Loop motions of triosephosphate isomerase observed with elastic networks

The internal dynamics of triosephosphate isomerase have been investigated with elastic networks, with and without a substrate bound. The slowest modes of motion involve large domain motions but also a loop motion that conforms to the changes observed between the crystal structures and . Our computations confirm that the different motions of this loop are important in several of the computed slowest modes. We have shown that elastic network computations on this protein system can combine atoms for the functional parts of the structure with coarse-grained (cg) representations of the remainder of the structure in several different ways. Similar loop motions are seen with elastic network models for atomistic and mixed cg models. The loop motions are reproduced with an overlap of 0.75-0.79 by combining the four slowest modes of motion for the free and complex forms of the enzyme.     

Vibrational Dynamics of Icosahedrally Symmetric Biomolecular Assemblies Compared with Predictions Based on Continuum Elasticity

Coarse-grained elastic network models elucidate the fluctuation dynamics of proteins around their native conformations. Low-frequency collective motions derived by simplified normal mode analysis are usually involved in biological function, and these motions often possess noteworthy symmetries related to the overall shape of the molecule. Here, insights into these motions and their frequencies are sought by considering continuum models with appropriate symmetry and boundary conditions to approximately represent the true atomistic molecular structure. We solve the elastic wave equations analytically for the case of spherical symmetry, yielding a symmetry-based classification of molecular motions together with explicit predictions for their vibrational frequencies. We address the case of icosahedral symmetry as a perturbation to the spherical case. Applications to lumazine synthase, satellite tobacco mosaic virus, and brome mosaic virus show that the spherical elastic model efficiently provides insights on collective motions that are otherwise obtained by detailed elastic network models. A major utility of the continuum models is the possibility of estimating macroscopic material properties such as the Young's modulus or Poisson's ratio for different types of viruses.     

Elastic network models capture the motions apparent within ensembles of RNA structures

The role of structure and dynamics in mechanisms for RNA becomes increasingly important. Computational approaches using simple dynamics models have been successful at predicting the motions of proteins and are often applied to ribonucleo-protein complexes but have not been thoroughly tested for well-packed nucleic acid structures. In order to characterize a true set of motions, we investigate the apparent motions from 16 ensembles of experimentally determined RNA structures. These indicate a relatively limited set of motions that are captured by a small set of principal components (PCs). These limited motions closely resemble the motions computed from low frequency normal modes from elastic network models (ENMs), either at atomic or coarse-grained resolution. Various ENM model types, parameters, and structure representations are tested here against the experimental RNA structural ensembles, exposing differences between models for proteins and for folded RNAs. Differences in performance are seen, depending on the structure alignment algorithm used to generate PCs, modulating the apparent utility of ENMs but not significantly impacting their ability to generate functional motions. The loss of dynamical information upon coarse-graining is somewhat larger for RNAs than for globular proteins, indicating, perhaps, the lower cooperativity of the less densely packed RNA. However, the RNA structures show less sensitivity to the elastic network model parameters than do proteins. These findings further demonstrate the utility of ENMs and the appropriateness of their application to well-packed RNA-only structures, justifying their use for studying the dynamics of ribonucleo-proteins, such as the ribosome and regulatory RNAs.     

Comparison of mode analyses at different resolutions applied to nucleic acid systems

More than two decades of different types of mode analyses has shown that these techniques can be useful in describing large-scale motions in protein systems. A number of mode analyses are available and include quasiharmonics, classical normal mode, block normal mode, and the elastic network model. Each of these methods has been validated for protein systems and this variety allows researchers to choose the technique that gives the best compromise between computational cost and the level of detail in the calculation. These same techniques have not been systematically tested for nucleic acid systems, however. Given the differences in interactions and structural features between nucleic acid and protein systems, the validity of these techniques in the protein regime cannot be directly translated into validity in the nucleic acid realm. In this work, we investigate the usefulness of the above mode analyses as applied to two RNA systems, i.e., the hammerhead ribozyme and a guanine riboswitch. We show that classical normal-mode analysis can match the magnitude and direction of residue fluctuations from the more detailed, anharmonic technique, quasiharmonic analysis of a molecular dynamics trajectory. The block normal-mode approximation is shown to hold in the nucleic acid systems studied. Only the mode analysis at the lowest level of detail, the elastic network model, produced mixed results in our calculations. We present data that suggest that the elastic network model, with the popular parameterization, is not best suited for systems that do not have a close packed structure; this observation also hints at why the elastic network model has been found to be valid for many globular protein systems. The different behaviors of block normal-mode analysis and the elastic network model, which invoke similar degrees of coarse-graining to the dynamics but use different potentials, suggest the importance of applying a heterogeneous potential function in a robust analysis of the dynamics of biomolecules, especially those that are not closely packed. In addition to these comparisons, we briefly discuss insights into the conformational space available to the hammerhead ribozyme.     

