Identifying conformational states of macromolecules by eigen-analysis of resampled cryo-EM images

Cryo-EM heterogeneity analysis using regularized covariance estimation and kernel regression

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 noncrystalline conditions and enables high-resolution reconstructions. However, analyzing the heterogeneous distributions of conformations from cryo-EM data is challenging. We present RECOVAR, a method for analyzing these distributions based on principal component analysis (PCA) computed using a REgularized COVARiance estimator. RECOVAR is fast, robust, interpretable, expressive, and competitive with state-of-the-art neural network methods on heterogeneous cryo-EM datasets. The regularized covariance method efficiently computes a large number of high-resolution principal components that can encode rich heterogeneous distributions of conformations and does so robustly thanks to an automatic regularization scheme. The reconstruction method based on adaptive kernel regression resolves conformational states to a higher resolution than all other tested methods on extensive independent benchmarks while remaining highly interpretable. Additionally, we exploit favorable properties of the PCA embedding to estimate the conformational density accurately. This density allows for better interpretability of the latent space by identifying stable states and low free-energy motions. Finally, we present a scheme to navigate the high-dimensional latent space by automatically identifying these low free-energy trajectories. We make the code freely available at https://github.com/ma-gilles/recovar.

CryoSTAR: leveraging structural priors and constraints for cryo-EM heterogeneous reconstruction

Resolving conformational heterogeneity in cryogenic electron microscopy datasets remains an important challenge in structural biology. Previous methods have often been restricted to working exclusively on volumetric densities, neglecting the potential of incorporating any preexisting structural knowledge as prior or constraints. Here we present cryoSTAR, which harnesses atomic model information as structural regularization to elucidate such heterogeneity. Our method uniquely outputs both coarse-grained models and density maps, showcasing the molecular conformational changes at different levels. Validated against four diverse experimental datasets, spanning large complexes, a membrane protein and a small single-chain protein, our results consistently demonstrate an efficient and effective solution to conformational heterogeneity with minimal human bias. By integrating atomic model insights with cryogenic electron microscopy data, cryoSTAR represents a meaningful step forward, paving the way for a deeper understanding of dynamic biological processes.

Cryo-EM Heterogeneity Analysis using Regularized Covariance Estimation and Kernel Regression

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 distributions of conformations from cryo-EM data is challenging. We present RECOVAR, a method for analyzing these distributions based on principal component analysis (PCA) computed using a REgularized COVARiance estimator. RECOVAR is fast, robust, interpretable, expressive, and competitive with the state-of-art neural network methods on heterogeneous cryo-EM datasets. The regularized covariance method efficiently computes a large number of high-resolution principal components that can encode rich heterogeneous distributions of conformations and does so robustly thanks to an automatic regularization scheme. The novel reconstruction method based on adaptive kernel regression resolves conformational states to a higher resolution than all other tested methods on extensive independent benchmarks while remaining highly interpretable. Additionally, we exploit favorable properties of the PCA embedding to estimate the conformational density accurately. This density allows for better interpretability of the latent space by identifying stable states and low free-energy motions. Finally, we present a scheme to navigate the high-dimensional latent space by automatically identifying these low free-energy trajectories. We make the code freely available at https://github.com/ma-gilles/recovar.

Deep generative priors for biomolecular 3D heterogeneous reconstruction from cryo-EM projections

Cryo-electron microscopy has become a powerful tool to determine three-dimensional (3D) structures of rigid biological macromolecules from noisy micrographs with single-particle reconstruction. Recently, deep neural networks, e.g., CryoDRGN, have demonstrated conformational and compositional heterogeneity of complexes. However, the lack of ground-truth conformations poses a challenge to assess the performance of heterogeneity analysis methods. In this work, variational autoencoders (VAE) with three types of deep generative priors were learned for latent variable inference and heterogeneous 3D reconstruction via Bayesian inference. More specifically, VAEs with "Variational Mixture of Posteriors" priors (VampPrior-SPR), non-parametric exemplar-based priors (ExemplarPrior-SPR) and priors from latent score-based generative models (LSGM-SPR) were quantitatively compared with CryoDRGN. We built four simulated datasets composed of hypothetical continuous conformation or discrete states of the hERG K + channel. Empirical and quantitative comparisons of inferred latent representations were performed with affine-transformation-based metrics. These models with more informative priors gave better regularized, interpretable factorized latent representations with better conserved pairwise distances, less deformed latent distributions and lower within-cluster variances. They were also tested on experimental datasets to resolve compositional and conformational heterogeneity (50S ribosome assembly, cowpea chlorotic mottle virus, and pre-catalytic spliceosome) with comparable high resolution. Codes and data are available: https://github.com/benjamin3344/DGP-SPR.

