Efficient determination of low-frequency normal modes of large protein structures by cluster-NMA

Protein dynamics developments for the large scale and cryoEM: case study of ProDy 2.0

New computational biophysics pipelines for analysing the global dynamics of structural ensembles and large, dynamic complexes resolved by cryoEM are reviewed.

Cryo-electron microscopy (cryoEM) has become a well established technique with the potential to produce structures of large and dynamic supramolecular complexes that are not amenable to traditional approaches for studying structure and dynamics. The size and low resolution of such molecular systems often make structural modelling and molecular dynamics simulations challenging and computationally expensive. This, together with the growing wealth of structural data arising from cryoEM and other structural biology methods, has driven a trend in the computational biophysics community towards the development of new pipelines for analysing global dynamics using coarse-grained models and methods. At the centre of this trend has been a return to elastic network models, normal mode analysis (NMA) and ensemble analyses such as principal component analysis, and the growth of hybrid simulation methodologies that make use of them. Here, this field is reviewed with a focus on ProDy, the Python application programming interface for protein dynamics, which has been developed over the last decade. Two key developments in this area are highlighted: (i) ensemble NMA towards extracting and comparing the signature dynamics of homologous structures, aided by the recent SignDy pipeline, and (ii) pseudoatom fitting for more efficient global dynamics analyses of large and low-resolution supramolecular assemblies from cryoEM, revisited in the CryoDy pipeline. It is believed that such a renewal and extension of old models and methods in new pipelines will be critical for driving the field forward into the next cryoEM revolution.

Entropy of Proteins Using Multiscale Cell Correlation

A new multiscale method is presented to calculate the entropy of proteins from molecular dynamics simulations. Termed Multiscale Cell Correlation (MCC), the method decomposes the protein into sets of rigid-body units based on their covalent-bond connectivity at three levels of hierarchy: molecule, residue, and united atom. It evaluates the vibrational and topographical entropy from forces, torques, and dihedrals at each level, taking into account correlations between sets of constituent units that together make up a larger unit at the coarser length scale. MCC gives entropies in close agreement with normal-mode analysis and smaller than those using quasiharmonic analysis as well as providing much faster convergence. Moreover, MCC provides an insightful decomposition of entropy at each length scale and for each type of amino acid according to their solvent exposure and whether they are terminal residues. While the residue entropy depends weakly on solvent exposure, there is greater variation in entropy components for larger, more polar amino acids, which have increased conformational entropy but reduced vibrational entropy with greater solvent exposure.

NOLB: Nonlinear Rigid Block Normal-Mode Analysis Method

We present a new conceptually simple and computationally efficient method for nonlinear normal-mode analysis called NOLB. It relies on the rotations-translations of blocks (RTB) theoretical basis developed by Y.-H. Sanejouand and colleagues [ Durand et al. Biopolymers 1994 , 34 , 759 - 771 . Tama et al. Proteins: Struct., Funct., Bioinf . 2000 , 41 , 1 - 7 ]. We demonstrate how to physically interpret the eigenvalues computed in the RTB basis in terms of angular and linear velocities applied to the rigid blocks and how to construct a nonlinear extrapolation of motion out of these velocities. The key observation of our method is that the angular velocity of a rigid block can be interpreted as the result of an implicit force, such that the motion of the rigid block can be considered as a pure rotation about a certain center. We demonstrate the motions produced with the NOLB method on three different molecular systems and show that some of the lowest frequency normal modes correspond to the biologically relevant motions. For example, NOLB detects the spiral sliding motion of the TALE protein, which is capable of rapid diffusion along its target DNA. Overall, our method produces better structures compared to the standard approach, especially at large deformation amplitudes, as we demonstrate by visual inspection, energy, and topology analyses and also by the MolProbity service validation. Finally, our method is scalable and can be applied to very large molecular systems, such as ribosomes. Standalone executables of the NOLB normal-mode analysis method are available at https://team.inria.fr/nano-d/software/nolb-normal-modes/ . A graphical user interface created for the SAMSON software platform will be made available at https://www.samson-connect.net .

