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International audience; Key steps in the life cycle of a virus, such as the fusion event as the virus infects a host cell and its maturation process, relate to an intricate interplay between the structure and the dynamics of its constituent proteins, especially those that define its capsid, much akin to an envelope that protects its genomic material. We present a comprehensive, comparative analysis of such interplay for the capsids of two viruses from the flaviviridae family, Dengue (DENV) and Zika (ZIKV). We use for that purpose our own software suite, DD-NMA, which is based on normal mode analysis. We describe the elements of DD-NMA that are relevant to the analysis of large systems, such as virus capsids. In particular, we introduce our implementation of simplified elastic networks and justify their parametrization. Using DD-NMA, we illustrate the importance of packing interactions within the virus capsids on the dynamics of the E proteins of DENV and ZIKV. We identify differences between the computed atomic fluctuations of the E proteins in DENV and ZIKV and relate those differences to changes observed in their high resolution structures. We conclude with a discussion on additional analyses that are needed to fully characterize the dynamics of the two viruses.

In this article we review the key modeling tools available for simulating biomolecular systems. We consider recent developments and representative applications of mixed quantum mechanics/molecular mechanics (QM/MM), elastic network models (ENMs), coarse-grained molecular dynamics, and grid-based tools for calculating interactions between essentially rigid protein assemblies. We consider how the different length scales can be coupled, both in a sequential fashion (e.g. a coarse-grained or grid model using parameterization from MD simulations), and via concurrent approaches, where the calculations are performed together and together control the progression of the simulation. We suggest how the concurrent coupling approach familiar in the context of QM/MM calculations can be generalized, and describe how this has been done in the CHARMM macromolecular simulation package.

Elastic network models (ENMs) based on simple harmonic potential energy function have been proven over the last decade to be reliable computational models for understanding the intrinsic dynamics of biomacromolecules. In the original ENMs, the spring constants for different contact pairs are assumed to be identical, while there are a number of recent developments to determine non-uniform spring constants from atomistic force fields or experimental information. In particular, the fluctuation matching approaches in coarse-grained modeling can be applied to build more realistic heterogeneous ENMs, using information from an atomistic force field or experimental B-factors. The same type of approaches is further implemented to parameterize heterogeneous structure-based models, which can be considered as a natural extension of ENMs in terms of the potential energy function. In this review, we give an overview of different fluctuation matching methods adopted for ENMs and structure-based models, including an improved formulation and algorithm based on the relative entropy scheme.

A multiscale coarse-graining method called the normal-mode analysis based fluctuation matching (NMA-FM) is developed for constructing coarse-grained models of biomolecular systems. In the framework of normal-mode analysis, an arbitrary fine-grained model can be systematically converted to a more coarse-grained model, while the crucial low-frequency motions of the fine-grained system are able to be reproduced in the coarse-grained modeling. The method relies on the technique of fluctuation matching that has been devised earlier for parametrizing heterogeneous elastic network models based on data from atomistic molecular dynamics simulations. The new approach is quite efficient since it avoids expensive atomistic molecular dynamics simulations and can start from already coarse-grained elastic network models. In the practical aspect, the method is suitable for conformational analyses of large biomacromolecules and calculations of mechanical properties of biomaterials, which is demonstrated by the studied systems including an amyloid dimer, lysozyme and adenylate kinase proteins, and the S2 subdomain of myosin.

Understanding the functions of biomolecules requires insight not only from structures but from dynamics as well. Often, the most interesting processes occur on time scales too slow for exploration by conventional molecular dynamics (MD) simulations. For this reason, alternative computational methods such as elastic network models (ENMs) have become increasingly popular. These simple, coarse-grained models represent molecules as beads connected by harmonic springs; the system's motions are solved analytically by normal-mode analysis. In the past few years, many different formalisms for performing ENM calculations have emerged, and several have been optimized using all-atom MD simulations. In contrast to other studies, we have compared the various formalisms in a systematic, quantitative way. In this study, we optimize many ENM functional forms using a uniform data set containing only long (>1 μs) all-atom MD simulations. Our results show that all models once optimized produce spring constants for immediate neighboring residues that are orders of magnitude stiffer than more distal contacts. In addition, the statistical significance of ENM performance varied with model resolution. We also show that fitting long trajectories does not improve ENM performance due to a problem inherent in all network models tested: they underestimate the relative importance of the most concerted motions. Finally, we characterize ENMs' resilience by tessellating the parameter space to show that broad ranges of parameters produce similar quality predictions. Taken together, our data reveal that the choice of spring function and parameters are not vital to the performance of a network model and that simple parameters can by derived "by hand" when no data are available for fitting, thus illustrating the robustness of these models.

Elastic networks have been used as simple models of proteins to study their slow structural dynamics. They consist of point-like particles connected by linear Hookean springs and hence are convenient for linear normal mode analysis around a given reference structure. Furthermore, dynamic simulations using these models can provide new insights. As the computational cost associated with these models is considerably lower compared to that of all-atom models, they are also convenient for comparative studies between multiple protein structures. In this review, we introduce examples of coarse-grained molecular dynamics studies using elastic network models and their derivatives, focusing on the nonlinear phenomena, and discuss their applicability to large-scale macromolecular assemblies.

