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.     

Combined approaches to flexible fitting and assessment in virus capsids undergoing conformational change

Fitting of atomic components into electron cryo-microscopy (cryoEM) density maps is routinely used to understand the structure and function of macromolecular machines. Many fitting methods have been developed, but a standard protocol for successful fitting and assessment of fitted models has yet to be agreed upon among the experts in the field. Here, we created and tested a protocol that highlights important issues related to homology modelling, density map segmentation, rigid and flexible fitting, as well as the assessment of fits. As part of it, we use two different flexible fitting methods (Flex-EM and iMODfit) and demonstrate how combining the analysis of multiple fits and model assessment could result in an improved model. The protocol is applied to the case of the mature and empty capsids of Coxsackievirus A7 (CAV7) by flexibly fitting homology models into the corresponding cryoEM density maps at 8.2 and 6.1 Å resolution. As a result, and due to the improved homology models (derived from recently solved crystal structures of a close homolog – EV71 capsid – in mature and empty forms), the final models present an improvement over previously published models. In close agreement with the capsid expansion observed in the EV71 structures, the new CAV7 models reveal that the expansion is accompanied by ∼5° counterclockwise rotation of the asymmetric unit, predominantly contributed by the capsid protein VP1. The protocol could be applied not only to viral capsids but also to many other complexes characterised by a combination of atomic structure modelling and cryoEM density fitting.     

Low-resolution structure refinement in electron microscopy

A real-space structure refinement method, originally developed for macromolecular X-ray crystallography, has been applied to protein structure analysis by electron microscopy (EM). This method simultaneously optimizes the fit of an atomic model to a density map and the stereo-chemical properties of the model by minimizing an energy function. The performance of this method is characterized at different resolution and signal-to-noise ratio conditions typical for EM electron density maps. A multi-resolution scheme is devised to improve the convergence of the refinement on the global energy minimum. Applications of the method to various model systems are demonstrated here. The first case is the arrangement of FlgE molecules in the helical filament of flagellar hook, in which refinement with segmented rigid bodies improves the density correlation and reduces severe van der Waals contacts among the symmetry-related subunits. The second case is a conformational analysis of the NSF AAA ATPase in which a multi-conformer model is used in the refinement to investigate the arrangement of the two ATPase domains in the molecule. The third case is a docking simulation in which the crystal structure of actin and the NOE data from NMR experiments on the dematin headpiece are combined with a low-resolution EM density map to generate an atomic model of the F-actin-dematin headpiece structure.     

Structural characterization of components of protein assemblies by comparative modeling and electron cryo-microscopy

We explore structural characterization of protein assemblies by a combination of electron cryo-microscopy (cryoEM) and comparative protein structure modeling. Specifically, our method finds an optimal atomic model of a given assembly subunit and its position within an assembly by fitting alternative comparative models into a cryoEM map. The alternative models are calculated by MODELLER [J. Mol. Biol. 234 (1993) 313] from different sequence alignments between the modeled protein and its template structures. The fitting of these models into a cryoEM density map is performed either by FOLDHUNTER [J. Mol. Biol. 308 (2001) 1033] or by a new density fitting module of MODELLER (Mod-EM). Identification of the most accurate model is based on the correlation between the model accuracy and the quality of fit into the cryoEM density map. To quantify this correlation, we created a benchmark consisting of eight proteins of different structural folds with corresponding density maps simulated at five resolutions from 5 to 15 angstroms, with three noise levels each. Each of the proteins in the set was modeled based on 300 different alignments to their remotely related templates (12-32% sequence identity), spanning the range from entirely inaccurate to essentially accurate alignments. The benchmark revealed that one of the most accurate models can usually be identified by the quality of its fit into the cryoEM density map, even for noisy maps at 15 angstroms resolution. Therefore, a cryoEM density map can be helpful in improving the accuracy of a comparative model. Moreover, a pseudo-atomic model of a component in an assembly may be built better with comparative models of the native subunit sequences than with experimentally determined structures of their homologs.     

