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Computing protein dynamics from protein structure with elastic network models
Abstract
Elastic network models ENMs allow to analytically predict the equilibrium dynamics of proteins without the need of lengthy simulations and force fields, and they depend on a small number of parameters and choices. Despite they are valid only for small fluctuations from the mean native structure, it was observed that large functional conformation changes are well described by a small number of low frequency normal modes. This observation has greatly stimulated the application of ENMs for studying the functional dynamics of proteins, and it is prompting the question whether this functional dynamics is a target of natural selection. From a physical point of view, the agreement between low frequency normal modes and large conformation changes is stimulating the study of anharmonicity in protein dynamics, probably one of the most interesting direction of development in ENMs . ENMs have many applications, of which we will review four general types: (1) the efficient sampling of native conformation space, with applications to molecular replacement in X‐ray spectroscopy, cryo electro‐miscroscopy, docking and homology modeling; (2) the prediction of paths of conformation change between two known end states; (3) the comparison of the dynamics of evolutionarily related proteins; (4) the prediction of dynamical couplings that allow the allosteric regulation of the active site from a distant control regions, with possible applications in the development of allosteric drugs. These goals have important biotechnological applications that are driving more and more attention on the analytical study of protein dynamics through ENMs . WIREs Comput Mol Sci 2014, 4:488–503.
This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods
... Back in 1987, Elber and Karplus first noted the similarity of MD fluctuations with evolutionary changes across the globin family [19], inaugurating a fruitful line of evolutionary and structural dynamics comparisons to this day. Since then, structural data have grown exponentially, and Elastic Network Models (ENMs) [20][21][22][23] have revealed that such fluctuations are largely defined by molecular shape and determine functional motions. Overall, this has led to a new structure-motion-function dogma, where molecular shape determines intrinsic motions, and motions make function, a concept increasingly supported via cryo-EM ensembles [24,25]. ...
... Soon after the first MD simulations, in 1982-1983 [29][30][31][32][33], NMA was applied for the first time to proteins to gain insight into their near-equilibrium dynamics. Instead of numerically solving Newton's equations as MD does, NMA assumes the harmonicity of the system around an energy minimum and, thus, through diagonalization of the mass-weighted Hessian matrix, allows the computation of a unique analytical solution, i.e., a set of linearly independent Normal Nodes (NMs) (see details in [21,34]). NMs are a series of eigenvectors (ν i ) ordered by their eigenvalues or frequencies (λ i ), that describe the natural motions of the system. ...
... It has been argued that proteins oscillate around the equilibrium, with energy increasing as they stretch along NMs' directions. This could elegantly agree with a dynamical systems perspective, as the Kolmogorov Arnold Moser (KAM) theorem assures the persistence of quasi-periodic motions under small perturbations [21,64]. Under this view, NMs would define major directions around a potential well, that hold relatively far from equilibrium. ...
At the very deepest molecular level, the mechanisms of life depend on the operation of proteins, the so-called “workhorses” of the cell. Proteins are nanoscale machines that transform energy into useful cellular work, such as ion or nutrient transport, information processing, or energy transformation. Behind every biological task, there is a nanometer-sized molecule whose shape and intrinsic motions, binding, and sensing properties have been evolutionarily polished for billions of years. With the emergence of structural biology, the most crucial property of biomolecules was thought to be their 3D shape, but how this relates to function was unclear. During the past years, Elastic Network Models have revealed that protein shape, motion and function are deeply intertwined, so that each structure displays robustly shape-encoded functional movements that can be extraordinarily conserved across the tree of life. Here, we briefly review the growing literature exploring the interplay between sequence evolution, protein shape, intrinsic motions and function, and highlight examples from our research in which fundamental movements are conserved from bacteria to mammals or selected by cancer cells to modulate function.
... Besides the MD simulation, another commonly used method to investigate the dynamical properties of proteins is the normal mode analysis of the elastic network model (ENM) [19][20][21][22] . ENM describes the protein structure as an elastic network, in which the interactions between residues are simplified as springs. ...
... ENM can also reproduce MD simulation data but does not require the high computational cost of MD simulation 23-25 . ENM has been proved in many applications to be a simple yet effective method for investigating large-scale conformational transitions, allosteric motions, equilibrium fluctuation of residues, decomposition of domains for proteins, as well as identification of key residues and refinement of low-resolution structural data [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] . However, at present, most of the application studies for ENM are focus on exploring the dynamical properties of the equilibrium state. ...
... Gaussian network model. Gaussian network model (GNM) 19,[21][22][23][24] , one type of the ENM, describes the protein tertiary structure as an elastic network, in which each residue is simplified as a node represented by its C α atom and the interactions between residues are simplified as springs. If the distance between two residues is less than the cutoff value (7.0 Å is adopted in this work), these two residues are considered to have interactions between them, and thus they are connected by a spring. ...
Intra-molecular energy transport between distant functional sites plays important roles in allosterically regulating the biochemical activity of proteins. How to identify the specific intra-molecular signaling pathway from protein tertiary structure remains a challenging problem. In the present work, a non-equilibrium dynamics method based on the elastic network model (ENM) was proposed to simulate the energy propagation process and identify the specific signaling pathways within proteins. In this method, a given residue was perturbed and the propagation of energy was simulated by non-equilibrium dynamics in the normal modes space of ENM. After that, the simulation results were transformed from the normal modes space to the Cartesian coordinate space to identify the intra-protein energy transduction pathways. The proposed method was applied to myosin and the third PDZ domain (PDZ3) of PSD-95 as case studies. For myosin, two signaling pathways were identified, which mediate the energy transductions form the nucleotide binding site to the 50 kDa cleft and the converter subdomain, respectively. For PDZ3, one specific signaling pathway was identified, through which the intra-protein energy was transduced from ligand binding site to the distant opposite side of the protein. It is also found that comparing with the commonly used cross-correlation analysis method, the proposed method can identify the anisotropic energy transduction pathways more effectively.
... 11 Elastic network model (ENM), as a coarsegrained normal mode analysis (NMA), 16,17 is a powerful tool to characterize dynamics of biomolecules based on their crystallographic structures. 11,18 Recently, dynamics were shown through ENM analyses to be important to further understand protein functions including catalysis and allostery, as well as evolution of proteins. [18][19][20][21][22] ENM predicts the dynamics of protein using much fewer parameters and lower computational cost than all atomic force field models. ...
... 11,18 Recently, dynamics were shown through ENM analyses to be important to further understand protein functions including catalysis and allostery, as well as evolution of proteins. [18][19][20][21][22] ENM predicts the dynamics of protein using much fewer parameters and lower computational cost than all atomic force field models. 18 The low frequency normal modes from NMA are sufficient to show the intrinsic collective motions in proteins 11 and thus ENM is suitable to describe the functional dynamics of proteins. ...
... [18][19][20][21][22] ENM predicts the dynamics of protein using much fewer parameters and lower computational cost than all atomic force field models. 18 The low frequency normal modes from NMA are sufficient to show the intrinsic collective motions in proteins 11 and thus ENM is suitable to describe the functional dynamics of proteins. 23 Previous studies indicated that the protein motions calculated by ENM show a good agreement with the results from experimental observation of protein dynamics. ...
Objectives:
PAS domains are widespread in archaea, bacteria and eukaryota, and play important roles in various functions. In this study, we aim to explore functional evolutionary relationship among proteins in the PAS domain superfamily in view of the sequence-structure-dynamics-function relationship.
Methods:
We collected protein sequences and crystal structures data from RCSB Protein Data Bank of the PAS domain superfamily belonging to three biological functions (nucleotide binding, photoreceptor activity and transferase activity). Protein sequences were aligned and then used to select sequence-conserved residues and build phylogenetic tree. Three-dimensional structure alignment was also applied to obtain structure-conserved residues. The protein dynamics were analyzed using elastic network model (ENM) and validated by molecular dynamics (MD) simulation.
Results:
The result showed that the proteins with same function could be grouped by sequence similarity, and proteins in different functional groups displayed statistically significant difference in their vibrational patterns. Interestingly, in all three functional groups, conserved amino acid residues identified by sequence and structure conservation analysis generally have a lower fluctuation than other residues. In addition, the fluctuation of conserved residues in each biological function group was strongly correlated with the corresponding biological function.
Conclusion:
This research suggested a direct connection in which the protein sequences were related to various functions through structural dynamics. This is a new attempt to delineate functional evolution of proteins using the integrated information of sequence, structure and dynamics. This article is protected by copyright. All rights reserved.
... A simplified mathematical approach is to consider the dominant vibrational modes using network models. Elastic network models have been successful in studying large scale motions of other biological systems and long distance communications between conformational states [78][79][80][81][82][83][84][85][86][87][88][89][90]. Normal mode analysis of these fluctuations reveal the low frequency modes corresponding to large scale conformational changes [91], which have been reported to resemble dynamics obtained from more accurate simulations [92,93]. ...
... The corresponding DnaK coordinates were added to the Hsp90 Ec -DnaK complex to create a symmetric complex. The Hsp90 Ec alone and Hsp90 Ec -DnaK complexes were then reduced from their all-atom representation to carbon-α only configuration for subsequent ENM calculations [78][79][80][81][82][83]95]. The structural overlap was computed using the ADP bound conformation of Hsp90 Ec , PDB ID 2IOP [30]. ...
The 70 kDa and 90 kDa heat shock proteins Hsp70 and Hsp90 are two abundant and highly conserved ATP-dependent molecular chaperones that participate in the maintenance of cellular homeostasis. In Escherichia coli, Hsp90 (Hsp90Ec) and Hsp70 (DnaK) directly interact and collaborate in protein remodeling. Previous work has produced a model of the direct interaction of both chaperones. The locations of the residues involved have been confirmed and the model has been validated. In this study, we investigate the allosteric communication between Hsp90Ec and DnaK and how the chaperones couple their conformational cycles. Using elastic network models (ENM), normal mode analysis (NMA), and a structural perturbation method (SPM) of asymmetric and symmetric DnaK-Hsp90Ec, we extract biologically relevant vibrations and identify residues involved in allosteric signaling. When one DnaK is bound, the dominant normal modes favor biological motions that orient a substrate protein bound to DnaK within the substrate/client binding site of Hsp90Ec and release the substrate from the DnaK substrate binding domain. The presence of one DnaK molecule stabilizes the entire Hsp90Ec protomer to which it is bound. Conversely, the symmetric model of DnaK binding results in steric clashes of DnaK molecules and suggests that the Hsp90Ec and DnaK chaperone cycles operate independently. Together, this data supports an asymmetric binding of DnaK to Hsp90Ec.