Dynamic communities in proteins comprise the cohesive structural units that individually exhibit rigid body motions. These can correspond to structural domains, but are usually smaller parts that move with respect to one another in a protein’s internal motions, key to its functional dynamics. Previous studies emphasized their importance to understand the nature of ligand-induced allosteric regulation. These studies reported that mutations to key community residues can hinder transmission of allosteric signals among the communities. Usually molecular dynamic (MD) simulations (~ 100 ns or longer) have been used to identify the communities—a demanding task for larger proteins. In the present study, we propose that dynamic communities obtained from MD simulations can also be obtained alternatively with simpler models–the elastic network models (ENMs). To verify this premise, we compare the specific communities obtained from MD and ENMs for 44 proteins. We evaluate the correspondence in communities from the two methods and compute the extent of agreement in the dynamic cross-correlation data used for community detection. Our study reveals a strong correspondence between the communities from MD and ENM and also good agreement for the residue cross-correlations. Importantly, we observe that the dynamic communities from MD can be closely reproduced with ENMs. With ENMs, we also compare the community structures of stable and unstable mutant forms of T4 Lysozyme with its wild-type. We find that communities for unstable mutants show substantially poorer agreement with the wild-type communities than do stable mutants, suggesting such ENM-based community structures can serve as a means to rapidly identify deleterious mutants.

We combine two methods to enable the prediction of the order in which contacts are broken under external stretching forces in single molecule experiments. These two methods are Gō-like models and elastic network models. The Gō-like models have shown remarkable success in representing many aspects of protein behavior, including the reproduction of experimental data obtained from atomic force microscopy. The simple elastic network models are often used successfully to predict the fluctuations of residues around their mean positions, comparing favorably with the experimentally measured crystallographic B-factors. The behavior of biomolecules under external forces has been demonstrated to depend principally on their elastic properties and the overall shape of their structure. We have studied in detail the muscle protein titin and green fluorescent protein and tested for ten other proteins. First, we stretch the proteins computationally by performing stochastic dynamics simulations with the Gō-like model. We obtain the force–displacement curves and unfolding scenarios of possible mechanical unfolding. We then use the elastic network model to calculate temperature factors (B-factors) and compare the slowest modes of motion for the stretched proteins and compare them with the predicted order of breaking contacts between residues in the Gō-like model. Our results show that a simple Gaussian network model is able to predict contacts that break in the next time stage of stretching. Additionally, we have found that the contact disruption is strictly correlated with the highest force exerted by the backbone on these residues. Our prediction of bond-breaking agrees well with the unfolding scenario obtained with the Gō-like model. We anticipate that this method will be a useful new tool for interpreting stretching experiments. Proteins 2008. © 2007 Wiley-Liss, Inc.

We have developed LOOS (Lightweight Object Oriented Structure-analysis) as a tool for making new tools to analyze molecular simulations. LOOS is an object-oriented library designed to facilitate the rapid development of new methods for structural analysis. LOOS is written in C++ and is easily extensible, only requiring knowledge of 4 core classes. A Python interface is also available, further facilitating rapid development of analysis tools and broadening the LOOS community. LOOS reads the native file formats of most common simulation packages and can write NAMD formats (PDB and DCD) and Gromacs XTC. A dynamic atom selection language, based on C expression syntax, is included and is easily accessible via a single function call. LOOS includes over 140 pre-built tools for common structural analysis tasks, including analyzing simulation convergence, 3D histograms, and elastic network models. Python-based packages include the optimal membrane generator and tools for the Voronoi tesselation of membranes. In addition, tool templates for common analysis work-flows are included to facilitate the development of new tools. LOOS is available for download at http://loos.sourceforge.net and is distributed under the GPLv3 license.

The role of ribonucleic acid (RNA) in molecular biology is shifting from a mere messenger between DNA and proteins to an important player in many cellular activities. The central role of RNA molecules calls for a precise characterization of their structural and dynamical properties. Nowadays, experiments can be efficiently complemented by computational approaches. Within this framework, elastic network models (ENMs) are valuable and efficient tools for characterizing the collective internal dynamics of biomolecules starting from the sole knowledge of their structures. The increasing evidence that the biological functionality of RNAs is often linked to their innate internal motions, poses the question of whether ENM approaches can be successfully extended to these biomolecules. This issue, which is still largely unexplored, is tackled here by considering various families of elastic networks for a representative set of RNAs. The large-scale motions predicted by the alternative ENMs are stringently validated by comparison against extensive molecular dynamics (MD) simulations and SHAPE experimental data. We propose a specific combination of three ENM centroids (sugar-base-phosphate) as an optimal compromise capable of reproducing simulations and experiments.

The intrinsic flexibility of proteins and nucleic acids can be grasped from remarkably simple mechanical models of particles connected by springs. In recent decades, Elastic Network Models (ENMs) combined with Normal Model Analysis widely confirmed their ability to predict biologically relevant motions of biomolecules and soon became a popular methodology to reveal large-scale dynamics in multiple structural biology scenarios. The simplicity, robustness, low computational cost, and relatively high accuracy are the reasons behind the success of ENMs. This review focuses on recent advances in the development and application of ENMs, paying particular attention to combinations with experimental data. Successful application scenarios include large macromolecular machines, structural refinement, docking, and evolutionary conservation.