Low-Resolution Density Maps from Atomic Models: How Stepping “Back” Can Be a Step “Forward”

Atomic-resolution structures have had a tremendous impact on modern biological science. Much useful information also has been gleaned by merging and correlating atomic-resolution structural details with lower-resolution (15–40 Å), three-dimensional (3D) reconstructions computed from images recorded with cryo-transmission electron microscopy (cryoTEM) procedures. One way to merge these structures involves reducing the resolution of an atomic model to a level comparable to a cryoTEM reconstruction. A low-resolution density map can be derived from an atomic-resolution structure by retrieving a set of atomic coordinates editing the coordinate file, computing structure factors from the model coordinates, and computing the inverse Fourier transform of the structure factors. This method is a useful tool for structural studies primarily in combination with 3D cryoTEM reconstructions. It has been used to assess the quality of 3D reconstructions, to determine corrections for the phase-contrast transfer function of the transmission electron microscope, to calibrate the dimensions and handedness of 3D reconstructions, to produce difference maps, to model features in macromolecules or macromolecular complexes, and to generate models to initiate model-based determination of particle orientation and origin parameters for 3D reconstruction.     

A rapid method for positioning small flexible molecules, nucleic acids, and large protein fragments in experimental electron density maps.

A Monte Carlo procedure, encoded in the program Blob, has been developed and tested for the purpose of positioning large molecular fragments or small flexible molecules in electron density maps. The search performed by the algorithm appears to be sufficiently thorough to accurately position a small flexible ligand in well-defined density while remaining sufficiently random to offer interesting alternate suggestions for density representing disordered binding modes of a ligand. Furthermore, the algorithm is shown to be efficient enough to accurately position large rigid molecular fragments. In the first of the test cases with large molecular fragments, Blob was surprisingly effective in positioning a poly-alanine model of a 53-residue domain in poor electron density resulting from molecular replacement with a partial model. At 3.0 A resolution the domain was positioned consistently within 0.2 A of its experimentally determined position. Even at 6.0 A resolution Blob could consistently position the domain to within 0.75 A of its actual position. A second set of tests with large molecular fragments revealed that Blob could correctly position large molecular fragments with quite significant deviations from the actual structure. In this test case, fragments ranging from a 170-residue protein domain with a 3.8 A rms deviation from the actual structure to a 22-base pair ideal B-form DNA duplex were positioned accurately in a 3.2 A electron density map derived from multiple isomorphous replacement methods. Even when decreasing the quality of the maps, from a figure of merit of 0.57 to as low as 0. 35, Blob could still effectively position the large protein domain and the DNA duplex. Since it is efficient, can handle large molecular fragments, and works in poor and low resolution maps, Blob could be a useful tool for interpreting electron density maps in de novo structure determinations and in molecular replacement studies. Proteins 1999;36:512-525.     

A core-weighted fitting method for docking atomic structures into low-resolution maps: application to cryo-electron microscopy

Cryo-electron microscopy of "single particles" is a powerful method to analyze structures of large macromolecular assemblies that are not amenable to investigation by traditional X-ray crystallographic methods. A key step in these studies is to obtain atomic interpretations of multiprotein complexes by fitting atomic structures of individual components into maps obtained from electron microscopic data. Here, we report the use of a "core-weighting" method, combined with a grid-threading Monte Carlo (GTMC) approach for this purpose. The "core" of an individual structure is defined to represent the part where the density distribution is least likely to be altered by other components that comprise the macromolecular assembly of interest. The performance of the method has been evaluated by its ability to determine the correct fit of (i) the alpha-chain of the T-cell receptor variable domain into a simulated map of the alphabeta complex at resolutions between 5 and 40 A, and (ii) the E2 catalytic domain of the pyruvate dehydrogenase into an experimentally determined map, at 14 A resolution, of the icosahedral complex formed by 60 copies of this enzyme. Using the X-ray structures of the two test cases as references, we demonstrate that, in contrast to more traditional methods, the combination of the core-weighting method and the grid-threading Monte Carlo approach can identify the correct fit reliably and rapidly from the low-resolution maps that are typical of structures determined with the use of single-particle electron microscopy.     

