Structural Variability from Noisy Tomographic Projections

Efficient high-resolution refinement in cryo-EM with stochastic gradient descent

Electron cryomicroscopy (cryo-EM) is an imaging technique widely used in structural biology to determine the three-dimensional structure of biological molecules from noisy two-dimensional projections with unknown orientations. As the typical pipeline involves processing large amounts of data, efficient algorithms are crucial for fast and reliable results. The stochastic gradient descent (SGD) algorithm has been used to improve the speed of ab initio reconstruction, which results in a first, low-resolution estimation of the volume representing the molecule of interest, but has yet to be applied successfully in the high-resolution regime, where expectation-maximization algorithms achieve state-of-the-art results, at a high computational cost. In this article, we investigate the conditioning of the optimization problem and show that the large condition number prevents the successful application of gradient descent-based methods at high resolution. Our results include a theoretical analysis of the condition number of the optimization problem in a simplified setting where the individual projection directions are known, an algorithm based on computing a diagonal preconditioner using Hutchinson's diagonal estimator, and numerical experiments showing the improvement in the convergence speed when using the estimated preconditioner with SGD. The preconditioned SGD approach can potentially enable a simple and unified approach to ab initio reconstruction and high-resolution refinement with faster convergence speed and higher flexibility, and our results are a promising step in this direction.

A Bayesian Framework for Cryo-EM Heterogeneity Analysis using Regularized Covariance Estimation

Proteins and the complexes they form are central to nearly all cellular processes. Their flexibility, expressed through a continuum of states, provides a window into their biological functions. Cryogenic-electron microscopy (cryo-EM) is an ideal tool to study these dynamic states as it captures specimens in non-crystalline conditions and enables high-resolution reconstructions. However, analyzing the heterogeneous distribution of conformations from cryo-EM data is challenging. Current methods face issues such as a lack of explainability, overfitting caused by lack of regularization, and a large number of parameters to tune; problems exacerbated by the lack of proper metrics to evaluate or compare heterogeneous reconstructions. To address these challenges, we present RECOVAR, a white-box method based on principal component analysis (PCA) computed via regularized covariance estimation that can resolve intricate heterogeneity with similar expressive power to neural networks with significantly lower computational demands. We extend the ubiquitous Bayesian framework used in homogeneous reconstruction to automatically regularize principal components, overcoming overfitting concerns and removing the need for most parameters. We further exploit the conservation of density and distances endowed by the embedding in PCA space, opening the door to reliable free energy computation. We leverage the predictable uncertainty of image labels to generate high-resolution reconstructions and identify high-density trajectories in latent space. We make the code freely available at https://github.com/ma-gilles/recovar.

MDTOMO method for continuous conformational variability analysis in cryo electron subtomograms based on molecular dynamics simulations

Cryo electron tomography (cryo-ET) allows observing macromolecular complexes in their native environment. The common routine of subtomogram averaging (STA) allows obtaining the three-dimensional (3D) structure of abundant macromolecular complexes, and can be coupled with discrete classification to reveal conformational heterogeneity of the sample. However, the number of complexes extracted from cryo-ET data is usually small, which restricts the discrete-classification results to a small number of enough populated states and, thus, results in a largely incomplete conformational landscape. Alternative approaches are currently being investigated to explore the continuity of the conformational landscapes that in situ cryo-ET studies could provide. In this article, we present MDTOMO, a method for analyzing continuous conformational variability in cryo-ET subtomograms based on Molecular Dynamics (MD) simulations. MDTOMO allows obtaining an atomic-scale model of conformational variability and the corresponding free-energy landscape, from a given set of cryo-ET subtomograms. The article presents the performance of MDTOMO on a synthetic ABC exporter dataset and an in situ SARS-CoV-2 spike dataset. MDTOMO allows analyzing dynamic properties of molecular complexes to understand their biological functions, which could also be useful for structure-based drug discovery.

3DFlex: determining structure and motion of flexible proteins from cryo-EM

Modeling flexible macromolecules is one of the foremost challenges in single-particle cryogenic-electron microscopy (cryo-EM), with the potential to illuminate fundamental questions in structural biology. We introduce Three-Dimensional Flexible Refinement (3DFlex), a motion-based neural network model for continuous molecular heterogeneity for cryo-EM data. 3DFlex exploits knowledge that conformational variability of a protein is often the result of physical processes that transport density over space and tend to preserve local geometry. From two-dimensional image data, 3DFlex enables the determination of high-resolution 3D density, and provides an explicit model of a flexible protein's motion over its conformational landscape. Experimentally, for large molecular machines (tri-snRNP spliceosome complex, translocating ribosome) and small flexible proteins (TRPV1 ion channel, αVβ8 integrin, SARS-CoV-2 spike), 3DFlex learns nonrigid molecular motions while resolving details of moving secondary structure elements. 3DFlex can improve 3D density resolution beyond the limits of existing methods because particle images contribute coherent signal over the conformational landscape.