Conformational changes and allosteric communications in human serum albumin due to ligand binding

It is well recognized that knowledge of structure alone is not sufficient to understand the fundamental mechanism of biomolecular recognition. Information of dynamics is necessary to describe motions involving relevant conformational states of functional importance. We carried out principal component analysis (PCA) of structural ensemble, derived from 84 crystal structures of human serum albumin (HSA) with different ligands and/or different conditions, to identify the functionally important collective motions, and compared with the motions along the low-frequency modes obtained from normal mode analysis of the elastic network model (ENM) of unliganded HSA. Significant overlap is observed in the collective motions derived from PCA and ENM. PCA and ENM analysis revealed that ligand selects the most favored conformation from accessible equilibrium structures of unliganded HSA. Further, we analyzed dynamic network obtained from molecular dynamics simulations of unliganded HSA and fatty acids- bound HSA. Our results show that fatty acids-bound HSA has more robust community network with several routes to communicate among different parts of the protein. Critical nodes (residues) identified from dynamic network analysis are in good agreement with allosteric residues obtained from sequence-based statistical coupling analysis method. This work underscores the importance of intrinsic structural dynamics of proteins in ligand recognition and can be utilized for the development of novel drugs with optimum activity.     

Collective motions in proteins: a covariance analysis of atomic fluctuations in molecular dynamics and normal mode simulations.

A method is described for identifying collective motions in proteins from molecular dynamics trajectories or normal mode simulations. The method makes use of the covariances of atomic positional fluctuations. It is illustrated by an analysis of the bovine pancreatic trypsin inhibitor. Comparison of the covariance and cross-correlation matrices shows that the relative motions have many similar features in the different simulations. Many regions of the protein, especially regions of secondary structure, move in a correlated manner. Anharmonic effects, which are included in the molecular dynamics simulations but not in the normal analysis, are of some importance in determining the larger scale collective motions, but not the more local fluctuations. Comparisons of molecular dynamics simulations in the present and absence of solvent indicate that the environment is of significance for the long-range motions.     

Multiscale modeling of macromolecular conformational changes combining concepts from rigidity and elastic network theory

The development of a two-step approach for multiscale modeling of macromolecular conformational changes is based on recent developments in rigidity and elastic network theory. In the first step, static properties of the macromolecule are determined by decomposing the molecule into rigid clusters by using the graph-theoretical approach FIRST and an all-atom representation of the protein. In this way, rigid clusters are not limited to consist of residues adjacent in sequence or secondary structure elements as in previous studies. Furthermore, flexible links between rigid clusters are identified and can be modeled as such subsequently. In the second step, dynamical properties of the molecule are revealed by the rotations-translations of blocks approach (RTB) using an elastic network model representation of the coarse-grained protein. In this step, only rigid body motions are allowed for rigid clusters, whereas links between them are treated as fully flexible. The approach was tested on a data set of 10 proteins that showed conformational changes on ligand binding. For efficiency, coarse-graining the protein results in a remarkable reduction of memory requirements and computational times by factors of 9 and 27 on average and up to 25 and 125, respectively. For accuracy, directions and magnitudes of motions predicted by our approach agree well with experimentally determined ones, despite embracing in extreme cases >50% of the protein into one rigid cluster. In fact, the results of our method are in general comparable with when no or a uniform coarse-graining is applied; and the results are superior if the movement is dominated by loop or fragment motions. This finding indicates that explicitly distinguishing between flexible and rigid regions is advantageous when using a simplified protein representation in the second step. Finally, motions of atoms in rigid clusters are also well predicted by our approach, which points to the need to consider mobile protein regions in addition to flexible ones when modeling correlated motions.     

