Damped-dynamics flexible fitting

In fitting atomic structures into EM maps, it often happens that the map corresponds to a different conformation of the structure. We have developed a new methodology to handle these situations that preserves the covalent geometry of the structure and allows the modeling of large deformations. The first goal is achieved by working in generalized coordinates (positional and internal coordinates), and the second by avoiding harmonic potentials. Instead, we use dampers (shock absorbers) between every pair of atoms, combined with a force field that attracts the atomic structure toward incompletely occupied regions of the EM map. The trajectory obtained by integrating the resulting equations of motion converges to a conformation that, in our validation cases, was very close to the target atomic structure. Compared to current methods, our approach is more efficient and robust against wrong solutions and to overfitting, and does not require user intervention or subjective decisions. Applications to the computation of transition pathways between known conformers, homology and loop modeling, as well as protein docking, are also discussed.     

Finding rigid bodies in protein structures: Application to flexible fitting into cryoEM maps

We present RIBFIND, a method for detecting flexibility in protein structures via the clustering of secondary structural elements (SSEs) into rigid bodies. To test the usefulness of the method in refining atomic structures within cryoEM density we incorporated it into our flexible fitting protocol (Flex-EM). Our benchmark includes 13 pairs of protein structures in two conformations each, one of which is represented by a corresponding cryoEM map. Refining the structures in simulated and experimental maps at the 5–15 Å resolution range using rigid bodies identified by RIBFIND shows a significant improvement over using individual SSEs as rigid bodies. For the 15 Å resolution simulated maps, using RIBFIND-based rigid bodies improves the initial fits by 40.64% on average, as compared to 26.52% when using individual SSEs. Furthermore, for some test cases we show that at the sub-nanometer resolution range the fits can be further improved by applying a two-stage refinement protocol (using RIBFIND-based refinement followed by an SSE-based refinement). The method is stand-alone and could serve as a general interactive tool for guiding flexible fitting into EM maps.     

Biased coarse-grained molecular dynamics simulation approach for flexible fitting of X-ray structure into cryo electron microscopy maps

Several approaches have been introduced to interpret, in terms of high-resolution structure, low-resolution structural data as obtained from cryo-EM. As conformational changes are often observed in biological molecules, these techniques need to take into account the flexibility of proteins. Flexibility has been described in terms of movement between rigid domains and between rigid secondary structure elements, which present some limitations for studying dynamical properties. Normal mode analysis has also been used, but is limited to medium resolution data. All-atom molecular dynamics fitting techniques are more appropriate to fit structures into higher-resolution data as full protein flexibility is considered, but are cumbersome in terms of computational time. Here, we introduce a coarse-grained approach; a Go-model was used to represent biological molecules, combined with biased molecular dynamics to reproduce accurately conformational transitions. Illustrative examples on simulated data are shown. Accurate fittings can be obtained for resolution ranging from 5 to 20 Å. The approach was also tested on experimental data of Elongation Factor G and Escherichia coli RNA polymerase, where its validity is compared to previous models obtained from different techniques. This comparison demonstrates that quantitative flexible techniques, as opposed to manual docking, need to be considered to interpret low-resolution data.     

Accurate flexible fitting of high-resolution protein structures into cryo-electron microscopy maps using coarse-grained pseudo-energy minimization

Cryo-electron microscopy (cryo-EM) has been widely used to explore conformational states of large biomolecular assemblies. The detailed interpretation of cryo-EM data requires the flexible fitting of a known high-resolution protein structure into a low-resolution cryo-EM map. To this end, we have developed what we believe is a new method based on a two-bead-per-residue protein representation, and a modified form of the elastic network model that allows large-scale conformational changes while maintaining pseudobonds and secondary structures. Our method minimizes a pseudo-energy which linearly combines various terms of the modified elastic network model energy with a cryo-EM-fitting score and a collision energy that penalizes steric collisions. Unlike previous flexible fitting efforts using the lowest few normal modes, our method effectively utilizes all normal modes so that both global and local structural changes can be fully modeled. We have validated our method for a diverse set of 10 pairs of protein structures using simulated cryo-EM maps with a range of resolutions and in the absence/presence of random noise. We have shown that our method is both accurate and efficient compared with alternative techniques, and its performance is robust to the addition of random noise. Our method is also shown to be useful for the flexible fitting of three experimental cryo-EM maps.     

