Self-organizing neural networks bridge the biomolecular resolution gap

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

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.

A new method for the construction of coarse-grained models of large biomolecules from low-resolution cryo-electron microscopy data

The rapid development of cryo-electron microscopy (cryo-EM) has led to the generation of significant low-resolution electron density data of biomolecules. However, the atomistic details of huge biomolecules usually cannot be obtained because it is very difficult to construct all-atom models for MD simulations. Thus, it is still a challenge to make use of the rich low-resolution cryo-EM data for computer simulation and functional study. In this study, we proposed a new method called Convolutional and K-means Coarse-Graining (CK-CG) for the efficient coarse-graining of large biological systems. Using the CK-CG method, we could directly map the cryo-EM data into coarse-grained (CG) beads. Furthermore, the CG beads were parameterized with an empirical harmonic potential to construct a new CG model. We subjected the CK-CG models of the fibrillar protein assemblies F-actin and collagen to external forces in pulling dynamic simulations to assess their mechanical response. The agreement between the estimated tensile stiffness between CG models and experiments demonstrates the validity of the CK-CG method. Thus, our method provides a practical strategy for the direct construction of a structural model from low-resolution data for biological function studies.

Survey of the analysis of continuous conformational variability of biological macromolecules by electron microscopy

Single-particle analysis by electron microscopy is a well established technique for analyzing the three-dimensional structures of biological macromolecules. Besides its ability to produce high-resolution structures, it also provides insights into the dynamic behavior of the structures by elucidating their conformational variability. Here, the different image-processing methods currently available to study continuous conformational changes are reviewed.

A complex multiscale virtual particle model based elastic network model (CMVP-ENM) for the normal mode analysis of biomolecular complexes

The CMVP-ENM for virus normal mode analysis. With a special ratio parameter, CMVP-ENM can characterize the multi-material properties of biomolecular complexes and systematically enhance or suppress the modes for different components.

Gaussian-input Gaussian mixture model for representing density maps and atomic models

A new Gaussian mixture model (GMM) has been developed for better representations of both atomic models and electron microscopy 3D density maps. The standard GMM algorithm employs an EM algorithm to determine the parameters. It accepted a set of 3D points with weights, corresponding to voxel or atomic centers. Although the standard algorithm worked reasonably well; however, it had three problems. First, it ignored the size (voxel width or atomic radius) of the input, and thus it could lead to a GMM with a smaller spread than the input. Second, the algorithm had a singularity problem, as it sometimes stopped the iterative procedure due to a Gaussian function with almost zero variance. Third, a map with a large number of voxels required a long computation time for conversion to a GMM. To solve these problems, we have introduced a Gaussian-input GMM algorithm, which considers the input atoms or voxels as a set of Gaussian functions. The standard EM algorithm of GMM was extended to optimize the new GMM. The new GMM has identical radius of gyration to the input, and does not suddenly stop due to the singularity problem. For fast computation, we have introduced a down-sampled Gaussian functions (DSG) by merging neighboring voxels into an anisotropic Gaussian function. It provides a GMM with thousands of Gaussian functions in a short computation time. We also have introduced a DSG-input GMM: the Gaussian-input GMM with the DSG as the input. This new algorithm is much faster than the standard algorithm.

Multiscale virtual particle based elastic network model (MVP-ENM) for normal mode analysis of large-sized biomolecules

In this paper, a multiscale virtual particle based elastic network model (MVP-ENM) is proposed for the normal mode analysis of large-sized biomolecules. The multiscale virtual particle (MVP) model is proposed for the discretization of biomolecular density data. With this model, large-sized biomolecular structures can be coarse-grained into virtual particles such that a balance between model accuracy and computational cost can be achieved. An elastic network is constructed by assuming "connections" between virtual particles. The connection is described by a special harmonic potential function, which considers the influence from both the mass distributions and distance relations of the virtual particles. Two independent models, i.e., the multiscale virtual particle based Gaussian network model (MVP-GNM) and the multiscale virtual particle based anisotropic network model (MVP-ANM), are proposed. It has been found that in the Debye-Waller factor (B-factor) prediction, the results from our MVP-GNM with a high resolution are as good as the ones from GNM. Even with low resolutions, our MVP-GNM can still capture the global behavior of the B-factor very well with mismatches predominantly from the regions with large B-factor values. Further, it has been demonstrated that the low-frequency eigenmodes from our MVP-ANM are highly consistent with the ones from ANM even with very low resolutions and a coarse grid. Finally, the great advantage of MVP-ANM model for large-sized biomolecules has been demonstrated by using two poliovirus virus structures. The paper ends with a conclusion.

Rigid-Body Fitting of Atomic Models on 3D Density Maps of Electron Microscopy

Cryo electron microscopy has revolutionarily evolved for the determination of the 3D structure of macromolecular complexes. The modeling procedures on the 3D density maps of electron microscopy are roughly classified into three categories: fitting, de novo modeling and refinement. The registered atomic models from the maps have mostly been hand-built and auto-refined. Several programs aiming at automatic modeling have also been developed using various kinds of molecular representations. Among these three classes of the modeling procedures, the rigid body fitting is reviewed here, because it is the most basic modeling process applied before the other steps. The fitting problems are classified as the fittings of single subunit or multiple subunits, and the fittings on global or local parts of maps. A higher resolution map enables more local fitting. Various molecular representations have been employed in the fitting programs. A point and digital image models are generally used to represent molecules, but new representations, such as the Gaussian mixture model, have been applied recently.