Models with energy penalty on interresidue rotation address insufficiencies of conventional elastic network models

In this study, I present a new elastic network model, to our knowledge, that addresses insufficiencies of two conventional models—the Gaussian network model (GNM) and the anisotropic network model (ANM). It has been shown previously that the GNM is not rotation-invariant due to its energy, which penalizes rigid-body rotation (external rotation). As a result, GNM models are found contaminated with rigid-body rotation, especially in the most collective ones. A new model (EPIRM) is proposed to remove such external component in modes. The extracted internal motions result from a potential that penalizes interresidue stretching and rotation in a protein. The new model is shown to pertinently describe crystallographic temperature factors (B-factors) and protein open↔closed transitions. Also, the capability of separating internal and external motions in GNM slow modes permits reexamining important mechanochemical properties in enzyme active sites. The results suggest that catalytic residues stay closer to rigid-body rotation axes than their immediate backbone neighbors. I show that the cumulative density of states for EPIRM and ANM follow different power laws as functions of low-mode frequencies. When using a cutoff distance of 7.5 Å, The cumulative density of states of EPIRM scales faster than that of all-atom normal mode analysis and slower than that of simple lattices.     

Defining Coarse-Grained Representations of Large Biomolecules and Biomolecular Complexes from Elastic Network Models

Coarse-grained (CG) models of large biomolecular complexes enable simulations of these systems over long timescales that are not accessible for atomistic molecular dynamics (MD) simulations. A systematic methodology, called essential dynamics coarse-graining (ED-CG), has been developed for defining coarse-grained sites in a large biomolecule. The method variationally determines the CG sites so that key dynamic domains in the protein are preserved in the CG representation. The original ED-CG method relies on a principal component analysis (PCA) of a MD trajectory. However, for many large proteins and multi-protein complexes such an analysis may not converge or even be possible. This work develops a new ED-CG scheme using an elastic network model (ENM) of the protein structure. In this procedure, the low-frequency normal modes obtained by ENM are used to define dynamic domains and to define the CG representation accordingly. The method is then applied to several proteins, such as the HIV-1 CA protein dimer, ATP-bound G-actin, and the Arp2/3 complex. Numerical results show that ED-CG with ENM (ENM-ED-CG) is much faster than ED-CG with PCA because no MD is necessary. The ENM-ED-CG models also capture functional essential dynamics of the proteins almost as well as those using full MD with PCA. Therefore, the ENM-ED-CG method may be better suited to coarse-grain a very large biomolecule or biomolecular complex that is too computationally expensive to be simulated by conventional MD, or when a high resolution atomic structure is not even available.     

Deriving correlated motions in proteins from X-ray structure refinement by using TLS parameters

Dynamic information in proteins may provide valuable information for understanding allosteric regulation of protein complexes or long-range effects of the mutations on enzyme activity. Experimental data such as X-ray B-factors or NMR order parameters provide a convenient estimate of atomic fluctuations (or atomic auto-correlated motions) in proteins. However, it is not as straightforward to obtain atomic cross-correlated motions in proteins — one usually resorts to more sophisticated computational methods such as Molecular Dynamics, normal mode analysis or atomic network models. In this report, we show that atomic cross-correlations can be reliably obtained directly from protein structure using X-ray refinement data. We have derived an analytic form of atomic correlated motions in terms of the original TLS parameters used to refine the B-factors of X-ray structures. The correlated maps computed using this equation are well correlated with those of the method based on a mechanical model (the correlation coefficient is 0.75) for a non-homologous dataset comprising 100 structures. We have developed an approach to compute atomic cross-correlations directly from X-ray protein structure. Being in analytic form, it is fast and provides a feasible way to compute correlated motions in proteins in a high throughput way. In addition, avoiding sophisticated computational operations; it provides a quick, reliable way, especially for non-computational biologists, to obtain dynamics information directly from protein structure relevant to its function.     

Reorientational contact-weighted elastic network model for the prediction of protein dynamics: comparison with NMR relaxation

A new model for the prediction of protein backbone motions is presented. The model, termed reorientational contact-weighted elastic network model, is based on a multidimensional reorientational harmonic potential of the backbone amide bond vector orientations and it is applied to the interpretation of dynamics parameters obtained from NMR relaxation data. The individual energy terms are weighted as a function of the intervector distances and by the contact strengths of each bond vector with respect to its local environment. Correlated reorientational motional properties of the bond vectors are obtained by means of normal mode analysis. Application to a set of proteins with known three-dimensional structures yields good to excellent agreement between predicted and experimental NMR order parameters presenting an improvement over the local contact model. The reorientational eigenmodes of the reorientational contact-weighted elastic network model method provide direct information on the collective nature of protein backbone motions. The dominant eigenmodes have a notably low collectivity, which is consistent with the behavior found for reorientational eigenmodes from molecular dynamics simulations.     