A better prediction of conformational changes of proteins using minimally connected network models

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.

New generation of elastic network models

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

Ribosome Mechanics Informs about Mechanism

The essential aspects of the ribosome's mechanism can be extracted from coarse-grained simulations, including the ratchet motion, the movement together of critical bases at the decoding center, and movements of the peptide tunnel lining that assist in the expulsion of the synthesized peptide. Because of its large size, coarse graining helps to simplify and to aid in the understanding of its mechanism. Results presented here utilize coarse-grained elastic network modeling to extract the dynamics, and both RNAs and proteins are coarse grained. We review our previous results, showing the well-known ratchet motions and the motions in the peptide tunnel and in the mRNA tunnel. The motions of the lining of the peptide tunnel appear to assist in the expulsion of the growing peptide chain, and clamps at the ends of the mRNA tunnel with three proteins ensure that the mRNA is held tightly during decoding and essential for the helicase activity at the entrance. The entry clamp may also assist in base recognition to ensure proper selection of the incoming tRNA. The overall precision of the ribosome machine-like motions is remarkable.

Exploring the conformational transitions of biomolecular systems using a simple two-state anisotropic network model

Biomolecular conformational transitions are essential to biological functions. Most experimental methods report on the long-lived functional states of biomolecules, but information about the transition pathways between these stable states is generally scarce. Such transitions involve short-lived conformational states that are difficult to detect experimentally. For this reason, computational methods are needed to produce plausible hypothetical transition pathways that can then be probed experimentally. Here we propose a simple and computationally efficient method, called ANMPathway, for constructing a physically reasonable pathway between two endpoints of a conformational transition. We adopt a coarse-grained representation of the protein and construct a two-state potential by combining two elastic network models (ENMs) representative of the experimental structures resolved for the endpoints. The two-state potential has a cusp hypersurface in the configuration space where the energies from both the ENMs are equal. We first search for the minimum energy structure on the cusp hypersurface and then treat it as the transition state. The continuous pathway is subsequently constructed by following the steepest descent energy minimization trajectories starting from the transition state on each side of the cusp hypersurface. Application to several systems of broad biological interest such as adenylate kinase, ATP-driven calcium pump SERCA, leucine transporter and glutamate transporter shows that ANMPathway yields results in good agreement with those from other similar methods and with data obtained from all-atom molecular dynamics simulations, in support of the utility of this simple and efficient approach. Notably the method provides experimentally testable predictions, including the formation of non-native contacts during the transition which we were able to detect in two of the systems we studied. An open-access web server has been created to deliver ANMPathway results.

Many biomolecules are like tiny molecular machines that need to change their shapes and visit many states to perform their biological functions. For a complete molecular understanding of a biological process, one needs to have information on the relevant stable states of the system in question, as well as the pathways by which the system travels from one state to another. We report here an efficient computational method that uses the knowledge of experimental structures of a pair of stable states in order to construct an energetically favoravle pathway between them. We adopt a simple representation of the molecular system by replacing the atoms with beads connected by springs and constructing an energy function with two minima around the end-states. We searched for the structure with highest energy that the system is most likely to visit during the transition and created two paths starting from this structure and proceeding toward the end-states. The combined result of these two paths is the minimum energy pathway between the two stable states. We apply this method to study important structural changes in one enzyme and three large proteins that transport small molecules and ions across the cell membrane.

Modeling protein conformational transitions by a combination of coarse-grained normal mode analysis and robotics-inspired methods

BACKGROUND

Obtaining atomic-scale information about large-amplitude conformational transitions in proteins is a challenging problem for both experimental and computational methods. Such information is, however, important for understanding the mechanisms of interaction of many proteins.

