Classification of cryo-electron sub-tomograms using constrained correlation

Cryo-electron tomography (CET) is currently the only three-dimensional imaging technique capable of visualizing macromolecules in their cellular context at close-to-native conditions with a resolution in the nanometer range. An important component for the analysis of the data is their classification, which should discriminate among various macromolecules, conformational changes and interaction partners. Missing structure factors, typically in a wedge-shaped region in Fourier space if single-axis tilting is performed, hamper classification of cryo-electron tomographic data. Here, we describe a classification method for three-dimensional (3D) sub-tomograms extracted from cryo-electron tomograms, which takes the missing wedge into account and provides reliable results. The similarity of the individually aligned sub-tomograms is scored by constrained correlation. Subsequently, they are clustered based on their pairwise correlation values. In order to demonstrate the feasibility of this approach, we apply the proposed method to simulated tomographic data of the chaperone thermosome in different conformations. By comparison of the principal components of the resulting matrix we show that the proposed metric is significantly less prone to the orientation of the missing wedge compared to the unconstrained correlation. Moreover, we apply our classification method to an experimental dataset of GroEL with and without GroES, where we achieve a distinct discrimination between the putative GroEL and GroEL/GroES complexes.     

Classification and 3D averaging with missing wedge correction in biological electron tomography

Strategies for the determination of 3D structures of biological macromolecules using electron crystallography and single-particle electron microscopy utilize powerful tools for the averaging of information obtained from 2D projection images of structurally homogeneous specimens. In contrast, electron tomographic approaches have often been used to study the 3D structures of heterogeneous, one-of-a-kind objects such as whole cells where image-averaging strategies are not applicable. Complex entities such as cells and viruses, nevertheless, contain multiple copies of numerous macromolecules that can individually be subjected to 3D averaging. Here we present a complete framework for alignment, classification, and averaging of volumes derived by electron tomography that is computationally efficient and effectively accounts for the missing wedge that is inherent to limited-angle electron tomography. Modeling the missing data as a multiplying mask in reciprocal space we show that the effect of the missing wedge can be accounted for seamlessly in all alignment and classification operations. We solve the alignment problem using the convolution theorem in harmonic analysis, thus eliminating the need for approaches that require exhaustive angular search, and adopt an iterative approach to alignment and classification that does not require the use of external references. We demonstrate that our method can be successfully applied for 3D classification and averaging of phantom volumes as well as experimentally obtained tomograms of GroEL where the outcomes of the analysis can be quantitatively compared against the expected results.     

Iterative reconstruction of cryo-electron tomograms using nonuniform fast Fourier transforms

Algorithms for three-dimensional (3D) reconstruction of objects based on their projections are essential in various biological and medical imaging modalities. In cryo-electron tomography (CET) a major challenge for reconstruction is the limited range of projection angles, which manifests itself as a “missing wedge” of data in Fourier space making the reconstruction problem ill-posed. Here, we apply an iterative reconstruction method that makes use of nonuniform fast Fourier transform (NUFFT) to the reconstruction of cryo-electron tomograms. According to several measures the reconstructions are superior to those obtained using conventional methods, most notably weighted backprojection. Most importantly, we show that it is possible to fill in partially the unsampled region in Fourier space with meaningful information without making assumptions about the data or applying prior knowledge. As a consequence, particles of known structure can be localized with higher confidence in cryotomograms and subtomogram averaging yields higher resolution densities.     

Clustering and variance maps for cryo-electron tomography using wedge-masked differences

Cryo-electron tomography provides 3D imaging of frozen hydrated biological samples with nanometer resolution. Reconstructed volumes suffer from low signal-to-noise-ratio (SNR)(1) and artifacts caused by systematically missing tomographic data. Both problems can be overcome by combining multiple subvolumes with varying orientations, assuming they contain identical structures. Clustering (unsupervised classification) is required to ensure or verify population homogeneity, but this process is complicated by the problems of poor SNR and missing data, the factors that led to consideration of multiple subvolumes in the first place. Here, we describe a new approach to clustering and variance mapping in the face of these difficulties. The combined subvolume is taken as an estimate of the true subvolume, and the effect of missing data is computed for individual subvolumes. Clustering and variance mapping then proceed based on differences between expected and observed subvolumes. We show that this new method is faster and more accurate than two current, widely used techniques.     

Computational Methods Toward Unbiased Pattern Mining and Structure Determination in Cryo-Electron Tomography Data

Cryo-electron tomography can uniquely probe the native cellular environment for macromolecular structures. Tomograms feature complex data with densities of diverse, densely crowded macromolecular complexes, low signal-to-noise, and artifacts such as the missing wedge effect. Post-processing of this data generally involves isolating regions or particles of interest from tomograms, organizing them into related groups, and rendering final structures through subtomogram averaging. Template-matching and reference-based structure determination are popular analysis methods but are vulnerable to biases and can often require significant user input. Most importantly, these approaches cannot identify novel complexes that reside within the imaged cellular environment. To reliably extract and resolve structures of interest, efficient and unbiased approaches are therefore of great value. This review highlights notable computational software and discusses how they contribute to making automated structural pattern discovery a possibility. Perspectives emphasizing the importance of features for user-friendliness and accessibility are also presented.     