Conformational changes of proteins during binding to other biomolecules play a vital role in many biological processes in a living body. There have been a lot of efforts to predict the conformational changes using normal modes of various elastic network models. In all the studies, usually a single or a few lowest modes are considered. In this study, we consider the contribution of all normal modes on the unbound structure of protein to predict the bound form. The results show that low frequency normal modes are not sufficient for describing these motions and there are contributing normal modes even in the high frequency range. The results also indicate that high decaying and low cutoff radii network models show similar behavior in their prediction of the conformational changes

We develop a mixed elastic network model (MENM) to study large-scale conformational transitions of proteins between two (or more) known structures. Elastic network potentials for the beginning and end states of a transition are combined, in effect, by adding their respective partition functions. The resulting effective MENM energy function smoothly interpolates between the original surfaces, and retains the beginning and end structures as local minima. Saddle points, transition paths, potentials of mean force, and partition functions can be found efficiently by largely analytic methods. To characterize the protein motions during a conformational transition, we follow "transition paths" on the MENM surface that connect the beginning and end structures and are invariant to parameterizations of the model and the mathematical form of the mixing scheme. As illustrations of the general formalism, we study large-scale conformation changes of the motor proteins KIF1A kinesin and myosin II. We generate possible transition paths for these two proteins that reveal details of their conformational motions. The MENM formalism is computationally efficient and generally applicable even for large protein systems that undergo highly collective structural changes.

Elastic network models (ENMs) are coarse-grained descriptions of proteins as networks of coupled harmonic oscillators. However, despite their widespread application to study collective movements, there is still no consensus parametrization for the ENMs. When compared to molecular dynamics (MD) flexibility in solution, the ENMs tend to disperse the important motions into multiple modes. We present here a new ENM, trained against a database of atomistic MD trajectories. The role of residue connectivity, the analytical form of the force constants, and the threshold for interactions were systematically explored. We found that contacts between the three nearest sequence neighbors are crucial determinants of the fundamental motions. We developed a new general potential function including both the sequential and spatial relationships between interacting residue pairs which is robust against size and fold variations. The proposed model provides a systematic improvement compared to standard ENMs: Not only do its results match the MD results-even for long time scales-but also the model is able to capture large X-ray conformational transitions as well as NMR ensemble diversity.

Elastic network models (ENMs) are a useful tool for describing large scale motions in protein systems. While they are well validated in the context of proteins, relatively little is known about their applicability to nucleic acids, whose different architecture does not necessarily warrant comparable performance. In this study we thoroughly evaluate and optimize the efficiency of popular ENMs for capturing RNA and DNA flexibility. We also introduce two alternative models in which the strength of elastic connections at a coarse-grained level is governed by distance distribution at atomic resolution. For each of the considered ENMs we report the optimal length of spring connections as well as the scaling of elastic force constants that provides the best agreement of vibrational frequencies with normal modes based on atomic force field. In order to determine the absolute values of force constants we introduce a novel method based on the overlap of pseudoinverse of Hessian matrices.

Recent structure-based computational studies suggest that, in contrast to the classical description of equilibrium fluctuations as wigglings and jigglings, proteins have access to well-defined spectra of collective motions, called intrinsic dynamics, encoded by their structure under native state conditions. In particular, the global modes of motions (at the low frequency end of the spectrum) are shown by multiple studies to be highly robust to minor differences in the structure or to detailed interactions at the atomic level. These modes, encoded by the overall fold, usually define the mechanisms of interactions with substrates. They can be estimated by low-resolution models such as the elastic network models (ENMs) exclusively based on interresidue contact topology. The ability of ENMs to efficiently assess the global motions intrinsically favored by the overall fold as well as the relevance of these predictions to the dominant changes in structure experimentally observed for a given protein in the presence of different substrates suggest that the intrinsic dynamics plays a role in mediating protein-substrate interactions. These observations underscore the functional significance of structure-encoded dynamics, or the importance of the predisposition to favor functional global modes in the evolutionary selection of structures.

Elastic network models (ENMs) are valuable and efficient tools for characterizing the collective internal dynamics of proteins based on the knowledge of their native structures. The increasing evidence that the biological functionality of RNAs is often linked to their innate internal motions, poses the question of whether ENM approaches can be successfully extended to this class of biomolecules. This issue is tackled here by considering various families of elastic networks of increasing complexity applied to a representative set of RNAs. The fluctuations predicted by the alternative ENMs are stringently validated by comparison against extensive molecular dynamics simulations and SHAPE experiments. We find that simulations and experimental data are systematically best reproduced by either an all-atom or a three-beads-per-nucleotide representation (sugar-base-phosphate), with the latter arguably providing the best balance of accuracy and computational complexity., This article has been accepted for publication in Nucleic Acids Research Published by Oxford University Press

With advances in structure genomics, it is now recognized that knowledge of structure alone is insufficient to understand and control the mechanisms of biomolecular function. Additional information in the form of dynamics is needed. As demonstrated in a large number of studies, the machinery of proteins and their complexes can be understood to a good approximation by adopting Gaussian (or elastic) network models (GNM) for simplified normal mode analyses. While this approximation lacks chemical details, it provides us with a means for assessing the collective motions of large structures/assemblies and perform a comparative analysis of a series of proteins, thus providing insights into the mechanical aspects of biomolecular dynamics. In this paper, we discuss recent applications of GNM to a series of enzymes as well as large structures such as the HK97 bacteriophage viral capsids. Understanding the dynamics of large protein structures can be computationally challenging. To this end, we introduce a new approach for building a hierarchical, reduced rank representation of the protein topology and consequently the fluctuation dynamics.