METHODS

This paper presents a computationally efficient approach, combining methods originating from robotics and computational biophysics, to model protein conformational transitions. The ability of normal mode analysis to predict directions of collective, large-amplitude motions is applied to bias the conformational exploration performed by a motion planning algorithm. To reduce the dimension of the problem, normal modes are computed for a coarse-grained elastic network model built on short fragments of three residues. Nevertheless, the validity of intermediate conformations is checked using the all-atom model, which is accurately reconstructed from the coarse-grained one using closed-form inverse kinematics.

RESULTS

Tests on a set of ten proteins demonstrate the ability of the method to model conformational transitions of proteins within a few hours of computing time on a single processor. These results also show that the computing time scales linearly with the protein size, independently of the protein topology. Further experiments on adenylate kinase show that main features of the transition between the open and closed conformations of this protein are well captured in the computed path.

CONCLUSIONS

The proposed method enables the simulation of large-amplitude conformational transitions in proteins using very few computational resources. The resulting paths are a first approximation that can directly provide important information on the molecular mechanisms involved in the conformational transition. This approximation can be subsequently refined and analyzed using state-of-the-art energy models and molecular modeling methods.

Density-cluster NMA: A new protein decomposition technique for coarse-grained normal mode analysis

Normal mode analysis has emerged as a useful technique for investigating protein motions on long time scales. This is largely due to the advent of coarse-graining techniques, particularly Hooke's Law-based potentials and the rotational-translational blocking (RTB) method for reducing the size of the force-constant matrix, the Hessian. Here we present a new method for domain decomposition for use in RTB that is based on hierarchical clustering of atomic density gradients, which we call Density-Cluster RTB (DCRTB). The method reduces the number of degrees of freedom by 85-90% compared with the standard blocking approaches. We compared the normal modes from DCRTB against standard RTB using 1-4 residues in sequence in a single block, with good agreement between the two methods. We also show that Density-Cluster RTB and standard RTB perform well in capturing the experimentally determined direction of conformational change. Significantly, we report superior correlation of DCRTB with B-factors compared with 1-4 residue per block RTB. Finally, we show significant reduction in computational cost for Density-Cluster RTB that is nearly 100-fold for many examples.

fSUB: normal mode analysis with flexible substructures

In this paper, we report a novel normal-mode analysis for supramolecular complexes, named fSUB. The method models a complex as a group of flexible substructures. The low-frequency substructure modes are first determined with substructures in isolation, and the motions of the whole complex are then calculated on the basis of substructure modes and substructure-substructure interactions. The calculation of modes in fSUB requires modal analysis without initial energy minimization, which is essential for maintaining energetic and structural consistency between substructures and whole complex. Compared with other coarse-grained methods, such as the RTB method, fSUB delivers much more accurate modes for the complex and allows for the choice of much larger substructures. The method can also accommodate any type of substructure arrangement including covalent bonds across the interface. In tests on molecular chaperonin GroEL (7350 residues) and HK97 capsid complex (118,092 residues), fSUB was shown to be much more efficient in terms of combined accuracy and demand of computing resources. Our results clearly demonstrated the vital importance of including substructure flexibility in complex modal analysis, as the deformational patterns of substructures were found to play an important role even in the lowest frequency modes of the whole complex.

The importance of slow motions for protein functional loops

Loops in proteins that connect secondary structures such as alpha-helix and beta-sheet, are often on the surface and may play a critical role in some functions of a protein. The mobility of loops is central for the motional freedom and flexibility requirements of active-site loops and may play a critical role for some functions. The structures and behaviors of loops have not been studied much in the context of the whole structure and its overall motions, especially how these might be coupled. Here we investigate loop motions by using coarse-grained structures (C(α) atoms only) to solve the motions of the system by applying Lagrange equations with elastic network models to learn about which loops move in an independent fashion and which move in coordination with domain motions, faster and slower, respectively. The normal modes of the system are calculated using eigen-decomposition of the stiffness matrix. The contribution of individual modes and groups of modes is investigated for their effects on all residues in each loop by using Fourier analyses. Our results indicate overall that the motions of functional sets of loops behave in similar ways as the whole structure. But overall only a relatively few loops move in coordination with the dominant slow modes of motion, and these are often closely related to function.

