BCL::EM-Fit: rigid body fitting of atomic structures into density maps using geometric hashing and real space refinement

Accurate global and local 3D alignment of cryo-EM density maps using local spatial structural features

Advances in cryo-electron microscopy (cryo-EM) imaging technologies have led to a rapidly increasing number of cryo-EM density maps. Alignment and comparison of density maps play a crucial role in interpreting structural information, such as conformational heterogeneity analysis using global alignment and atomic model assembly through local alignment. Here, we present a fast and accurate global and local cryo-EM density map alignment method called CryoAlign, that leverages local density feature descriptors to capture spatial structure similarities. CryoAlign is a feature-based cryo-EM map alignment tool, in which the employment of feature-based architecture enables the rapid establishment of point pair correspondences and robust estimation of alignment parameters. Extensive experimental evaluations demonstrate the superiority of CryoAlign over the existing methods in terms of both alignment accuracy and speed.

Application of conformational space annealing to the protein structure modeling using cryo-EM maps

Conformational space annealing (CSA), a global optimization method, has been applied to various protein structure modeling tasks. In this paper, we applied CSA to the cryo-EM structure modeling task by combining the python subroutine of CSA (PyCSA) and the fast relax (FastRelax) protocol of PyRosetta. Refinement of initial structures generated from two methods, rigid fitting of predicted structures to the Cryo-EM map and de novo protein modeling by tracing the Cryo-EM map, was performed by CSA. In the refinement of the rigid-fitted structures, the final models showed that CSA can generate reliable atomic structures of proteins, even when large movements of protein domains were required. In the de novo modeling case, although the overall structural qualities of the final models were rather dependent on the initial models, the final models generated by CSA showed improved MolProbity scores and cross-correlation coefficients to the maps. These results suggest that CSA can accomplish flexible fitting and refinement together by sampling diverse conformations effectively and thus can be utilized for cryo-EM structure modeling.

Benchmarking applicability of medium-resolution cryo-EM protein structures for structure-based drug design

Cryo-electron microscopy (cryo-EM) is gaining large attention for high-resolution protein structure determination in solutions. However, a very high percentage of cryo-EM structures correspond to resolutions of 3-5 Å, making the structures difficult to be used in in silico drug design. In this study, we analyze how useful cryo-EM protein structures are for in silico drug design by evaluating ligand docking accuracy. From realistic cross-docking scenarios using medium resolution (3-5 Å) cryo-EM structures and a popular docking tool Autodock-Vina, only 20% of docking succeeded, when the success rate doubles in the same kind of cross-docking but using high-resolution (<2 Å) crystal structures instead. We decipher the reason for failures by decomposing the contribution from resolution-dependent and independent factors. The heterogeneity in the protein side-chain and backbone conformations is identified as the major resolution-dependent factor causing docking difficulty from our analysis, while intrinsic receptor flexibility mainly comprises the resolution-independent factor. We demonstrate the flexibility implementation in current ligand docking tools is able to rescue only a portion of failures (10%), and the limited performance was majorly due to potential structural errors than conformational changes. Our work suggests the strong necessity of more robust method developments on ligand docking and EM modeling techniques in order to fully utilize cryo-EM structures for in silico drug design.

MarkovFit: Structure Fitting for Protein Complexes in Electron Microscopy Maps Using Markov Random Field

An increasing number of protein complex structures are determined by cryo-electron microscopy (cryo-EM). When individual protein structures have been determined and are available, an important task in structure modeling is to fit the individual structures into the density map. Here, we designed a method that fits the atomic structures of proteins in cryo-EM maps of medium to low resolutions using Markov random fields, which allows probabilistic evaluation of fitted models. The accuracy of our method, MarkovFit, performed better than existing methods on datasets of 31 simulated cryo-EM maps of resolution 10 Å , nine experimentally determined cryo-EM maps of resolution less than 4 Å , and 28 experimentally determined cryo-EM maps of resolution 6 to 20 Å .