... However, such calculations are still computationally expensive for large macromolecular assemblies and slow/large-amplitude motions. See also: Molecular Dynamics A less time-consuming alternative to simulate large or slow conformational rearrangements for large biological molecules is normal mode analysis (NMA) (Bastolla, 2014;Cui and Bahar, 2010). This approach, commonly used in physics, was introduced in structural biology around 30 years ago to study the dynamics of the biological macromolecules (Brooks and Karplus, 1983;Go et al., 1983;Levitt et al., 1985). ...
... Using the classical mechanics formulation of NMA (Goldstein et al., 2002), the complex dynamical behaviour of a macromolecule can be approximated as a simple set of harmonic oscillators vibrating around a given equilibrium conformation (Bastolla, 2014;Cui and Bahar, 2010). This mechanical system consists of N atoms under a given force field and located at positions r=(r 1 , _ , r n , _ , r N ), where r n represents the Cartesian coordinates (x n ,y n ,z n ) of the atom n. ...
The dynamic simulation of macromolecular systems with biologically relevant sizes and time scales is critical for understanding macromolecular function. In this context, normal mode analysis (NMA) approximates the complex dynamical behaviour of a macromolecule as a simple set of harmonic oscillators vibrating around a given equilibrium conformation. This technique, originated from classical mechanics, was first applied to investigate the dynamical properties of small biological systems more than 30 years ago. During this time, a wealth of evidence has accumulated to support NMA as a successful tool for simulating macromolecular motions at extended length scales. Today, NMA combined with coarse‐grained representations has become an efficient alternative to molecular dynamics simulations for studying the slow and large‐amplitude motions of macromolecular machines. Interesting insights into these systems can be obtained very quickly with NMA to characterise their flexibility, to predict the directions of their collective conformational changes, or to help in the interpretation of experimental structural data. The recently developed methods and applications of NMA together with an introduction of the underlying theory will be briefly reviewed here.
... A common criterion for determining the optimal rotation consists in maximizing the template-model (TM) score (Zhang and Skolnick 2004) (see Materials and methods) that trades off small rootmean-square deviation (RMSD) and large number of superimposed residues. The average protein coordinates in the native state allow predicting through the structure-based elastic network model (ENM) (Tirion 1996, Bastolla 2014a) native dynamical fluctuations that agree reasonably with experiments and correlate with observed large-scale functional motions (Tama and Sanejouand 2001). Therefore, we expect that proteins with high TM-score present similar native dynamics, as predicted through their ENM. ...
Motivation:
Evolutionary inference depends crucially on the quality of multiple sequence alignments (MSA), which is problematic for distantly related proteins. Since protein structure is more conserved than sequence, it seems natural to use structure alignments for distant homologs. However, structure alignments may not be suitable for inferring evolutionary relationships.
Results:
Here we examined four protein similarity measures that depend on sequence and structure (fraction of aligned residues, sequence identity, fraction of superimposed residues and contact overlap), finding that they are intimately correlated but none of them provides a complete and unbiased picture of conservation in proteins. Therefore, we propose the new hybrid protein sequence and structure similarity score PC_sim based on their main Principal Component. The corresponding divergence measure PC_div shows the strongest correlation with divergences obtained from individual similarities, suggesting that it infers accurate evolutionary divergences. We developed the program PC_ali that constructs protein MSAs either de novo or modifying an input MSA, using a similarity matrix based on PC_sim. tructs a starting MSA based on the maximal cliques of the graph of these PAs and it refines it through progressive alignments along the tree reconstructed with PC_div. Compared with eight state-of-the-art multiple structure or sequence alignment tools, PC_ali achieve higher or equal aligned fraction and structural scores, sequence identity higher than structure aligners although lower than sequence aligners, highest score PC_sim and highest similarity with the MSAs produced by other tools and with the reference MSA Balibase.
Availability:
https://github.com/ugobas/PC_ali.
Supplementary information:
Supplementary data are available at Bioinformatics online.
... The idea was taken further by Tirion [41], who approximates the physical energy by a Hookean potential to resolve the issue of possible negative eigenvalues of the Hessian of the physical energy. The latter approximation has many extensions [4], and has been highly successful in structural biology [31,40]. ...
An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich structure to account for a wide range of geometries that can be modelled after an energy landscape. Second, many standard data analysis tools initially developed for data in Euclidean space can also be generalised to data on a Riemannian manifold. In the context of protein dynamics, a conceptual challenge comes from the lack of a suitable smooth manifold and the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a novel local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold of point clouds modulo rigid body group actions and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. It performs exceptionally well on coarse-grained molecular dynamics simulated data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.
... A commonly applied criterion for determining this optimal rotation consists in numerically maximizing the template-model score [25] (TM, see Methods) that superimposes pairs of residues that are closer than expected by chance. The average protein coordinates in the native state allow predicting native dynamical fluctuations in reasonable agreement with experiments through the structure based Elastic network model (ENM) [26,27]. These predicted fluctuations correlate with observed large-scale functional motions [28]. ...
Motivation
Evolutionary inferences depend crucially on the quality of multiple sequence alignments (MSA), which is problematic for distantly related proteins. Since protein structure is more conserved than protein sequence, it seems natural to use structure alignments for distant homologs. However, structure alignments may not be suitable for inferring evolutionary relationships at the sequence level.
Results
Here we investigate the mutual relationships between four protein similarity measures that depend on sequence and structure (fraction of aligned residues, sequence similarity, fraction of superimposed backbones and contact overlap) and the corresponding alignments. Changes in protein sequences and structures are intimately correlated, but our results suggest that no individual measure can provide a complete and unbiased picture of changes in protein sequences and structure. Therefore, we propose a new hybrid measure of protein sequence and structure similarity based on Principal Components (PC_sim). Starting from an MSA, we obtain modified pairwise alignments (PA) based on PC_sim, and from them we construct a new MSA based on the maximal cliques of the PA graph. These alignments yield larger protein similarities and agree better with the Balibase “reference” MSA and with consensus MSA than alignments that target individual similarity measures. Moreover, PC_sim is associated with a divergence measure that correlates strongest with divergences obtained from individual similarities, which suggests that it can infer more accurate evolutionary divergences for the reconstruction of phylogenetic trees with distance methods.
Availability
https://github.com/ugobas/Evol_div
Contact
ubastolla@cbm.csic.es
... We employed a graph-based representation of protein structures, where residues are nodes and the couplings between them are edges [67][68][69][70][71]. Here, g_correlation program of GROMACS v3.3.4 was employed for the calculation of generalized correlation coefficient (GC) of Ca i -Ca j [72]. ...
During their life cycle, Leishmania parasites display a fine-tuned regulation of the mRNA translation through the differential expression of isoforms of eukaryotic translation initiation factor 4E (LeishIF4Es). The interaction between allosteric modulators such as 4E-interacting proteins (4E-IPs) and LeishIF4E affects the affinity of this initiation factor for the mRNA cap. Here, several computational approaches were employed to elucidate the molecular bases of the previously-reported allosteric modulation in L. major exerted by 4E-IP1 (Lm4E-IP1) on eukaryotic translation initiation factor 4E 1 (LmIF4E-1). Molecular dynamics simulations and accurate binding free energy calculations (ΔGbind) were combined with network-based modeling of residue-residue correlations. We also describe the differences in internal motions of LmIF4E-1 apo form, cap-bound, and Lm4E-IP1-bound systems. Through the community network calculations, the differences in the allosteric pathways of allosterically-inhibited and active forms of LmIF4E-1 were revealed. The ΔGbind values show significant differences between the active and inhibited systems, which are in agreement with the available experimental data. Our study thoroughly describes the dynamical perturbations of LmIF4E-1 cap-binding site triggered by Lm4E-IP1. These findings are not only essential for the understanding of a critical process of trypanosomatids’ gene expression but also for gaining insight into the allostery of eIF4Es, which could be useful for structure-guided design efforts against this protein family.
... Since the graph spectral approach was proposed to detect a variety of side-chain clusters of proteins [31], network theory has been adopted in studying protein folding, identifying functional residues, analyzing allosteric communications, as well as in understanding proteinprotein interactions [32]. Last but not least, network theory can be applied not only to the analysis of protein structures but also extended to the analysis and modeling of dynamics data obtained from nuclear magnetic resonance experiments [33], NMA [34] and MD simulations for systematic data mining [35]. It should be noted that dynamical network model (DNM) or so-called dynamic residues interaction networks (DRINs) has become the valuable one of the major methods in the field [36]. ...
Proteins are dynamical entities that undergo a plethora of conformational changes, accomplishing their biological functions. Molecular dynamics simulation and normal mode analysis methods have become the gold standard for studying protein dynamics, analyzing molecular mechanism and allosteric regulation of biological systems. The enormous amount of the ensemble-based experimental and computational data on protein structure and dynamics has presented a major challenge for the high-throughput modeling of protein regulation and molecular mechanisms. In parallel, bioinformatics and systems biology approaches including genomic analysis, coevolution and network-based modeling have provided an array of powerful tools that complemented and enriched biophysical insights by enabling high-throughput analysis of biological data and dissection of global molecular signatures underlying mechanisms of protein function and interactions in the cellular environment. These developments have provided a powerful interdisciplinary framework for quantifying the relationships between protein dynamics and allosteric regulation, allowing for high-throughput modeling and engineering of molecular mechanisms. Here, we review fundamental advances in protein dynamics, network theory and coevolutionary analysis that have provided foundation for rapidly growing computational tools for modeling of allosteric regulation. We discuss recent developments in these interdisciplinary areas bridging computational biophysics and network biology, focusing on promising applications in allosteric regulations, including the investigation of allosteric communication pathways, protein-DNA/RNA interactions and disease mutations in genomic medicine. We conclude by formulating and discussing future directions and potential challenges facing quantitative computational investigations of allosteric regulatory mechanisms in protein systems.