Computational methods for identifying a layered allosteric regulatory mechanism for ALS-causing mutations of Cu-Zn superoxide dismutase 1

The most prominent form of familial amyotrophic lateral sclerosis (fALS, Lou Gehrig's Disease) is caused by mutations of Cu-Zn superoxide dismutase 1 (SOD1). SOD1 maintains antioxidant activity under fALS causing mutations, suggesting that the mutations introduce a new, toxic, function. There are 100+ such known mutations that are chemically diverse and spatially distributed across the structure. The common phenotype leads us to propose an allosteric regulatory mechanism hypothesis: SOD1 mutants alter the correlated dynamics of the structure and differentially signal across an inherent allosteric network, thereby driving the disease mechanism at varying rates of efficiency. Two recently developed computational methods for identifying allosteric control sites are applied to the wild type crystal structure, 4 fALS mutant crystal structures, 20 computationally generated fALS mutants and 1 computationally generated non-fALS mutant. The ensemble of mutant structures is used to generate an ensemble of dynamics, from which two allosteric control networks are identified. One network is connected to the catalytic site and thus may be involved in the natural antioxidant function. The second allosteric control network has a locus bordering the dimer interface and thus may serve as a mechanism to modulate dimer stability. Though the toxic function of mutated SOD1 is unknown and likely due to several contributing factors, this study explains how diverse mutations give rise to a common function. This new paradigm for allostery controlled function has broad implications across allosteric systems and may lead to the identification of the key chemical activity of SOD1-linked ALS.

Immunoglobulin Structure Exhibits Control over CDR Motion

Motions of the IgG structure are evaluated using normal mode analysis of an elastic network model to detect hinges, the dominance of low frequency modes, and the most important internal motions. One question we seek to answer is whether or not IgG hinge motions facilitate antigen binding. We also evaluate the protein crystal and packing effects on the experimental temperature factors and disorder predictions. We find that the effects of the protein environment on the crystallographic temperature factors may be misleading for evaluating specific functional motions of IgG. The extent of motion of the antigen binding domains is computed to show their large spatial sampling. We conclude that the IgG structure is specifically designed to facilitate large excursions of the antigen binding domains. Normal modes are shown as capable of computationally evaluating the hinge motions and the spatial sampling by the structure. The antigen binding loops and the major hinge appear to behave similarly to the rest of the structure when we consider the dominance of the low frequency modes and the extent of internal motion. The full IgG structure has a lower spectral dimension than individual F_ab domains, pointing to more efficient information transfer through the antibody than through each domain. This supports the claim that the IgG structure is specifically constructed to facilitate antigen binding by coupling motion of the antigen binding loops with the large scale hinge motions.

Rationale for more diverse inhibitors in competition with substrates in HIV-1 protease

The structural fluctuations of HIV-1 protease in interaction with its substrates versus inhibitors were analyzed using the anisotropic network model. The directions of fluctuations in the most cooperative functional modes differ mainly around the dynamically key regions, i.e., the hinge axes, which appear to be more flexible in substrate complexes. The flexibility of HIV-1 protease is likely optimized for the substrates' turnover, resulting in substrate complexes being dynamic. In contrast, in an inhibitor complex, the inhibitor should bind and lock down to inactivate the active site. Protease and ligands are not independent. Substrates are also more flexible than inhibitors and have the potential to meet the dynamic distributions that are inherent in the protease. This may suggest a rationale and guidelines for designing inhibitors that can better fit the ensemble of binding sites that are dynamically accessible to the protease.

Group theory and biomolecular conformation: I. Mathematical and computational models

Biological macromolecules, and the complexes that they form, can be described in a variety of ways ranging from quantum mechanical and atomic chemical models, to coarser grained models of secondary structure and domains, to continuum models. At each of these levels, group theory can be used to describe both geometric symmetries and conformational motion. In this survey, a detailed account is provided of how group theory has been applied across computational structural biology to analyze the conformational shape and motion of macromolecules and complexes.