Novel Artificial Intelligence-Based Approaches for Ab Initio Structure Determination and Atomic Model Building for Cryo-Electron Microscopy

Single particle cryo-electron microscopy (cryo-EM) has emerged as the prevailing method for near-atomic structure determination, shedding light on the important molecular mechanisms of biological macromolecules. However, the inherent dynamics and structural variability of biological complexes coupled with the large number of experimental images generated by a cryo-EM experiment make data processing nontrivial. In particular, ab initio reconstruction and atomic model building remain major bottlenecks that demand substantial computational resources and manual intervention. Approaches utilizing recent innovations in artificial intelligence (AI) technology, particularly deep learning, have the potential to overcome the limitations that cannot be adequately addressed by traditional image processing approaches. Here, we review newly proposed AI-based methods for ab initio volume generation, heterogeneous 3D reconstruction, and atomic model building. We highlight the advancements made by the implementation of AI methods, as well as discuss remaining limitations and areas for future development.

Conformational heterogeneity and probability distributions from single-particle cryo-electron microscopy

Single-particle cryo-electron microscopy (cryo-EM) is a technique that takes projection images of biomolecules frozen at cryogenic temperatures. A major advantage of this technique is its ability to image single biomolecules in heterogeneous conformations. While this poses a challenge for data analysis, recent algorithmic advances have enabled the recovery of heterogeneous conformations from the noisy imaging data. Here, we review methods for the reconstruction and heterogeneity analysis of cryo-EM images, ranging from linear-transformation-based methods to nonlinear deep generative models. We overview the dimensionality-reduction techniques used in heterogeneous 3D reconstruction methods and specify what information each method can infer from the data. Then, we review the methods that use cryo-EM images to estimate probability distributions over conformations in reduced subspaces or predefined by atomistic simulations. We conclude with the ongoing challenges for the cryo-EM community.

Beyond ManifoldEM: geometric relationships between manifold embeddings of a continuum of 3D molecular structures and their 2D projections

ManifoldEM is an established method of geometric machine learning developed to extract information on conformational motions of molecules from their projections obtained by cryogenic electron microscopy (cryo-EM). In a previous work, in-depth analysis of the properties of manifolds obtained for simulated ground-truth data from molecules exhibiting domain motions has led to improvements of this method, as demonstrated in selected applications of single-particle cryo-EM. In the present work this analysis has been extended to investigate the properties of manifolds constructed by embedding data from synthetic models represented by atomic coordinates in motion, or three-dimensional density maps from biophysical experiments other than single-particle cryo-EM, with extensions to cryo-electron tomography and single-particle imaging with a X-ray free-electron laser. Our theoretical analysis revealed interesting relationships between all these manifolds, which can be exploited in future work.

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.

A method for restoring signals and revealing individual macromolecule states in cryo-ET, REST

Cryo-electron tomography (cryo-ET) is widely used to explore the 3D density of biomacromolecules. However, the heavy noise and missing wedge effect prevent directly visualizing and analyzing the 3D reconstructions. Here, we introduced REST, a deep learning strategy-based method to establish the relationship between low-quality and high-quality density and transfer the knowledge to restore signals in cryo-ET. Test results on the simulated and real cryo-ET datasets show that REST performs well in denoising and compensating the missing wedge information. The application in dynamic nucleosomes, presenting either in the form of individual particles or in the context of cryo-FIB nuclei section, indicates that REST has the capability to reveal different conformations of target macromolecules without subtomogram averaging. Moreover, REST noticeably improves the reliability of particle picking. These advantages enable REST to be a powerful tool for the straightforward interpretation of target macromolecules by visual inspection of the density and of a broad range of other applications in cryo-ET, such as segmentation, particle picking, and subtomogram averaging.

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.

Equilibrium Dynamics of a Biomolecular Complex Analyzed at Single-amino Acid Resolution by Cryo-electron Microscopy

The biological function of macromolecular complexes depends not only on large-scale transitions between conformations, but also on small-scale conformational fluctuations at equilibrium. Information on the equilibrium dynamics of biomolecular complexes could, in principle, be obtained from local resolution (LR) data in cryo-electron microscopy (cryo-EM) maps. However, this possibility had not been validated by comparing, for a same biomolecular complex, LR data with quantitative information on equilibrium dynamics obtained by an established solution technique. In this study we determined the cryo-EM structure of the minute virus of mice (MVM) capsid as a model biomolecular complex. The LR values obtained correlated with crystallographic B factors and with hydrogen/deuterium exchange (HDX) rates obtained by mass spectrometry (HDX-MS), a gold standard for determining equilibrium dynamics in solution. This result validated a LR-based cryo-EM approach to investigate, with high spatial resolution, the equilibrium dynamics of biomolecular complexes. As an application of this approach, we determined the cryo-EM structure of two mutant MVM capsids and compared their equilibrium dynamics with that of the wild-type MVM capsid. The results supported a previously suggested linkage between mechanical stiffening and impaired equilibrium dynamics of a virus particle. Cryo-EM is emerging as a powerful approach for simultaneously acquiring information on the atomic structure and local equilibrium dynamics of biomolecular complexes.