Data-driven coarse graining of large biomolecular structures

Advances in experimental and computational techniques allow us to study the structure and dynamics of large biomolecular assemblies at increasingly higher resolution. However, with increasing structural detail it can be challenging to unravel the mechanism underlying the function of molecular machines. One reason is that atomistic simulations become computationally prohibitive. Moreover it is difficult to rationalize the functional mechanism of systems composed of tens of thousands to millions of atoms by following each atom’s movements. Coarse graining (CG) allows us to understand biological structures from a hierarchical perspective and to gradually zoom into the adequate level of structural detail. This article introduces a Bayesian approach for coarse graining biomolecular structures. We develop a probabilistic model that aims to represent the shape of an experimental structure as a cloud of bead particles. The particles interact via a pairwise potential whose parameters are estimated along with the bead positions and the CG mapping between atoms and beads. Our model can also be applied to density maps obtained by cryo-electron microscopy. We illustrate our approach on various test systems.

SAXS Structural Studies of Dps from Deinococcus radiodurans Highlights the Conformation of the Mobile N-Terminal Extensions

The radiation-resistant bacterium Deinococcus radiodurans contains two DNA-binding proteins from starved cells (Dps): Dps1 (DR2263) and Dps2 (DRB0092). These are suggested to play a role in DNA interaction and manganese and iron storage. The proteins assemble as a conserved dodecameric structure with structurally uncharacterised N-terminal extensions. In the case of DrDps1, these extensions have been proposed to be involved in DNA interactions, while in DrDps2, their function has yet to be established. The reported data reveal the relative position of the N-terminal extensions to the dodecameric sphere in solution for both Dps. The low-resolution small angle X-ray scattering (SAXS) results show that the N-terminal extensions protrude from the spherical shell of both proteins. The SAXS envelope of a truncated form of DrDps1 without the N-terminal extensions appears as a dodecameric sphere, contrasting strongly with the protrusions observed in the full-length models. The effect of iron incorporation into DrDps2 was investigated by static and stopped-flow SAXS measurements, revealing dynamic structural changes upon iron binding and core formation, as reflected by a quick alteration of its radius of gyration. The truncated and full-length versions of DrDps were also compared on the basis of their interaction with DNA to analyse functional roles of the N-terminal extensions. DrDps1 N-terminal protrusions appear to be directly involved with DNA, whilst those from DrDps2 are indirectly associated with DNA binding. Furthermore, detection of DrDps2 in the D. radiodurans membrane fraction suggests that the N-terminus of the protein interacts with the membrane.

Versatility of Approximating Single-Particle Electron Microscopy Density Maps Using Pseudoatoms and Approximation-Accuracy Control

Three-dimensional Gaussian functions have been shown useful in representing electron microscopy (EM) density maps for studying macromolecular structure and dynamics. Methods that require setting a desired number of Gaussian functions or a maximum number of iterations may result in suboptimal representations of the structure. An alternative is to set a desired error of approximation of the given EM map and then optimize the number of Gaussian functions to achieve this approximation error. In this article, we review different applications of such an approach that uses spherical Gaussian functions of fixed standard deviation, referred to as pseudoatoms. Some of these applications use EM-map normal mode analysis (NMA) with elastic network model (ENM) (applications such as predicting conformational changes of macromolecular complexes or exploring actual conformational changes by normal-mode-based analysis of experimental data) while some other do not use NMA (denoising of EM density maps). In applications based on NMA and ENM, the advantage of using pseudoatoms in EM-map coarse-grain models is that the ENM springs are easily assigned among neighboring grains thanks to their spherical shape and uniformed size. EM-map denoising based on the map coarse-graining was so far only shown using pseudoatoms as grains.

Denoising of high-resolution single-particle electron-microscopy density maps by their approximation using three-dimensional Gaussian functions

Cryo-electron microscopy (cryo-EM) of frozen-hydrated preparations of isolated macromolecular complexes is the method of choice to obtain the structure of complexes that cannot be easily studied by other experimental methods due to their flexibility or large size. An increasing number of macromolecular structures are currently being obtained at subnanometer resolution but the interpretation of structural details in such EM-derived maps is often difficult because of noise at these high-frequency signal components that reduces their contrast. In this paper, we show that the method for EM density-map approximation using Gaussian functions can be used for denoising of single-particle EM maps of high (typically subnanometer) resolution. We show its denoising performance using simulated and experimental EM density maps of several complexes.

Omokage search: shape similarity search service for biomolecular structures in both the PDB and EMDB

Summary:Omokage search is a service to search the global shape similarity of biological macromolecules and their assemblies, in both the Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB). The server compares global shapes of assemblies independent of sequence order and number of subunits. As a search query, the user inputs a structure ID (PDB ID or EMDB ID) or uploads an atomic model or 3D density map to the server. The search is performed usually within 1 min, using one-dimensional profiles (incremental distance rank profiles) to characterize the shapes. Using the gmfit (Gaussian mixture model fitting) program, the found structures are fitted onto the query structure and their superimposed structures are displayed on the Web browser. Our service provides new structural perspectives to life science researchers.

Availability and implementation:Omokage search is freely accessible at http://pdbj.org/omokage/.

Contact:hirofumi@protein.osaka-u.ac.jp

Supplementary information:Supplementary data are available at Bioinformatics online.