Coarse-grained models for proteins and biomolecular aggregates have recently enjoyed renewed interest. Coarse-grained representations combined with enhanced computer power currently allow the simulation of systems of biologically relevant size (submicrometric) and timescale (microsecond or millisecond). Although these techniques still cannot be considered as predictive as all-atom simulations, noticeable advances have recently been achieved, mainly concerning the use of more rigorous parameterization techniques and novel algorithms for sampling configurational space. Moreover, the simulation size scales and timescales coincide with those that can be reached with the most advanced spectroscopic techniques, making it possible to directly compare simulation and experiment.

Enzymes have evolved during millions of years to become efficient catalysts for specific biochemical reactions within a specific range of working conditions in the cellular environment. The activity of an enzyme is directly related to its folded structure and even slight changes in its 3D conformation may cause irreversible, negative effects on its activity. The structure of enzymes are very sensitive to the environmental conditions and changes from their optimal conditions, like higher temperature, salt concentration, and pH, might result in denaturation and subsequently inactivation. In the past decades, natural enzymes extracted from different organisms have found a wide range of applications in the industrial and biotechnological setting. For the majority of the applications their activity at high temperatures is more favorable, however enzymes have evolved, in most of the cases, to optimally work in limited range of temperature of their native cellular environment. Therefore, enhancing enzyme thermostability will not only increase their application range, but could also shed light into new aspects of their evolution and chemical activity. The physical chemical principles underlying enzymatic thermostability are keys in fact to understand the way evolution has shaped proteins to adapt to a broad range of temperatures. Understanding the molecular determinants at the basis of protein thermostability, using both in silico methods and in vitro experiments, is also an important way for engineering more thermo-resistant enzymes to be used in the industrial setting, as for instance DNA ligases, which are important for DNA replication and repair and have been long used in molecular biology and biotechnology. In this thesis I used in silico techniques, like molecular modeling and simulation coupled with bioinformatics analyses, to assess existing methods and predict potential thermo-stabilizing mutations for target proteins. First, I studied a thermophilic protein and after exploring the origins of its thermostability I proposed mutations to further increase its thermostability. Then, I took advantage of what learned from this study to explore further thermostability engineering methods in order to develop faster, accurate, and easy-to-use methods that can be generally used for a broad array of proteins. 1. Understanding and engineering thermostability in the DNA ligase from Thermococcus sp. 1519 (LigTh1519). In this thesis, I first addressed the origin of thermostability in the thermophilic DNA ligase from archaeon Thermococcus sp. 1519, and identified thermo-sensitive regions using molecular modeling and simulations. In addition, I predicted mutations that can enhance thermostability of the enzyme through bioinformatics analyses. I showed that thermo-sensitive regions of this enzyme are stabilized at higher temperatures by optimization of charged groups on the surface, and predicted that thermostability can be further increased by further optimization of the network among these charged groups. Engineering this DNA ligase by introducing selected mutations (i.e., A287K, G304D, S364I and A387K) produced eventually a significant and additive increase in the half-life time of the enzyme when compared to the wild-type. Then, based on what I learned from thermostability analyses and improvement of LigTh1519, my aim was to design a general-purpose protein thermostability engineering protocol that can enable thermostability engineering [...]

It is often valuable to compare protein structures to determine how similar they are. Structure comparison methods such as RMSD and GDT-TS are based solely on fixed geometry and do not take into account the intrinsic flexibility or energy landscape of the protein. We propose a method, which we call FlexE, that is based on a simple elastic network model and uses the deformation energy as measure of the similarity between two structures. FlexE can distinguish biologically relevant conformational changes from random changes, while existing geometry-based methods cannot. Additionally, FlexE incorporates the concept of thermal energy, which provides a rational way to determine when two models are “the same”. FlexE provides a unique measure of the similarity between protein structures that is complementary to existing methods.

The Gaussian network model (GNM) and anisotropic network model (ANM) are two elastic network models that have been widely used to study protein fluctuation dynamics. Both models have strengths and weaknesses. Attempts have been made in the past to unify the two models but they had severe limitations. This work presents a novel theoretical result that shows how GNM and ANM can be unified through taking into account the effect of inter-residue forces. The unified model, called the force spring model, or FSM, is reduced to ANM when all the inter-residue forces are set to zero. Moreover, the unification reveals the role of inter-residue forces in protein fluctuation dynamics. Specifically, the effect of inter-residue forces is closely examined by studying the changes in mean-square fluctuations when inter-residue forces are present.

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.

Protein structures are dynamic entities with a myriad of atomic fluctuations, side-chain rotations, and collective domain movements. Although the importance of these dynamics to proper functioning of proteins is emerging in the studies of many protein families, there is a lack of broad evidence for the critical role of protein dynamics in shaping the biological functions of a substantial fraction of residues for a large number of proteins in the human proteome. Here, we propose a novel dynamic flexibility index (dfi) to quantify the dynamic properties of individual residues in any protein and use it to assess the importance of protein dynamics in 100 human proteins. Our analyses involving functionally critical positions, disease-associated and putatively neutral population variations, and the rate of interspecific substitutions per residue produce concordant patterns at a proteome scale. They establish that the preservation of dynamic properties of residues in a protein structure is critical for maintaining the protein/biological function. Therefore, structural dynamics needs to become a major component of the analysis of protein function and evolution. Such analyses will be facilitated by the dfi, which will also enable the integrative use of structural dynamics with evolutionary conservation in genomic medicine as well as functional genomics investigations. © 2013 The Authors. Published by Blackwell Publishing Ltd.