Fast principal component analysis for cryo-electron microscopy images

Principal component analysis (PCA) plays an important role in the analysis of cryo-electron microscopy (cryo-EM) images for various tasks such as classification, denoising, compression, and ab initio modeling. We introduce a fast method for estimating a compressed representation of the 2-D covariance matrix of noisy cryo-EM projection images affected by radial point spread functions that enables fast PCA computation. Our method is based on a new algorithm for expanding images in the Fourier-Bessel basis (the harmonics on the disk), which provides a convenient way to handle the effect of the contrast transfer functions. For N images of size L × L, our method has time complexity O(NL3 + L4) and space complexity O(NL2 + L3). In contrast to previous work, these complexities are independent of the number of different contrast transfer functions of the images. We demonstrate our approach on synthetic and experimental data and show acceleration by factors of up to two orders of magnitude.

Exploring the Structural Variability of Dynamic Biological Complexes by Single-Particle Cryo-Electron Microscopy

Biological macromolecules and assemblies precisely rearrange their atomic 3D structures to execute cellular functions. Understanding the mechanisms by which these molecular machines operate requires insight into the ensemble of structural states they occupy during the functional cycle. Single-particle cryo-electron microscopy (cryo-EM) has become the preferred method to provide near-atomic resolution, structural information about dynamic biological macromolecules elusive to other structure determination methods. Recent advances in cryo-EM methodology have allowed structural biologists not only to probe the structural intermediates of biochemical reactions, but also to resolve different compositional and conformational states present within the same dataset. This article reviews newly developed sample preparation and single-particle analysis (SPA) techniques for high-resolution structure determination of intrinsically dynamic and heterogeneous samples, shedding light upon the intricate mechanisms employed by molecular machines and helping to guide drug discovery efforts.

New insights into the structure and function of class B1 GPCRs

G protein-coupled receptors (GPCRs) are the largest family of cell surface receptors. Class B1 GPCRs constitute a subfamily of 15 receptors that characteristically contain large extracellular domains (ECDs) and respond to long polypeptide hormones. Class B1 GPCRs are critical regulators of homeostasis, and as such, many are important drug targets. While most transmembrane proteins, including GPCRs, are recalcitrant to crystallization, recent advances in electron cryo-microscopy (cryo-EM) have facilitated a rapid expansion of the structural understanding of membrane proteins. As a testament to this success, structures for all the class B1 receptors bound to G proteins have been determined by cryo-EM in the past five years. Further advances in cryo-EM have uncovered dynamics of these receptors, ligands, and signalling partners. Here, we examine the recent structural underpinnings of the class B1 GPCRs with an emphasis on structure-function relationships.

Artificial Intelligence in Cryo-Electron Microscopy

Cryo-electron microscopy (cryo-EM) has become an unrivaled tool for determining the structure of macromolecular complexes. The biological function of macromolecular complexes is inextricably tied to the flexibility of these complexes. Single particle cryo-EM can reveal the conformational heterogeneity of a biochemically pure sample, leading to well-founded mechanistic hypotheses about the roles these complexes play in biology. However, the processing of increasingly large, complex datasets using traditional data processing strategies is exceedingly expensive in both user time and computational resources. Current innovations in data processing capitalize on artificial intelligence (AI) to improve the efficiency of data analysis and validation. Here, we review new tools that use AI to automate the data analysis steps of particle picking, 3D map reconstruction, and local resolution determination. We discuss how the application of AI moves the field forward, and what obstacles remain. We also introduce potential future applications of AI to use cryo-EM in understanding protein communities in cells.