Flexible fitting of atomic structures into electron microscopy maps using molecular dynamics

A novel method to flexibly fit atomic structures into electron microscopy (EM) maps using molecular dynamics simulations is presented. The simulations incorporate the EM data as an external potential added to the molecular dynamics force field, allowing all internal features present in the EM map to be used in the fitting process, while the model remains fully flexible and stereochemically correct. The molecular dynamics flexible fitting (MDFF) method is validated for available crystal structures of protein and RNA in different conformations; measures to assess and monitor the fitting process are introduced. The MDFF method is then used to obtain high-resolution structures of the E. coli ribosome in different functional states imaged by cryo-EM.     

Refining the structure of the strongly bound actin-myosin complex by protein docking

The structure of the strongly bound complex of the globular myosin head and F-actin is a key for understanding some important details of the mechanism of the actin-myosin motor. Current knowledge about the structure is based on the docking of known atomic structures of actin and myosin heads into low-resolution EM electron density maps. To refine the structure, we suggested a new approach based on energy minimization using the ICM-Pro software. The minimization includes rigid-body movement of protein backbone and side chain optimization on the protein interface. Our best model structure is similar to that obtained from EM. It also provides the highest calculated interaction energy and agrees with a number of mutagenesis experiments. Using the structure, we suggest molecular explanations for actin activation of product release from myosin and actin-induced myosin dissociation.     

Protein structure fitting and refinement guided by cryo-EM density

For many macromolecular assemblies, both a cryo-electron microscopy map and atomic structures of its component proteins are available. Here we describe a method for fitting and refining a component structure within its map at intermediate resolution (<15 A). The atomic positions are optimized with respect to a scoring function that includes the crosscorrelation coefficient between the structure and the map as well as stereochemical and nonbonded interaction terms. A heuristic optimization that relies on a Monte Carlo search, a conjugate-gradients minimization, and simulated annealing molecular dynamics is applied to a series of subdivisions of the structure into progressively smaller rigid bodies. The method was tested on 15 proteins of known structure with 13 simulated maps and 3 experimentally determined maps. At approximately 10 A resolution, Calpha rmsd between the initial and final structures was reduced on average by approximately 53%. The method is automated and can refine both experimental and predicted atomic structures.     

Cryo‐EM map interpretation and protein model‐building using iterative map segmentation

A procedure for building protein chains into maps produced by single‐particle electron cryo‐microscopy (cryo‐EM) is described. The procedure is similar to the way an experienced structural biologist might analyze a map, focusing first on secondary structure elements such as helices and sheets, then varying the contour level to identify connections between these elements. Since the high density in a map typically follows the main‐chain of the protein, the main‐chain connection between secondary structure elements can often be identified as the unbranched path between them with the highest minimum value along the path. This chain‐tracing procedure is then combined with finding side‐chain positions based on the presence of density extending away from the main path of the chain, allowing generation of a C_α model. The C_α model is converted to an all‐atom model and is refined against the map. We show that this procedure is as effective as other existing methods for interpretation of cryo‐EM maps and that it is considerably faster and produces models with fewer chain breaks than our previous methods that were based on approaches developed for crystallographic maps.     

Scoring functions for cryoEM density fitting

In fitting atomic structures into cryoEM density maps of macromolecular assemblies, the cross-correlation function (CCF) is the most prevalent method of scoring the goodness-of-fit. However, there are still many possible, less studied ways of scoring fits. In this paper, we introduce four scores new to cryoEM fitting and compare their performance to three known scores. Our benchmark consists of (a) 4 protein assemblies with simulated maps at 5-20 ? resolution, including the heptameric ring of GroEL; and (b) 4 experimental maps of GroEL at ～6-23 ? resolution with corresponding fitted atomic models. We perturb each fit 1000 times and assess each new fit with each score. The correlation between a score and the Cα RMSD of each fit from the "correctly" fitted structure shows that the CCF is one of the best scores, but in certain situations could be augmented or even replaced by other scores. For instance, our implementation of a score based on mutual information outperforms or is comparable to the CCF in almost all test cases, and our new "envelope score" works as well as the CCF at sub-nanometer resolution but is an order of magnitude faster to calculate. The results also suggest that the width of the Gaussian function used to blur the atomic structure into a density map can significantly affect the fitting process. Finally, we show that our score-testing method, when combined with the Laplacian CCF or the mutual information scores, can be used as a statistical tool for improving cryoEM density fitting.     