Background: Normal Mode Analysis is one of the most successful techniques for studying motions in proteins and macromolecules. It can provide information on the mechanism of protein functions, used to aid crystallography and NMR data reconstruction, and calculate protein free energies. Results: ΔΔPT is a toolbox allowing calculation of elastic network models and principle component analysis. It allows the analysis of pdb files or trajectories taken from; Gromacs, Amber, and DL_POLY. As well as calculation of the normal modes it also allows comparison of the modes with experimental protein motion, variation of modes with mutation or ligand binding, and calculation of molecular dynamic entropies. Conclusions: This toolbox makes the respective tools available to a wide community of potential NMA users, and allows them unrivalled ability to analyse normal modes using a variety of techniques and current software.

Protein structures are dynamic entities with a myriad of atomic fluctuations, side-chain rotations, and collective domain movements. Although the importance of these dynamics to proper functioning of proteins is emerging in the studies of many protein families, there is a lack of broad evidence for the critical role of protein dynamics in shaping the biological functions of a substantial fraction of residues for a large number of proteins in the human proteome. Here, we propose a novel dynamic flexibility index (dfi) to quantify the dynamic properties of individual residues in any protein and use it to assess the importance of protein dynamics in 100 human proteins. Our analyses involving functionally critical positions, disease-associated and putatively neutral population variations, and the rate of interspecific substitutions per residue produce concordant patterns at a proteome scale. They establish that the preservation of dynamic properties of residues in a protein structure is critical for maintaining the protein/biological function. Therefore, structural dynamics needs to become a major component of the analysis of protein function and evolution. Such analyses will be facilitated by the dfi, which will also enable the integrative use of structural dynamics with evolutionary conservation in genomic medicine as well as functional genomics investigations.

International audience; This paper presents a method, inspired by robot motion planning algorithms, to model conformational transitions in proteins. The capacity of normal mode analysis to predict directions of collective large-amplitude motions is exploited to bias the conformational exploration. A coarse-grained elastic network model built on short fragments of three residues is proposed for the rapid computation of normal modes. The accurate reconstruction of the all-atom model from the coarse-grained one is achieved using closed-form inverse kinematics. Results show the capacity of the method to model conformational transitions of proteins within a few hours of computing time on a single processor. Tests on a set of ten proteins demonstrate that the computing time scales linearly with the protein size, independently of the protein topology. Further experiments on adenylate kinase show that main features of the transition between the open and closed conformations of this protein are well captured in the computed path.

Fifteen years ago, Monique Tirion showed that the low-frequency normal modes of a protein are not significantly altered when nonbonded interactions are replaced by Hookean springs, for all atom pairs whose distance is smaller than a given cutoff value. Since then, it has been shown that coarse-grained versions of Tirion's model are able to provide fair insights on many dynamical properties of biological macromolecules. In this chapter, theoretical tools required for studying these so-called Elastic Network Models are described, focusing on practical issues and, in particular, on possible artifacts. Then, an overview of some typical results that have been obtained by studying such models is given.

Identifying the functional motions of membrane proteins is difficult because they range from large-scale collective dynamics to local small atomic fluctuations at different timescales that are difficult to measure experimentally due to the hydrophobic nature of these proteins. Elastic Network Models, and in particular their most widely used implementation, the Anisotropic Network Model (ANM), have proven to be useful computational methods in many recent applications to predict membrane protein dynamics. These models are based on the premise that biomolecules possess intrinsic mechanical characteristics uniquely defined by their particular architectures. In the ANM, interactions between residues in close proximity are represented by harmonic potentials with a uniform spring constant. The slow mode shapes generated by the ANM provide valuable information on the global dynamics of biomolecules that are relevant to their function. In its recent extension in the form of ANM-guided molecular dynamics (MD), this coarse-grained approach is augmented with atomic detail. The results from ANM and its extensions can be used to guide experiments and thus speedup the process of quantifying motions in membrane proteins. Testing the predictions can be accomplished through (a) direct observation of motions through studies of structure and biophysical probes, (b) perturbation of the motions by, e.g., cross-linking or site-directed mutagenesis, and (c) by studying the effects of such perturbations on protein function, typically through ligand binding and activity assays. To illustrate the applicability of the combined computational ANM-experimental testing framework to membrane proteins, we describe-alongside the general protocols-here the application of ANM to rhodopsin, a prototypical member of the pharmacologically relevant G-protein coupled receptor family.

Elastic network models (ENMs) offer the capability to study collective motions of large systems without the overwhelming computational cost of all-atom molecular dynamics (MD). They accomplish this by replacing the standard force field terms with simple harmonic connections, typically between neighboring alpha carbons; the collective motions can then be computed using normal mode analysis. We investigate new methods for performing ENM calculations to improve their sensitivity and fidelity. First, we have explored using a higher resolution representation for the protein, supplementing the standard 1-node per residue approach by adding a second node to represent the side chain. Our aim is to utilize this increase in spatial resolution, enabling the ENMs to predict the effects of subtler changes, such as the binding of small ligands. Second, we have explored the efficacy of different choices for the spring constants in the harmonic representation. Following our previous work (Romo and Grossfield. Proteins, 2010), we have implemented a new functional form, which distinguishes bonded and nonbonded interactions. In both cases, we have validated our approaches via quantitative comparison to several microsecond-scale all-atom simulations of G protein-coupled receptors.

