Bridging between normal mode analysis and elastic network models

Normal mode analysis (NMA) has been a powerful tool for studying protein dynamics. Elastic network models (ENM), through their simplicity, have made normal mode computations accessible to a much broader research community and for many more biomolecular systems. The drawback of ENMs, however, is that they are less accurate than NMA. In this work, through steps of simplification that starts with NMA and ends with ENMs we build a tight connection between NMA and ENMs. In the process of bridging between the two, we have also discovered several high‐quality simplified models. Our best simplified model has a mean correlation with the original NMA that is as high as 0.88. In addition, the model is force‐field independent and does not require energy minimization, and thus can be applied directly to experimental structures. Another benefit of drawing the connection is a clearer understanding why ENMs work well and how it can be further improved. We discovered that can be greatly enhanced by including an additional torsional term and a geometry term. Proteins 2014; 82:2157–2168. © 2014 Wiley Periodicals, Inc.     

A mass weighted chemical elastic network model elucidates closed form domain motions in proteins

An elastic network model (ENM), usually Cα coarse‐grained one, has been widely used to study protein dynamics as an alternative to classical molecular dynamics simulation. This simple approach dramatically saves the computational cost, but sometimes fails to describe a feasible conformational change due to unrealistically excessive spring connections. To overcome this limitation, we propose a mass‐weighted chemical elastic network model (MWCENM) in which the total mass of each residue is assumed to be concentrated on the representative alpha carbon atom and various stiffness values are precisely assigned according to the types of chemical interactions. We test MWCENM on several well‐known proteins of which both closed and open conformations are available as well as three α‐helix rich proteins. Their normal mode analysis reveals that MWCENM not only generates more plausible conformational changes, especially for closed forms of proteins, but also preserves protein secondary structures thus distinguishing MWCENM from traditional ENMs. In addition, MWCENM also reduces computational burden by using a more sparse stiffness matrix.     

The performance of fine‐grained and coarse‐grained elastic network models and its dependence on various factors

In a recent work we developed a method for deriving accurate simplified models that capture the essentials of conventional all‐atom NMA and identified two best simplified models: ssNMA and eANM, both of which have a significantly higher correlation with NMA in mean square fluctuation calculations than existing elastic network models such as ANM and ANMr2, a variant of ANM that uses the inverse of the squared separation distances as spring constants. Here, we examine closely how the performance of these elastic network models depends on various factors, namely, the presence of hydrogen atoms in the model, the quality of input structures, and the effect of crystal packing. The study reveals the strengths and limitations of these models. Our results indicate that ssNMA and eANM are the best fine‐grained elastic network models but their performance is sensitive to the quality of input structures. When the quality of input structures is poor, ANMr2 is a good alternative for computing mean‐square fluctuations while ANM model is a good alternative for obtaining normal modes. Proteins 2015; 83:1273–1283. © 2015 Wiley Periodicals, Inc.     

Flexible fitting to cryo-electron microscopy maps with coarse-grained elastic network models

Cryo-electron microscopy has become an important tool for protein structure determination in recent decades. Since proteins may exist in multiple conformational states, combining high resolution X-ray or NMR structures with cryo-electron microscopy maps is a useful approach to obtain proteins in different functional states. Flexible fitting methods used in cryo-electron microscopy aim to obtain an unknown protein conformation from a high resolution structure and a cryo-electron microscopy map. Since all-atom flexible fitting is computationally expensive, many efficient flexible fitting algorithms that utilize coarse-grained elastic network models have been proposed. In this study, we investigated performance of three coarse-grained elastic network model-based flexible fitting methods (EMFF, iModFit, NMFF) using 25 protein pairs at four resolutions. This study shows that the application of coarse-grained elastic network models to flexible fitting of cryo-electron microscopy maps can provide fast and fruitful models of various conformational states of proteins.     