Understanding the apparent stator-rotor connections in the rotary ATPase family using coarse-grained computer modeling

Advances in structural biology, such as cryo-electron microscopy (cryo-EM) have allowed for a number of sophisticated protein complexes to be characterized. However, often only a static snapshot of a protein complex is visualized despite the fact that conformational change is frequently inherent to biological function, as is the case for molecular motors. Computer simulations provide valuable insights into the different conformations available to a particular system that are not accessible using conventional structural techniques. For larger proteins and protein complexes, where a fully atomistic description would be computationally prohibitive, coarse-grained simulation techniques such as Elastic Network Modeling (ENM) are often employed, whereby each atom or group of atoms is linked by a set of springs whose properties can be customized according to the system of interest. Here we compare ENM with a recently proposed continuum model known as Fluctuating Finite Element Analysis (FFEA), which represents the biomolecule as a viscoelastic solid subject to thermal fluctuations. These two complementary computational techniques are used to answer a critical question in the rotary ATPase family; implicit within these motors is the need for a rotor axle and proton pump to rotate freely of the motor domain and stator structures. However, current single particle cryo-EM reconstructions have shown an apparent connection between the stators and rotor axle or pump region, hindering rotation. Both modeling approaches show a possible role for this connection and how it would significantly constrain the mobility of the rotary ATPase family.

Flexibility within the rotor and stators of the vacuolar H+-ATPase

The V-ATPase is a membrane-bound protein complex which pumps protons across the membrane to generate a large proton motive force through the coupling of an ATP-driven 3-stroke rotary motor (V1) to a multistroke proton pump (Vo). This is done with near 100% efficiency, which is achieved in part by flexibility within the central rotor axle and stator connections, allowing the system to flex to minimise the free energy loss of conformational changes during catalysis. We have used electron microscopy to reveal distinctive bending along the V-ATPase complex, leading to angular displacement of the V1 domain relative to the Vo domain to a maximum of ~30°. This has been complemented by elastic network normal mode analysis that shows both flexing and twisting with the compliance being located in the rotor axle, stator filaments, or both. This study provides direct evidence of flexibility within the V-ATPase and by implication in related rotary ATPases, a feature predicted to be important for regulation and their high energetic efficiencies.

NONUNIFORM FOURIER TRANSFORMS FOR RIGID-BODY AND MULTI-DIMENSIONAL ROTATIONAL CORRELATIONS

The task of evaluating correlations is central to computational structural biology. The rigid-body correlation problem seeks the rigid-body transformation (R, t), R ∈ SO(3), t ∈ ℝ³ that maximizes the correlation between a pair of input scalar-valued functions representing molecular structures. Exhaustive solutions to the rigid-body correlation problem take advantage of the fast Fourier transform to achieve a speedup either with respect to the sought translation or rotation. We present PFcorr, a new exhaustive solution, based on the non-equispaced SO(3) Fourier transform, to the rigid-body correlation problem; unlike previous solutions, ours achieves a combination of translational and rotational speedups without requiring equispaced grids. PFcorr can be straightforwardly applied to a variety of problems in protein structure prediction and refinement that involve correlations under rigid-body motions of the protein. Additionally, we show how it applies, along with an appropriate flexibility model, to analogs of the above problems in which the flexibility of the protein is relevant.

3DEM Loupe: Analysis of macromolecular dynamics using structures from electron microscopy

Electron microscopy (EM) provides access to structural information of macromolecular complexes in the 3-20 Å resolution range. Normal mode analysis has been extensively used with atomic resolution structures and successfully applied to EM structures. The major application of normal modes is the identification of possible conformational changes in proteins. The analysis can throw light on the mechanism following ligand binding, protein-protein interactions, channel opening and other functional macromolecular movements. In this article, we present a new web server, 3DEM Loupe, which allows normal mode analysis of any uploaded EM volume using a user-friendly interface and an intuitive workflow. Results can be fully explored in 3D through animations and movies generated by the server. The application is freely available at http://3demloupe.cnb.csic.es.

Outcome of the first electron microscopy validation task force meeting

This Meeting Review describes the proceedings and conclusions from the inaugural meeting of the Electron Microscopy Validation Task Force organized by the Unified Data Resource for 3DEM (http://www.emdatabank.org) and held at Rutgers University in New Brunswick, NJ on September 28 and 29, 2010. At the workshop, a group of scientists involved in collecting electron microscopy data, using the data to determine three-dimensional electron microscopy (3DEM) density maps, and building molecular models into the maps explored how to assess maps, models, and other data that are deposited into the Electron Microscopy Data Bank and Protein Data Bank public data archives. The specific recommendations resulting from the workshop aim to increase the impact of 3DEM in biology and medicine.

Conventions and workflows for using Situs

Situs is a modular program package for the multi-scale modeling of atomic resolution structures and low-resolution biophysical data from electron microscopy, tomography or small-angle X-ray scattering. This article provides an overview of recent developments in the Situs package, with an emphasis on workflows and conventions that are important for practical applications. The modular design of the programs facilitates scripting in the bash shell that allows specific programs to be combined in creative ways that go beyond the original intent of the developers. Several scripting-enabled functionalities, such as flexible transformations of data type, the use of symmetry constraints or the creation of two-dimensional projection images, are described. The processing of low-resolution biophysical maps in such workflows follows not only first principles but often relies on implicit conventions. Situs conventions related to map formats, resolution, correlation functions and feature detection are reviewed and summarized. The compatibility of the Situs workflow with CCP4 conventions and programs is discussed.

Coarse-Grained Representations of Large Biomolecular Complexes from Low-Resolution Structural Data

High-resolution atomistic structures of many large biomolecular complexes have not yet been solved by experiments, such as X-ray crystallography or NMR. Often however low-resolution information is obtained by alternative techniques, such as cryo-electron microscopy or small-angle X-ray scattering. Coarse-grained (CG) models are an appropriate choice to computationally study these complexes given the limited resolution experimental data. One of the important questions therefore is how to define CG representations from these low-resolution density maps. This work provides a space-based essential dynamics coarse-graining (ED-CG) method to define a CG representation from a density map without detailed knowledge of its underlying atomistic structure and primary sequence information. This method is demonstrated on G-actin (both the atomic structure and its density map). It is then applied to the density maps of the Escherichia coli 70S ribosome and the microtubule. The results indicate that the method can define highly CG models that still preserve functionally important dynamics of large biomolecular complexes.