Interface refinement of low- to medium-resolution Cryo-EM complexes using HADDOCK2.4

A wide range of cellular processes requires the formation of multimeric protein complexes. The rise of cryo-electron microscopy (cryo-EM) has enabled the structural characterization of these protein assemblies. The density maps produced can, however, still suffer from limited resolution, impeding the process of resolving structures at atomic resolution. In order to solve this issue, monomers can be fitted into low- to medium-resolution maps. Unfortunately, the models produced frequently contain atomic clashes at the protein-protein interfaces (PPIs), as intermolecular interactions are typically not considered during monomer fitting. Here, we present a refinement approach based on HADDOCK2.4 to remove intermolecular clashes and optimize PPIs. A dataset of 14 cryo-EM complexes was used to test eight protocols. The best-performing protocol, consisting of a semi-flexible simulated annealing refinement with centroid restraints on the monomers, was able to decrease intermolecular atomic clashes by 98% without significantly deteriorating the quality of the cryo-EM density fit.

Leveraging crosslinking mass spectrometry in structural and cell biology

Crosslinking mass spectrometry (crosslinking-MS) is a versatile tool providing structural insights into protein conformation and protein-protein interactions. Its medium-resolution residue-residue distance restraints have been used to validate protein structures proposed by other methods and have helped derive models of protein complexes by integrative structural biology approaches. The use of crosslinking-MS in integrative approaches is underpinned by progress in estimating error rates in crosslinking-MS data and in combining these data with other information. The flexible and high-throughput nature of crosslinking-MS has allowed it to complement the ongoing resolution revolution in electron microscopy by providing system-wide residue-residue distance restraints, especially for flexible regions or systems. Here, we review how crosslinking-MS information has been leveraged in structural model validation and integrative modeling. Crosslinking-MS has also been a key technology for cell biology studies and structural systems biology where, in conjunction with cryoelectron tomography, it can provide structural and mechanistic insights directly in situ.

Fitting Low-Resolution Protein Structures into Cryo-EM Density Maps by Multiobjective Optimization of Global and Local Correlations

The rigid-body fitting of predicted structural models into cryo-electron microscopy (cryo-EM) density maps is a necessary procedure for density map-guided protein structure determination and prediction. We proposed a novel multiobjective optimization protocol, MOFIT, which performs a rigid-body density-map fitting based on particle swarm optimization (PSO). MOFIT was tested on a large set of 292 nonhomologous single-domain proteins. Starting from structural models predicted by I-TASSER, MOFIT achieved an average coordinate root-mean-square deviation of 2.46 Å, which was 1.57, 2.79, and 3.95 Å lower than three leading single-objective function-based methods, where the differences were statistically significant with p-values of 1.65 × 10^-6, 6.36 × 10^-8, and 6.44 × 10^-11 calculated using two-tail Student's t tests. Detailed analyses showed that the major advantages of MOFIT lie in the multiobjective protocol and the extensive PSO search simulations guided by the composite objective functions, which integrates complementary correlation coefficients from the global structure, local fragments, and individual residues with the cryo-EM density maps.

Hybrid methods for combined experimental and computational determination of protein structure

Knowledge of protein structure is paramount to the understanding of biological function, developing new therapeutics, and making detailed mechanistic hypotheses. Therefore, methods to accurately elucidate three-dimensional structures of proteins are in high demand. While there are a few experimental techniques that can routinely provide high-resolution structures, such as x-ray crystallography, nuclear magnetic resonance (NMR), and cryo-EM, which have been developed to determine the structures of proteins, these techniques each have shortcomings and thus cannot be used in all cases. However, additionally, a large number of experimental techniques that provide some structural information, but not enough to assign atomic positions with high certainty have been developed. These methods offer sparse experimental data, which can also be noisy and inaccurate in some instances. In cases where it is not possible to determine the structure of a protein experimentally, computational structure prediction methods can be used as an alternative. Although computational methods can be performed without any experimental data in a large number of studies, inclusion of sparse experimental data into these prediction methods has yielded significant improvement. In this Perspective, we cover many of the successes of integrative modeling, computational modeling with experimental data, specifically for protein folding, protein-protein docking, and molecular dynamics simulations. We describe methods that incorporate sparse data from cryo-EM, NMR, mass spectrometry, electron paramagnetic resonance, small-angle x-ray scattering, Förster resonance energy transfer, and genetic sequence covariation. Finally, we highlight some of the major challenges in the field as well as possible future directions.