Contact and impact in the multibody dynamics of motor protein locomotion

This paper presents the development of a rigid multibody dynamics approach to modeling, simulation, and analysis of motor proteins. A key element of this new model is that it retains the mass properties, in contrast to many commonly used models that do not. The mass properties are usually omitted because their inclusion yields a model with multiple time scales whose simulation requires a significant amount of time. However, the proposed model can be numerically integrated in a reasonable amount of time. Thus this approach represents a new method for treating multiple scale models. In addition, retaining the mass properties allows a detailed study of contact and impact between the protein and substrate, which is critical for protein processivity. The new model also provides insights into the characteristics of the protein and its environment, specifically, the effective damping experienced by the protein moving through its fluid environment may be quite small yielding under or critically damped motion. This conclusion runs contrary to the widely accepted notion that the protein's motion is strictly over damped. Herein, the differences between the motion predicted by the old and new modeling approaches are compared using a simplified model of Myosin V.

Approximate normal mode analysis based on vibrational subsystem analysis with high accuracy and efficiency

Normal mode analysis (NMA) has been proven valuable in modeling slow conformational dynamics of biomolecular structures beyond the reach of direct molecular simulations. However, it remains computationally expensive to directly solve normal modes for large biomolecular systems. In this study, we have evaluated the accuracy and efficiency of two approximate NMA protocols-one based on our recently proposed vibrational subsystem analysis (VSA), the other based on the rotation translation block (RTB), in comparison with standard NMA that directly solves a full Hessian matrix. By properly accounting for flexibility within blocks of residues or atoms based on a subsystem-environment partition, VSA-based NMA has attained a much higher accuracy than RTB and much lower computing cost than standard NMA. Therefore, VSA enables accurate and efficient calculations of normal modes from all-atom or coarse-grained potential functions, which promise to improve conformational sampling driven by low-frequency normal modes.

Computational methods for predicting sites of functionally important dynamics

Understanding and controlling biological function of proteins at the atomic level is of great importance; allosteric mechanisms provide such an interface. Experimental and computational methods have been developed to search for residue mutations that produce changes in function by altering sites of correlated motion. These methods are often observational in that altered motions are achieved by random sampling without revealing the underlying mechanism(s). We present two deterministic methods founded on structure-function relationships that predict dynamic control sites (i.e., locations that experience correlated motions as a result of altered dynamics). The first method ("static") is based on a single structure conformation (e.g., the wild type (WT)) and utilizes a graph description of atomic connectivity. The local atomic interactions are used to compute the propagation of contact paths. This description of structure connectivity reveals flexible locations that are susceptible to altered dynamics. The second method ("dynamic") is a comparative analysis between the normal modes of a WT structure and a mutant structure. A mapping function is defined that quantifies the significance of the motions in one structure projected onto the motions of the other. Each mode is considered up- or down-regulated according to its change in relative significance. This description of altered dynamics is the basis for a motion correlation analysis, from which the dynamic control sites are readily identified. The methods are theoretically derived and applied using the canonical system dihydrofolate reductase (DHFR). Both methods demonstrate a very high predictive value (p<0.005) in identifying known dynamic control sites. The dynamic method also produces a new hypothesis regarding the mechanism by which the DHFR mutant achieves hyperactivity. These tools are suitable for allosteric investigations and may greatly enhance the speed and effectiveness of other computational and experimental methods.

The small mass assumption applied to the multibody dynamics of motor proteins

This paper presents a multibody dynamics approach to the modeling, simulation and analysis of motor protein dynamic behavior. Usually proteins are modeled as a collection of a small number of particles, oftentimes only two particles, with spring-like connections between them. However, these simple particle models potentially omit interactions and dynamics between the unmodeled particles and the environment that may be important. Multibody dynamics models can be used to address this issue. However, widely used techniques for analyzing particle models may not be applicable to multibody models. Herein the validity of the small mass assumption, which is central to the analysis of particle models, is examined in detail using a simplified multibody model of the molecular motor protein Myosin V.