Endoglucanases are crucial enzymes used in the production of biofuels from cellulosic biomass, a process which requires thermostability at high processing temperatures. Despite the economic importance of these industrial proteins, we currently lack a basic understanding of how some endoglucanases can efficiently function at elevated processing temperatures, while others with the same fold have substantial reduction in activity.Here we explore the origins of thermostability in endoglucanases from sequence, structure, and dynamics perspectives using thermostable and mesostable protein sets. We performed a comparative sequence and structure analysis for thermophilic and mesophilic endoglucanases in (α/β)8, β-jelly roll, and (α/α)6 folds, followed by a dynamics analysis of the (α/β)8 fold using elastic network models. We observed that thermophilic endoglucanases and their mesophilic counterparts differ significantly in their amino acid compositions. Interestingly, these compositional differences are specific to protein folds and enzyme families and lead to modification in hydrophobic, aromatic, and ionic interactions in a fold-dependent fashion.We then focused specifically on a pair of thermostable and mesostable endoglucanases for a detailed dynamics analysis. It is often the case that thermophiles have shorter loops than their mesophilic counterparts, which was suggested to impart thermostability. In our case, however, the thermophile surprisingly possessed three insertions in the mesophilic loop regions and therefore has longer loops. The comparative structural dynamics analysis using elastic network models of (α/β)8 fold indicate that these three loops may contribute to the thermostability by modulating the direction of correlated motions between the catalytic residues (acid/base donor and nucleophile). We also observed that the thermostable protein showed larger dynamic domains than its mesostable counterpart, which suggests that cooperative dynamics is a critical contributing factor to thermostability.

Flexibility and dynamics of protein structures are reflected in the B-factors and order parameters obtained experimentally with X-ray crystallography and Nuclear Magnetic Resonance (NMR). Methods such as Normal Mode Analysis (NMA) and Elastic Network Models (ENM) can be used to predict the fluctuations of protein structures for either atomic level or coarse-grained structures. Here, we introduce the Local-Distance Matrix Error (DME), an efficient and simple analytic method to study the fluctuations of protein structures, especially for the ensembles of NMR-determined protein structures. Comparisons with the fluctuations obtained by experiments and other by computations show strong correlations.

Elastic network models of biomolecules have proved to be relatively good at predicting global conformational changes particularly in large systems. Software that facilitates rapid and intuitive exploration of conformational change in elastic network models of large biomolecules in response to externally applied forces would therefore be of considerable use, particularly if the forces mimic those that arise in the interaction with a functional ligand. We have developed software that enables a user to apply forces to individual atoms of an elastic network model of a biomolecule through a haptic feedback device or a mouse. With a haptic feedback device the user feels the response to the applied force whilst seeing the biomolecule deform on the screen. Prior to the interactive session normal mode analysis is performed, or pre-calculated normal mode eigenvalues and eigenvectors are loaded. For large molecules this allows the memory and number of calculations to be reduced by employing the idea of the important subspace, a relatively small space of the first M lowest frequency normal mode eigenvectors within which a large proportion of the total fluctuation occurs. Using this approach it was possible to study GroEL on a standard PC as even though only 2.3% of the total number of eigenvectors could be used, they accounted for 50% of the total fluctuation. User testing has shown that the haptic version allows for much more rapid and intuitive exploration of the molecule than the mouse version.

SummaryMany proteins require bound metals to achieve their function. We take advantage of increasing structural data on metal-binding proteins to elucidate three properties: the involvement of metal-binding sites in the global dynamics of the protein, predicted by elastic network models, their exposure/burial to solvent, and their signal-processing properties indicated by Markovian stochastics analysis. Systematic analysis of a data set of 145 structures reveals that the residues that coordinate metal ions enjoy remarkably efficient and precise signal transduction properties. These properties are rationalized in terms of their physical properties: participation in hinge sites that control the softest modes collectively accessible to the protein and occupancy of central positions minimally exposed to solvent. Our observations suggest that metal-binding sites may have been evolutionary selected to achieve optimum allosteric communication. They also provide insights into basic principles for designing metal-binding sites, which are verified to be met by recently designed de novo metal-binding proteins.

The decryption of sequence of structural events during protein conformational transitions is essential to a detailed understanding of molecular functions of various biological nanomachines. Coarse-grained models have proven useful by allowing highly efficient simulations of protein conformational dynamics. By combining two coarse-grained elastic network models constructed based on the beginning and end conformations of a transition, we have developed an interpolated elastic network model to generate a transition pathway between the two protein conformations. For validation, we have predicted the order of local and global conformational changes during key ATP-driven transitions in three important biological nanomachines (myosin, F(1) ATPase and chaperonin GroEL). We have found that the local conformational change associated with the closing of active site precedes the global conformational change leading to mechanical motions. Our finding is in good agreement with the distribution of intermediate experimental structures, and it supports the importance of local motions at active site to drive or gate various conformational transitions underlying the workings of a diverse range of biological nanomachines.