Simulation-Based Methods for Model Building and Refinement in Cryoelectron Microscopy

Advances in cryoelectron microscopy (cryo-EM) have revolutionized the structural investigation of large macromolecular assemblies. In this review, we first provide a broad overview of modeling methods used for flexible fitting of molecular models into cryo-EM density maps. We give special attention to approaches rooted in molecular simulations-atomistic molecular dynamics and Monte Carlo. Concise descriptions of the methods are given along with discussion of their advantages, limitations, and most popular alternatives. We also describe recent extensions of the widely used molecular dynamics flexible fitting (MDFF) method and discuss how different model-building techniques could be incorporated into new hybrid modeling schemes and simulation workflows. Finally, we provide two illustrative examples of model-building and refinement strategies employing MDFF, cascade MDFF, and RosettaCM. These examples come from recent cryo-EM studies that elucidated transcription preinitiation complexes and shed light on the functional roles of these assemblies in gene expression and gene regulation.

Principles for Integrative Structural Biology Studies

Integrative structure determination is a powerful approach to modeling the structures of biological systems based on data produced by multiple experimental and theoretical methods, with implications for our understanding of cellular biology and drug discovery. This Primer introduces the theory and methods of integrative approaches, emphasizing the kinds of data that can be effectively included in developing models and using the nuclear pore complex as an example to illustrate the practice and challenges involved. These guidelines are intended to aid the researcher in understanding and applying integrative structural methods to systems of their interest and thus take advantage of this rapidly evolving field.

Advances in Structure Modeling Methods for Cryo-Electron Microscopy Maps

Cryo-electron microscopy (cryo-EM) has now become a widely used technique for structure determination of macromolecular complexes. For modeling molecular structures from density maps of different resolutions, many algorithms have been developed. These algorithms can be categorized into rigid fitting, flexible fitting, and de novo modeling methods. It is also observed that machine learning (ML) techniques have been increasingly applied following the rapid progress of the ML field. Here, we review these different categories of macromolecule structure modeling methods and discuss their advances over time.

SSEThread: Integrative threading of the DNA-PKcs sequence based on data from chemical cross-linking and hydrogen deuterium exchange

X-ray crystallography and electron microscopy maps resolved to 3-8 Å are generally sufficient for tracing the path of the polypeptide chain in space, while often insufficient for unambiguously registering the sequence on the path (i.e., threading). Frequently, however, additional information is available from other biophysical experiments, physical principles, statistical analyses, and other prior models. Here, we formulate an integrative approach for sequence assignment to a partial backbone model as an optimization problem, which requires three main components: the representation of the system, the scoring function, and the optimization method. The method is implemented in the open source Integrative Modeling Platform (IMP) (https://integrativemodeling.org), allowing a number of different terms in the scoring function. We apply this method to localizing the sequence assignment within a 199-residue disordered region of three structured and sequence unassigned helices in the DNA-PKcs crystallographic structure, using chemical crosslinks, hydrogen deuterium exchange, and sequence connectivity. The resulting ensemble of threading models provides two major solutions, one of which suggests that the crucial ABCDE cluster of phosphorylation sites cannot undergo intra-molecular autophosphorylation without a conformational rearrangement. The ensemble of solutions embodies the most accurate and precise sequence threading given the available information.