Visualizing Conformational Space of Functional Biomolecular Complexes by Deep Manifold Learning

The cellular functions are executed by biological macromolecular complexes in nonequilibrium dynamic processes, which exhibit a vast diversity of conformational states. Solving the conformational continuum of important biomolecular complexes at the atomic level is essential to understanding their functional mechanisms and guiding structure-based drug discovery. Here, we introduce a deep manifold learning framework, named AlphaCryo4D, which enables atomic-level cryogenic electron microscopy (cryo-EM) reconstructions that approximately visualize the conformational space of biomolecular complexes of interest. AlphaCryo4D integrates 3D deep residual learning with manifold embedding of pseudo-energy landscapes, which simultaneously improves 3D classification accuracy and reconstruction resolution via an energy-based particle-voting algorithm. In blind assessments using simulated heterogeneous datasets, AlphaCryo4D achieved 3D classification accuracy three times those of alternative methods and reconstructed continuous conformational changes of a 130-kDa protein at sub-3 Å resolution. By applying this approach to analyze several experimental datasets of the proteasome, ribosome and spliceosome, we demonstrate its potential generality in exploring hidden conformational space or transient states of macromolecular complexes that remain hitherto invisible. Integration of this approach with time-resolved cryo-EM further allows visualization of conformational continuum in a nonequilibrium regime at the atomic level, thus potentially enabling therapeutic discovery against highly dynamic biomolecular targets.

Data-driven determination of number of discrete conformations in single-particle cryo-EM

BACKGROUND AND OBJECTIVE

One of the strengths of single-particle cryo-EM compared to other structural determination techniques is its ability to image heterogeneous samples containing multiple molecular species, different oligomeric states or distinct conformations. This is achieved using routines for in-silico 3D classification that are now well established in the field and have successfully been used to characterize the structural heterogeneity of important biomolecules. These techniques, however, rely on expert-user knowledge and trial-and-error experimentation to determine the correct number of conformations, making it a labor intensive, subjective, and difficult to reproduce procedure.

METHODS

We propose an approach to address the problem of automatically determining the number of discrete conformations present in heterogeneous single-particle cryo-EM datasets. We do this by systematically evaluating all possible partitions of the data and selecting the result that maximizes the average variance of similarities measured between particle images and the corresponding 3D reconstructions.

RESULTS

Using this strategy, we successfully analyzed datasets of heterogeneous protein complexes, including: 1) in-silico mixtures obtained by combining closely related antibody-bound HIV-1 Env trimers and other important membrane channels, and 2) naturally occurring mixtures from diverse and dynamic protein complexes representing varying degrees of structural heterogeneity and conformational plasticity.

CONCLUSIONS

The availability of unsupervised strategies for 3D classification combined with existing approaches for fully automatic pre-processing and 3D refinement, represents an important step towards converting single-particle cryo-EM into a high-throughput technique.

Recovery of Conformational Continuum From Single-Particle Cryo-EM Images: Optimization of ManifoldEM Informed by Ground Truth

This work is based on the manifold-embedding approach to study biological molecules exhibiting continuous conformational changes. Previous work established a method-now termed ManifoldEM-capable of reconstructing 3D movies and accompanying free-energy landscapes from single-particle cryo-EM images of macromolecules exercising multiple conformational degrees of freedom. While ManifoldEM has proven its viability in several experimental studies, critical limitations and uncertainties have been found throughout its extended development and use. Guided by insights from studies with cryo-EM ground-truth data, simulated from atomic structures undergoing conformational changes, we have built a novel framework, ESPER, able to retrieve the free-energy landscape and respective 3D Coulomb potential maps for all states simulated. As shown by a direct comparison of ground truth vs. recovered maps, and analysis of experimental data from the 80S ribosome and ryanodine receptor, ESPER offers substantial improvements relative to the previous work.

Hierarchical autoclassification of cryo-EM samples and macromolecular energy landscape determination

BACKGROUND AND OBJECTIVE

Cryo-electron microscopy using single particle analysis is a powerful technique for obtaining 3D reconstructions of macromolecules in near native conditions. One of its major advances is its capacity to reveal conformations of dynamic molecular complexes. Most popular and successful current approaches to analyzing heterogeneous complexes are founded on Bayesian inference. However, these 3D classification methods require the tuning of specific parameters by the user and the use of complicated 3D re-classification procedures for samples affected by extensive heterogeneity. Thus, the success of these approaches highly depends on the user experience. We introduce a robust approach to identify many different conformations presented in a cryo-EM dataset based on Bayesian inference through Relion classification methods that does not require tuning of parameters and reclassification strategies.

METHODS

The algorithm allows both 2D and 3D classification and is based on a hierarchical clustering approach that runs automatically without requiring typical inputs, such as the number of conformations present in the dataset or the required classification iterations. This approach is applied to robustly determine the energy landscapes of macromolecules.

RESULTS

We tested the performance of the methods proposed here using four different datasets, comprising structurally homogeneous and highly heterogeneous cases. In all cases, the approach provided excellent results. The routines are publicly available as part of the CryoMethods plugin included in the Scipion package.