A fast mathematical programming procedure for simultaneous fitting of assembly components into cryoEM density maps

MOTIVATION

Single-particle cryo electron microscopy (cryoEM) typically produces density maps of macromolecular assemblies at intermediate to low resolution (approximately 5-30 A). By fitting high-resolution structures of assembly components into these maps, pseudo-atomic models can be obtained. Optimizing the quality-of-fit of all components simultaneously is challenging due to the large search space that makes the exhaustive search over all possible component configurations computationally unfeasible.

RESULTS

We developed an efficient mathematical programming algorithm that simultaneously fits all component structures into an assembly density map. The fitting is formulated as a point set matching problem involving several point sets that represent component and assembly densities at a reduced complexity level. In contrast to other point matching algorithms, our algorithm is able to match multiple point sets simultaneously and not only based on their geometrical equivalence, but also based on the similarity of the density in the immediate point neighborhood. In addition, we present an efficient refinement method based on the Iterative Closest Point registration algorithm. The integer quadratic programming method generates an assembly configuration in a few seconds. This efficiency allows the generation of an ensemble of candidate solutions that can be assessed by an independent scoring function. We benchmarked the method using simulated density maps of 11 protein assemblies at 20 A, and an experimental cryoEM map at 23.5 A resolution. Our method was able to generate assembly structures with root-mean-square errors <6.5 A, which have been further reduced to <1.8 A by the local refinement procedure.

AVAILABILITY

The program is available upon request as a Matlab code package.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics Online.

Evolutionary tabu search strategies for the simultaneous registration of multiple atomic structures in cryo-EM reconstructions

A structural characterization of multi-component cellular assemblies is essential to explain the mechanisms governing biological function. Macromolecular architectures may be revealed by integrating information collected from various biophysical sources - for instance, by interpreting low-resolution electron cryomicroscopy reconstructions in relation to the crystal structures of the constituent fragments. A simultaneous registration of multiple components is beneficial when building atomic models as it introduces additional spatial constraints to facilitate the native placement inside the map. The high-dimensional nature of such a search problem prevents the exhaustive exploration of all possible solutions. Here we introduce a novel method based on genetic algorithms, for the efficient exploration of the multi-body registration search space. The classic scheme of a genetic algorithm was enhanced with new genetic operations, tabu search and parallel computing strategies and validated on a benchmark of synthetic and experimental cryo-EM datasets. Even at a low level of detail, for example 35-40 A, the technique successfully registered multiple component biomolecules, measuring accuracies within one order of magnitude of the nominal resolutions of the maps. The algorithm was implemented using the Sculptor molecular modeling framework, which also provides a user-friendly graphical interface and enables an instantaneous, visual exploration of intermediate solutions.

Defining coarse-grained representations of large biomolecules and biomolecular complexes from elastic network models

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

Four distinct structural domains in Clostridium difficile toxin B visualized using SAXS

Clostridium difficile is a nosocomial bacterial pathogen causing antibiotic-associated diarrhea and fatal pseudomembranous colitis. Key virulence factors are toxin A and toxin B (TcdB), two highly related toxins that are members of the large clostridial toxin family. These large multifunctional proteins disrupt cell function using a glucosyltransferase domain that is translocated into the cytosol after vesicular internalization of intact holotoxin. Although substantial information about the biochemical mechanisms of intoxication exists, research has been hampered by limited structural information, particularly of intact holotoxin. Here, we used small-angle X-ray scattering (SAXS) methods to obtain an ab initio low-resolution structure of native TcdB, which demonstrated that this molecule is monomeric in solution and possesses a highly asymmetric shape with a maximum dimension of approximately 275 A. Combining this SAXS information with crystallographic or modeled structures of individual functional domains of TcdB reveals for the first time that the three-dimensional structure of TcdB is organized into four distinct structural domains. Structures of the N-terminal glucosyltransferase, the cysteine protease, and the C-terminal repeat region can be aligned within three domains of the SAXS envelope. A fourth domain, predicted to be involved in the translocation of the glucosyltransferase, appears as a large solvent-exposed protrusion. Knowledge of the shapes and relative orientations of toxin domains provides new insight into defining functional domain boundaries and provides a framework for understanding how potential intra-domain interactions enable conformational changes to propagate between domains to facilitate intoxication processes.

Bend-twist-stretch model for coarse elastic network simulation of biomolecular motion