De novo main-chain modeling with MAINMAST in 2015/2016 EM Model Challenge

Protein tertiary structure modeling is a critical step for the interpretation of three dimensional (3D) election microscopy density. Our group participated the 2015/2016 EM Model Challenge using the MAINMAST software for a de novo main chain modeling. The software generates local dense points using the mean shifting algorithm, and connects them into Cα models by calculating the minimum spanning tree and the longest path. Subsequently, full atom structure models are generated, which are subject to structural refinement. Here, we summarize the qualities of our submitted models and examine successful and unsuccessful models, including 3D models we did not submit to the Challenge. Our protocol using the MAINMAST software was sometimes able to build correct conformations with 3.4–5.1 Å RMSD. Unsuccessful models had failure of chain traces, however, their Cα positions and some local structures were quite correctly built. For evaluate the quality of the models, the MAINMAST software provides a confidence score for each Cα position from the consensus of top 100 scoring models.

Development of a Prototype System for Archiving Integrative/Hybrid Structure Models of Biological Macromolecules

Essential processes in biology are carried out by large macromolecular assemblies, whose structures are often difficult to determine by traditional methods. Increasingly, researchers combine measured data and computed information from several complementary methods to obtain "hybrid" or "integrative" structural models of macromolecules and their assemblies. These integrative/hybrid (I/H) models are not archived in the PDB because of the absence of standard data representations and processing mechanisms. Here we present the development of data standards and a prototype system for archiving I/H models. The data standards provide the definitions required for representing I/H models that span multiple spatiotemporal scales and conformational states, as well as spatial restraints derived from different experimental techniques. Based on these data definitions, we have built a prototype system called PDB-Dev, which provides the infrastructure necessary to archive I/H structural models. PDB-Dev is now accepting structures and is open to the community for new submissions.

Finding the needle in the haystack: towards solving the protein-folding problem computationally

Prediction of protein tertiary structures from amino acid sequence and understanding the mechanisms of how proteins fold, collectively known as "the protein folding problem," has been a grand challenge in molecular biology for over half a century. Theories have been developed that provide us with an unprecedented understanding of protein folding mechanisms. However, computational simulation of protein folding is still difficult, and prediction of protein tertiary structure from amino acid sequence is an unsolved problem. Progress toward a satisfying solution has been slow due to challenges in sampling the vast conformational space and deriving sufficiently accurate energy functions. Nevertheless, several techniques and algorithms have been adopted to overcome these challenges, and the last two decades have seen exciting advances in enhanced sampling algorithms, computational power and tertiary structure prediction methodologies. This review aims at summarizing these computational techniques, specifically conformational sampling algorithms and energy approximations that have been frequently used to study protein-folding mechanisms or to de novo predict protein tertiary structures. We hope that this review can serve as an overview on how the protein-folding problem can be studied computationally and, in cases where experimental approaches are prohibitive, help the researcher choose the most relevant computational approach for the problem at hand. We conclude with a summary of current challenges faced and an outlook on potential future directions.

A fragment based method for modeling of protein segments into cryo-EM density maps

BACKGROUND

Single-particle analysis of electron cryo-microscopy (cryo-EM) is a key technology for elucidation of macromolecular structures. Recent technical advances in hardware and software developments significantly enhanced the resolution of cryo-EM density maps and broadened the applicability and the circle of users. To facilitate modeling of macromolecules into cryo-EM density maps, fast and easy to use methods for modeling are now demanded.

RESULTS

Here we investigated and benchmarked the suitability of a classical and well established fragment-based approach for modeling of segments into cryo-EM density maps (termed FragFit). FragFit uses a hierarchical strategy to select fragments from a pre-calculated set of billions of fragments derived from structures deposited in the Protein Data Bank, based on sequence similarly, fit of stem atoms and fit to a cryo-EM density map. The user only has to specify the sequence of the segment and the number of the N- and C-terminal stem-residues in the protein. Using a representative data set of protein structures, we show that protein segments can be accurately modeled into cryo-EM density maps of different resolution by FragFit. Prediction quality depends on segment length, the type of secondary structure of the segment and local quality of the map.

CONCLUSION

Fast and automated calculation of FragFit renders it applicable for implementation of interactive web-applications e.g. to model missing segments, flexible protein parts or hinge-regions into cryo-EM density maps.