Methods for Cryo-EM Single Particle Reconstruction of Macromolecules Having Continuous Heterogeneity

Macromolecules change their shape (conformation) in the process of carrying out their functions. The imaging by cryo-electron microscopy of rapidly-frozen, individual copies of macromolecules (single particles) is a powerful and general approach to understanding the motions and energy landscapes of macromolecules. Widely-used computational methods already allow the recovery of a few distinct conformations from heterogeneous single-particle samples, but the treatment of complex forms of heterogeneity such as the continuum of possible transitory states and flexible regions remains largely an open problem. In recent years there has been a surge of new approaches for treating the more general problem of continuous heterogeneity. This paper surveys the current state of the art in this area.

Machine Learning for Structure Determination in Single-Particle Cryo-Electron Microscopy: A Systematic Review

Recently, single-particle cryo-electron microscopy (cryo-EM) has become an indispensable method for determining macromolecular structures at high resolution to deeply explore the relevant molecular mechanism. Its recent breakthrough is mainly because of the rapid advances in hardware and image processing algorithms, especially machine learning. As an essential support of single-particle cryo-EM, machine learning has powered many aspects of structure determination and greatly promoted its development. In this article, we provide a systematic review of the applications of machine learning in this field. Our review begins with a brief introduction of single-particle cryo-EM, followed by the specific tasks and challenges of its image processing. Then, focusing on the workflow of structure determination, we describe relevant machine learning algorithms and applications at different steps, including particle picking, 2-D clustering, 3-D reconstruction, and other steps. As different tasks exhibit distinct characteristics, we introduce the evaluation metrics for each task and summarize their dynamics of technology development. Finally, we discuss the open issues and potential trends in this promising field.

Cryo-EM Analyses Permit Visualization of Structural Polymorphism of Biological Macromolecules

The functions of biological macromolecules are often associated with conformational malleability of the structures. This phenomenon of chemically identical molecules with different structures is coined structural polymorphism. Conventionally, structural polymorphism is observed directly by structural determination at the density map level from X-ray crystal diffraction. Although crystallography approach can report the conformation of a macromolecule with the position of each atom accurately defined in it, the exploration of structural polymorphism and interpreting biological function in terms of crystal structures is largely constrained by the crystal packing. An alternative approach to studying the macromolecule of interest in solution is thus desirable. With the advancement of instrumentation and computational methods for image analysis and reconstruction, cryo-electron microscope (cryo-EM) has been transformed to be able to produce “in solution” structures of macromolecules routinely with resolutions comparable to crystallography but without the need of crystals. Since the sample preparation of single-particle cryo-EM allows for all forms co-existing in solution to be simultaneously frozen, the image data contain rich information as to structural polymorphism. The ensemble of structure information can be subsequently disentangled through three-dimensional (3D) classification analyses. In this review, we highlight important examples of protein structural polymorphism in relation to allostery, subunit cooperativity and function plasticity recently revealed by cryo-EM analyses, and review recent developments in 3D classification algorithms including neural network/deep learning approaches that would enable cryo-EM analyese in this regard. Finally, we brief the frontier of cryo-EM structure determination of RNA molecules where resolving the structural polymorphism is at dawn.

A Bayesian approach to extracting free-energy profiles from cryo-electron microscopy experiments

Cryo-electron microscopy (cryo-EM) extracts single-particle density projections of individual biomolecules. Although cryo-EM is widely used for 3D reconstruction, due to its single-particle nature it has the potential to provide information about a biomolecule's conformational variability and underlying free-energy landscape. However, treating cryo-EM as a single-molecule technique is challenging because of the low signal-to-noise ratio (SNR) in individual particles. In this work, we propose the cryo-BIFE method (cryo-EM Bayesian Inference of Free-Energy profiles), which uses a path collective variable to extract free-energy profiles and their uncertainties from cryo-EM images. We test the framework on several synthetic systems where the imaging parameters and conditions were controlled. We found that for realistic cryo-EM environments and relevant biomolecular systems, it is possible to recover the underlying free energy, with the pose accuracy and SNR as crucial determinants. We then use the method to study the conformational transitions of a calcium-activated channel with real cryo-EM particles. Interestingly, we recover not only the most probable conformation (used to generate a high-resolution reconstruction of the calcium-bound state) but also a metastable state that corresponds to the calcium-unbound conformation. As expected for turnover transitions within the same sample, the activation barriers are on the order of [Formula: see text]. We expect our tool for extracting free-energy profiles from cryo-EM images to enable more complete characterization of the thermodynamic ensemble of biomolecules.