CONCLUSIONS

Our results show that the proposed method can be used to align and classify homogeneous and heterogeneous datasets without requiring previous alignment information or any prior knowledge about the number of co-existing conformations. The approach can be used for both 2D and 3D autoclassification and only requires an initial volume. In addition, the approach is robust to the "attractor" problem providing many different conformations/views for samples affected by extensive heterogeneity. The obtained 3D classes can render high resolution 3D structures, while the obtained energy landscapes can be used to determine structural trajectories.

Catalytic trajectory of a dimeric nonribosomal peptide synthetase subunit with an inserted epimerase domain

Nonribosomal peptide synthetases (NRPSs) are modular assembly-line megaenzymes that synthesize diverse metabolites with wide-ranging biological activities. The structural dynamics of synthetic elongation has remained unclear. Here, we present cryo-EM structures of PchE, an NRPS elongation module, in distinct conformations. The domain organization reveals a unique “H”-shaped head-to-tail dimeric architecture. The capture of both aryl and peptidyl carrier protein-tethered substrates and intermediates inside the heterocyclization domain and l-cysteinyl adenylate in the adenylation domain illustrates the catalytic and recognition residues. The multilevel structural transitions guided by the adenylation C-terminal subdomain in combination with the inserted epimerase and the conformational changes of the heterocyclization tunnel are controlled by two residues. Moreover, we visualized the direct structural dynamics of the full catalytic cycle from thiolation to epimerization. This study establishes the catalytic trajectory of PchE and sheds light on the rational re-engineering of domain-inserted dimeric NRPSs for the production of novel pharmaceutical agents.

The catalytic domains in nonribosomal peptide synthetases (NRPSs) are responsible for a choreography of events that elongates substrates into natural products. Here, the authors present cryo-EM structures of a siderophore-producing dimeric NRPS elongation module in multiple distinct conformations, which provides insight into the mechanisms of catalytic trajectory.

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.

TomoFlow: Analysis of Continuous Conformational Variability of Macromolecules in Cryogenic Subtomograms based on 3D Dense Optical Flow

Cryogenic Electron Tomography (cryo-ET) allows structural and dynamics studies of macromolecules in situ. Averaging different copies of imaged macromolecules is commonly used to obtain their structure at higher resolution and discrete classification to analyze their dynamics. Instrumental and data processing developments are progressively equipping cryo-ET studies with the ability to escape the trap of classification into a complete continuous conformational variability analysis. In this work, we propose TomoFlow, a method for analyzing macromolecular continuous conformational variability in cryo-ET subtomograms based on a three-dimensional dense optical flow (OF) approach. The resultant lower-dimensional conformational space allows generating movies of macromolecular motion and obtaining subtomogram averages by grouping conformationally similar subtomograms. The animations and the subtomogram group averages reveal accurate trajectories of macromolecular motion based on a novel mathematical model that makes use of OF properties. This paper describes TomoFlow with tests on simulated datasets generated using different techniques, namely Normal Mode Analysis and Molecular Dynamics Simulation. It also shows an application of TomoFlow on a dataset of nucleosomes in situ, which provided promising results coherent with previous findings using the same dataset but without imposing any prior knowledge on the analysis of the conformational variability. The method is discussed with its potential uses and limitations.

Deep learning-based mixed-dimensional Gaussian mixture model for characterizing variability in cryo-EM

Structural flexibility and/or dynamic interactions with other molecules is a critical aspect of protein function. CryoEM provides direct visualization of individual macromolecules sampling different conformational and compositional states. While numerous methods are available for computational classification of discrete states, characterization of continuous conformational changes or large numbers of discrete state without human supervision remains challenging. Here we present e2gmm, a machine learning algorithm to determine a conformational landscape for proteins or complexes using a 3-D Gaussian mixture model mapped onto 2-D particle images in known orientations. Using a deep neural network architecture, e2gmm can automatically resolve the structural heterogeneity within the protein complex and map particles onto a small latent space describing conformational and compositional changes. This system presents a more intuitive and flexible representation than other manifold methods currently in use. We demonstrate this method on both simulated data as well as three biological systems, to explore compositional and conformational changes at a range of scales. The software is distributed as part of EMAN2.

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.

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.