Iterative Molecular Dynamics-Rosetta Membrane Protein Structure Refinement Guided by Cryo-EM Densities

Knowing atomistic details of proteins is essential not only for the understanding of protein function but also for the development of drugs. Experimental methods such as X-ray crystallography, NMR, and cryo-electron microscopy (cryo-EM) are the preferred forms of protein structure determination and have achieved great success over the most recent decades. Computational methods may be an alternative when experimental techniques fail. However, computational methods are severely limited when it comes to predicting larger macromolecule structures with little sequence similarity to known structures. The incorporation of experimental restraints in computational methods is becoming increasingly important to more reliably predict protein structure. One such experimental input used in structure prediction and refinement is cryo-EM densities. Recent advances in cryo-EM have arguably revolutionized the field of structural biology. Our previously developed cryo-EM-guided Rosetta-MD protocol has shown great promise in the refinement of soluble protein structures. In this study, we extended cryo-EM density-guided iterative Rosetta-MD to membrane proteins. We also improved the methodology in general by picking models based on a combination of their score and fit-to-density during the Rosetta model selection. By doing so, we have been able to pick models superior to those with the previous selection based on Rosetta score only and we have been able to further improve our previously refined models of soluble proteins. The method was tested with five membrane spanning protein structures. By applying density-guided Rosetta-MD iteratively we were able to refine the predicted structures of these membrane proteins to atomic resolutions. We also showed that the resolution of the density maps determines the improvement and quality of the refined models. By incorporating high-resolution density maps (∼4 Å), we were able to more significantly improve the quality of the models than when medium-resolution maps (6.9 Å) were used. Beginning from an average starting structure root mean square deviation (RMSD) to native of 4.66 Å, our protocol was able to refine the structures to bring the average refined structure RMSD to 1.66 Å when 4 Å density maps were used. The protocol also successfully refined the HIV-1 CTD guided by an experimental 5 Å density map.

Improved metrics for comparing structures of macromolecular assemblies determined by 3D electron-microscopy

Recent developments in 3-dimensional electron microcopy (3D-EM) techniques and a concomitant drive to look at complex molecular structures, have led to a rapid increase in the amount of volume data available for biomolecules. This creates a demand for better methods to analyse the data, including improved scores for comparison, classification and integration of data at different resolutions. To this end, we developed and evaluated a set of scoring functions that compare 3D-EM volumes. To test our scores we used a benchmark set of volume alignments derived from the Electron Microscopy Data Bank. We find that the performance of different scores vary with the map-type, resolution and the extent of overlap between volumes. Importantly, adding the overlap information to the local scoring functions can significantly improve their precision and accuracy in a range of resolutions. A combined score involving the local mutual information and overlap (LMI_OV) performs best overall, irrespective of the map category, resolution or the extent of overlap, and we recommend this score for general use. The local mutual information score itself is found to be more discriminatory than cross-correlation coefficient for intermediate-to-low resolution maps or when the map size and density distribution differ significantly. For comparing map surfaces, we implemented two filters to detect the surface points, including one based on the 'extent of surface exposure'. We show that scores that compare surfaces are useful at low resolutions and for maps with evident surface features. All the scores discussed are implemented in TEMPy (http://tempy.ismb.lon.ac.uk/).

3DIANA: 3D Domain Interaction Analysis: A Toolbox for Quaternary Structure Modeling