MCSS functionality maps for a flexible protein

The Multiple Copy Simultaneous Search (MCSS) methodology for finding energetically favorable positions and orientations of small functional groups in a binding site is extended to include flexibility of the target. This makes possible the finding of novel minima not present in a fixed structure and so extends the diversity of inhibitors that can be constructed starting with the MCSS procedure. Quenched molecular dynamics is used to generate energetically favorable positions and orientations of the functional groups in the field of a flexible protein. The method is applied to the viral protein HIV-1 protease with methanol and methyl ammonium as a test case. If the protein is quenched with many copies of functional groups randomly distributed in the binding site, the resulting minima have ligand-protein interaction energies that are, on average, less favorable than those obtained with standard MCSS. This is a consequence of the renormalized potential function employed in the Locally Enhanced Sampling (LES) approximation. However, local optimizations of existing MCSS minima with a flexible protein results in lower energy minima in regions of the protein that are of particular interest. Their use in constructing a consensus protein model for ligand design is discussed.     

SymmRef: a flexible refinement method for symmetric multimers

Symmetric protein complexes are abundant in the living cell. Predicting their atomic structure can shed light on the mechanism of many important biological processes. Symmetric docking methods aim to predict the structure of these complexes given the unbound structure of a single monomer, or its model. Symmetry constraints reduce the search-space of these methods and make the prediction easier compared to asymmetric protein-protein docking. However, the challenge of modeling the conformational changes that the monomer might undergo is a major obstacle. In this article, we present SymmRef, a novel method for refinement and reranking of symmetric docking solutions. The method models backbone and side-chain movements and optimizes the rigid-body orientations of the monomers. The backbone movements are modeled by normal modes minimization and the conformations of the side-chains are modeled by selecting optimal rotamers. Since solved structures of symmetric multimers show asymmetric side-chain conformations, we do not use symmetry constraints in the side-chain optimization procedure. The refined models are re-ranked according to an energy score. We tested the method on a benchmark of unbound docking challenges. The results show that the method significantly improves the accuracy and the ranking of symmetric rigid docking solutions. SymmRef is available for download at http:// bioinfo3d.cs.tau.ac.il/SymmRef/download.html.     

Geometry-based flexible and symmetric protein docking

We present a set of geometric docking algorithms for rigid, flexible, and cyclic symmetry docking. The algorithms are highly efficient and have demonstrated very good performance in CAPRI Rounds 3-5. The flexible docking algorithm, FlexDock, is unique in its ability to handle any number of hinges in the flexible molecule, without degradation in run-time performance, as compared to rigid docking. The algorithm for reconstruction of cyclically symmetric complexes successfully assembles multimolecular complexes satisfying C(n) symmetry for any n in a matter of minutes on a desktop PC. Most of the algorithms presented here are available at the Tel Aviv University Structural Bioinformatics Web server (http://bioinfo3d.cs.tau.ac.il/).     

Precise circuitry links bilaterally symmetric olfactory maps

Olfactory sensory neurons expressing a common receptor gene converge onto one or a few glomeruli with stereotyped positions within the mouse main olfactory bulb (MOB), producing a map of approximately 1800 olfactory columns representing approximately 1000 odorant receptors. Here, we report that this precise olfactory map is maintained over several synapses that ultimately cross MOB hemispheres to link bilateral isofunctional olfactory columns. Focal injection of tracer into genetically identified glomeruli revealed an exquisite topography that involves a bilateral connection via the anterior olfactory nucleus pars externa (AONpE) that links isofunctional olfactory columns in the contralateral MOB. Physiological and behavioral assays revealed an important role for the AONpE in bilateral exchange of odorant-specific information. These results indicate that the interbulbar link through the AONpE integrates bilateral olfactory sensory maps and exchanges olfactory information, positioning it as a unique model system for studying interhemispheric connections.     