CryoDRGN: reconstruction of heterogeneous cryo-EM structures using neural networks

Cryo-electron microscopy (cryo-EM) single-particle analysis has proven powerful in determining the structures of rigid macromolecules. However, many imaged protein complexes exhibit conformational and compositional heterogeneity that poses a major challenge to existing three-dimensional reconstruction methods. Here, we present cryoDRGN, an algorithm that leverages the representation power of deep neural networks to directly reconstruct continuous distributions of 3D density maps and map per-particle heterogeneity of single-particle cryo-EM datasets. Using cryoDRGN, we uncovered residual heterogeneity in high-resolution datasets of the 80S ribosome and the RAG complex, revealed a new structural state of the assembling 50S ribosome, and visualized large-scale continuous motions of a spliceosome complex. CryoDRGN contains interactive tools to visualize a dataset's distribution of per-particle variability, generate density maps for exploratory analysis, extract particle subsets for use with other tools and generate trajectories to visualize molecular motions. CryoDRGN is open-source software freely available at http://cryodrgn.csail.mit.edu .

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.

Structural Heterogeneities of the Ribosome: New Frontiers and Opportunities for Cryo-EM

The extent of ribosomal heterogeneity has caught increasing interest over the past few years, as recent studies have highlighted the presence of structural variations of the ribosome. More precisely, the heterogeneity of the ribosome covers multiple scales, including the dynamical aspects of ribosomal motion at the single particle level, specialization at the cellular and subcellular scale, or evolutionary differences across species. Upon solving the ribosome atomic structure at medium to high resolution, cryogenic electron microscopy (cryo-EM) has enabled investigating all these forms of heterogeneity. In this review, we present some recent advances in quantifying ribosome heterogeneity, with a focus on the conformational and evolutionary variations of the ribosome and their functional implications. These efforts highlight the need for new computational methods and comparative tools, to comprehensively model the continuous conformational transition pathways of the ribosome, as well as its evolution. While developing these methods presents some important challenges, it also provides an opportunity to extend our interpretation and usage of cryo-EM data, which would more generally benefit the study of molecular dynamics and evolution of proteins and other complexes.

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.

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.

Modelling structures in cryo-EM maps

Recent advances in structure determination of sub-cellular structures using cryo-electron microscopy and tomography have enabled us to understand their architecture in a more detailed manner and gain insight into their function. The choice of approach to use for atomic model building, fitting, refinement and validation in the 3D map resulting from these experiments depends primarily on the resolution of the map and the prior information on the corresponding model. Here, we survey some of such methods and approaches and highlight their uses in specific recent examples.

Mechanics and pharmacology of substrate selection and transport by eukaryotic ABC exporters

Much structural information has been amassed on ATP-binding cassette (ABC) transporters, including hundreds of structures of isolated domains and an increasing array of full-length transporters. The structures capture different steps in the transport cycle and have aided in the design and interpretation of computational simulations and biophysics experiments. These data provide a maturing, although still incomplete, elucidation of the protein dynamics and mechanisms of substrate selection and transit through the transporters. We present an updated view of the classical alternating-access mechanism as it applies to eukaryotic ABC transporters, focusing on type I exporters. Our model helps frame the progress in, and remaining questions about, transporter energetics, how substrates are selected and how ATP is consumed to perform work at the molecular scale. Many human ABC transporters are associated with disease; we highlight progress in understanding their pharmacology through the lens of structural biology and describe how this knowledge suggests approaches to pharmacologically targeting these transporters.

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.

Structural Variability from Noisy Tomographic Projections

In cryo-electron microscopy, the three-dimensional (3D) electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy two-dimensional images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For n images of size N-by-N pixels, we propose an algorithm for calculating this covariance estimator with computational complexity O ( n N 4 + κ N 6 log N ) , where the condition number κ is empirically in the range 10-200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in six dimensions. This is then solved by the conjugate gradient method with an appropriate circulant preconditioner. The result is the first computationally efficient algorithm for consistent estimation of the 3D covariance from noisy projections. It also compares favorably in runtime with respect to previously proposed nonconsistent estimators. Motivated by the recent success of eigenvalue shrinkage procedures for high-dimensional covariance matrix estimation, we incorporate a shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We evaluate our methods on simulated datasets and achieve classification results comparable to state-of-the-art methods in shorter running time. We also present results on clustering volumes in an experimental dataset, illustrating the power of the proposed algorithm for practical determination of structural variability.