... Typically, the analysis of (infinitesimal) motions involves diagonalization of H to determine the normal modes of the protein. While real potential energy surfaces of proteins are complex, highly nonlinear, and containing many minima [7], elastic models have been surprisingly effective for the analysis of slow equilibrium motions of proteins [8,9]. Another common use of ENMs is for the calculation of node fluctuations, which have shown good agreement with crystallographic B factors [10,11]. ...
We present an edge-based framework for the study of geometric elastic network models to model mechanical interactions in physical systems. We use a formulation in the edge space, instead of the usual node-centric approach, to characterize edge fluctuations of geometric networks defined in d-dimensional space and define the edge mechanical embeddedness, an edge mechanical susceptibility measuring the force felt on each edge given a force applied on the whole system. We further show that this formulation can be directly related to the infinitesimal rigidity of the network, which additionally permits three- and four-center forces to be included in the network description. We exemplify the approach in protein systems, at both the residue and atomistic levels of description.
... Typically, the analysis of (infinitesimal) motions involves diagonalisation of H to determine the normal modes of the protein. Whilst real potential energy surfaces of proteins are complex, highly nonlinear and containing many minima [7], elastic models have been surprisingly effective for the analysis of slow equilibrium motions of proteins [8,9]. Another common use of ENMs is for the calculation of node fluctuations, which have shown good agreement with crystallographic B-factors [10,11]. ...
We present an edge-based framework for the study of geometric elastic network models to model mechanical interactions in physical systems. We use a formulation in the edge space, instead of the usual node-centric approach, to characterise edge fluctuations of geometric networks defined in d- dimensional space and define the edge mechanical embeddedness, an edge mechanical susceptibility measuring the force felt on each edge given a force applied on the whole system. We further show that this formulation can be directly related to the infinitesimal rigidity of the network, which additionally permits three- and four-centre forces to be included in the network description. We exemplify the approach in protein systems, at both the residue and atomistic levels of description.
... This raises a important question: how is it possible that such simple C-alpha based harmonic models like eBDIMS, can predict the directions of non-equilibrium conformational changes, while MD often requires powerful computing or enhanced sampling? On one hand, it has been suggested that dynamical systems theory assures the conservation of quasi-periodic motions upon small perturbations (Bastolla, 2014), and thus, ENMs are valid beyond the equilibrium, and in a wider set of conditions than was previously thought. On the other hand, the evident power of CG-methods to predict large-scale transitions and intermediates trapped by cryo-EM and crystallography, not only demonstrates such validity, but more importantly, it confirms that the collective shape-encoded dynamics of proteins, is maybe an essential determinant driving their underlying biologically functional transitions. ...
Large-scale conformational changes are essential to link protein structures with their function at the cell and organism scale, but have been elusive both experimentally and computationally. Over the past few years developments in cryo-electron microscopy and crystallography techniques have started to reveal multiple snapshots of increasingly large and flexible systems, deemed impossible only short time ago. As structural information accumulates, theoretical methods become central to understand how different conformers interconvert to mediate biological function. Here we briefly survey current in silico methods to tackle large conformational changes, reviewing recent examples of cross-validation of experiments and computational predictions, which show how the integration of different scale simulations with biological information is already starting to break the barriers between the in silico, in vitro, and in vivo worlds, shedding new light onto complex biological problems inaccessible so far.
... In order to further enhance the sampling of global motions we also performed Hamiltonian replica exchange (H-REMD) simulations coupled with an elastic network model (ENM) description of the MC dimer. A low resolution representation of protein dynamics can be obtained using coarse-grained elastic network models (ENM) to extract directions of global mobility (Bahar and Rader, 2005;Bastolla, 2014). Recently, we have developed a H-REMD approach that uses information from an ENM analysis and combines it with atomistic MD simulations in explicit solvent (Ostermeir and Zacharias, 2014). ...
The Hsp90 protein complex is one of the most abundant molecular chaperone proteins that assists in folding of a variety of client proteins. During its functional cycle it undergoes large domain rearrangements coupled to the hydrolysis of ATP and association or dissociation of domain interfaces. In order to better understand the domain dynamics comparative Molecular Dynamics (MD) simulations of a sub-structure of Hsp90, the dimer formed by the middle (M) and C-terminal domain (C), were performed. Since this MC dimer lacks the ATP-binding N-domain it allows studying global motions decoupled from ATP binding and hydrolysis. Conventional (c)MD simulations starting from several different closed and open conformations resulted in only limited sampling of global motions. However, the application of a Hamiltonian Replica exchange (H-REMD) method based on the addition of a biasing potential extracted from a coarse-grained elastic network description of the system allowed much broader sampling of domain motions than the cMD simulations. With this multiscale approach it was possible to extract the main directions of global motions and to obtain insight into the molecular mechanism of the global structural transitions of the MC dimer.
... Here, the protein is modeled as an elastic network with residues represented by a node at their alpha-carbon atom and the interacting residues are connected by harmonic springs. Using linear response theory [25,[40][41][42][43][44] and a Brownian kick as a perturbative force, the displacements of the nodes can be calculated as: ...
β-lactamases are enzymes produced by bacteria to hydrolyze β-lactam antibiotics as a common mechanism of resistance. Evolution in such enzymes has been rendering a wide variety of antibiotics impotent, therefore posing a major threat. Clinical and in vitro studies of evolution in TEM-1 β-lactamase have revealed a large number of single point mutations that are responsible for driving resistance to antibiotics and/or inhibitors. The distal locations of these mutations from the active sites suggest that these allosterically modulate the antibiotic resistance. We investigated the effects of resistance driver mutations on the conformational dynamics of the enzyme to provide insights about the mechanism of their long-distance interactions. Through all-atom molecular dynamics (MD) simulations, we obtained the dynamic flexibility profiles of the variants and compared those with that of the wild type TEM-1. While the mutational sites in the variants did not have any direct van der Waals interactions with the active site position S70 and E166, we observed a change in the flexibility of these sites, which play a very critical role in hydrolysis. Such long distance dynamic interactions were further confirmed by dynamic coupling index (DCI) analysis as the sites involved in resistance driving mutations exhibited high dynamic coupling with the active sites. A more exhaustive dynamic analysis, using a selection pressure for ampicillin and cefotaxime resistance on all possible types of substitutions in the amino acid sequence of TEM-1, further demonstrated the observed mechanism. Mutational positions that play a crucial role for the emergence of resistance to new antibiotics exhibited high dynamic coupling with the active site irrespective of their locations. These dynamically coupled positions were neither particularly rigid nor particularly flexible, making them more evolvable positions. Nature utilizes these sites to modulate the dynamics of the catalytic sites instead of mutating the highly rigid positions around the catalytic site.
... Elastic Network Model: There is a large variety of ENMs (Bastolla 2014, Fugebakk et al. 2013, López-Blanco and Chacón 2016. Here, I used the ENM of Ming and Wall (Ming and Wall 2005): amino acids are represented by single nodes; nodes are connected if they are within R 0 = 10.5Å; ...
The rate of evolution varies among sites within proteins. In enzymes, two rate gradients are observed: rate decreases with increasing local packing and it increases with increasing distance from catalytic residues. The rate-packing gradient would be mainly due to stability constraints and is well reproduced by biophysical models with selection for protein stability. However, stability constraints are unlikely to account for the rate-distance gradient. Here, to explore the mechanistic underpinnings of the rate gradients observed in enzymes, I propose a stability-activity model of enzyme evolution, MSA. This model is based on a two-dimensional fitness function that depends on stability, quantified by DG, the enzyme's folding free energy, and activity, quantified by DG* , the activation energy barrier of the enzymatic reaction. I test MSA on a diverse data set of enzymes, comparing it with two simpler models: MS , which depends only on DG, and MA , which depends only on DG*. I found that MSA clearly outperforms both MS and MA and it accounts for both the rate-packing and rate-distance gradients. Thus, MSA captures the distribution of stability and activity constraints within enzymes, explaining the resulting patterns of rate variation among sites.
... One of the major hurdles in the development of allosteric drugs lies in the finding of allosteric sites [5,[15][16][17], for which a repertoire of experimental and computational methods is being developed. High-throughput fragment-based screening using a large chemical library formed the main thrust in the identification of potential allosteric sites and lead compounds in pharmaceutical research [18][19][20]. ...
The omnipresence of allosteric regulation together with the fundamental role of structural dynamics in this phenomenon have initiated a great interest to the detection of regulatory exosites and design of corresponding effectors. However, despite a general consensus on the key role of dynamics most of the earlier efforts on the prediction of allosteric sites are heavily crippled by the static nature of the underlying methods, which are either structure-based approaches seeking for deep surface pockets typical for “traditional” orthosteric drugs or sequence-based techniques exploiting the conservation of protein sequences. Because of the critical role of global protein dynamics in allosteric signaling, we investigate the hypothesis of reversibility in allosteric communication, according to which allosteric sites can be detected via the perturbation of the functional sites. The reversibility hypothesis is tested here using our structure-based perturbation model of allostery, which allows one to analyze the causality and energetics of allosteric communication. We validate the “reverse perturbation” hypothesis and its predictive power on a set of classical allosteric proteins, then, on the independent extended benchmark set. We also show that, in addition to known allosteric sites, the perturbation of the functional sites unravels rather extended protein regions, which can host latent regulatory exosites. These protein parts that are dynamically coupled with functional sites can also be used for inducing and tuning allosteric communication, and an exhaustive exploration of the per-residue contributions to allosteric effects can eventually lead to the optimal modulation of protein activity. The site-effector interactions necessary for a specific mode and level of allosteric communication can be fine-tuned by adjusting the site’s structure to an available effector molecule and by the design or selection of an appropriate ligand.
... Since accurate ensemble preparation may require manual curation and often there are not enough conformers solved for a protein, the eBDIMS web server also presents as default reaction coordinates the first two NMs of the starting structure computed by the ED-ENM force-field (7), which was calibrated against a MD library (8,9) and experimental data from X-ray and NMR, and has been used to successfully assess CASP predictions (10). Low frequency NMs have been shown not only to describe efficiently conformational changes 3 (15) involving large rigid-body motions (5,11,12) but also to correlate with PCs from experimental ensembles or MD simulations (7,13,14). Therefore, they provide alternative axes useful to monitor how a given transition is proceeding in terms of molecular movements in case not enough representative conformers are sampled in the ensemble. ...