The empirical harmonic potential function of elastic network models (ENMs) is augmented by three- and four-body interactions as well as by a parameter-free connection rule. In the new bend-twist-stretch (BTS) model the complexity of the parametrization is shifted from the spatial level of detail to the potential function, enabling an arbitrary coarse graining of the network. Compared to distance cutoff-based Hookean springs, the approach yields a more stable parametrization of coarse-grained ENMs for biomolecular dynamics. Traditional ENMs give rise to unbounded zero-frequency vibrations when (pseudo)atoms are connected to fewer than three neighbors. A large cutoff is therefore chosen in an ENM (about twice the average nearest-neighbor distance), resulting in many false-positive connections that reduce the spatial detail that can be resolved. More importantly, the required three-neighbor connectedness also limits the coarse graining, i.e., the network must be dense, even in the case of low-resolution structures that exhibit few spatial features. The new BTS model achieves such coarse graining by extending the ENM potential to include three-and four-atom interactions (bending and twisting, respectively) in addition to the traditional two-atom stretching. Thus, the BTS model enables reliable modeling of any three-dimensional graph irrespective of the atom connectedness. The additional potential terms were parametrized using continuum elastic theory of elastic rods, and the distance cutoff was replaced by a competitive Hebb connection rule, setting all free parameters in the model. We validate the approach on a carbon-alpha representation of adenylate kinase and illustrate its use with electron microscopy maps of E. coli RNA polymerase, E. coli ribosome, and eukaryotic chaperonin containing T-complex polypeptide 1, which were difficult to model with traditional ENMs. For adenylate kinase, we find excellent reproduction (>90% overlap) of the ENM modes and B factors when BTS is applied to the carbon-alpha representation as well as to coarser descriptions. For the volumetric maps, coarse BTS yields similar motions (70%-90% overlap) to those obtained from significantly denser representations with ENM. Our Python-based algorithms of ENM and BTS implementations are freely available.

Inferential optimization for simultaneous fitting of multiple components into a CryoEM map of their assembly

Models of macromolecular assemblies are essential for a mechanistic description of cellular processes. Such models are increasingly obtained by fitting atomic-resolution structures of components into a density map of the whole assembly. Yet, current density-fitting techniques are frequently insufficient for an unambiguous determination of the positions and orientations of all components. Here, we describe MultiFit, a method used for simultaneously fitting atomic structures of components into their assembly density map at resolutions as low as 25 A. The component positions and orientations are optimized with respect to a scoring function that includes the quality-of-fit of components in the map, the protrusion of components from the map envelope, and the shape complementarity between pairs of components. The scoring function is optimized by our exact inference optimizer DOMINO (Discrete Optimization of Multiple INteracting Objects) that efficiently finds the global minimum in a discrete sampling space. MultiFit was benchmarked on seven assemblies of known structure, consisting of up to seven proteins each. The input atomic structures of the components were obtained from the Protein Data Bank, as well as by comparative modeling based on a 16-99% sequence identity to a template structure. A near-native configuration was usually found as the top-scoring model. Therefore, MultiFit can provide initial configurations for further refinement of many multicomponent assembly structures described by electron microscopy.

Biomolecular pleiomorphism probed by spatial interpolation of coarse models

In low resolution structures of biological assemblies one can often observe conformational deviations that require a flexible rearrangement of structural domains fitted at the atomic level. We are evaluating interpolation methods for the flexible alignment of atomic models based on coarse models. Spatial interpolation is well established in image-processing and visualization to describe the overall deformation or warping of an object or an image. Combined with a coarse representation of the biological system by feature vectors, such methods can provide a flexible approximation of the molecular structure. We have compared three well-known interpolation techniques and evaluated the results by comparing them with constrained molecular dynamics. One method, inverse distance weighting interpolation, consistently produced models that were nearly indistinguishable on the alpha carbon level from the molecular dynamics results. The method is simple to apply and enables flexing of structures by non-expert modelers. This is useful for the basic interpretation of volumetric data in biological applications such as electron microscopy. The method can be used as a general interpretation tool for sparsely sampled motions derived from coarse models.

Multiple subunit fitting into a low-resolution density map of a macromolecular complex using a gaussian mixture model

Recently, electron microscopy measurement of single particles has enabled us to reconstruct a low-resolution 3D density map of large biomolecular complexes. If structures of the complex subunits can be solved by x-ray crystallography at atomic resolution, fitting these models into the 3D density map can generate an atomic resolution model of the entire large complex. The fitting of multiple subunits, however, generally requires large computational costs; therefore, development of an efficient algorithm is required. We developed a fast fitting program, “gmfit”, which employs a Gaussian mixture model (GMM) to represent approximated shapes of the 3D density map and the atomic models. A GMM is a distribution function composed by adding together several 3D Gaussian density functions. Because our model analytically provides an integral of a product of two distribution functions, it enables us to quickly calculate the fitness of the density map and the atomic models. Using the integral, two types of potential energy function are introduced: the attraction potential energy between a 3D density map and each subunit, and the repulsion potential energy between subunits. The restraint energy for symmetry is also employed to build symmetrical origomeric complexes. To find the optimal configuration of subunits, we randomly generated initial configurations of subunit models, and performed a steepest-descent method using forces and torques of the three potential energies. Comparison between an original density map and its GMM showed that the required number of Gaussian distribution functions for a given accuracy depended on both resolution and molecular size. We then performed test fitting calculations for simulated low-resolution density maps of atomic models of homodimer, trimer, and hexamer, using different search parameters. The results indicated that our method was able to rebuild atomic models of a complex even for maps of 30 Å resolution if sufficient numbers (eight or more) of Gaussian distribution functions were employed for each subunit, and the symmetric restraints were assigned for complexes with more than three subunits. As a more realistic test, we tried to build an atomic model of the GroEL/ES complex by fitting 21-subunit atomic models into the 3D density map obtained by cryoelectron microscopy using the C7 symmetric restraints. A model with low root mean-square deviations (14.7 Å) was obtained as the lowest-energy model, showing that our fitting method was reasonably accurate. Inclusion of other restraints from biological and biochemical experiments could further enhance the accuracy.