Allosteric effects in bacteriophage HK97 procapsids revealed directly from covariance analysis of cryo EM data

The information content of cryo EM data sets exceeds that of the electron scattering potential (cryo EM) density initially derived for structure determination. Previously we demonstrated the power of data variance analysis for characterizing regions of cryo EM density that displayed functionally important variance anomalies associated with maturation cleavage events in Nudaurelia Omega Capensis Virus and the presence or absence of a maturation protease in bacteriophage HK97 procapsids. Here we extend the analysis in two ways. First, instead of imposing icosahedral symmetry on every particle in the data set during the variance analysis, we only assume that the data set as a whole has icosahedral symmetry. This change removes artifacts of high variance along icosahedral symmetry axes, but retains all of the features previously reported in the HK97 data set. Second we present a covariance analysis that reveals correlations in structural dynamics (variance) between the interior of the HK97 procapsid with the protease and regions of the exterior (not seen in the absence of the protease). The latter analysis corresponds well with hydrogen deuterium exchange studies previously published that reveal the same correlation.

Particle segmentation algorithm for flexible single particle reconstruction

As single particle cryo-electron microscopy has evolved to a new era of atomic resolution, sample heterogeneity still imposes a major limit to the resolution of many macromolecular complexes, especially those with continuous conformational flexibility. Here, we describe a particle segmentation algorithm towards solving structures of molecules composed of several parts that are relatively flexible with each other. In this algorithm, the different parts of a target molecule are segmented from raw images according to their alignment information obtained from a preliminary 3D reconstruction and are subjected to single particle processing in an iterative manner. This algorithm was tested on both simulated and experimental data and showed improvement of 3D reconstruction resolution of each segmented part of the molecule than that of the entire molecule.

Cryo-electron microscopy and X-ray crystallography: complementary approaches to structural biology and drug discovery

The invention of the electron microscope has greatly enhanced the view scientists have of small structural details. Since its implementation, this technology has undergone considerable evolution and the resolution that can be obtained for biological objects has been extended. In addition, the latest generation of cryo-electron microscopes equipped with direct electron detectors and software for the automated collection of images, in combination with the use of advanced image-analysis methods, has dramatically improved the performance of this technique in terms of resolution. While calculating a sub-10 Å resolution structure was an accomplishment less than a decade ago, it is now common to generate structures at sub-5 Å resolution and even better. It is becoming possible to relatively quickly obtain high-resolution structures of biological molecules, in particular large ones (>500 kDa) which, in some cases, have resisted more conventional methods such as X-ray crystallography or nuclear magnetic resonance (NMR). Such newly resolved structures may, for the first time, shed light on the precise mechanisms that are essential for cellular physiological processes. The ability to attain atomic resolution may support the development of new drugs that target these proteins, allowing medicinal chemists to understand the intimacy of the relationship between their molecules and targets. In addition, recent developments in cryo-electron microscopy combined with image analysis can provide unique information on the conformational variability of macromolecular complexes. Conformational flexibility of macromolecular complexes can be investigated using cryo-electron microscopy and multiconformation reconstruction methods. However, the biochemical quality of the sample remains the major bottleneck to routine cryo-electron microscopy-based determination of structures at very high resolution.

Structural Study of Heterogeneous Biological Samples by Cryoelectron Microscopy and Image Processing

In living organisms, biological macromolecules are intrinsically flexible and naturally exist in multiple conformations. Modern electron microscopy, especially at liquid nitrogen temperatures (cryo-EM), is able to visualise biocomplexes in nearly native conditions and in multiple conformational states. The advances made during the last decade in electronic technology and software development have led to the revelation of structural variations in complexes and also improved the resolution of EM structures. Nowadays, structural studies based on single particle analysis (SPA) suggests several approaches for the separation of different conformational states and therefore disclosure of the mechanisms for functioning of complexes. The task of resolving different states requires the examination of large datasets, sophisticated programs, and significant computing power. Some methods are based on analysis of two-dimensional images, while others are based on three-dimensional studies. In this review, we describe the basic principles implemented in the various techniques that are currently used in the analysis of structural conformations and provide some examples of successful applications of these methods in structural studies of biologically significant complexes.

Computational methods for analyzing conformational variability of macromolecular complexes from cryo-electron microscopy images

Thanks to latest technical advances in cryo-electron microscopy (cryo-EM), structures of macromolecular complexes (viruses, ribosomes, etc.) are now often obtained at near-atomic resolution. Also, studies of conformational changes of complexes, in connection with their function, are gaining ground. Conformational variability analysis is usually done by classifying images in a number of discrete classes supposedly representing all conformational states present in the specimen. However, discrete classes cannot be meaningfully defined when the conformational change is continuous (the specimen contains a continuum of states instead of a few discrete states). For such cases, first image analysis methods that explicitly consider continuous conformational changes were recently developed. The latest developments in cryo-EM image analysis methods for conformational variability analysis are the focus of this review.