On the relationship between the protein structure and protein dynamics

Recently, we have developed a method (Shih et al., Proteins: Structure, Function, and Bioinformatics 2007;68: 34-38) to compute correlation of fluctuations of proteins. This method, referred to as the protein fixed-point (PFP) model, is based on the positional vectors of atoms issuing from the fixed point, which is the point of the least fluctuations in proteins. One corollary from this model is that atoms lying on the same shell centered at the fixed point will have the same thermal fluctuations. In practice, this model provides a convenient way to compute the average dynamical properties of proteins directly from the geometrical shapes of proteins without the need of any mechanical models, and hence no trajectory integration or sophisticated matrix operations are needed. As a result, it is more efficient than molecular dynamics simulation or normal mode analysis. Though in the previous study the PFP model has been successfully applied to a number of proteins of various folds, it is not clear to what extent this model will be applied. In this article, we have carried out the comprehensive analysis of the PFP model for a dataset comprising 972 high-resolution X-ray structures with pairwise sequence identity or=0.5. Our result shows that the fixed-point model is indeed quite general and will be a useful tool for high throughput analysis of dynamical properties of proteins.     

Validating and improving elastic network models with molecular dynamics simulations

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

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

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

Opening and closing of a toroidal group II chaperonin revealed by a symmetry constrained elastic network model

Recently, the atomic structures of both the closed and open forms of Group 2 chaperonin protein Mm‐cpn were revealed through crystallography and cryo‐electron microscopy. This toroidal‐like chaperonin is composed of two eightfold rings that face back‐to‐back. To gain a computational advantage, we used a symmetry constrained elastic network model (SCENM), which requires only a repeated subunit structure and its symmetric connectivity to neighboring subunits to simulate the entire system. In the case of chaperonin, only six subunits (i.e., three from each ring) were used out of the eight subunits comprising each ring. A smooth and symmetric pathway between the open and closed conformations was generated by elastic network interpolation (ENI). To support this result, we also performed a symmetry‐constrained normal mode analysis (NMA), which revealed the intrinsic vibration features of the given structures. The NMA and ENI results for the representative single subunit were duplicated according to the symmetry pattern to reconstruct the entire assembly. To test the feasibility of the symmetry model, its results were also compared with those obtained from the full model. This study allowed the folding mechanism of chaperonin Mm‐cpn to be elucidated by SCENM in a timely manner.     

Inferring a weighted elastic network from partial unfolding with coarse‐grained simulations

A number of studies have demonstrated that simple elastic network models can reproduce experimental B‐factors, providing insights into the structure–function properties of proteins. Here, we report a study on how to improve an elastic network model and explore its performance by predicting the experimental B‐factors. Elastic network models are built on the experimental coordinates, and they only take the pairs of atoms within a given cutoff distance r_c into account. These models describe the interactions by elastic springs with the same force constant. We have developed a method based on numerical simulations with a simple coarse‐grained force field, to attribute weights to these spring constants. This method considers the time that two atoms remain connected in the network during partial unfolding, establishing a means of measuring the strength of each link. We examined two different coarse‐grained force fields and explored the computation of these weights by unfolding the native structures. Proteins 2014; 82:119–129. © 2013 Wiley Periodicals, Inc.     

Analysis of conformational motions and related key residue interactions responsible for a specific function of proteins with elastic network model

Protein collective motions play a critical role in many biochemical processes. How to predict the functional motions and the related key residue interactions in proteins is important for our understanding in the mechanism of the biochemical processes. Normal mode analysis (NMA) of the elastic network model (ENM) is one of the effective approaches to investigate the structure-encoded motions in proteins. However, the motion modes revealed by the conventional NMA approach do not necessarily correspond to a specific function of protein. In the present work, a new analysis method was proposed to identify the motion modes responsible for a specific function of proteins and then predict the key residue interactions involved in the functional motions by using a perturbation approach. In our method, an internal coordinate that accounts for the specific function was introduced, and the Cartesian coordinate space was transformed into the internal/Cartesian space by using linear approximation, where the introduced internal coordinate serves as one of the axes of the coordinate space. NMA of ENM in this internal/Cartesian space was performed and the function-relevant motion modes were identified according to their contributions to the specific function of proteins. Then the key residue interactions important for the functional motions of the protein were predicted as the interactions whose perturbation largely influences the fluctuation along the internal coordinate. Using our proposed methods, the maltose transporter (MalFGK₂) from E. Coli was studied. The functional motions and the key residue interactions that are related to the channel-gating function of this protein were successfully identified.     