Using situs for flexible and rigid-body fitting of multiresolution single-molecule data

We describe here a set of multiresolution visualization and docking procedures that we refer to as the Situs package. The package was developed to provide an efficient and robust method for the fitting of atomic structures into low-resolution data. The current release was optimized specifically for the visualization and docking of single molecules. A novel 3D graphics viewer, volslice3d, was developed for the package to provide an immersive virtual reality environment for measuring and rendering volumetric data sets. The precision of single-molecule, rigid-body docking was tested on simulated (noise-free) low-resolution density maps. For spatial resolutions near 20 A typically arising in electron microscopy image reconstructions, a docking precision on the order of 1 A can be achieved. The shape-matching score captured the correct solutions in all 10 trial cases and was sufficiently stringent to yield unique matches in 8 systems. Novel routines were developed for the flexible docking of atomic structures whose shape deviates from the corresponding low-resolution shape. Test calculations on isoforms of actin and lactoferrin demonstrate that the flexible docking faithfully reproduces conformational differences with a precision < 2 A if atomic structures are locally conserved.     

Applications of the molecular dynamics flexible fitting method

In recent years, cryo-electron microscopy (cryo-EM) has established itself as a key method in structural biology, permitting the structural characterization of large biomolecular complexes in various functional states. The data obtained through single-particle cryo-EM has recently seen a leap in resolution thanks to landmark advances in experimental and computational techniques, resulting in sub-nanometer resolution structures being obtained routinely. The remaining gap between these data and revealing the mechanisms of molecular function can be closed through hybrid modeling tools that incorporate known atomic structures into the cryo-EM data. One such tool, molecular dynamics flexible fitting (MDFF), uses molecular dynamics simulations to combine structures from X-ray crystallography with cryo-EM density maps to derive atomic models of large biomolecular complexes. The structures furnished by MDFF can be used subsequently in computational investigations aimed at revealing the dynamics of the complexes under study. In the present work, recent applications of MDFF are presented, including the interpretation of cryo-EM data of the ribosome at different stages of translation and the structure of a membrane-curvature-inducing photosynthetic complex.     

Dynamics in cryo EM reconstructions visualized with maximum-likelihood derived variance maps

CryoEM data capture the dynamic character associated with biological macromolecular assemblies by preserving the various conformations of the individual specimens at the moment of flash freezing. Regions of high variation in the data set are apparent in the image reconstruction due to the poor density that results from the lack of superposition of these regions. These observations are qualitative and, to date, only preliminary efforts have been made to quantitate the heterogeneity in the ensemble of particles that are individually imaged. We developed and tested a quantitative method for simultaneously computing a reconstruction of the particle and a map of the space-varying heterogeneity of the particle based on an entire data set. The method uses a maximum likelihood algorithm that explicitly takes into account the continuous variability from one instance to another instance of the particle. The result describes the heterogeneity of the particle as a variance to be plotted at every voxel of the reconstructed density. The test, employing time resolved data sets of virus maturation, not only recapitulated local variations obtained with difference map analysis, but revealed a remarkable time dependent reduction in the overall particle dynamics that was unobservable with classical methods of analysis.     

A design criterion for symmetric model discrimination based on flexible nominal sets

Experimental design applications for discriminating between models have been hampered by the assumption to know beforehand which model is the true one, which is counter to the very aim of the experiment. Previous approaches to alleviate this requirement were either symmetrizations of asymmetric techniques, or Bayesian, minimax, and sequential approaches. Here we present a genuinely symmetric criterion based on a linearized distance between mean-value surfaces and the newly introduced tool of flexible nominal sets. We demonstrate the computational efficiency of the approach using the proposed criterion and provide a Monte-Carlo evaluation of its discrimination performance on the basis of the likelihood ratio. An application for a pair of competing models in enzyme kinetics is given.     

Cryo-electron microscopy modeling by the molecular dynamics flexible fitting method

The increasing power and popularity of cryo-electron microscopy (cryo-EM) in structural biology brought about the development of so-called hybrid methods, which permit the interpretation of cryo-EM density maps beyond their nominal resolution in terms of atomic models. The Cryo-EM Modeling Challenge 2010 is the first community effort to bring together developers of hybrid methods as well as cryo-EM experimentalists. Participating in the challenge, the molecular dynamics flexible fitting (MDFF) method was applied to a number of cryo-EM density maps. The results are described here with special emphasis on the use of symmetry-based restraints to improve the quality of atomic models derived from density maps of symmetric complexes; on a comparison of the stereochemical quality of atomic models resulting from different hybrid methods; and on application of MDFF to electron crystallography data.     