REACH coarse-grained biomolecular simulation: transferability between different protein structural classes

Coarse graining of protein interactions provides a means of simulating large biological systems. The REACH (Realistic Extension Algorithm via Covariance Hessian) coarse-graining method, in which the force constants of a residue-scale elastic network model are calculated from the variance-covariance matrix obtained from atomistic molecular dynamics (MD) simulation, involves direct mapping between scales without the need for iterative optimization. Here, the transferability of the REACH force field is examined between protein molecules of different structural classes. As test cases, myoglobin (all α), plastocyanin (all β), and dihydrofolate reductase (α/β) are taken. The force constants derived are found to be closely similar in all three proteins. An MD version of REACH is presented, and low-temperature coarse-grained (CG) REACH MD simulations of the three proteins are compared with atomistic MD results. The mean-square fluctuations of the atomistic MD are well reproduced by the CGMD. Model functions for the CG interactions, derived by averaging over the three proteins, are also shown to produce fluctuations in good agreement with the atomistic MD. The results indicate that, similarly to the use of atomistic force fields, it is now possible to use a single, generic REACH force field for all protein studies, without having first to derive parameters from atomistic MD simulation for each individual system studied. The REACH method is thus likely to be a reliable way of determining spatiotemporal motion of a variety of proteins without the need for expensive computation of long atomistic MD simulations.     

Systematic multiscale parameterization of heterogeneous elastic network models of proteins

We present a method to parameterize heterogeneous elastic network models (heteroENMs) of proteins to reproduce the fluctuations observed in atomistic simulations. Because it is based on atomistic simulation, our method allows the development of elastic coarse-grained models of proteins under different conditions or in different environments. The method is simple and applicable to models at any level of coarse-graining. We validated the method in three systems. First, we computed the persistence length of ADP-bound F-actin, using a heteroENM model. The value of 6.1 ± 1.6 μm is consistent with the experimentally measured value of 9.0 ± 0.5 μm. We then compared our method to a uniform elastic network model and a realistic extension algorithm via covariance Hessian (REACH) model of carboxy myoglobin, and found that the heteroENM method more accurately predicted mean-square fluctuations of α-carbon atoms. Finally, we showed that the method captures critical differences in effective harmonic interactions for coarse-grained models of the N-terminal Bin/amphiphysin/Rvs (N-BAR) domain of amphiphysin, by building models of N-BAR both bound to a membrane and free in solution.     

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.     

hdANM: a new comprehensive dynamics model for protein hinges

Hinge motions are essential for many protein functions, and their dynamics are important to understand underlying biological mechanisms. The ways that these motions are represented by various computational methods differ significantly. By focusing on a specific class of motion, we have developed a new hinge-domain anisotropic network model (hdANM) that is based on the prior identification of flexible hinges and rigid domains in the protein structure and the subsequent generation of global hinge motions. This yields a set of motions in which the relative translations and rotations of the rigid domains are modulated and controlled by the deformation of the flexible hinges, leading to a more restricted, specific view of these motions. hdANM is the first model, to our knowledge, that combines information about protein hinges and domains to model the characteristic hinge motions of a protein. The motions predicted with this new elastic network model provide important conceptual advantages for understanding the underlying biological mechanisms. As a matter of fact, the generated hinge movements are found to resemble the expected mechanisms required for the biological functions of diverse proteins. Another advantage of this model is that the domain-level coarse graining makes it significantly more computationally efficient, enabling the generation of hinge motions within even the largest molecular assemblies, such as those from cryo-electron microscopy. hdANM is also comprehensive as it can perform in the same way as the well-known protein dynamics models (anisotropic network model, rotations-translations of blocks, and nonlinear rigid block normal mode analysis), depending on the definition of flexible and rigid parts in the protein structure and on whether the motions are extrapolated in a linear or nonlinear fashion. Furthermore, our results indicate that hdANM produces more realistic motions as compared to the anisotropic network model. hdANM is an open-source software, freely available, and hosted on a user-friendly website.     

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.     