Principal component analysis is limited to low-resolution analysis in cryoEM

Principal component analysis (PCA) has been widely proposed to analyze flexibility and heterogeneity in cryo-electron microscopy (cryoEM). In this paper, it is argued that (i) PCA is an excellent technique to describe continuous flexibility at low resolution (but not so much at high resolution) and (ii) PCA components should be analyzed in a concerted manner (and not independently).

HEMNMA-3D: Cryo Electron Tomography Method Based on Normal Mode Analysis to Study Continuous Conformational Variability of Macromolecular Complexes

Cryogenic electron tomography (cryo-ET) allows structural determination of biomolecules in their native environment (in situ). Its potential of providing information on the dynamics of macromolecular complexes in cells is still largely unexploited, due to the challenges of the data analysis. The crowded cell environment and continuous conformational changes of complexes make difficult disentangling the data heterogeneity. We present HEMNMA-3D, which is, to the best of our knowledge, the first method for analyzing cryo electron subtomograms in terms of continuous conformational changes of complexes. HEMNMA-3D uses a combination of elastic and rigid-body 3D-to-3D iterative alignments of a flexible 3D reference (atomic structure or electron microscopy density map) to match the conformation, orientation, and position of the complex in each subtomogram. The elastic matching combines molecular mechanics simulation (Normal Mode Analysis of the 3D reference) and experimental, subtomogram data analysis. The rigid-body alignment includes compensation for the missing wedge, due to the limited tilt angle of cryo-ET. The conformational parameters (amplitudes of normal modes) of the complexes in subtomograms obtained through the alignment are processed to visualize the distribution of conformations in a space of lower dimension (typically, 2D or 3D) referred to as space of conformations. This allows a visually interpretable insight into the dynamics of the complexes, by calculating 3D averages of subtomograms with similar conformations from selected (densest) regions and by recording movies of the 3D reference's displacement along selected trajectories through the densest regions. We describe HEMNMA-3D and show its validation using synthetic datasets. We apply HEMNMA-3D to an experimental dataset describing in situ nucleosome conformational variability. HEMNMA-3D software is available freely (open-source) as part of ContinuousFlex plugin of Scipion V3.0 (http://scipion.i2pc.es).

3D variability analysis: Resolving continuous flexibility and discrete heterogeneity from single particle cryo-EM

Single particle cryo-EM excels in determining static structures of protein molecules, but existing 3D reconstruction methods have been ineffective in modelling flexible proteins. We introduce 3D variability analysis (3DVA), an algorithm that fits a linear subspace model of conformational change to cryo-EM data at high resolution. 3DVA enables the resolution and visualization of detailed molecular motions of both large and small proteins, revealing new biological insight from single particle cryo-EM data. Experimental results demonstrate the ability of 3DVA to resolve multiple flexible motions of α-helices in the sub-50 kDa transmembrane domain of a GPCR complex, bending modes of a sodium ion channel, five types of symmetric and symmetry-breaking flexibility in a proteasome, large motions in a spliceosome complex, and discrete conformational states of a ribosome assembly. 3DVA is implemented in the cryoSPARC software package.

A set of common movements within GPCR-G-protein complexes from variability analysis of cryo-EM datasets

G-protein coupled receptors (GPCRs) are among the most versatile signal transducers in the cell. Once activated, GPCRs sample a large conformational space and couple to G-proteins to initiate distinct signaling pathways. The dynamical behavior of GPCR-G-protein complexes is difficult characterize structurally, and it might hinder obtaining routine high-resolution density maps in single-particle reconstructions. Here, we used variability analysis on the rhodopsin-G_i-Fab16 complex cryo-EM dataset, and the results provide insights into the dynamic nature of the receptor-complex interaction. We compare the outcome of this analysis with recent results obtained on the cannabinoid-G_i- and secretin-G_s-receptor complexes. Despite differences related to the biochemical compositions of the three samples, a set of consensus movements emerges. We anticipate that systematic variability analysis on GPCR-G-protein complexes may provide useful information not only at the biological level, but also for improving the preparation of more stable samples for cryo-EM single-particle analysis.

Computational Methods for Single-Particle Electron Cryomicroscopy

Single-particle electron cryomicroscopy (cryo-EM) is an increasingly popular technique for elucidating the three-dimensional structure of proteins and other biologically significant complexes at near-atomic resolution. It is an imaging method that does not require crystallization and can capture molecules in their native states. In single-particle cryo-EM, the three-dimensional molecular structure needs to be determined from many noisy two-dimensional tomographic projections of individual molecules, whose orientations and positions are unknown. The high level of noise and the unknown pose parameters are two key elements that make reconstruction a challenging computational problem. Even more challenging is the inference of structural variability and flexible motions when the individual molecules being imaged are in different conformational states. This review discusses computational methods for structure determination by single-particle cryo-EM and their guiding principles from statistical inference, machine learning, and signal processing that also play a significant role in many other data science applications.