Influence of nonuniform geometry on nanoindentation of viral capsids

A series of recent nanoindentation experiments on the protein shells (capsids) of viruses has established atomic force microscopy (AFM) as a useful framework for probing the mechanics of large protein assemblies. Specifically these experiments provide an opportunity to study the coupling of the global assembly response to local conformational changes. AFM experiments on cowpea chlorotic mottle virus, known to undergo a pH-controlled swelling conformational change, have revealed a pH-dependent mechanical response. Previous theoretical studies have shown that homogeneous changes in shell geometry can play a significant role in the mechanical response. This article develops a method for accurately capturing the heterogeneous geometry of a viral capsid and explores its effect on mechanical response with a nonlinear continuum elasticity model. Models of both native and swollen cowpea chlorotic mottle virus capsids are generated from x-ray crystal structures, and are used in finite element simulations of AFM indentation along two-, three-, and fivefold icosahedral symmetry orientations. The force response of the swollen capsid model is observed to be softer by roughly a factor of two, significantly more nonlinear, and more orientation-dependent than that of a native capsid with equivalent elastic moduli, demonstrating that capsid geometric heterogeneity can have significant effects on the global structural response.

Multi-resolution anchor-point registration of biomolecular assemblies and their components

An atomic scale interpretation facilitates the assignment of functional properties to 3D reconstructions of macromolecular assemblies in electron microscopy (EM). Such a high-resolution interpretation is typically achieved by docking the known atomic structures of components into the volumetric EM maps. Docking locations are often determined by maximizing the cross-correlation coefficient of the two objects in a slow, exhaustive search. If time is of essence, such as in related visualization and image processing fields, the matching of data is accelerated by incorporating feature points that form a compact description of 3D objects. The complexity reduction afforded by the feature point representation enables a near-instantaneous matching. We show that such reduced matching can also deliver robust and accurate results in the presence of noise or artifacts. We therefore propose a novel multi-resolution registration technique employing feature-based shape descriptions of the volumetric and structural data. The pattern-matching algorithm carries out a hierarchical alignment of the point sets generated by vector quantization. The search-space complexity is reduced by an integrated tree-pruning technique, which permits the detection of subunits in large macromolecular assemblies in real-time. The efficiency and accuracy of the novel algorithm are validated on a standard test system of homo-oligomeric assemblies.

SYMMETRY, FORM, AND SHAPE: Guiding Principles for Robustness in Macromolecular Machines

Computational studies of large macromolecular assemblages have come a long way during the past 10 years. With the explosion of computer power and parallel computing, timescales of molecular dynamics simulations have been extended far beyond the hundreds of picoseconds timescale. However, limitations remain for studies of large-scale conformational changes occurring on timescales beyond nanoseconds, especially for large macromolecules. In this review, we describe recent methods based on normal mode analysis that have enabled us to study dynamics on the microsecond timescale for large macromolecules using different levels of coarse graining, from atomically detailed models to those employing only low-resolution structural information. Emerging from such studies is a control principle for robustness in Nature's machines. We discuss this idea in the context of large-scale functional reorganization of the ribosome, virus particles, and the muscle protein myosin.

Visualizing the dual space of biological molecules

An important part of protein structure characterization is the determination of excluded space such as fissures in contact interfaces, pores, inaccessible cavities, and catalytic pockets. We introduce a general tessellation method for visualizing the dual space around, within, and between biological molecules. Using Delaunay triangulation, a three-dimensional graph is constructed to provide a displayable discretization of the continuous volume. This graph structure is also used to compare the dual space of a system in two different states. Tessellator, a cross-platform implementation of the algorithm, is used to analyze the cavities within myoglobin, the protein-RNA docking interface between aspartyl-tRNA synthetase and tRNA(Asp), and the ammonia channel in the hisH-hisF complex of imidazole glycerol phosphate synthase.

The effect of salt on self-assembled actin-lysozyme complexes

We present a combined experimental and computational study of the bundling of F-actin filaments induced by lysozyme proteins. Synchrotron small-angle x-ray scattering results show that these bundles consist of close-packed columnar complexes in which the actin is held together by incommensurate, one-dimensional arrays of lysozyme macroions. Molecular dynamics simulations of a coarse-grained model confirm the arrangement of the lysozyme and the stability of this structure. In addition, we find that these complexes remain stable even in the presence of significant concentrations of monovalent salt. The simulations show that this arises from partitioning of the salt between the aqueous and the condensed phases. The osmotic pressure resulting from the excess concentration of the salt in the aqueous phase balances the osmotic pressure increase in the bundle. These results are relevant for a variety of biological and biomedical problems in which electrostatic complexation between anionic polyelectrolytes and cationic globular proteins takes place, such as the pathological self-assembly of endogenous antibiotic polypeptides and inflammatory polymers in cystic fibrosis.

Voronoi and Voronoi-related tessellations in studies of protein structure and interaction

The three-dimensional structure of a protein can be modeled by a set of polyhedra drawn around its atoms or residues. The tessellation invented by Voronoi in 1908, and other tessellations of space derived from it, provide versatile representations of three-dimensional structures. In recent years, they have been used to investigate a series of issues relating to proteins: atom and residue volumes, packing, folding, interactions and binding.

Topology representing network enables highly accurate classification of protein images taken by cryo electron-microscope without masking

In single-particle analysis, a three-dimensional (3-D) structure of a protein is constructed using electron microscopy (EM). As these images are very noisy in general, the primary process of this 3-D reconstruction is the classification of images according to their Euler angles, the images in each classified group then being averaged to reduce the noise level. In our newly developed strategy of classification, we introduce a topology representing network (TRN) method. It is a modified method of a growing neural gas network (GNG). In this system, a network structure is automatically determined in response to the images input through a growing process. After learning without a masking procedure, the GNG creates clear averages of the inputs as unit coordinates in multi-dimensional space, which are then utilized for classification. In the process, connections are automatically created between highly related units and their positions are shifted where the inputs are distributed in multi-dimensional space. Consequently, several separated groups of connected units are formed. Although the interrelationship of units in this space are not easily understood, we succeeded in solving this problem by converting the unit positions into two-dimensional (2-D) space, and by further optimizing the unit positions with the simulated annealing (SA) method. In the optimized 2-D map, visualization of the connections of units provided rich information about clustering. As demonstrated here, this method is clearly superior to both the multi-variate statistical analysis (MSA) and the self-organizing map (SOM) as a classification method and provides a first reliable classification method which can be used without masking for very noisy images.