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

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

A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution

A popular way to represent clustered binary, count, or other data is via the generalized linear mixed model framework, which accommodates correlation through incorporation of random effects. A standard assumption is that the random effects follow a parametric family such as the normal distribution; however, this may be unrealistic or too restrictive to represent the data. We relax this assumption and require only that the distribution of random effects belong to a class of 'smooth' densities and approximate the density by the seminonparametric (SNP) approach of Gallant and Nychka (1987). This representation allows the density to be skewed, multi-modal, fat- or thin-tailed relative to the normal and includes the normal as a special case. Because an efficient algorithm to sample from an SNP density is available, we propose a Monte Carlo EM algorithm using a rejection sampling scheme to estimate the fixed parameters of the linear predictor, variance components and the SNP density. The approach is illustrated by application to a data set and via simulation.     

Consensus among flexible fitting approaches improves the interpretation of cryo-EM data

Cryo-elecron microscopy (cryo-EM) can provide important structural information of large macromolecular assemblies in different conformational states. Recent years have seen an increase in structures deposited in the Protein Data Bank (PDB) by fitting a high-resolution structure into its low-resolution cryo-EM map. A commonly used protocol for accommodating the conformational changes between the X-ray structure and the cryo-EM map is rigid body fitting of individual domains. With the emergence of different flexible fitting approaches, there is a need to compare and revise these different protocols for the fitting. We have applied three diverse automated flexible fitting approaches on a protein dataset for which rigid domain fitting (RDF) models have been deposited in the PDB. In general, a consensus is observed in the conformations, which indicates a convergence from these theoretically different approaches to the most probable solution corresponding to the cryo-EM map. However, the result shows that the convergence might not be observed for proteins with complex conformational changes or with missing densities in cryo-EM map. In contrast, RDF structures deposited in the PDB can represent conformations that not only differ from the consensus obtained by flexible fitting but also from X-ray crystallography. Thus, this study emphasizes that a "consensus" achieved by the use of several automated flexible fitting approaches can provide a higher level of confidence in the modeled configurations. Following this protocol not only increases the confidence level of fitting, but also highlights protein regions with uncertain fitting. Hence, this protocol can lead to better interpretation of cryo-EM data.     

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.     

Computational prediction of atomic structures of helical membrane proteins aided by EM maps

Integral membrane proteins pose a major challenge for protein-structure prediction because only approximately 100 high-resolution structures are available currently, thereby impeding the development of rules or empirical potentials to predict the packing of transmembrane alpha-helices. However, when an intermediate-resolution electron microscopy (EM) map is available, it can be used to provide restraints which, in combination with a suitable computational protocol, make structure prediction feasible. In this work we present such a protocol, which proceeds in three stages: 1), generation of an ensemble of alpha-helices by flexible fitting into each of the density rods in the low-resolution EM map, spanning a range of rotational angles around the main helical axes and translational shifts along the density rods; 2), fast optimization of side chains and scoring of the resulting conformations; and 3), refinement of the lowest-scoring conformations with internal coordinate mechanics, by optimizing the van der Waals, electrostatics, hydrogen bonding, torsional, and solvation energy contributions. In addition, our method implements a penalty term through a so-called tethering map, derived from the EM map, which restrains the positions of the alpha-helices. The protocol was validated on three test cases: GpA, KcsA, and MscL.     

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.     

Ecological fitting by phenotypically flexible genotypes: implications for species associations, community assembly and evolution

Ecological fitting is the process whereby organisms colonize and persist in novel environments, use novel resources or form novel associations with other species as a result of the suites of traits that they carry at the time they encounter the novel condition. This paper has four major aims. First, we review the original concept of ecological fitting and relate it to the concept of exaptation and current ideas on the positive role of phenotypic plasticity in evolution. Second, we propose phenotypic plasticity, correlated trait evolution and phylogenetic conservatism as specific mechanisms behind ecological fitting. Third, we attempt to operationalize the concept of ecological fitting by providing explicit definitions for terms. From these definitions, we propose a simple conceptual model of ecological fitting. Using this model, we demonstrate the differences and similarities between ecological fitting and ecological resource tracking and illustrate the process in the context of species colonizing new areas and forming novel associations with other species. Finally, we discuss how ecological fitting can be both a precursor to evolutionary diversity or maintainer of evolutionary stasis, depending on conditions. We conclude that ecological fitting is an important concept for understanding topics ranging from the assembly of ecological communities and species associations, to biological invasions, to the evolution of biodiversity.