Hyper-Molecules: on the Representation and Recovery of Dynamical Structures for Applications in Flexible Macro-Molecules in Cryo-EM

Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for obtaining 3-D reconstructions of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. These molecules are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori; one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in reconstructing rigid molecules based on homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge. We introduce the "hyper-molecule" theoretical framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as high-dimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for reconstructing such heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a preliminary prototype implementation, applied to synthetic data.

Need for Cross-Validation of Single Particle Cryo-EM

Cross-validation is used to determine the validity of a model on unseen data by assessing if the model is overfitted to noise. It is widely used in many fields, from artificial intelligence to structural biology in X-ray crystallography and nuclear magnetic resonance. Although there are concerns of map overfitting in cryo-electron microscopy (cryo-EM), cross-validation is rarely used. The problem is that establishing a performance metric of the maps over unseen data (given by 2D-projection images) is difficult due to the low signal-to-noise ratios in the individual particles. Here, I present recent advances for cryo-EM map reconstruction. I highlight that the gold-standard procedure can fail to detect map overfitting in certain cases, showing the necessity of assessing the map quality on unbiased data. Finally, I describe the challenges and advantages of developing a robust cross-validation methodology for cryo-EM.

Cryo-EM reconstruction of continuous heterogeneity by Laplacian spectral volumes

Single-particle electron cryomicroscopy is an essential tool for high-resolution 3D reconstruction of proteins and other biological macromolecules. An important challenge in cryo-EM is the reconstruction of non-rigid molecules with parts that move and deform. Traditional reconstruction methods fail in these cases, resulting in smeared reconstructions of the moving parts. This poses a major obstacle for structural biologists, who need high-resolution reconstructions of entire macromolecules, moving parts included. To address this challenge, we present a new method for the reconstruction of macromolecules exhibiting continuous heterogeneity. The proposed method uses projection images from multiple viewing directions to construct a graph Laplacian through which the manifold of three-dimensional conformations is analyzed. The 3D molecular structures are then expanded in a basis of Laplacian eigenvectors, using a novel generalized tomographic reconstruction algorithm to compute the expansion coefficients. These coefficients, which we name spectral volumes, provide a high-resolution visualization of the molecular dynamics. We provide a theoretical analysis and evaluate the method empirically on several simulated data sets.

Hybrid Electron Microscopy Normal Mode Analysis with Scipion

Hybrid Electron Microscopy Normal Mode Analysis (HEMNMA) method was introduced in 2014. HEMNMA computes normal modes of a reference model (an atomic structure or an electron microscopy map) of a molecular complex and uses this model and its normal modes to analyze single-particle images of the complex to obtain information on its continuous conformational changes, by determining the full distribution of conformational variability from the images. An advantage of HEMNMA is a simultaneous determination of all parameters of each image (particle conformation, orientation, and shift) through their iterative optimization, which allows applications of HEMNMA even when the effects of conformational changes dominate those of orientational changes. HEMNMA was first implemented in Xmipp and was using MATLAB for statistical analysis of obtained conformational distributions and for fitting of underlying trajectories of conformational changes. A HEMNMA implementation independent of MATLAB is now available as part of a plugin of Scipion V2.0 (http://scipion.i2pc.es). This plugin, named ContinuousFlex, can be installed by following the instructions at https://pypi.org/project/scipion-em-continuousflex. In this article, we present this new HEMNMA software, which is user-friendly, totally free, and open-source. STATEMENT FOR A BROADER AUDIENCE: This article presents Hybrid Electron Microscopy Normal Mode Analysis (HEMNMA) software that allows analyzing single-particle images of a complex to obtain information on continuous conformational changes of the complex, by determining the full distribution of conformational variability from the images. The HEMNMA software is user-friendly, totally free, open-source, and available as part of ContinuousFlex plugin (https://pypi.org/project/scipion-em-continuousflex) of Scipion V2.0 (http://scipion.i2pc.es).

Reconstruction of Stochastic 3D Signals With Symmetric Statistics From 2D Projection Images Motivated by Cryo-Electron Microscopy

Cryo-electron microscopy provides 2D projection images of the 3D electron scattering intensity of many instances of the particle under study (e.g., a virus). Both symmetry (rotational point groups) and heterogeneity are important aspects of biological particles and both aspects can be combined by describing the electron scattering intensity of the particle as a stochastic process with a symmetric probability law and, therefore, symmetric moments. A maximum likelihood estimator implemented by an expectation-maximization algorithm is described, which estimates the unknown statistics of the electron scattering intensity stochastic process from the images of instances of the particle. The algorithm is demonstrated on the bacteriophage HK97 and the virus [Formula: see text]. The results are contrasted with the existing algorithms, which assume that each instance of the particle has the symmetry rather than the less restrictive assumption that the probability law has the symmetry.

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.