Understanding how proteins transition between different conformers, and how conformers relate to each other in terms of structure and function, is not trivial. Here, we present an online tool for transition pathway generation between two protein conformations using Elastic Network Driven Brownian Dynamics Importance Sampling, a coarse-grained simulation algorithm, which spontaneously predicts transition intermediates trapped experimentally. In addition to path-generation, the server provides an interactive 2D-motion landscape graphical representation of the transitions or any additional conformers to explore their structural relationships.
Availability and implementation
eBDIMS is available online: http://ebdims.biophysics.se/ or as standalone software: https://github.com/laura-orellana/eBDIMS, https://github.com/cabergh/eBDIMS.
Supplementary information
Supplementary data are available at Bioinformatics online.
... Elastic network models (ENMs) have recently been made popular as, despite their simplicity, provide valuable information on large-scale functional motions of proteins [1][2][3][4][5][6]. These motions usually occur when proteins undergo conformational changes upon binding to other biomolecules. ...
Elastic network models have recently been used for studying low-frequency collective motions of proteins. These models simplify the complexity that arises from normal mode analysis by considering a simplified potential involving a few parameters. Two common parameters in most of the elastic network models are cutoff radius and force constant. Although the latter has been studied extensively and even elaborate new models were introduced, for the former usually an ad-hoc cutoff radius is considered. Moreover, the quality of the network models is usually assessed by evaluating their prediction against experimental B-factors. In this work, we consider various common elastic network models with different cutoff radii and assess them by their ability to predict conformational changes of proteins in complexes from unbound to bound state. This prediction is performed by perturbing the unbound structure using a number of low-frequency normal modes of its network model to optimally fit the bound structure. We evaluated a dataset of 30 proteins with distinct unbound and bound structures using this criterion. The results showed that, opposed to the common calibration process based on B-factors, a meaningful relationship exists between the quality of the prediction and model parameters. It was shown that the cutoff radius has a major role in this prediction and minimally connected network models, which use the shortest cutoff radius for which the network is stable, give the best results. It was also shown that by considering the first ten normal modes, the conformational changes can be predicted by about 25 percent. Hence, the evaluation process was extended to the case of considering the contribution of all normal modes in the prediction. The results indicated that minimally connected network models are superior, despite their simplicity, when any number of modes are considered in the prediction.
... Elastic network model is a network composed of a set of nodes which are represented by the atoms (or alpha carbons, Cα) of the protein and every two nodes are connected by a link, if their distance are within a specific cutoff radius [12][13][14]. By using these kinds of models, the complicated potential is simplified to an easy-to-understand quadratic potential described as ...
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
... Based on the work of Tirion [25], Bahar suggested an ENM with harmonic potentials connecting the Cα pairs [26], a concept that was emphasized by the same group a few months later [27]. ENMs thus became a very popular method of studying the dynamic behavior of protein conformations, and could be designed in several ways according to their different schemes of connectivity or various functional forms, as reported in several reviews [29][30][31][32][33]. Hence, in addition to performing NMA calculations, several software products, such as the Bio3D package, now include suites of tools to relate functional dynamics information coming from NMA with protein structural data as well as evolutionary analyses [34]. ...
Despite progress in computer modeling, most biological processes are still out of reach when using all-atom (AA) models. Coarse-grained (CG) models allow classical molecular dynamics (MD) simulations to be accelerated. Although simplification of spatial resolution at different levels is often investigated, simplification of the CG potential in itself has been less common. CG potentials are often similar to AA potentials. In this work, we consider the design and reliability of purely mechanical CG models of the μ opioid receptor (μOR), a G protein-coupled receptor (GPCR). In this sense, CG force fields (FF) consist of a set of holonomic constraints guided by an elastic network model (ENM). Even though ENMs are used widely to perform normal mode analysis (NMA), they are not often implemented as a single FF in the context of MD simulations. In this work, various ENM-like potentials were investigated by varying their force constant schemes and connectivity patterns. A method was established to systematically parameterize ENM-like potentials at different spatial resolutions by using AA data. To do so, new descriptors were introduced. The choice of conformation descriptors that also include flexibility information is important for a reliable parameterization of ENMs with different degrees of sensitivity. Hence, ENM-like potentials, with specific parameters, can be sufficient to accurately reproduce AA MD simulations of μOR at highly coarse-grained resolutions. Therefore, the essence of the flexibility properties of μOR can be captured with simple models at different CG spatial resolutions, opening the way to mechanical approaches to understanding GPCR functions. Graphical AbstractAll atom structure, residue interaction network and coarse-grained elastic network models of the μ opioid receptor (μOR)
... CG-models 29,30 , where each residue is reduced to a few beads interacting by simple potentials, minimize computational costs. Among CG-methods, ENMs 31,32 are conceptually simple but capable of predicting accurately conformational changes 33 . Despite reducing protein architecture to a minimalist network of Ca-carbons connected by springs, NMs computed from the ENM potential describe transitions between X-ray pairs with surprising precision 5,[34][35][36] and reproduce the flexibility from experimental ensembles or long MD simulations [8][9][10]37,38 . ...
Protein conformational changes are at the heart of cell functions, from signalling to ion transport. However, the transient nature of the intermediates along transition pathways hampers their experimental detection, making the underlying mechanisms elusive. Here we retrieve dynamic information on the actual transition routes from principal component analysis (PCA) of structurally-rich ensembles and, in combination with coarse-grained simulations, explore the conformational landscapes of five well-studied proteins. Modelling them as elastic networks in a hybrid elastic-network Brownian dynamics simulation (eBDIMS), we generate trajectories connecting stable end-states that spontaneously sample the crystallographic motions, predicting the structures of known intermediates along the paths. We also show that the explored non-linear routes can delimit the lowest energy passages between end-states sampled by atomistic molecular dynamics. The integrative methodology presented here provides a powerful framework to extract and expand dynamic pathway information from the Protein Data Bank, as well as to validate sampling methods in general.
... (ENM), which explores fluctuations around the native structure, and has been used to explain allosteric regulation and conformational changes. We suggest a couple of recent reviews (Dehouck and Mikhailov, 2013;Bastolla, 2014) for the combination of ENM and NMR, for interested readers. ...
Proteins participate in information pathways in cells, both as links in the chain of signals, and as the ultimate effectors. Upon ligand binding, proteins undergo conformation and motion changes, which can be sensed by the following link in the chain of information. Nuclear magnetic resonance (NMR) spectroscopy and molecular dynamics (MD) simulations represent powerful tools for examining the time-dependent function of biological molecules. The recent advances in NMR and the availability of faster computers have opened the door to more detailed analyses of structure, dynamics, and interactions. Here we briefly describe the recent applications that allow NMR spectroscopy and MD simulations to offer unique insight into the basic motions that underlie information transfer within and between cells.
... Such Markov state modeling has been previously applied to obtain pathways for the conformational changes of proteins [61][62][63][64]. Other methods to explore rare conformational transitions of proteins with atomistic molecular dynamic simulations identify reaction coordinates or collective variables for such transitions [65][66][67][68][69]. Conformational changes of proteins have also been investigated in molecular dynamics simulations with coarse-grained models [70][71][72] and by normal mode analysis of elastic network models [73][74][75][76][77]. model follow from the equations p(P 1 ) + p(P 2 ) + p(P 1 L) + p(P 2 L) = 1 (23) p(P 1 )u 12 − p(P 2 )u 21 = 0 (24) ...
Protein binding and function often involves conformational changes. Advanced NMR experiments indicate that these conformational changes can occur in the absence of ligand molecules (or with bound ligands), and that the ligands may ‘select’ protein conformations for binding (or unbinding). In this review, we argue that this conformational selection requires transition times for ligand binding and unbinding that are small compared to the dwell times of proteins in different conformations, which is plausible for small ligand molecules. Such a separation of timescales leads to a decoupling and temporal ordering of binding/unbinding events and conformational changes. We propose that conformational-selection and induced-change processes (such as induced fit) are two sides of the same coin, because the temporal ordering is reversed in binding and unbinding direction. Conformational-selection processes can be characterized by a conformational excitation that occurs prior to a binding or unbinding event, while induced-change processes exhibit a characteristic conformational relaxation that occurs after a binding or unbinding event. We discuss how the ordering of events can be determined from relaxation rates and effective on- and off-rates determined in mixing experiments, and from the conformational exchange rates measured in advanced NMR or single-molecule FRET experiments. For larger ligand molecules such as peptides, conformational changes and binding events can be intricately coupled and exhibit aspects of conformational-selection and induced-change processes in both binding and unbinding direction.
Protein dynamics and conformational transitions are essential for most biological functions. They are the necessary link to connect atomic-level structural details with cellular processes ranging from enzymatic catalysis to signaling, solute transport, or synaptic transmission. In this chapter, we review standard atomistic and coarse-grained techniques of increasing complexity to simulate protein motions and conformational transitions. We present the key theoretical foundations and compare how different standard methods explore the conformational landscape in the case of a protein with known intermediates for a simple conformational change. The goal is to provide nonspecialist readers with a broad overview of available approaches that can be used to obtain a basic assessment of protein flexibility.
Complex I functions as a primary redox-driven proton pump in aerobic respiratory chains, establishing proton motive force that powers ATP synthesis and active transport. Recent cryo-electron microscopy (cryo-EM) experiments have resolved the mammalian complex I in the biomedically relevant active (A) and deactive (D) states(1–3) that could regulate enzyme turnover, but it still remains unclear how the conformational state and activity are linked. We show here how global motion along the A/D transition accumulates molecular strain at specific coupling regions important for both redox chemistry and proton pumping. Our data suggest that the A/D motion modulates force propagation pathways between the substrate-binding site and the proton pumping machinery that could alter electrostatic and conformational coupling across large distances. Our findings provide a molecular basis to understand how global protein dynamics can modulate the biological activity of large molecular complexes.