Electron microscopy (EM) is experiencing a revolution with the advent of a new generation of Direct Electron Detectors, enabling a broad range of large and flexible structures to be resolved well below 1 nm resolution. Although EM techniques are evolving to the point of directly obtaining structural data at near-atomic resolution, for many molecules the attainable resolution might not be enough to propose high-resolution structural models. However, accessing information on atomic coordinates is a necessary step toward a deeper understanding of the molecular mechanisms that allow proteins to perform specific tasks. For that reason, methods for the integration of EM three-dimensional maps with x-ray and NMR structural data are being developed, a modeling task that is normally referred to as fitting, resulting in the so called hybrid models. In this work, we present a novel application—3DIANA—specially targeted to those cases in which the EM map resolution is medium or low and additional experimental structural information is scarce or even lacking. In this way, 3DIANA statistically evaluates proposed/potential contacts between protein domains, presents a complete catalog of both structurally resolved and predicted interacting regions involving these domains and, finally, suggests structural templates to model the interaction between them. The evaluation of the proposed interactions is computed with DIMERO, a new method that scores physical binding sites based on the topology of protein interaction networks, which has recently shown the capability to increase by 200% the number of domain-domain interactions predicted in interactomes as compared to previous approaches. The new application displays the information at a sequence and structural level and is accessible through a web browser or as a Chimera plugin at http://3diana.cnb.csic.es.

PF2fit: Polar Fast Fourier Matched Alignment of Atomistic Structures with 3D Electron Microscopy Maps

There continue to be increasing occurrences of both atomistic structure models in the PDB (possibly reconstructed from X-ray diffraction or NMR data), and 3D reconstructed cryo-electron microscopy (3D EM) maps (albeit at coarser resolution) of the same or homologous molecule or molecular assembly, deposited in the EMDB. To obtain the best possible structural model of the molecule at the best achievable resolution, and without any missing gaps, one typically aligns (match and fits) the atomistic structure model with the 3D EM map. We discuss a new algorithm and generalized framework, named PF(2) fit (Polar Fast Fourier Fitting) for the best possible structural alignment of atomistic structures with 3D EM. While PF(2) fit enables only a rigid, six dimensional (6D) alignment method, it augments prior work on 6D X-ray structure and 3D EM alignment in multiple ways: Scoring. PF(2) fit includes a new scoring scheme that, in addition to rewarding overlaps between the volumes occupied by the atomistic structure and 3D EM map, rewards overlaps between the volumes complementary to them. We quantitatively demonstrate how this new complementary scoring scheme improves upon existing approaches. PF(2) fit also includes two scoring functions, the non-uniform exterior penalty and the skeleton-secondary structure score, and implements the scattering potential score as an alternative to traditional Gaussian blurring. Search. PF(2) fit utilizes a fast polar Fourier search scheme, whose main advantage is the ability to search over uniformly and adaptively sampled subsets of the space of rigid-body motions. PF(2) fit also implements a new reranking search and scoring methodology that considerably improves alignment metrics in results obtained from the initial search.

Integrative Modeling of Macromolecular Assemblies from Low to Near-Atomic Resolution

While conventional high-resolution techniques in structural biology are challenged by the size and flexibility of many biological assemblies, recent advances in low-resolution techniques such as cryo-electron microscopy (cryo-EM) and small angle X-ray scattering (SAXS) have opened up new avenues to define the structures of such assemblies. By systematically combining various sources of structural, biochemical and biophysical information, integrative modeling approaches aim to provide a unified structural description of such assemblies, starting from high-resolution structures of the individual components and integrating all available information from low-resolution experimental methods. In this review, we describe integrative modeling approaches, which use complementary data from either cryo-EM or SAXS. Specifically, we focus on the popular molecular dynamics flexible fitting (MDFF) method, which has been widely used for flexible fitting into cryo-EM maps. Second, we describe hybrid molecular dynamics, Rosetta Monte-Carlo and minimum ensemble search (MES) methods that can be used to incorporate SAXS into pseudoatomic structural models. We present concise descriptions of the two methods and their most popular alternatives, along with select illustrative applications to protein/nucleic acid assemblies involved in DNA replication and repair.

Atomic-accuracy models from 4.5-Å cryo-electron microscopy data with density-guided iterative local refinement

We describe a general approach for refining protein structure models on the basis of cryo-electron microscopy maps with near-atomic resolution. The method integrates Monte Carlo sampling with local density-guided optimization, Rosetta all-atom refinement and real-space B-factor fitting. In tests on experimental maps of three different systems with 4.5-Å resolution or better, the method consistently produced models with atomic-level accuracy largely independently of starting-model quality, and it outperformed the molecular dynamics-based MDFF method. Cross-validated model quality statistics correlated with model accuracy over the three test systems.