Residue Level Three-dimensional Workspace Maps for Conformational Trajectory Planning of Proteins

The function of a protein macromolecule often requires conformational transitions between two native conformations. Understanding these transitions is essential to the understanding of how proteins function, as well as to the ability to design and manipulate protein-based nanomechanical systems. In this paper we propose a set of 3D Cartesian workspace maps for exploring protein pathways. These 3D maps are constructed in the Euclidean space for triads of chain segments of protein molecules that have been shown to have a high probability of occurrence in naturally observed proteins based on data obtained from more than 38,600 proteins from the Protein Data Bank (PDB). We show that the proposed 3D propensity maps are more effective navigation tools than the propensity maps constructed in the angle space. We argue that the main reason for this improved efficiency is the fact that, although there is a one-to-one mapping between the 2D dihedral angle maps and the 3D Cartesian maps, the propensity distributions are significantly different in the two spaces. Hence, the 3D maps allow the pathway planning to be performed directly in the 3D Euclidean space based on propensities computed in the same space, which is where protein molecules change their conformations.

Allosteric communication occurs via networks of tertiary and quaternary motions in proteins

Allosteric proteins bind an effector molecule at one site resulting in a functional change at a second site. We hypothesize that allosteric communication in proteins relies upon networks of quaternary (collective, rigid-body) and tertiary (residue-residue contact) motions. We argue that cyclic topology of these networks is necessary for allosteric communication. An automated algorithm identifies rigid bodies from the displacement between the inactive and the active structures and constructs "quaternary networks" from these rigid bodies and the substrate and effector ligands. We then integrate quaternary networks with a coarse-grained representation of contact rearrangements to form "global communication networks" (GCNs). The GCN reveals allosteric communication among all substrate and effector sites in 15 of 18 multidomain and multimeric proteins, while tertiary and quaternary networks exhibit such communication in only 4 and 3 of these proteins, respectively. Furthermore, in 7 of the 15 proteins connected by the GCN, 50% or more of the substrate-effector paths via the GCN are "interdependent" paths that do not exist via either the tertiary or the quaternary network. Substrate-effector "pathways" typically are not linear but rather consist of polycyclic networks of rigid bodies and clusters of rearranging residue contacts. These results argue for broad applicability of allosteric communication based on structural changes and demonstrate the utility of the GCN. Global communication networks may inform a variety of experiments on allosteric proteins as well as the design of allostery into non-allosteric proteins.

Iterative cluster-NMA: A tool for generating conformational transitions in proteins

Computational models provide insight into the structure-function relationship in proteins. These approaches, especially those based on normal mode analysis, can identify the accessible motion space around a given equilibrium structure. The large magnitude, collective motions identified by these methods are often well aligned with the general direction of the expected conformational transitions. However, these motions cannot realistically be extrapolated beyond the local neighborhood of the starting conformation. In this article, the iterative cluster-NMA (icNMA) method is presented for traversing the energy landscape from a starting conformation to a desired goal conformation. This is accomplished by allowing the evolving geometry of the intermediate structures to define the local accessible motion space, and thus produce an appropriate displacement. Following the derivation of the icNMA method, a set of sample simulations are performed to probe the robustness of the model. A detailed analysis of beta1,4-galactosyltransferase-T1 is also given, to highlight many of the capabilities of icNMA. Remarkably, during the transition, a helix is seen to be extended by an additional turn, emphasizing a new unknown role for secondary structures to absorb slack during transitions. The transition pathway for adenylate kinase, which has been frequently studied in the literature, is also discussed.

Group theory of icosahedral virus capsid vibrations: a top-down approach

We explore the use of a top-down approach to analyse the dynamics of icosahedral virus capsids and complement the information obtained from bottom-up studies of viral vibrations available in the literature. A normal mode analysis based on protein association energies is used to study the frequency spectrum, in which we reveal a universal plateau of low-frequency modes shared by a large class of Caspar-Klug capsids. These modes break icosahedral symmetry and are potentially relevant to the genome release mechanism. We comment on the role of viral tiling theory in such dynamical considerations.