The long-ranged coupling between residues that gives rise to allostery in a protein is built up from short-ranged physical interactions. Computational tools used to predict this coupling and its functional relevance have relied on the application of graph theoretical metrics to residue-level correlations measured from all-atom molecular dynamics (aaMD) simulations. The short-ranged interactions that yield these long-ranged residue-level correlations are quantified by the effective coarse-grained Hessian. Here we compute an effective harmonic coarse-grained Hessian from aaMD simulations of a benchmark allosteric protein, IGPS, and demonstrate the improved locality of this graph Laplacian over two other connectivity matrices. Additionally, two centrality metrics are developed that indicate the direct and indirect importance of each residue at producing the covariance between the effector binding pocket and the active site. The residue importance indicated by these two metrics is corroborated by previous mutagenesis experiments and leads to unique functional insights; in contrast to previous computational analyses, our results suggest that fP76-hK181 is the most important contact for conveying direct allosteric paths across the HisF-HisH interface. The connectivity around fD98 is found to be important at affecting allostery through indirect means.
The long-ranged coupling between sites that gives rise to allostery in a protein is built up from short-ranged physical interactions. Computational tools used to predict this coupling and its functional relevance have relied heavily on the application of graph theoretical metrics to residue-level correlations measured from all-atom molecular dynamics simulations. The short-ranged interactions that yield these residue-level correlations, and thus the appropriate graph Laplacian, are quantified by the effective coarse-grained Hessian. Here we compute an effective harmonic coarse-grained Hessian for a benchmark allosteric protein, IGPS, and demonstrate the improved locality of this Laplacian over two other connectivity matrices. Additionally, two centrality metrics are developed that indicate the direct and indirect importance of each residue at producing the covariance between the effector binding pocket and the active site. The results from this procedure are corroborated by previous mutagenesis experiments and lead to unique functional insights. In contrast to previous computational analyses, our results suggest that fP76-hK181, not fD98-hK181, is the most important contact for conveying direct allosteric paths across the HisF-HisH interface. fD98 is found to play a minor role in paths and contribute greatly to indirect allostery between the effector binding pocket and the glutaminase active site.
Membrane ion channels are the fundamental electrical components in the nervous system. Recent developments in X-ray crystallography and cryo-EM microscopy have revealed what these proteins look like in atomic detail but do not tell us how they function. Molecular dynamics simulations have progressed to the point that we can now simulate realistic molecular assemblies to produce quantitative calculations of the thermodynamic and kinetic quantities that control function. In this review, we summarize the state of atomistic simulation methods for ion channels to understand their conduction, activation, and drug modulation mechanisms. We are at a crossroads in atomistic simulation, where long time scale observation can provide unbiased exploration of mechanisms, supplemented by biased free energy methodologies. We illustrate the use of these approaches to describe ion conduction and selectivity in voltage-gated sodium and acid-sensing ion channels. Studies of channel gating present a significant challenge, as activation occurs on longer time scales. Enhanced sampling approaches can ensure convergence on minimum free energy pathways for activation, as illustrated here for pentameric ligand-gated ion channels that are principal to nervous system function and the actions of general anesthetics. We also examine recent studies of local anesthetic and antiepileptic drug binding to a sodium channel, revealing sites and pathways that may offer new targets for drug development. Modern simulations thus offer a range of molecular-level insights into ion channel function and modulation as a learning platform for mechanistic discovery and drug development.
We propose here a working unit for teaching basic concepts of structural bioinformatics and evolution through the example of a wooden snake puzzle, strikingly similar to toy models widely used in the literature of protein folding. In our experience, developed at a Master’s course at the Universidad Autónoma de Madrid (Spain), the concreteness of this example helps to overcome difficulties caused by the interdisciplinary nature of this field and its high level of abstraction, in particular for students coming from traditional disciplines. The puzzle will allow us discussing a simple algorithm for finding folded solutions, through which we will introduce the concept of the configuration space and the contact matrix representation. This is a central tool for comparing protein structures, for studying simple models of protein energetics, and even for a qualitative discussion of folding kinetics, through the concept of the Contact Order. It also allows a simple representation of misfolded conformations and their free energy. These concepts will motivate evolutionary questions, which we will address by simulating a structurally constrained model of protein evolution, again modelled on the snake puzzle. In this way, we can discuss the analogy between evolutionary concepts and statistical mechanics that facilitates the understanding of both concepts. The proposed examples and literature are accessible, and we provide supplementary material (see ‘Data Availability’) to reproduce the numerical experiments. We also suggest possible directions to expand the unit. We hope that this work will further stimulate the adoption of games in teaching practice.
Contemporary protein structure is a result of the trade off between the laws of physics and the evolutionary selection. The polymer nature of proteins played a decisive role in establishing the basic structural and functional units of soluble proteins. We discuss how these elementary building blocks work in the hierarchy of protein domain structure, co-translational folding, as well as in enzymatic activity and molecular interactions. Next, we consider modulators of the protein function, such as intermolecular interactions, disorder-to-order transitions, and allosteric signaling, acting via interference with the protein's structural dynamics. We also discuss the post-translational modifications, which is a complementary intricate mechanism evolved for regulation of protein functions and interactions. In conclusion, we assess an anticipated contribution of discussed topics to the future advancements in the field. Addresses
We present a theoretical analysis that associates the resonances of extraordinary acoustic Raman (EAR) spectroscopy [Wheaton et al., Nat Photon 9, 68 (2015)] with the collective modes of proteins. The theory uses the anisotropic elastic network model to find the protein acoustic modes, and calculates Raman intensity by treating the protein as a polarizable ellipsoid. Reasonable agreement is found between EAR spectra and our theory. Protein acoustic modes have been extensively studied theoretically to assess the role they play in protein function; this result suggests EAR as a new experimental tool for studies of protein acoustic modes.
The presence of multiple allosteric sites in proteins motivates development of allosteric drugs-modulators of protein activity with potentially higher specificity and less toxicity than traditional orthosteric compounds. A quest for allosteric control of any protein starts from the identification and characterization of allosteric sites. Protein dynamics is the basis for allosteric communication. Binding of effector molecules to allosteric sites modulates structural dynamics, thus affecting activity of remote functional sites. We review here theoretical concepts and experimental approaches for exploring allosteric sites, their role in allosteric regulation, and ways to assess their druggability. Key steps of the design procedure aimed at obtaining allosteric drugs with required agonistic/antagonistic effect are proposed, and their computational and experimental elements are discussed.
Author Summary
Decades of experimental evidence have underlined the fact that protein structures can hardly be considered as static objects. To understand how a protein achieves its biological purpose, it is therefore quite often necessary to unravel the complexity of its dynamical behavior. However, the definition of accurate and computationally tractable descriptions of protein dynamics remains a highly challenging task. Indeed, even though proteins are all built from a limited set of amino acids and local conformational arrangements, the specific nature of biologically relevant motions may vary widely from one protein to another, which constitutes a serious obstacle to the identification of common rules and properties. Here, instead of focusing on the study of a single protein, we adopt a more general perspective by condensing the information contained in a multitude of NMR conformational ensembles. This approach allows us to characterize the dynamical behavior of residues and residue pairs in a mean protein environment, independently of each protein's specific architecture. We describe how this analysis can be exploited to assess the performances of coarse-grained models of protein dynamics, to take advantage of existing experimental data for a more rational and efficient parametrization of these models and, ultimately, to improve our understanding of the intrinsic dynamical properties of amino acid chains.
Here, we employed the collective motions extracted from normal mode analysis (NMA) in internal coordinates (torsional space) for the flexible fitting of atomic-resolution structures into electron microscopy (EM) density maps. The proposed methodology was validated using a benchmark of simulated cases, highlighting its robustness over the full range of EM resolutions and even over coarse-grained representations. A systematic comparison with other methods further showcased the advantages of this proposed methodology, especially at medium to lower resolutions. Using this method, computational costs and potential overfitting problems are naturally reduced by constraining the search in low-frequency NMA space, where covalent geometry is implicitly maintained. This method also effectively captures the macromolecular changes of a representative set of experimental test cases. We believe that this novel approach will extend the currently available EM hybrid methods to the atomic-level interpretation of large conformational changes and their functional implications.
The use of allosteric modulators as preferred therapeutic agents against classic orthosteric ligands has colossal advantages, including higher specificity, fewer side effects and lower toxicity. Therefore, the computational prediction of allosteric sites in proteins is receiving increased attention in the field of drug discovery. Allosite is a newly developed automatic tool for the prediction of allosteric sites in proteins of interest and is now available through a web server.
The Allosite server and tutorials are freely available at http://mdl.shsmu.edu.cn/AST CONTACT: jian.zhang@sjtu.edu.cn SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.
The SPACER server provides an interactive framework for exploring allosteric communication in proteins with different sizes,
degrees of oligomerization and function. SPACER uses recently developed theoretical concepts based on the thermodynamic view
of allostery. It proposes easily tractable and meaningful measures that allow users to analyze the effect of ligand binding
on the intrinsic protein dynamics. The server shows potential allosteric sites and allows users to explore communication between
the regulatory and functional sites. It is possible to explore, for instance, potential effector binding sites in a given
structure as targets for allosteric drugs. As input, the server only requires a single structure. The server is freely available
at http://allostery.bii.a-star.edu.sg/.
Background
Allostery is one of the most powerful and common ways of regulation of protein activity. However, for most allosteric proteins identified to date the mechanistic details of allosteric modulation are not yet well understood. Uncovering common mechanistic patterns underlying allostery would allow not only a better academic understanding of the phenomena, but it would also streamline the design of novel therapeutic solutions. This relatively unexplored therapeutic potential and the putative advantages of allosteric drugs over classical active-site inhibitors fuel the attention allosteric-drug research is receiving at present. A first step to harness the regulatory potential and versatility of allosteric sites, in the context of drug-discovery and design, would be to detect or predict their presence and location. In this article, we describe a simple computational approach, based on the effect allosteric ligands exert on protein flexibility upon binding, to predict the existence and position of allosteric sites on a given protein structure.