Protein dynamics and enzymatic catalysis: investigating the peptidyl-prolyl cis-trans isomerization activity of cyclophilin A

A growing body of evidence suggests a connection between protein dynamics and enzymatic catalysis. In this paper, we present a variety of computational studies designed to investigate the role of protein dynamics in the detailed mechanism of peptidyl-prolyl cis-trans isomerization catalyzed by human cyclophilin A. The results identify a network of protein vibrations, extending from surface regions of the enzyme to the active site and coupled to substrate turnover. Indications are that this network may have a role in promoting catalysis. Crucial parts of this network are found to be conserved in 10 cyclophilin structures from six different species. Experimental evidence for the existence of this network comes from previous NMR relaxation studies, where motions in several residues, forming parts of this network, were detected only during substrate turnover. The high temperature factors (from X-ray crystal structures) associated with the network residues provide further evidence of these vibrations. Along with the knowledge of enzyme structure, this type of network could provide new insights into enzymatic catalysis and the effect of distant ligand binding on protein function. The procedure outlined in this paper is general and can be applied to other enzymatic systems as well. This presents an interesting opportunity; collaborative experimental and theoretical investigations designed to characterize in detail the nature and function of this type of network could enhance the understanding of protein dynamics in enzymatic catalysis.     

Coarse-grained normal mode analysis in structural biology

The realization that experimentally observed functional motions of proteins can be predicted by coarse-grained normal mode analysis has renewed interest in applications to structural biology. Notable applications include the prediction of biologically relevant motions of proteins and supramolecular structures driven by their structure-encoded collective dynamics; the refinement of low-resolution structures, including those determined by cryo-electron microscopy; and the identification of conserved dynamic patterns and mechanically key regions within protein families. Additionally, hybrid methods that couple atomic simulations with deformations derived from coarse-grained normal mode analysis are able to sample collective motions beyond the range of conventional molecular dynamics simulations. Such applications have provided great insight into the underlying principles linking protein structures to their dynamics and their dynamics to their functions.     

Coarse-grained biomolecular simulation with REACH: realistic extension algorithm via covariance Hessian

Coarse-graining of protein interactions provides a means of simulating large biological systems. Here, a coarse-graining method, REACH, is introduced, in which the force constants of a residue-scale elastic network model are calculated from the variance-covariance matrix obtained from atomistic molecular dynamics (MD) simulation. In test calculations, the C(alpha)-atoms variance-covariance matrices are calculated from the ensembles of 1-ns atomistic MD trajectories in monomeric and dimeric myoglobin, and used to derive coarse-grained force constants for the local and nonbonded interactions. Construction of analytical model functions of the distance-dependence of the interresidue force constants allows rapid calculation of the REACH normal modes. The model force constants from monomeric and dimeric myoglobin are found to be similar in magnitude to each other. The MD intra- and intermolecular mean-square fluctuations and the vibrational density of states are well reproduced by the residue-scale REACH normal modes without requiring rescaling of the force constant parameters. The temperature-dependence of the myoglobin REACH force constants reveals that the dynamical transition in protein internal fluctuations arises principally from softening of the elasticity in the nonlocal interactions. The REACH method is found to be a reliable way of determining spatiotemporal protein motion without the need for expensive computations of long atomistic MD simulations.     

ProMode: a database of normal mode analyses on protein molecules with a full-atom model

MOTIVATION: Although information from protein dynamics simulation is important to understand principles of architecture of a protein structure and its function, simulations such as molecular dynamics and Monte Carlo are very CPU-intensive. Although the ability of normal mode analysis (NMA) is limited because of the need for a harmonic approximation on which NMA is based, NMA is adequate to carry out routine analyses on many proteins to compute aspects of the collective motions essential to protein dynamics and function. Furthermore, it is hoped that realistic animations of the protein dynamics can be observed easily without expensive software and hardware, and that the dynamic properties for various proteins can be compared with each other. RESULTS: ProMode, a database collecting NMA results on protein molecules, was constructed. The NMA calculations are performed with a full-atom model, by using dihedral angles as independent variables, faster and more efficiently than the calculations using Cartesian coordinates. In ProMode, an animation of the normal mode vibration is played with a free plug-in, Chime (MDL Information Systems, Inc.). With the full-atom model, the realistic three-dimensional motions at an atomic level are displayed with Chime. The dynamic domains and their mutual screw motions defined from the NMA results are also displayed. Properties for each normal mode vibration and their time averages, e.g. fluctuations of atom positions, fluctuations of dihedral angles and correlations between the atomic motions, are also presented graphically for characterizing the collective motions in more detail. AVAILABILITY: http://promode.socs.waseda.ac.jp