How orientational order governs collectivity of folded proteins

The past decade has witnessed the development and success of coarse‐grained network models of proteins for predicting many equilibrium properties related to collective modes of motion. Curiously, the results are usually robust toward the different cutoff distances used for constructing the residue networks from the knowledge of the experimental coordinates. In this study, we present a systematical study of network construction and their effect on the predicted properties. Probing bond orientational order around each residue, we propose a natural partitioning of the interactions into an essential and a residual set. In this picture, the robustness originates from the way with which new contacts are added, so that an unusual local orientational order builds up. These residual interactions have a vanishingly small effect on the force vectors on each residue. The stability of the overall force balance then translates into the Hessian as small shifts in the slow modes of motion and an invariance of the corresponding eigenvectors. We introduce a rescaled version of the Hessian matrix and point out a link between the matrix Frobenius norm based on spectral stability arguments and orientational local order. A recipe for the optimal choice of partitioning the interactions into essential and residual components is prescribed. Implications for the study of biologically relevant properties of proteins are discussed with specific examples. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

Tensorial elastic network model for protein dynamics: Integration of the anisotropic network model with bond‐bending and twist elasticities

We present a tensorial elastic network model (TNM) to describe the equilibrium fluctuations of proteins near their native fold structure. The model combines the anisotropic network model (ANM), bond bending elasticity, and backbone twist elasticity, and can predict both the isotropic fluctuations, similar to the Gaussian network model (GNM), and anisotropic fluctuations, similar to the ANM. TNM performs equally well for B‐factor predictions as GNM and predicts the anisotropy of B‐factors better than ANM. The model also outperforms the ANM in its predictability of the complete anisotropic displacement parameters. Proteins 2012; © 2012 Wiley Periodicals, Inc.     

Identification of dynamical correlations within the myosin motor domain by the normal mode analysis of an elastic network model

In order to systematically analyze functionally relevant dynamical correlations within macromolecular complexes, we have developed computational methods based on the normal mode analysis of an elastic network model. First, we define two types of dynamical correlations (fluctuation-based and density-based), which are computed by summing up contributions from all low-frequency normal modes up to a given cutoff. Then we use them to select dynamically important "hinge residues" whose elastic distortion affects the fluctuations of a large number of residues. Second, in order to clarify long-range dynamical correlations, we decompose the dynamical correlations to individual normal modes to identify the most relevant modes. We have applied these methods to the analysis of the motor domain of Dictyostelium myosin and have obtained the following three interesting results that shed light on its mechanism of force generation: first, we find the hinge residues are distributed over several key inter-subdomain joints (including the nucleotide-binding pocket, the relay helix, the SH1 helix, the strut between the upper 50 kDa and the lower 50 kDa subdomains), which is consistent with their hypothesized roles in modulating functionally relevant inter-subdomain conformational changes; second, a single mode 7 (for structure 1VOM) is found to dominate the fluctuation-based correlations between the converter/strut and the nucleotide-binding pocket, revealing a surprising simplicity for their intriguing roles in the force generation mechanism; finally, we find a negative density-based correlation between the strut and the nucleotide-binding pocket, which is consistent with the hypothesized signaling pathway that links the actin-binding site's opening/closing with the nucleotide-binding pocket's closing/opening.     