Results
By querying the literature and a recently available database of allosteric sites, we gathered 213 allosteric proteins with structural information that we further filtered into a non-redundant set of 91 proteins. We performed normal-mode analysis and observed significant changes in protein flexibility upon allosteric-ligand binding in 70% of the cases. These results agree with the current view that allosteric mechanisms are in many cases governed by changes in protein dynamics caused by ligand binding. Furthermore, we implemented an approach that achieves 65% positive predictive value in identifying allosteric sites within the set of predicted cavities of a protein (stricter parameters set, 0.22 sensitivity), by combining the current analysis on dynamics with previous results on structural conservation of allosteric sites. We also analyzed four biological examples in detail, revealing that this simple coarse-grained methodology is able to capture the effects triggered by allosteric ligands already described in the literature.
Conclusions
We introduce a simple computational approach to predict the presence and position of allosteric sites in a protein based on the analysis of changes in protein normal modes upon the binding of a coarse-grained ligand at predicted cavities. Its performance has been demonstrated using a newly curated non-redundant set of 91 proteins with reported allosteric properties. The software developed in this work is available upon request from the authors.
KOSMOS is the first online morph server to be able to address the structural dynamics of DNA/RNA, proteins and even their complexes, such as ribosomes. The key functions of KOSMOS are the harmonic and anharmonic analyses of macromolecules. In the harmonic analysis, normal mode analysis (NMA) based on an elastic network model (ENM) is performed, yielding vibrational modes and B-factor calculations, which provide insight into the potential biological functions of macromolecules based on their structural features. Anharmonic analysis involving elastic network interpolation (ENI) is used to generate plausible transition pathways between two given conformations by optimizing a topology-oriented cost function that guarantees a smooth transition without steric clashes. The quality of the computed pathways is evaluated based on their various facets, including topology, energy cost and compatibility with the NMA results. There are also two unique features of KOSMOS that distinguish it from other morph servers: (i) the versatility in the coarse-graining methods and (ii) the various connection rules in the ENM. The models enable us to analyze macromolecular dynamics with the maximum degrees of freedom by combining a variety of ENMs from full-atom to coarse-grained, backbone and hybrid models with one connection rule, such as distance-cutoff, number-cutoff or chemical-cutoff. KOSMOS is available at http://bioengineering.skku.ac.kr/kosmos.
It is well recognized that thermal motions of atoms in the protein native state, the fluctuations about the minimum of the global free energy, are well reproduced by the simple elastic network models (ENMs) such as the anisotropic network model (ANM). Elastic network models represent protein dynamics as vibrations of a network of nodes (usually represented by positions of the heavy atoms or by the C(α) atoms only for coarse-grained representations) in which the spatially close nodes are connected by harmonic springs. These models provide a reliable representation of the fluctuational dynamics of proteins and RNA, and explain various conformational changes in protein structures including those important for ligand binding. In the present paper, we study the problem of protein structure refinement by analyzing thermal motions of proteins in non-native states. We represent the conformational space close to the native state by a set of decoys generated by the I-TASSER protein structure prediction server utilizing template-free modeling. The protein substates are selected by hierarchical structure clustering. The main finding is that thermal motions for some substates, overlap significantly with the deformations necessary to reach the native state. Additionally, more mobile residues yield higher overlaps with the required deformations than do the less mobile ones. These findings suggest that structural refinement of poorly resolved protein models can be significantly enhanced by reduction of the conformational space to the motions imposed by the dominant normal modes.
We propose a dissipative electro-elastic network model to describe the dynamics and statistics of electrostatic fluctuations at active sites of proteins. The model combines the harmonic network of residue beads with overdamped dynamics of the normal modes of the network characterized by two friction coefficients. The electrostatic component is introduced to the model through atomic charges of the protein force field. The overall effect of the electrostatic fluctuations of the network is recorded through the frequency-dependent response functions of the electrostatic potential and electric field at the protein active site. We also consider the dynamics of displacements of individual residues in the network and the dynamics of distances between pairs of residues. The model is tested against loss spectra of residue displacements and the electrostatic potential and electric field at the heme's iron from all-atom molecular dynamics simulations of three hydrated globular proteins.
Network-based models of proteins are popular tools employed to determine dynamic features related to the folded structure. They encompass all topological and geometric computational approaches idealizing proteins as directly interacting nodes. Topology makes use of neighborhood information of residues, and geometry includes relative placement of neighbors. Coarse-grained approaches efficiently predict alternative conformations because of inherent collectivity in the protein structure. Such collectivity is moderated by topological characteristics that also tune neighborhood structure: That rich residues have richer neighbors secures robustness toward random loss of interactions/nodes due to environmental fluctuations/mutations. Geometry conveys the additional information of force balance to network models, establishing the local shape of the energy landscape. Here, residue and/or bond perturbations are critically evaluated to suggest new experiments, as network-based computational techniques prove useful in capturing domain movements and conformational shifts resulting from environmental alterations. Evolutionarily conserved residues are optimally connected, defining a subnetwork that may be utilized for further coarsening.
Author Summary
Allosteric protein regulation is the mechanism by which binding of a molecule to one site in a protein affects the activity at another site. Although the two classical phenomenological models, Monod-Wyman-Changeux (MWC) and Koshland-Némethy-Filmer (KNF), span from the case of hemoglobin to membrane receptors, they do not describe the intramolecular interactions involved. The coupling between two allosterically connected sites commonly takes place through coherent collective motion involving the whole protein. We therefore introduce a quantity called binding leverage to measure the strength of the coupling between particular binding sites and such motions. We show that high binding leverage is a characteristic of both allosteric sites and catalytic sites, emphasizing that both enzymatic function and allosteric regulation require a coupling between ligand binding and protein dynamics. We also consider the first known case of purely entropic allostery, where ligand binding only affects the amplitudes of fluctuations. We find that the binding site in this protein does not primarily connect to collective motions – instead the modulation of fluctuations is controlled from a deeply buried and highly connected site. Finally, sites with high binding leverage but no known biological function could be latent allosteric sites, and thus drug targets.
Dynamic simulations of systems with biologically relevant sizes and time scales are critical for understanding macromolecular functioning. Coarse-grained representations combined with normal mode analysis (NMA) have been established as an alternative to atomistic simulations. The versatility and efficiency of current approaches normally based on Cartesian coordinates can be greatly enhanced with internal coordinates (IC).
Here, we present a new versatile tool chest to explore conformational flexibility of both protein and nucleic acid structures using NMA in IC. Consideration of dihedral angles as variables reduces the computational cost and non-physical distortions of classical Cartesian NMA methods. Our proposed framework operates at different coarse-grained levels and offers an efficient framework to conduct NMA-based conformational studies, including standard vibrational analysis, Monte-Carlo simulations or pathway exploration. Examples of these approaches are shown to demonstrate its applicability, robustness and efficiency.
pablo@chaconlab.org
Supplementary data are available at Bioinformatics online.
The ability to generate, visualize, and analyze motions of biomolecules has made a significant impact upon modern biology. Molecular Dynamics has gained substantial use, but remains computationally demanding and difficult to setup for many biologists. Elastic network models (ENMs) are an alternative and have been shown to generate the dominant equilibrium motions of biomolecules quickly and efficiently. These dominant motions have been shown to be functionally relevant and also to indicate the likely direction of conformational changes. Most structures have a small number of dominant motions. Comparing computed motions to the structure's conformational ensemble derived from a collection of static structures or frames from an MD trajectory is an important way to understand functional motions as well as evaluate the models. Modes of motion computed from ENMs can be visualized to gain functional and mechanistic understanding and to compute useful quantities such as average positional fluctuations, internal distance changes, collectiveness of motions, and directional correlations within the structure.
Our new software, MAVEN, aims to bring ENMs and their analysis to a broader audience by integrating methods for their generation and analysis into a user friendly environment that automates many of the steps. Models can be constructed from raw PDB files or density maps, using all available atomic coordinates or by employing various coarse-graining procedures. Visualization can be performed either with our software or exported to molecular viewers. Mixed resolution models allow one to study atomic effects on the system while retaining much of the computational speed of the coarse-grained ENMs. Analysis options are available to further aid the user in understanding the computed motions and their importance for its function.
MAVEN has been developed to simplify ENM generation, allow for diverse models to be used, and facilitate useful analyses, all on the same platform. This represents an integrated approach that incorporates all four levels of the modeling process - generation, evaluation, analysis, visualization - and also brings to bear multiple ENM types. The intension is to provide a versatile modular suite of programs to a broader audience. MAVEN is available for download at http://maven.sourceforge.net.
Molecular dynamics (MD) simulations have dramatically improved the atomistic understanding of protein motions, energetics and function. These growing datasets have necessitated a corresponding emphasis on trajectory analysis methods for characterizing simulation data, particularly since functional protein motions and transitions are often rare and/or intricate events. Observing that such events give rise to long-tailed spatial distributions, we recently developed a higher-order statistics based dimensionality reduction method, called quasi-anharmonic analysis (QAA), for identifying biophysically-relevant reaction coordinates and substates within MD simulations. Further characterization of conformation space should consider the temporal dynamics specific to each identified substate.
Our model uses hierarchical clustering to learn energetically coherent substates and dynamic modes of motion from a 0.5 μs ubiqutin simulation. Autoregressive (AR) modeling within and between states enables a compact and generative description of the conformational landscape as it relates to functional transitions between binding poses. Lacking a predictive component, QAA is extended here within a general AR model appreciative of the trajectory's temporal dependencies and the specific, local dynamics accessible to a protein within identified energy wells. These metastable states and their transition rates are extracted within a QAA-derived subspace using hierarchical Markov clustering to provide parameter sets for the second-order AR model. We show the learned model can be extrapolated to synthesize trajectories of arbitrary length.
ramanathana@ornl.gov; chakracs@pitt.edu.
Internal motions enable proteins to explore a range of conformations, even in the vicinity of native state. The role of conformational fluctuations in the designated function of a protein is widely debated. Emerging evidence suggests that sub-groups within the range of conformations (or sub-states) contain properties that may be functionally relevant. However, low populations in these sub-states and the transient nature of conformational transitions between these sub-states present significant challenges for their identification and characterization.
To overcome these challenges we have developed a new computational technique, quasi-anharmonic analysis (QAA). QAA utilizes higher-order statistics of protein motions to identify sub-states in the conformational landscape. Further, the focus on anharmonicity allows identification of conformational fluctuations that enable transitions between sub-states. QAA applied to equilibrium simulations of human ubiquitin and T4 lysozyme reveals functionally relevant sub-states and protein motions involved in molecular recognition. In combination with a reaction pathway sampling method, QAA characterizes conformational sub-states associated with cis/trans peptidyl-prolyl isomerization catalyzed by the enzyme cyclophilin A. In these three proteins, QAA allows identification of conformational sub-states, with critical structural and dynamical features relevant to protein function.