Characterization and modeling of the oligomeric state and ligand binding behavior of purified translocator protein 18 kDa from Rhodobacter sphaeroides

Translocator Protein 18 kDa (TSPO), previously known as the peripheral-type benzodiazepine receptor (PBR), is a mitochondrial outer membrane protein that has been identified as a key player in cholesterol and porphyrin transport, apoptotic signaling, and cancer development, as well as neurological inflammation and disease. Despite a number of TSPO ligands whose effects have been studied with respect to these varied biological activities, the nature of their interactions with TSPO and the molecular mechanism of their effects remain controversial, in part because of the lack of an atomic-resolution structure. We expressed and purified the homologue of mammalian TSPO from Rhodobacter sphaeroides (RsTSPO), as well as a mutant form in a proposed drug binding loop, RsTSPOW38C. We characterized their binding behaviors with endogenous ligands and a series of compounds that affect apoptosis by using a sensitive tryptophan fluorescence quenching assay. Our results show that RsTSPO behaves as a dimer in the purified state and binds with low micromolar affinity to many of these ligands, including retinoic acid, curcumin, and a known Bcl-2 inhibitor, gossypol, suggesting a possible direct role for TSPO in their regulation of apoptosis. A computational model of the RsTSPO dimer is constructed using EM-Fold, Rosetta, and a cryo-electron microscopy density map. Binding behaviors of known ligands are discussed in the context of the model with respect to regions that may be involved in binding.

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.

Computational methods for constructing protein structure models from 3D electron microscopy maps

Protein structure determination by cryo-electron microscopy (EM) has made significant progress in the past decades. Resolutions of EM maps have been improving as evidenced by recently reported structures that are solved at high resolutions close to 3Å. Computational methods play a key role in interpreting EM data. Among many computational procedures applied to an EM map to obtain protein structure information, in this article we focus on reviewing computational methods that model protein three-dimensional (3D) structures from a 3D EM density map that is constructed from two-dimensional (2D) maps. The computational methods we discuss range from de novo methods, which identify structural elements in an EM map, to structure fitting methods, where known high resolution structures are fit into a low-resolution EM map. A list of available computational tools is also provided.

EM-fold: de novo atomic-detail protein structure determination from medium-resolution density maps

Electron density maps of membrane proteins or large macromolecular complexes are frequently only determined at medium resolution between 4 Å and 10 Å, either by cryo-electron microscopy or X-ray crystallography. In these density maps, the general arrangement of secondary structure elements (SSEs) is revealed, whereas their directionality and connectivity remain elusive. We demonstrate that the topology of proteins with up to 250 amino acids can be determined from such density maps when combined with a computational protein folding protocol. Furthermore, we accurately reconstruct atomic detail in loop regions and amino acid side chains not visible in the experimental data. The EM-Fold algorithm assembles the SSEs de novo before atomic detail is added using Rosetta. In a benchmark of 27 proteins, the protocol consistently and reproducibly achieves models with root mean square deviation values <3 Å.

Fitting multimeric protein complexes into electron microscopy maps using 3D Zernike descriptors

A novel computational method for fitting high-resolution structures of multiple proteins into a cryoelectron microscopy map is presented. The method named EMLZerD generates a pool of candidate multiple protein docking conformations of component proteins, which are later compared with a provided electron microscopy (EM) density map to select the ones that fit well into the EM map. The comparison of docking conformations and the EM map is performed using the 3D Zernike descriptor (3DZD), a mathematical series expansion of three-dimensional functions. The 3DZD provides a unified representation of the surface shape of multimeric protein complex models and EM maps, which allows a convenient, fast quantitative comparison of the three-dimensional structural data. Out of 19 multimeric complexes tested, near native complex structures with a root-mean-square deviation of less than 2.5 Å were obtained for 14 cases while medium range resolution structures with correct topology were computed for the additional 5 cases.