Biomolecular systems possess unique, structure-encoded dynamic properties that underlie their biological functions. Recent studies indicate that these dynamic properties are determined to a large extent by the topology of native contacts. In recent years, elastic network models used in conjunction with normal mode analyses have proven to be useful for elucidating the collective dynamics intrinsically accessible under native state conditions, including in particular the global modes of motions that are robustly defined by the overall architecture. With increasing availability of structural data for well-studied proteins in different forms (liganded, complexed, or free), there is increasing evidence in support of the correspondence between functional changes in structures observed in experiments and the global motions predicted by these coarse-grained analyses. These observed correlations suggest that computational methods may be advantageously employed for assessing functional changes in structure and allosteric mechanisms intrinsically favored by the native fold.

Recent years have seen an explosion in the number of studies that explore the collective dynamics of biomolecular systems using coarse-grained (CG) models along with methods based on principal component analysis (PCA). Among them, elastic network models (ENMs) and normal mode analyses (NMAs) have

In this study, the variance-covariance matrix of protein motions is used to compare several elastic network models within the theoretical framework of x-ray scattering from crystals. A set of 33 ultra-high resolution structures is used to characterize the average scaling behavior of the vibrational density of states and make comparisons between experimental and theoretical temperature factors. Detailed investigations of the vibrational density of states, correlations, and predicted diffuse x-ray scatter are carried out for crystalline Staphylococcal nuclease; correlations and diffuse x-ray scatter are also compared to predictions from the translation, libration, screw model and a liquid-like dynamics model. We show that elastic network models developed to best predict temperature factors without regard for the crystal environment have relatively strong long-range interactions that yield very short-ranged atom-atom correlations. Further, we find that the low-frequency modes dominate the variance-covariance matrix only for those models with a physically reasonable vibrational density of states, and the fraction of modes required to converge the correlations is higher than that typically used for elastic network model studies. The practical implications are explored using computed diffuse x-ray scatter, which can be measured experimentally.

Elastic network models (ENMs) are a class of simple models intended to represent the collective motions of proteins. In contrast to all-atom molecular dynamics simulations, the low computational investment required to use an ENM makes them ideal for speculative hypothesis-testing situations. Historically, ENMs have been validated via comparison to crystallographic B-factors, but this comparison is relatively low-resolution and only tests the predictions of relative flexibility. In this work, we systematically validate and optimize a number of ENM-type models by quantitatively comparing their predictions to microsecond-scale all-atom simulations of three different G protein coupled receptors. We show that, despite their apparent simplicity, well-optimized ENMs perform remarkably well, reproducing the protein fluctuations with an accuracy comparable to what one would expect from all-atom simulations run for several hundred nanoseconds. Proteins 2010. © 2010 Wiley-Liss, Inc.

The activity within a living cell is based on a complex network of interactions among biomolecules, exchanging information and energy through biochemical processes. These events occur on different scales, from the nano- to the macroscale, spanning about 10 orders of magnitude in the space domain and 15 orders of magnitude in the time domain. Consequently, many different modeling techniques, each proper for a particular time or space scale, are commonly used. In addition, a single process often spans more than a single time or space scale. Thus, the necessity arises for combining the modeling techniques in multiscale approaches. In this Account, I first review the different modeling methods for bio-systems, from quantum mechanics to the coarse-grained and continuum-like descriptions, passing through the atomistic force field simulations. Special attention is devoted to their combination in different possible multiscale approaches and to the questions and problems related to their coherent matching in the space and time domains. These aspects are often considered secondary, but in fact, they have primary relevance when the aim is the coherent and complete description of bioprocesses. Subsequently, applications are illustrated by means of two paradigmatic examples: (i) the green fluorescent protein (GFP) family and (ii) the proteins involved in the human immunodeficency virus (HIV) replication cycle. The GFPs are currently one of the most frequently used markers for monitoring protein trafficking within living cells; nanobiotechnology and cell biology strongly rely on their use in fluorescence microscopy techniques. A detailed knowledge of the actions of the virus-specific enzymes of HIV (specifically HIV protease and integrase) is necessary to study novel therapeutic strategies against this disease. Thus, the insight accumulated over years of intense study is an excellent framework for this Account. The foremost relevance of these two biomolecular systems was recently confirmed by the assignment of two of the Nobel prizes in 2008: in chemistry for the discovery of GFP and in medicine for the discovery of HIV. Accordingly, these proteins were studied with essentially all of the available modeling techniques, making them ideal examples for studying the details of multiscale approaches in protein modeling.

A stable parameterization of biomolecular elastic network models (ENMs) is proposed to enable coarse graining of the representation and to model any 3D graph irrespective of the atom connectedness of a system. Traditional ENMs rely on a distance cutoff which is unforgiving in the presence of false negatives in the connectivity, giving rise to unbounded zero-frequency motions when atoms are connected to fewer than three neighbors. A large cutoff is therefore chosen in an ENM, resulting in many false positives in the connectivity that reduce the spatial detail that can be resolved. The required connectivity also has the undesired effect of limiting the coarse-graining, i.e. the network must be dense even in the case of low-resolution structures that exhibit few spatial features. To facilitate such a coarse graining, the newly proposed potential includes 3- and 4-atom interactions (bending and twisting, respectively), in addition to the traditional 2-atom stretching. Thus, in our new Bend-Twist-Stretch (BTS) model the complexity of the parameterization is shifted from the spatial level of detail to the potential function. The additional potential terms were parameterized using continuum elastic theory, and the distance cutoff was replaced by a parameter free competitive Hebb connection rule. We validate the approach on a carbon-alpha representation of adenylate kinase, and illustrate its use with electron microscopy maps of RNA polymerase, ribosome and CCT chaperonin, which were difficult to model with traditional ENMs. For adenylate kinase, we find excellent reproduction (>95% overlap) of the ENM modes and B-factors when BTS is applied to the carbon-alpha representation as well as to coarser descriptions. For the volumetric maps, coarse BTS yields similar motions (75-90% overlap) to those obtained from denser representations with ENM.