Overall, QAA provides a novel framework to intuitively understand the biophysical basis of conformational diversity and its relevance to protein function.
We develop a general framework for the analysis of residue fluctuations that simultaneously incorporates anharmonicity and mode-coupling in a unified formalism. We show that both deviations from the Gaussian model are important for modeling the multidimensional energy landscape of the protein Crambin (1EJG) in the vicinity of its native state. The effect of anharmonicity and mode-coupling on the fluctuational entropy is in the order of a few percent.
We introduce the torsional network model (TNM), an elastic network model whose degrees of freedom are the torsion angles of the protein backbone. Normal modes of the TNM displace backbone atoms including C(β) maintaining their covalent geometry. For many proteins, low frequency TNM modes are localized in torsion space yet collective in Cartesian space, reminiscent of hinge motions. A smaller number of TNM modes than anisotropic network model modes are enough to represent experimentally observed conformation changes. We observed significant correlation between the contribution of each normal mode to equilibrium fluctuations and to conformation changes, and defined the excess correlation with respect to a simple neutral model. The stronger this excess correlation, the lower the predicted free energy barrier of the conformation change and the fewer modes contribute to the change.
Bacterial outer membrane porins (Omp) that have robust β -barrel structures, show potential applications for nanomedicine devices in synthetic membranes and single molecule detection biosensors. Here, we explore the conformational dynamics of a set of 22 outer membrane porins, classified into five major groups: general porins, specific porins, transport Omps, poreless Omps and composed pores. Normal mode analysis, based on mechanical vibration theory and elastic network model, is performed to study the fluctuations of residues of aforementioned porins around their equilibrium positions. We find that a simple modification in this model considering weak interaction between protein and membrane, dramatically enhance the stability of results and improve the correlation coefficient between computational output and experimental results.
Normal mode analysis of proteins of various sizes, ranging from 46 (crambin) up to 858 residues (dimeric citrate synthase) were performed, by using standard approaches, as well as a recently proposed method that rests on the hypothesis that low-frequency normal modes of proteins can be described as pure rigid-body motions of blocks of consecutive amino-acid residues. Such a hypothesis is strongly supported by our results, because we show that the latter method, named RTB, yields very accurate approximations for the low-frequency normal modes of all proteins considered. Moreover, the quality of the normal modes thus obtained depends very little on the way the polypeptidic chain is split into blocks. Noteworthy, with six amino-acids per block, the normal modes are almost as accurate as with a single amino-acid per block. In this case, for a protein of n residues and N atoms, the RTB method requires the diagonalization of an n × n matrix, whereas standard procedures require the diagonalization of a 3N × 3N matrix. Being a fast method, our approach can be useful for normal mode analyses of large systems, paving the way for further developments and applications in contexts for which the normal modes are needed frequently, as for example during molecular dynamics calculations. Proteins 2000;41:1–7. © 2000 Wiley-Liss, Inc.
Normal mode analysis of proteins of various sizes, ranging from 46 (crambin) up to 858 residues (dimeric citrate synthase) were performed, by using standard approaches, as well as a recently proposed method that rests on the hypothesis that low-frequency normal modes of proteins can be described as pure rigid-body motions of blocks of consecutive amino-acid residues. Such a hypothesis is strongly supported by our results, because we show that the latter method, named RTB, yields very accurate approximations for the low-frequency normal modes of all proteins considered. Moreover, the quality of the normal modes thus obtained depends very little on the way the polypeptidic chain is split into blocks. Noteworthy, with six amino-acids per block, the normal modes are almost as accurate as with a single amino-acid per block. In this case, for a protein of n residues and N atoms, the RTB method requires the diagonalization of an n × n matrix, whereas standard procedures require the diagonalization of a 3N × 3N matrix. Being a fast method, our approach can be useful for normal mode analyses of large systems, paving the way for further developments and applications in contexts for which the normal modes are needed frequently, as for example during molecular dynamics calculations. Proteins 2000;41:1–7. © 2000 Wiley-Liss, Inc.
The identification of dynamical domains in proteins and the description of the low-frequency domain motions are one of the important applications of numerical simulation techniques. The application of these techniques to large proteins requires a substantial computational effort and therefore cannot be performed routinely, if at all. This article shows how physically motivated approximations permit the calculation of low-frequency normal modes in a few minutes on standard desktop computers. The technique is based on the observation that the low-frequency modes, which describe domain motions, are independent of force field details and can be obtained with simplified mechanical models. These models also provide a useful measure for rigidity in proteins, allowing the identification of quasi-rigid domains. The methods are validated by application to three well-studied proteins, crambin, lysozyme, and ATCase. In addition to being useful techniques for studying domain motions, the success of the approximations provides new insight into the relevance of normal mode calculations and the nature of the potential energy surface of proteins. Proteins 33:417–429, 1998. © 1998 Wiley-Liss, Inc.
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A coarse-grained protein model implemented in the ATTRACT protein-protein docking program has been employed to predict protein-protein complex structures in CAPRI rounds 22-27. For six targets acceptable or better quality solutions have been submitted corresponding to ~60% of all targets. For one target promising results on the prediction of the hydration structure at the protein-protein interface have been achieved. New approaches for the rapid flexible refinement have been developed based on a combination of atomistic representation of the bonded geometry and a coarse-grained description of non-bonded interactions. Possible further improvements of the docking approach in particular at the scoring and the flexible refinement steps are discussed. © Proteins 2013;. © 2013 Wiley Periodicals, Inc.
Theoretical methods are developed and applied to the protein crambin as a model system to characterize collective normal mode dynamics and their effects on correlations between torsion angle fluctuations and heteronuclear NMR relaxation parameters. Backbone N–H NMR S2 order parameters are found to be predominantly determined by local φ and ψ torsion angle fluctuations induced by collective protein modes. The ratio between Cβ–Hβ and Cα–Hα order parameters directly yields fluctuation amplitudes of the sidechain χ1 torsion angles. The results allow a more direct interpretation of motional effects monitored by nuclear spin relaxation.
The evolutionary divergence of protein structure is mostly contained within a subspace spanned by the evolving protein’s lowest vibrational normal modes, a remarkable recent result, so far unexplained. To understand the mechanism underlying this behavior, here I introduce a linearly forced elastic network model (LFENM) of protein structural evolution. For a test case of globins, LFENM results are in very good agreement with observations. Moreover, in contrast with tentative biological explanations, the model predicts that protein structures will evolve along the lowest normal modes even under unselected random mutations, as a result of the chemical physics of the response of elastic networks of oscillators to perturbations.
We have developed a computer program, named PDBETA, that performs normal mode analysis (NMA) based on an elastic network model that uses dihedral angles as independent variables. Taking advantage of the relatively small number of degrees of freedom required to describe a molecular structure in dihedral angle space and a simple potential-energy function independent of atom types, we aimed to develop a program applicable to a full-atom system of any molecule in the Protein Data Bank (PDB). The algorithm for NMA used in PDBETA is the same as the computer program FEDER/2, developed previously. Therefore, the main challenge in developing PDBETA was to find a method that can automatically convert PDB data into molecular structure information in dihedral angle space. Here, we illustrate the performance of PDBETA with a protein-DNA complex, a protein-tRNA complex, and some non-protein small molecules, and show that the atomic fluctuations calculated by PDBETA reproduce the temperature factor data of these molecules in the PDB. A comparison was also made with elastic-network-model based NMA in a Cartesian-coordinate system.
The relationship between functional conformation changes and thermal dynamics of proteins is investigated with the help of the torsional network model (TNM), an elastic network model in torsion angle space that we recently introduced. We propose and test a null-model of "random" conformation changes that assumes that the contributions of normal modes to conformation changes are proportional to their contributions to thermal fluctuations. Deviations from this null model are generally small. When they are large and significant, they consist in conformation changes that are represented by very few low frequency normal modes and overcome small energy barriers. We interpret these features as the result of natural selection favoring the intrinsic protein dynamics consistent with functional conformation changes. These "selected" conformation changes are more frequently associated to ligand binding, and in particular phosphorylation, than to pairs of conformations with the same ligands. This deep relationship between the thermal dynamics of a protein, represented by its normal modes, and its functional dynamics can reconcile in a unique framework the two models of conformation changes, conformational selection and induced fit. The program TNM that computes torsional normal modes and analyzes conformation changes is available in writing from ubastolla@cbm.uam.es. This article is part of a Special Issue entitled: The emerging dynamic view of proteins: Protein plasticity in allostery, evolution and self-assembly.
The growing interest for comparing protein internal dynamics owes much to the realisation that protein function can be accompanied or assisted by structural fluctuations and conformational changes. Analogously to the case of functional structural elements, those aspects of protein flexibility and dynamics that are functionally oriented should be subject to evolutionary conservation. Accordingly, dynamics-based protein comparisons or alignments could be used to detect protein relationships that are more elusive to sequence and structural alignments. Here we provide an account of the progress that has been made in recent years towards developing and applying general methods for comparing proteins in terms of their internal dynamics and advance the understanding of the structure-function relationship.
Substrate transport in sodium-coupled amino acid symporters involves a large-scale conformational change that shifts the access to the substrate-binding site from one side of the membrane to the other. The structural change is particularly substantial and entails a unique piston-like quaternary rearrangement in glutamate transporters, as evidenced by the difference between the outward-facing and inward-facing structures resolved for the archaeal aspartate transporter Glt(Ph). These structural changes occur over time and length scales that extend beyond the reach of current fully atomic models, but are regularly explored with the use of elastic network models (ENMs). Despite their success with other membrane proteins, ENM-based approaches for exploring the collective dynamics of Glt(Ph) have fallen short of providing a plausible mechanism. This deficiency is attributed here to the anisotropic constraints imposed by the membrane, which are not incorporated into conventional ENMs. Here we employ two novel (to our knowledge) ENMs to demonstrate that one can largely capture the experimentally observed structural change using only the few lowest-energy modes of motion that are intrinsically accessible to the transporter, provided that the surrounding lipid molecules are incorporated into the ENM. The presence of the membrane reduces the overall energy of the transition compared with conventional models, showing that the membrane not only guides the selected mechanism but also acts as a facilitator. Finally, we show that the dynamics of Glt(Ph) is biased toward transitions of individual subunits of the trimer rather than cooperative transitions of all three subunits simultaneously, suggesting a mechanism of transport that exploits the intrinsic dynamics of individual subunits. Our software is available online at http://www.membranm.csb.pitt.edu.