Protein fold recognition and dynamics in the space of contact maps

We introduce an energy function for contact maps of proteins. In addition to the standard term, that takes into account pair-wise interactions between amino acids, our potential contains a new hydrophobic energy term. Parameters of the energy function were obtained from a statistical analysis of the contact maps of known structures. The quality of our energy function was tested extensively in a variety of ways. In particular, fold recognition experiments revealed that for a fixed sequence the native map is identified correctly in an overwhelming majority of the cases tested. We succeeded in identifying the structure of some proteins that are known to pose difficulties for such tests (BPTI, spectrin, and cro-protein). In addition, many known pairs of homologous structures were correctly identified, even when the two sequences had relatively low sequence homology. We also introduced a dynamic Monte Carlo procedure in the space of contact maps, taking topological and polymeric constraints into account by restrictive dynamic rules. Various aspects of protein dynamics, including high-temperature melting and refolding, were simulated. Perspectives of application of the energy function and the method for structure checking and fold prediction are discussed. Proteins 26:391–410 © 1996 Wiley-Liss, Inc.     

Practical tools for molecular modeling of complex carbohydrates and their interactions with proteins

Computer modeling has become a valuable component of studies of carbohydrate three-dimensional structures and their relationship to function and properties. In this paper we examine the methods required for conformational modeling of carbohydrates, and we present a series of tools that have been developed to this end. These tools can be integrated into three-dimensional real-time molecular modeling software. A data base of pre-optimized carbohydrate fragments has been established to be used further in the construction of much more complex molecules. In addition we describe some possible uses of a data base of three dimensional structures of the disaccharide fragments present in the glycan moiety ofN-glycoprotein. A molecular mechanical force field appropriate for the conformational analysis of oligosaccharides has been derived by the addition of new parameters to the Tripos force field and is compatible with protein simulations. The new parametrization has been assessed in three stages of increasing complexity: computations of potential energy surfaces, conformational refinement of relevant oligosaccharides, modeling at the atomic level of a protein/carbohydrate complex.     

Improving integrative 3D modeling into low‐ to medium‐resolution electron microscopy structures with evolutionary couplings

Electron microscopy (EM) continues to provide near‐atomic resolution structures for well‐behaved proteins and protein complexes. Unfortunately, structures of some complexes are limited to low‐ to medium‐resolution due to biochemical or conformational heterogeneity. Thus, the application of unbiased systematic methods for fitting individual structures into EM maps is important. A method that employs co‐evolutionary information obtained solely from sequence data could prove invaluable for quick, confident localization of subunits within these structures. Here, we incorporate the co‐evolution of intermolecular amino acids as a new type of distance restraint in the integrative modeling platform in order to build three‐dimensional models of atomic structures into EM maps ranging from 10–14 Å in resolution. We validate this method using four complexes of known structure, where we highlight the conservation of intermolecular couplings despite dynamic conformational changes using the BAM complex. Finally, we use this method to assemble the subunits of the bacterial holo‐translocon into a model that agrees with previous biochemical data. The use of evolutionary couplings in integrative modeling improves systematic, unbiased fitting of atomic models into medium‐ to low‐resolution EM maps, providing additional information to integrative models lacking in spatial data.     