An NMA-guided path planning approach for computing large-amplitude conformational changes in proteins

This paper presents a new method for computing macromolecular motions based on the combination of path planning algorithms, originating from robotics research, and elastic network normal mode analysis. The low-frequency normal modes are regarded as the collective degrees of freedom of the molecule. Geometric path planning algorithms are used to explore these collective degrees of freedom in order to find possible large-amplitude conformational changes. To overcome the limits of the harmonic approximation, which is valid in the vicinity of the minimum energy structure, and to get larger conformational changes, normal mode calculations are iterated during the exploration. Initial results show the efficiency of our method, which requires a small number of normal mode calculations to compute large-amplitude conformational transitions in proteins. A detailed analysis is presented for the computed transition between the open and closed structures of adenylate kinase. This transition, important for its biological function, involves large-amplitude domain motions. The obtained motion correlates well with results presented in related works.

Coarse-graining of protein structures for the normal mode studies

The coarse-grained structural model such as Gaussian network has played a vital role in the normal mode studies for understanding protein dynamics related to biological functions. However, for the large proteins, the Gaussian network model is computationally unfavorable for diagonalization of Hessian (stiffness) matrix for the normal mode studies. In this article, we provide the coarse-graining method, referred to as "dynamic model condensation," which enables the further coarse-graining of protein structures consisting of small number of residues. It is shown that the coarser-grained structures reconstructed by dynamic model condensation exhibit the dynamic characteristics, such as low-frequency normal modes, qualitatively comparable to original structures. This sheds light on that dynamic model condensation and may enable one to study the large protein dynamics for gaining insight into biological functions of proteins.

Dynamic protein domains: identification, interdependence, and stability

Existing methods of domain identification in proteins usually provide no information about the degree of domain independence and stability. However, this information is vital for many areas of protein research. The recently developed hierarchical clustering of correlation patterns (HCCP) technique provides machine-based domain identification in a computationally simple and physically consistent way. Here we present the modification of this technique, which not only allows determination of the most plausible number of dynamic domains but also makes it possible to estimate the degree of their independence (the extent of correlated motion) and stability (the range of environmental conditions, where domains remain intact). With this technique we provided domain assignments and calculated intra- and interdomain correlations and interdomain energies for >2500 test proteins. It is shown that mean intradomain correlation of motions can serve as a quantitative criterion of domain independence, and the HCCP stability gap is a measure of their stability. Our data show that the motions of domains with high stability are usually independent. In contrast, the domains with moderate stability usually exhibit a substantial degree of correlated motions. It is shown that in multidomain proteins the domains are most stable if they are of similar size, and this correlates with the observed abundance of such proteins.

A method for finding candidate conformations for molecular replacement using relative rotation between domains of a known structure

This paper presents a methodology to obtain candidate conformations of multidomain proteins for use in molecular replacement. For each separate domain, the orientational relationship between the template and the target structure is obtained using standard molecular replacement. The orientational relationships of the domains are then used to calculate the relative rotation between the domains in the target conformation by using pose-estimation techniques from the field of robotics and computer vision. With the angle of relative rotation between the domains as a cost function, iterative normal-mode analysis is used to drive the template structure to a candidate conformation that matches the X-ray crystallographic data obtained for the target conformation. The selection of the correct intra-protein domain orientations from among the many spurious maxima in the rotation function (including orientations obtained from domains in symmetry mates rather than within the same copy of the protein) presents a challenge. This problem is resolved by checking R factors of each domain, measuring the absolute value of relative rotation between domains, and evaluating the cost value after each candidate conformation is driven to convergence with iterative NMA. As a validation, the proposed method is applied to three test proteins: ribose-binding protein, lactoferrin and calcium ATPase. In each test case, the orientation and translation of the final candidate conformation in the unit cell are generated correctly from the suggested procedure. The results show that the proposed method can yield viable candidate conformations for use in molecular replacement and can reveal the structural details and pose of the target conformation in the crystallographic unit cell.