An algorithm is presented for determining multi-dimensional reaction coordinates between two known conformers. Only the energy function and its gradient are required. The resulting paths follow the adiabatic energy valleys and have energy maxima that are true saddle points, which can be multiple along each path. The method is suitable for the study of complex isomerization reactions, including allosteric transitions in proteins and more general conformational changes of macromolecules.
Currently, there are two types of drugs on the market: orthosteric, which bind at the active site; and allosteric, which bind elsewhere on the protein surface, and allosterically change the conformation of the protein binding site. In this perspective we argue that the different mechanisms through which the two drug types affect protein activity and their potential pitfalls call for different considerations in drug design. The key problem facing orthosteric drugs is side effects which can occur by drug binding to homologous proteins sharing a similar binding site. Hence, orthosteric drugs should have very high affinity to the target; this would allow a low dosage to selectively achieve the goal of target-only binding. By contrast, allosteric drugs work by shifting the free energy landscape. Their binding to the protein surface perturbs the protein surface atoms, and the perturbation propagates like waves, finally reaching the binding site. Effective drugs should have atoms in good contact with the 'right' protein atoms; that is, the contacts should elicit propagation waves optimally reaching the protein binding site target. While affinity is important, the design should consider the protein conformational ensemble and the preferred propagation states. We provide examples from functional in vivo scenarios for both types of cases, and suggest how high potency can be achieved in allosteric drug development.
Elastic network models provide an efficient way to quickly calculate protein global dynamics from experimentally determined structures. The model's single parameter, its force constant, determines the physical extent of equilibrium fluctuations. The values of force constants can be calculated by fitting to experimental data, but the results depend on the type of experimental data used. Here, we investigate the differences between calculated values of force constants and data from NMR and X-ray structures. We find that X-ray B factors carry the signature of rigid-body motions, to the extent that B factors can be almost entirely accounted for by rigid motions alone. When fitting to more refined anisotropic temperature factors, the contributions of rigid motions are significantly reduced, indicating that the large contribution of rigid motions to B factors is a result of over-fitting. No correlation is found between force constants fit to NMR data and those fit to X-ray data, possibly due to the inability of NMR data to accurately capture protein dynamics.
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.
We present what to our knowledge is a new method of optimized torsion-angle normal-mode analysis, in which the normal modes move along curved paths in Cartesian space. We show that optimized torsion-angle normal modes reproduce protein conformational changes more accurately than Cartesian normal modes. We also show that orthogonalizing the displacement vectors from torsion-angle normal-mode analysis and projecting them as straight lines in Cartesian space does not lead to better performance than Cartesian normal modes. Clearly, protein motion is more naturally described by curved paths in Cartesian space.
Modeling protein flexibility constitutes a major challenge in accurate prediction of protein-ligand and protein-protein interactions in docking simulations. The lack of a reliable method for predicting the conformational changes relevant to substrate binding prevents the productive application of computational docking to proteins that undergo large structural rearrangements. Here, we examine how coarse-grained normal mode analysis has been advantageously applied to modeling protein flexibility associated with ligand binding. First, we highlight recent studies that have shown that there is a close agreement between the large-scale collective motions of proteins predicted by elastic network models and the structural changes experimentally observed upon ligand binding. Then, we discuss studies that have exploited the predicted soft modes in docking simulations. Two general strategies are noted: pregeneration of conformational ensembles that are then utilized as input for standard fixed-backbone docking and protein structure deformation along normal modes concurrent to docking. These studies show that the structural changes apparently "induced" upon ligand binding occur selectively along the soft modes accessible to the protein prior to ligand binding. They further suggest that proteins offer suitable means of accommodating/facilitating the recognition and binding of their ligand, presumably acquired by evolutionary selection of the suitable three-dimensional structure.
In this paper, we report a new method for coarse-grained elastic normal-mode analysis. The purpose is to overcome a long-standing problem in the conventional analysis called the tip effect that makes the motional patterns (eigenvectors) of some low-frequency modes irrational. The new method retains the merits of a conventional method such as not requiring lengthy initial energy minimization, which always distorts structures, and also delivers substantially more accurate low-frequency modes with no tip effect for proteins of any size. This improvement of modes is crucial for certain types of applications such as structural refinement or normal-mode-based sampling.
Allosteric regulation of protein function is a mechanism by which an event in one place of a protein structure causes an effect at another site, much like the behavior of a telecommunications network in which a collection of transmitters, receivers and transceivers communicate with each other across long distances. For example, ligand binding or an amino acid mutation at an allosteric site can alter enzymatic activity or binding affinity in a distal region such as the active site or a second binding site. The mechanism of this site-to-site communication is of great interest, especially since allosteric effects must be considered in drug design and protein engineering. In this review, conformational mobility as the common route between allosteric regulation and catalysis is discussed. We summarize recent experimental data and the resulting insights into allostery within proteins, and we discuss the nature of future studies and the new applications that may result from increased understanding of this regulatory mechanism.
Understanding protein interactions has broad implications for the mechanism of recognition, protein design, and assigning putative functions to uncharacterized proteins. Studying protein flexibility is a key component in the challenge of describing protein interactions. In this work, we characterize the observed conformational change for a set of 20 proteins that undergo large conformational change upon association (>2 Å Cα RMSD) and ask what features of the motion are successfully reproduced by the normal modes of the system. We demonstrate that normal modes can be used to identify mobile regions and, in some proteins, to reproduce the direction of conformational change. In 35% of the proteins studied, a single low-frequency normal mode was found that describes well the direction of the observed conformational change. Finally, we find that for a set of 134 proteins from a docking benchmark that the characteristic frequencies of normal modes can be used to predict reliably the extent of observed conformational change. We discuss the implications of the results for the mechanics of protein recognition.
• conformational selection
• elastic network model
• induced fit
• protein interactions
• protein recognition
Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins.
Prion proteins (PrP) are the infectious agent in transmissible spongiform encephalopathies (i.e., mad cow disease). To be infectious, prion proteins must undergo a conformational change involving a decrease in α-helical content along with an increase in β-strand content. This conformational change was evaluated by means of elastic normal modes. Elastic normal modes show a diminution of two α-helices by one and two residues, as well as an extension of two β-strands by three residues each, which could instigate the conformational change. The conformational change occurs in a region that is compatible with immunological studies, and it is observed more frequently in mutant prions that are prone to conversion than in wild-type prions because of differences in their starting structures, which are amplified through normal modes. These findings are valuable for our comprehension of the conversion mechanism associated with the conformational change in prion proteins.
Cryo-electron microscopy (cryo-EM) has been widely used to explore conformational states of large biomolecular assemblies. The detailed interpretation of cryo-EM data requires the flexible fitting of a known high-resolution protein structure into a low-resolution cryo-EM map. To this end, we have developed what we believe is a new method based on a two-bead-per-residue protein representation, and a modified form of the elastic network model that allows large-scale conformational changes while maintaining pseudobonds and secondary structures. Our method minimizes a pseudo-energy which linearly combines various terms of the modified elastic network model energy with a cryo-EM-fitting score and a collision energy that penalizes steric collisions. Unlike previous flexible fitting efforts using the lowest few normal modes, our method effectively utilizes all normal modes so that both global and local structural changes can be fully modeled. We have validated our method for a diverse set of 10 pairs of protein structures using simulated cryo-EM maps with a range of resolutions and in the absence/presence of random noise. We have shown that our method is both accurate and efficient compared with alternative techniques, and its performance is robust to the addition of random noise. Our method is also shown to be useful for the flexible fitting of three experimental cryo-EM maps.
Mode coupling and anharmonicity in a native fluctuating protein are investigated in modal space by projecting the motion along the eigenvectors of the fluctuation correlation matrix. The probability distribution of mode fluctuations is expressed in terms of tensorial Hermite polynomials. Molecular dynamics trajectories of Crambin are generated and used to evaluate the terms of the polynomials and to obtain the modal energies. The energies of a few modes exhibit large deviations from the harmonic energy of kT/2 per mode, resulting from coupling to the surroundings, or to another specific mode or to several other modes. Slowest modes have energies that are below that of the harmonic, and a few fast modes have energies significantly larger than the harmonic. Detailed analysis of the coupling of these modes to others is presented in terms of the lowest order two-mode coupling terms. Finally, the effects of mode coupling on conformational properties of the protein are investigated.
A large-scale comparison of essential dynamics (ED) modes from molecular dynamic simulations and normal modes from coarse-grained normal mode methods (CGNM) was performed on a dataset of 335 proteins. As CGNM methods, the elastic network model (ENM) and the rigid cluster normal mode analysis (RCNMA) were used. Low-frequency normal modes from ENM correlate very well with ED modes in terms of directions of motions and relative amplitudes of motions. Notably, a similar performance was found if normal modes from RCNMA were used, despite a higher level of coarse graining. On average, the space spanned by the first quarter of ENM modes describes 84% of the space spanned by the five ED modes. Furthermore, no prominent differences for ED and CGNM modes among different protein structure classes (CATH classification) were found. This demonstrates the general potential of CGNM approaches for describing intrinsic motions of proteins with little computational cost. For selected cases, CGNM modes were found to be more robust among proteins that have the same topology or are of the same homologous superfamily than ED modes. In view of recent evidence regarding evolutionary conservation of vibrational dynamics, this suggests that ED modes, in some cases, might not be representative of the underlying dynamics that are characteristic of a whole family, probably due to insufficient sampling of some of the family members by MD.
The purpose of this lecture is to describe the KAM theorem in its most basic form and to give a complete and detailed proof. This proof essentially follows the traditional lines laid out by the inventors of this theory, and the emphasis is more on the underlying ideas than on the sharpness of the arguments. Comment: 33 pages. Small corrections and updated references