Multi-resolution contour-based fitting of macromolecular structures

A novel contour-based matching criterion is presented for the quantitative docking of high-resolution structures of components into low-resolution maps of macromolecular complexes. The proposed Laplacian filter is combined with a six-dimensional search using fast Fourier transforms to rapidly scan the rigid-body degrees of freedom of a probe molecule relative to a fixed target density map. A comparison of the docking performance with the standard cross-correlation criterion demonstrates that contour matching with the Laplacian filter significantly extends the viable resolution range of correlation-based fitting to resolutions as low as 30 A. The gain in docking precision at medium to low resolution (15-30 A) is critical for image reconstructions from electron microscopy (EM). The new algorithm enables for the first time the reliable docking of smaller molecular components into EM densities of large biomolecular assemblies at such low resolutions. As an example of the practical effectiveness of contour-based fitting, a new pseudo-atomic model of a microtubule was constructed from a 20 A resolution EM map and from atomic structures of alpha and beta tubulin subunits.     

A probabilistic approach to protein backbone tracing in electron density maps

One particularly time-consuming step in protein crystallography is interpreting the electron density map; that is, fitting a complete molecular model of the protein into a 3D image of the protein produced by the crystallographic process. In poor-quality electron density maps, the interpretation may require a significant amount of a crystallographer's time. Our work investigates automating the time-consuming initial backbone trace in poor-quality density maps. We describe ACMI (Automatic Crystallographic Map Interpreter), which uses a probabilistic model known as a Markov field to represent the protein. Residues of the protein are modeled as nodes in a graph, while edges model pairwise structural interactions. Modeling the protein in this manner allows the model to be flexible, considering an almost infinite number of possible conformations, while rejecting any that are physically impossible. Using an efficient algorithm for approximate inference--belief propagation--allows the most probable trace of the protein's backbone through the density map to be determined. We test ACMI on a set of ten protein density maps (at 2.5 to 4.0 A resolution), and compare our results to alternative approaches. At these resolutions, ACMI offers a more accurate backbone trace than current approaches.     

Determination of chemical identity and occupancy from experimental density maps

Three basic electronic properties of molecules, electron density (ED), charge density (CD), and electrostatic potentials (ESP), are dependent on both atomic mobility and occupancy of components in the molecules. Small protein subunits may bind large macromolecular complexes with a reduced occupancy or an increased atomic mobility or both due to affinity‐based functional regulation, and so may substrates, products, cofactors, ions or solvent molecule to the active sites of enzymes. A quantitative theory is presented in this study that describes the dependence of atomic functions on atomic B‐factor in Fourier transforms of the corresponding maps. An application of this theory is described to an experimental ED map at 1.73‐Å resolution, and to an experimental CD map at 2.2‐Å resolution. All the three density functions are linearly proportional to occupancy when the structure factor F(000) term of Fourier transforms of experimental density maps is included. Upon application of this theory to both experimental CD and ESP maps recently reported for photosystem II‐light harvesting complex II supercomplex at 3.2‐Å resolution, the occupancy of two extrinsic protein subunits PsbQ and PsbP is determined to be 20.4 ± 0.2%, and the negative mean ESP value of vitreous ice displaced by the supercomplex on electron scattering path is estimated to be 3% of the mean ESP value of protein α‐helices.     

Automated electron-density sampling reveals widespread conformational polymorphism in proteins

Although proteins populate large structural ensembles, X-ray diffraction data are traditionally interpreted using a single model. To search for evidence of alternate conformers, we developed a program, Ringer, which systematically samples electron density around the dihedral angles of protein side chains. In a diverse set of 402 structures, Ringer identified weak, nonrandom electron-density features that suggest of the presence of hidden, lowly populated conformations for >18% of uniquely modeled residues. Although these peaks occur at electron-density levels traditionally regarded as noise, statistically significant (P < 10⁻⁵) enrichment of peaks at successive rotameric χ angles validates the assignment of these features as unmodeled conformations. Weak electron density corresponding to alternate rotamers also was detected in an accurate electron density map free of model bias. Ringer analysis of the high-resolution structures of free and peptide-bound calmodulin identified shifts in ensembles and connected the alternate conformations to ligand recognition. These results show that the signal in high-resolution electron density maps extends below the traditional 1 σ cutoff, and crystalline proteins are more polymorphic than current crystallographic models. Ringer provides an objective, systematic method to identify previously undiscovered alternate conformations that can mediate protein folding and function.