Low-resolution reconstruction of a synthetic DNA holliday junction

We have studied the low-resolution solution conformation of a Holliday (or four-way) DNA junction by using small-angle x-ray scattering, sedimentation velocity, and computational modeling techniques. The scattering data were analyzed in two independent ways: firstly, by rigid-body modeling of the scattering data using previously suggested models for the Holliday junction (HJ), and secondly, by ab initio reconstruction methods. The models found by both methods agree with experimentally determined sedimentation coefficients and are compatible with the results of previous studies using different techniques, but provide a more direct and accurate determination of the solution conformation of the HJ. Our results show that addition of Mg(2+) alters the conformation of the HJ from an extended to a stacked arrangement.

Automated three-dimensional reconstruction of keyhole limpet hemocyanin type 1

We have reconstructed a three-dimensional map of keyhole limpet hemocyanin isoform 1 (KLH1), using our automated data collection software, Leginon, integrated with particle selection algorithms, and the SPIDER reconstruction package. KLH1, a 7.9 MDa macromolecule, is an extracellular respiratory pigment composed of two asymmetric decamers, and presents an overall D(5) point-group symmetry. The reconstruction is in agreement with previous data published on molluscan hemocyanins. The reconstructed map (11.3A resolution, 3sigma criterion) was used to fit an available X-ray crystallography structure of Octopus dofleini Odg, solved at 2.3A [J. Mol. Biol. 278 (4) (1998) 855], with satisfactory results. The results validate the approach of automating the cryoEM process and demonstrate that the quality of the images acquired and the particles selected is comparable to those obtained using manual methods. Several problems remain to be solved however before these results can be generalized.

Conformational flexibility of pyruvate dehydrogenase complexes: a computational analysis by quantized elastic deformational model

Pyruvate dehydrogenase complex (PDC) is one of the largest multienzyme complexes known and consists of a dodecahedral E2 core to which other components are attached. We report the results of applying a new computational method, quantized elastic deformational model, to simulating the conformational fluctuations of the truncated E2 core, using low-resolution electron cryomicroscopy density maps. The motional features are well reproduced; especially, the symmetric breathing mode revealed in simulation is nearly identical with what was observed experimentally. Structural details of the motions of the trimeric building blocks, which are critical to facilitating the global expansion and contraction of the complex, were revealed. Using the low-resolution maps from electron cryomicroscopy reconstructions, the simulations showed a picture of the motional mechanism of the PDC core, which is an example without precedent of thermally activated global dynamics. Moreover, the current results support an earlier suggestion that, at low resolution and without the use of amino acid sequence and atomic coordinates, it is possible for computer simulations to provide an accurate description of protein dynamics.

Minimum energy configurations of the 2-dimensional HP-model of proteins by self-organizing networks

We use self-organizing maps (SOM) as an efficient tool to find the minimum energy configurations of the 2-dimensional HP-models of proteins. The usage of the SOM for the protein folding problem is similar to that for the Traveling Salesman Problem. The lattice nodes represent the cities whereas the neurons in the network represent the amino acids moving towards the closest cities, subject to the HH interactions. The valid path that maximizes the HH contacts corresponds to the minimum energy configuration of the protein. We report promising results for the cases when the protein completely fills a lattice and discuss the current problems and possible extensions. In all the test sequences up to 36 amino acids, the algorithm was able to find the global minimum and its degeneracies.

Mega-Dalton biomolecular motion captured from electron microscopy reconstructions

The vibrational analysis of elastic models suggests that the essential motions of large biomolecular assemblies can be captured efficiently at an intermediate scale without requiring knowledge of the atomic structure. While prior work has established a theoretical foundation for this analysis, we demonstrate here on experimental electron microscopy maps that vibrational modes indeed describe functionally relevant movements of macromolecular machines. The clamp closure in bacterial RNA polymerase, the ratcheting of 30S and 50S subunits of the ribosome, and the dynamic flexibility of chaperonin CCT are extracted directly from single electron microscopy structures at 15-27 A resolution. The striking agreement of the presented results with experimentally observed motions suggests that the motion of the large scale machinery in the cell is surprisingly independent of detailed atomic interactions and can be quite reasonably described as a motion of elastic bodies.

Exploring global distortions of biological macromolecules and assemblies from low-resolution structural information and elastic network theory

A theory of elastic normal modes is described for the exploration of global distortions of biological structures and their assemblies based upon low-resolution image data. Structural information at low resolution, e.g. from density maps measured by cryogenic electron microscopy (cryo-EM), is used to construct discrete multi-resolution models for the electron density using the techniques of vector quantization. The elastic normal modes computed based on these discretized low-resolution models are found to compare well with the normal modes obtained at atomic resolution. The quality of the normal modes describing global displacements of the molecular system is found to depend on the resolution of the synthetic EM data and the extent of reductionism in the discretized representation. However, models that reproduce the functional rearrangements of our test set of molecules are achieved for realistic values of experimental resolution. Thus large conformational changes as occur during the functioning of biological macromolecules and assemblies can be elucidated directly from low-resolution structural data through the application of elastic normal mode theory and vector quantization.

