotations, whereas rotations are slightly damped in the binary complex with nicotinamide adenine dinucleotide phosphate (NADPH), and remarkably distorted in the presence of NADP⁺, showing that NADP⁺ is solely responsible for the loss of correlation of the domains in DHFR. Although mean square fluctuations of MD simulations in the same PC subspaces are similar for different ligation states, linear stochastic time series models show that backbone flexibility along the first five PCs is decreased upon NADPH and NADP⁺ binding in subpicosecond scale. This shows that mobility of the protein along the PCs is closely related with intraminimum dynamics, and alterations in ligation states may change the intraminimum dynamics significantly. Low vibrational frequencies of the alpha‐carbon atoms of DHFR are determined from the time series models of a larger number of low indexed PCs, and it is found that number of modes in the lowest frequencies is reduced upon ligand binding. A similar result is obtained for TIM in the unliganded and dihydroxyacetone phosphate bound states. We suggest that stochastic time series modeling is a promising method to be used in determining subtle perturbations in protein dynamics. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

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.     

Dynamics of large proteins through hierarchical levels of coarse-grained structures

Elastic network models have been successful in elucidating the largest scale collective motions of proteins. These models are based on a set of highly coupled springs, where only the close neighboring amino acids interact, without any residue specificity. Our objective here is to determine whether the equivalent cooperative motions can be obtained upon further coarse-graining of the protein structure along the backbone. The influenza virus hemagglutinin A (HA), composed of N = 1509 residues, is utilized for this analysis. Elastic network model calculations are performed for coarse-grained HA structures containing only N/2, N/10, N/20, and N/40 residues along the backbone. High correlations (>0.95) between residue fluctuations are obtained for the first dominant (slowest) mode of motion between the original model and the coarse-grained models. In the case of coarse-graining by a factor of 1/40, the slowest mode shape for HA is reconstructed for all residues by successively selecting different subsets of residues, shifting one residue at a time. The correlation for this reconstructed first mode shape with the original all-residue case is 0.73, while the computational time is reduced by about three orders of magnitude. The reduction in computational time will be much more significant for larger targeted structures. Thus, the dominant motions of protein structures are robust enough to be captured at extremely high levels of coarse-graining. And more importantly, the dynamics of extremely large complexes are now accessible with this new methodology.     

A computational docking study on the pH dependence of peptide binding to HLA-B27 sub-types differentially associated with ankylosing spondylitis

A single amino acid difference (Asp116His), having a key role in a pathogenesis pathway, distinguishes HLA-B*27:05 and HLA-B*27:09 sub-types as associated and non-associated with ankylosing spondylitis, respectively. In this study, molecular docking simulations were carried out with the aim of comprehending the differences in the binding behavior of both alleles at varying pH conditions. A library of modeled peptides was formed upon single point mutations aiming to address the effect of 20 naturally occurring amino acids at the binding core peptide positions. For both alleles, computational docking was applied using Autodock 4.2. Obtained free energies of binding (FEB) were compared within the peptide library and between the alleles at varying pH conditions. The amino acid preferences of each position were studied enlightening the role of each on binding. The preferred amino acids for each position of pVIPR were found to be harmonious with experimental studies. Our results indicate that, as the pH is lowered, the capacity of HLA-B*27:05 to bind peptides in the library is largely lost. Hydrogen bonding analysis suggests that the interaction between the main anchor positions of pVIPR and their respective binding pocket residues are affected from the pH the most, causing an overall shift in the FEB profiles.     

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.