Toward the mechanism of dynamical couplings and translocation in hepatitis C virus NS3 helicase using elastic network model

Hepatitis C virus NS3 helicase is an enzyme that unwinds double-stranded polynucleotides in an ATP-dependent reaction. It provides a promising target for small molecule therapeutic agents against hepatitis C. Design of such drugs requires a thorough understanding of the dynamical nature of the mechanochemical functioning of the helicase. Despite recent progress, the detailed mechanism of the coupling between ATPase activity and helicase activity remains unclear. Based on an elastic network model (ENM), we apply two computational analysis tools to probe the dynamical mechanism underlying the allosteric coupling between ATP binding and polynucleotide binding in this enzyme. The correlation analysis identifies a network of hot-spot residues that dynamically couple the ATP-binding site and the polynucleotide-binding site. Several of these key residues have been found by mutational experiments as functionally important, while our analysis also reveals previously unexplored hot-spot residues that are potential targets for future mutational studies. The conformational changes between different crystal structures of NS3 helicase are found to be dominated by the lowest frequency mode solved from the ENM. This mode corresponds to a hinge motion of the highly flexible domain 2. This motion simultaneously modulates the opening/closing of the domains 1-2 cleft where ATP binds, and the domains 2-3 cleft where the polynucleotide binds. Additionally, a small twisting motion of domain 1, observed in both mode 1 and the computed ATP binding induced conformational change, fine-tunes the binding affinity of the domains 1-3 interface for the polynucleotide. The combination of these motions facilitates the translocation of a single-stranded polynucleotide in an inchworm-like manner.     

A simple way to compute protein dynamics without a mechanical model

We found that in proteins the average atomic fluctuation is linearly related to the square of the atomic distance from the center of mass of the protein. Using this simple relation, we can accurately compute the temperature factors of proteins of a wide range of sizes and folds, and the correlation of the fluctuations in proteins. This simple relation provides a direct link between protein dynamics and the static protein's geometrical shape and offers a simple way to compute protein dynamics without either long time trajectory integration or any matrix operations.     

Protein conformational transitions explored by mixed elastic network models

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

The effects of rigid motions on elastic network model force constants

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.     

Multiple Gaussian network modes alignment reveals dynamically variable regions: The hemoglobin case

Gaussian network model (GNM) modes of motion are calculated to a dataset of h emoglobin (Hb) structures and modes with dynamics similarity to the T state are multiply aligned. The sole criterion for the alignment is the mode shape itself and not sequence or structural similarity. Standard deviation (SD) of the GNM value score along the alignment is calculated, regions with high SD are defined as dynamically variable. The analysis shows that the α1β1/α2β2 interface is a dynamically variable region but not the α1β2/α2β1 and the α1α2/β1β2 interfaces. The results are in accordance with the T → R2 transition of Hb. We suggest that dynamically variable regions are regions that are likely to undergo structural change in the protein upon binding, conformational transition, or any other relevant chemical event. The represented technique of multiple dynamics ‐ based alignment of modes is novel and may offer a new insight in proteins ' dynamics to function relation. Proteins 2014; 82:2097–2105. © 2014 Wiley Periodicals, Inc.     

A network of dynamically conserved residues deciphers the motions of maltose transporter

The maltose transporter of Escherichia coli is a member of the ATP‐binding cassette (ABC) transporter superfamily. The crystal structures of maltose transporter MalK have been determined for distinct conformations in the presence and absence of the ligand ATP, and other interacting proteins. Using the distinct MalK structures, normal mode analysis was performed to understand the dynamics behavior of the system. A network of dynamically important residues was obtained from the normal mode analysis and the analysis of point mutation on the normal modes. Our results suggest that the intradomain rotation occurs earlier than the interdomain rotation during the maltose‐binding protein (MBP)‐driven conformational changes of MalK. We inquire if protein motion and functional‐driven evolutionary conservation are related. The sequence conservation of MalK was analyzed to derive a network of evolutionarily important residues. There are highly significant correlations between protein sequence and protein dynamics in many regions on the maltose transporter MalK, suggesting a link between the protein evolution and dynamics. The significant overlaps between the network of dynamically important residues and the network of evolutionarily important residues form a network of dynamically conserved residues. Proteins 2009. © 2009 Wiley‐Liss, Inc.