Elastic network models (ENMs) are entropic models that have demonstrated in many previous studies their abilities to capture overall the important internal motions, with comparisons having been made against crystallographic B-factors and NMR conformational variabilities. ENMs have become an increasingly important tool and have been widely used to comprehend protein dynamics, function, and even conformational changes. However, reliance upon an arbitrary cutoff distance to delimit the range of interactions has presented a drawback for these models, because the optimal cutoff values can differ somewhat from protein to protein and can lead to quirks such as some shuffling in the order of the normal modes when applied to structures that differ only slightly. Here, we have replaced the requirement for a cutoff distance and introduced the more physical concept of inverse power dependence for the interactions, with a set of elastic network models that are parameter-free, with the distance cutoff removed. For small fluctuations about the native forms, the power dependence is the inverse square, but for larger deformations, the power dependence may become inverse 6th or 7th power. These models maintain and enhance the simplicity and generality of the original ENMs, and at the same time yield better predictions of crystallographic B-factors (both isotropic and anisotropic) and of the directions of conformational transitions. Thus, these parameter-free ENMs can be models of choice whenever elastic network models are used.

With recent advances in single-molecule manipulation techniques, it is now possible to measure the mechanical resistance of proteins to external pulling forces applied at specific positions. Remarkably, such recent studies demonstrated that the pulling/stretching forces required to initiate unfolding vary considerably depending on the location of the application of the forces, unraveling residue/position-specific response of proteins to uniaxial tension. Here we show that coarse-grained elastic network models based on the topology of interresidue contacts in the native state can satisfactory explain the relative sizes of such stretching forces exerted on different residue pairs. Despite their simplicity, such models presumably capture a fundamental property that dominates the observed behavior: deformations that can be accommodated by the relatively lower frequency modes of motions intrinsically favored by the structure require weaker forces and vice versa. The mechanical response of proteins to external stress is therefore shown to correlate with the anisotropic fluctuation dynamics intrinsically accessible in the folded state. The dependence on the overall fold implies that evolutionarily related proteins sharing common structural features tend to possess similar mechanical properties. However, the theory cannot explain the differences observed in a number of structurally similar but sequentially distant domains, such as the fibronectin domains.

We introduce a topology-based nonlinear network model of protein dynamics with the aim of investigating the interplay of spatial disorder and nonlinearity. We show that spontaneous localization of energy occurs generically and is a site-dependent process. Localized modes of nonlinear origin form spontaneously in the stiffest parts of the structure and display site-dependent activation energies. Our results provide a straightforward way for understanding the recently discovered link between protein local stiffness and enzymatic activity. They strongly suggest that nonlinear phenomena may play an important role in enzyme function, allowing for energy storage during the catalytic process., 4 pages, 5 figures. Minor changes

Coarse-grained elastic network models have been successful in determining functionally relevant collective motions. The level of coarse-graining, however, has usually focused on the level of one point per residue. In this work, we compare the applicability of elastic network models over a broader range of representational scales. We apply normal mode analysis for multiple scales on a high-resolution protein data set using various cutoff radii to define the residues considered to be interacting, or the extent of cooperativity of their motions. These scales include the residue-, atomic-, proton-, and explicit solvent-levels. Interestingly, atomic, proton, and explicit solvent level calculations all provide similar results at the same cutoff value, with the computed mean-square fluctuations showing only a slightly higher correlation (0.61) with the experimental temperature factors from crystallography than the results of the residue-level coarse-graining. The qualitative behavior of each level of coarse graining is similar at different cutoff values. The correlations between these fluctuations and the number of internal contacts improve with increased cutoff values. Our results demonstrate that atomic level elastic network models provide an improved representation for the collective motions of proteins compared to the coarse-grained models.

Normal mode analysis offers an efficient way of modeling the conformational flexibility of protein structures. Simple models defined by contact topology, known as elastic network models, have been used to model a variety of systems, but the validation is typically limited to individual modes for a single protein. We use anisotropic displacement parameters from crystallography to test the quality of prediction of both the magnitude and directionality of conformational variance. Normal modes from four simple elastic network model potentials and from the CHARMM forcefield are calculated for a data set of 83 diverse, ultrahigh resolution crystal structures. While all five potentials provide good predictions of the magnitude of flexibility, the methods that consider all atoms have a clear edge at prediction of directionality, and the CHARMM potential produces the best agreement. The low-frequency modes from different potentials are similar, but those computed from the CHARMM potential show the greatest difference from the elastic network models. This was illustrated by computing the dynamic correlation matrices from different potentials for a PDZ domain structure. Comparison of normal mode results with anisotropic temperature factors opens the possibility of using ultrahigh resolution crystallographic data as a quantitative measure of molecular flexibility. The comprehensive evaluation demonstrates the costs and benefits of using normal mode potentials of varying complexity. Comparison of the dynamic correlation matrices suggests that a combination of topological and chemical potentials may help identify residues in which chemical forces make large contributions to intramolecular coupling., Comment: 17 pages, 4 figures