Modeling shape and topology of low-resolution density maps of biological macromolecules

In the present work we develop an efficient way of representing the geometry and topology of volumetric datasets of biological structures from medium to low resolution, aiming at storing and querying them in a database framework. We make use of a new vector quantization algorithm to select the points within the macromolecule that best approximate the probability density function of the original volume data. Connectivity among points is obtained with the use of the alpha shapes theory. This novel data representation has a number of interesting characteristics, such as 1) it allows us to automatically segment and quantify a number of important structural features from low-resolution maps, such as cavities and channels, opening the possibility of querying large collections of maps on the basis of these quantitative structural features; 2) it provides a compact representation in terms of size; 3) it contains a subset of three-dimensional points that optimally quantify the densities of medium resolution data; and 4) a general model of the geometry and topology of the macromolecule (as opposite to a spatially unrelated bunch of voxels) is easily obtained by the use of the alpha shapes theory.

Single-particle imaging of macromolecules by cryo-electron microscopy

Cryo-electron microscopy (cryo-EM) of biological molecules in single-particle (i.e., unordered, nonaggregated) form is a new approach to the study of molecular assemblies, which are often too large and flexible to be amenable to X-ray crystallography. New insights into biological function on the molecular level are expected from cryo-EM applied to the study of such complexes "trapped" at different stages of their conformational changes and dynamical interactions. Important molecular machines involved in the fundamental processes of transcription, mRNA splicing, and translation are examples for successful applications of the new technique, combined with structural knowledge gained by conventional techniques of structure determination, such as X-ray crystallography and NMR.

How to describe protein motion without amino acid sequence and atomic coordinates

This paper reports a computational method, the quantized elastic deformational model, that can reliably describe the conformational flexibility of a protein in the absence of the amino acid sequence and atomic coordinates. The essence of this method lies in the fact that, in modeling the functionally important conformational changes such as domain movements, it is possible to abandon the traditional concepts of protein structure (bonds, angles, dihedrals, etc.) and treat the protein as an elastic object. The shape and mass distribution of the object are described by the electron density maps, at various resolutions, from methods such as x-ray diffraction or cryo-electron microscopy. The amplitudes and directionality of the elastic deformational modes of a protein, whose patterns match the biologically relevant conformational changes, can then be derived solely based on the electron density map. The method yields an accurate description of protein dynamics over a wide range of resolutions even as low as 15-20 A at which there is nearly no visually distinguishable internal structures. Therefore, this method dramatically enhances the capability of studying protein motions in structural biology. It is also expected to have ample applications in related fields such as bioinformatics, structural genomics, and proteomics, in which one's ability to extract functional information from the not-so-well-defined structural models is vitally important.

Domain movements in human fatty acid synthase by quantized elastic deformational model

This paper reports the results of applying a computational method called the quantized elastic deformational model, to the determination of conformational flexibility of the supermolecular complex of human fatty acid synthase. The essence of this method is the ability to model large-scale conformational changes such as domain movements by treating the protein as an elastic object without the knowledge of protein primary sequence and atomic coordinates. The calculation was based on the electron density maps of the synthase at 19 A. The results suggest that the synthase is a very flexible molecule. Two types of flexible hinges in the structure were identified. One is an intersubunit hinge formed by the intersubunit connection and the other is an intrasubunit hinge located between domains I and II. Despite the fact that the dimeric synthase has a chemically symmetric structure, large domain movements around the hinge region occur in various directions and allow the molecule to adopt a wide range of conformations. These domain movements are likely to be important in facilitating and regulating the entire palmitate synthesis by coordinating the communication between components of the molecule, for instance, adjusting the distance between various active sites inside the catalytic reaction center. Finally, the ability to describe protein motions of a supermolecular complex, without the information of protein sequence and atomic coordinates, is a major advance in computational modeling of protein dynamics. The method provides an unprecedented ability to model protein motions at such a low resolution of structure.

Striking conformational change suspected within the phosphoribulokinase dimer induced by interaction with GAPDH

A multitechnique approach was used to study the [glyceraldehyde-3-phosphate dehydrogenase](2 x 4)-[phosphoribulokinase](2 x 2) multienzymatic complex of the alga Chlamydomonas reinhardtii. On the one hand, each component of the complex was compared with known atomic structures of related enzymes or of similar enzymes originating from different organisms. On the other hand, the overall low resolution architecture of the whole complex was studied using cryoelectron microscopy and image processing techniques. The dimers of phosphoribulokinase are suspected to undergo a dramatic change in activity during a cycle of binding and detaching from tetramers of glyceraldehyde-3-phosphate dehydrogenase. This is likely supported by strong structural differences between the modeled phosphoribulokinase dimers and the counterpart in the three-dimensional reconstruction volume of the whole complex obtained from cryoelectron microscope images.

Cryo-electron microscopy as an investigative tool: the ribosome as an example

Cryo-electron microscopy allows the visualization of macromolecules in their native state. Combined with techniques of three-dimensional reconstruction, cryo-EM images of single molecules can be used to study macromolecular interactions. The ribosome, a large RNA-protein complex with multiple binding interactions, is an excellent test case illustrating the power of these new techniques. Conformational changes during the binding of tRNA and protein factors to the ribosome can now be studied without the interference of crystal packing. Now that the first X-ray structures of ribosomal subunits have become available, conformational changes observed by cryo-EM in different functional states can be traced back to internal rearrangements of the underlying structural framework. Electron microscopy, X-ray crystallography, and modeling should be used together in the endeavor to understand the functioning of the translational machinery.