Unsupervised Cryo-EM Data Clustering through Adaptively Constrained K-Means Algorithm

In single-particle cryo-electron microscopy (cryo-EM), K-means clustering algorithm is widely used in unsupervised 2D classification of projection images of biological macromolecules. 3D ab initio reconstruction requires accurate unsupervised classification in order to separate molecular projections of distinct orientations. Due to background noise in single-particle images and uncertainty of molecular orientations, traditional K-means clustering algorithm may classify images into wrong classes and produce classes with a large variation in membership. Overcoming these limitations requires further development on clustering algorithms for cryo-EM data analysis. We propose a novel unsupervised data clustering method building upon the traditional K-means algorithm. By introducing an adaptive constraint term in the objective function, our algorithm not only avoids a large variation in class sizes but also produces more accurate data clustering. Applications of this approach to both simulated and experimental cryo-EM data demonstrate that our algorithm is a significantly improved alterative to the traditional K-means algorithm in single-particle cryo-EM analysis.

Cryo-electron Microscopy Analysis of Structurally Heterogeneous Macromolecular Complexes

Cryo-electron microscopy (cryo-EM) has for a long time been a technique of choice for determining structure of large and flexible macromolecular complexes that were difficult to study by other experimental techniques such as X-ray crystallography or nuclear magnetic resonance. However, a fast development of instruments and software for cryo-EM in the last decade has allowed that a large range of complexes can be studied by cryo-EM, and that their structures can be obtained at near-atomic resolution, including the structures of small complexes (e.g., membrane proteins) whose size was earlier an obstacle to cryo-EM. Image analysis to identify multiple coexisting structures in the same specimen (multiconformation reconstruction) is now routinely done both to solve structures at near-atomic resolution and to study conformational dynamics. Methods for multiconformation reconstruction and latest examples of their applications are the focus of this review.

While the revolution will not be crystallized, biochemistry reigns supreme

Single-particle cryo-electron microscopy (EM) is currently gaining attention for the ability to calculate structures that reach sub-5 Å resolutions; however, the technique is more than just an alternative approach to X-ray crystallography. Molecular machines work via dynamic conformational changes, making structural flexibility the hallmark of function. While the dynamic regions in molecules are essential, they are also the most challenging to structurally characterize. Single-particle EM has the distinct advantage of being able to directly visualize purified molecules without the formation of ordered arrays of molecules locked into identical conformations. Additionally, structures determined using single-particle EM can span resolution ranges from very low- to atomic-levels (>30-1.8 Å), sometimes even in the same structure. The ability to accommodate various resolutions gives single-particle EM the unique capacity to structurally characterize dynamic regions of biological molecules, thereby contributing essential structural information needed for the development of molecular models that explain function. Further, many important molecular machines are intrinsically dynamic and compositionally heterogeneous. Structures of these complexes may never reach sub-5 Å resolutions due to this flexibility required for function. Thus, the biochemical quality of the sample, as well as, the calculation and interpretation of low- to mid-resolution cryo-EM structures (30-8 Å) remains critical for generating insights into the architecture of many challenging biological samples that cannot be visualized using alternative techniques.

Extraction of Protein Conformational Modes from Distance Distributions Using Structurally Imputed Bayesian Data Augmentation

Protein conformational changes are known to play important roles in assorted biochemical and biological processes. Driven by thermal motions of surrounding solvent molecules, such a structural remodeling often occurs stochastically. Yet, regardless of how random the conformational reconfiguration may appear, it could in principle be described by a linear combination of a set of orthogonal modes which, in turn, are contained in the intramolecular distance distributions. The central challenge is how to obtain the distribution. This contribution proposes a Bayesian data-augmentation scheme to extract the predominant modes from only few distance distributions, be they from computational sampling or directly from experiments such as single-molecule Förster-type resonance energy transfer (smFRET). The inference of the complete protein structure from insufficient data was recognized as isomorphic to the missing-data problem in Bayesian statistical learning. Using smFRET data as an example, the missing coordinates were deduced, given protein structural constraints and multiple but limited number of smFRET distances; the Boltzmann weighing of each inferred protein structure was then evaluated using computational modeling to numerically construct the posterior density for the global protein conformation. The conformational modes were then determined from the iteratively converged overall conformational distribution using principal component analysis. Two examples were presented to illustrate these basic ideas as well as their practical implementation. The scheme described herein was based on the theory behind the powerful Tanner-Wang algorithm that guarantees convergence to the true posterior density. However, instead of assuming a mathematical model to calculate the likelihood as in conventional statistical inference, here the protein structure was treated as a statistical parameter and was imputed from the numerical likelihood function based on structural information, a probability model-free method. The framework put forth here is anticipated to be generally applicable, offering a new way to articulate protein conformational changes in a quantifiable manner.