Computational exploration of structural information from cryo-electron tomograms

Cryo-electron tomography aims to act as an interface between in vivo cell imaging and techniques achieving atomic resolution. This attempt to bridge the resolution gap is facilitated by recent software and hardware advances. Information provided by atomically resolved macromolecules and molecular interaction data need to be put into a common framework in order to create a hybrid multidimensional cellular image. A major partner in this enterprise is the development of regularization and pattern recognition techniques, which try to identify macromolecular complexes as a function of their structural signature in cryo-electron tomograms of living cells.     

High-throughput subtomogram alignment and classification by Fourier space constrained fast volumetric matching

Cryo-electron tomography allows the visualization of macromolecular complexes in their cellular environments in close-to-live conditions. The nominal resolution of subtomograms can be significantly increased when individual subtomograms of the same kind are aligned and averaged. A vital step for such a procedure are algorithms that speedup subtomogram alignment and improve its accuracy to allow reference-free subtomogram classifications. Such methods will facilitate automation of tomography analysis and overall high throughput in the data processing. Building on previous work, here we propose a fast rotational alignment method that uses the Fourier equivalent form of a popular constrained correlation measure that considers missing wedge corrections and density variances in the subtomograms. The fast rotational search is based on 3D volumetric matching, which improves the rotational alignment accuracy in particular for highly distorted subtomograms with low SNR and tilt angle ranges in comparison to fast rotational matching of projected 2D spherical images. We further integrate our fast rotational alignment method in a reference-free iterative subtomogram classification scheme, and propose a local feature enhancement strategy in the classification process. As a proof of principle, we can demonstrate that the automatic method can successfully classify a large number of experimental subtomograms without the need of a reference structure.     

A 3D image filter for parameter-free segmentation of macromolecular structures from electron tomograms

3D image reconstruction of large cellular volumes by electron tomography (ET) at high (≤ 5 nm) resolution can now routinely resolve organellar and compartmental membrane structures, protein coats, cytoskeletal filaments, and macromolecules. However, current image analysis methods for identifying in situ macromolecular structures within the crowded 3D ultrastructural landscape of a cell remain labor-intensive, time-consuming, and prone to user-bias and/or error. This paper demonstrates the development and application of a parameter-free, 3D implementation of the bilateral edge-detection (BLE) algorithm for the rapid and accurate segmentation of cellular tomograms. The performance of the 3D BLE filter has been tested on a range of synthetic and real biological data sets and validated against current leading filters-the pseudo 3D recursive and Canny filters. The performance of the 3D BLE filter was found to be comparable to or better than that of both the 3D recursive and Canny filters while offering the significant advantage that it requires no parameter input or optimisation. Edge widths as little as 2 pixels are reproducibly detected with signal intensity and grey scale values as low as 0.72% above the mean of the background noise. The 3D BLE thus provides an efficient method for the automated segmentation of complex cellular structures across multiple scales for further downstream processing, such as cellular annotation and sub-tomogram averaging, and provides a valuable tool for the accurate and high-throughput identification and annotation of 3D structural complexity at the subcellular level, as well as for mapping the spatial and temporal rearrangement of macromolecular assemblies in situ within cellular tomograms.     

Parakeet: a digital twin software pipeline to assess the impact of experimental parameters on tomographic reconstructions for cryo-electron tomography

In cryo-electron tomography (cryo-ET) of biological samples, the quality of tomographic reconstructions can vary depending on the transmission electron microscope (TEM) instrument and data acquisition parameters. In this paper, we present Parakeet, a ‘digital twin’ software pipeline for the assessment of the impact of various TEM experiment parameters on the quality of three-dimensional tomographic reconstructions. The Parakeet digital twin is a digital model that can be used to optimize the performance and utilization of a physical instrument to enable in silico optimization of sample geometries, data acquisition schemes and instrument parameters. The digital twin performs virtual sample generation, TEM image simulation, and tilt series reconstruction and analysis within a convenient software framework. As well as being able to produce physically realistic simulated cryo-ET datasets to aid the development of tomographic reconstruction and subtomogram averaging programs, Parakeet aims to enable convenient assessment of the effects of different microscope parameters and data acquisition parameters on reconstruction quality. To illustrate the use of the software, we present the example of a quantitative analysis of missing wedge artefacts on simulated planar and cylindrical biological samples and discuss how data collection parameters can be modified for cylindrical samples where a full 180° tilt range might be measured.     

Montage electron tomography of vitrified specimens

Cryo-electron tomography provides detailed views of macromolecules in situ. However, imaging a large field of view to provide more cellular context requires reducing magnification during data collection, which in turn restricts the resolution. To circumvent this trade-off between field of view and resolution, we have developed a montage data collection scheme that uniformly distributes the dose throughout the specimen. In this approach, sets of slightly overlapping circular tiles are collected at high magnification and stitched to form a composite projection image at each tilt angle. These montage tilt-series are then reconstructed into massive tomograms with a small pixel size but a large field of view. For proof-of-principle, we applied this method to the thin edge of HeLa cells. Thon rings to better than 10 Å were detected in the montaged tilt-series, and diverse cellular features were observed in the resulting tomograms. These results indicate that the additional dose required by this technique is not prohibitive to performing structural analysis to intermediate resolution across a large field of view. We anticipate that montage tomography will prove particularly useful for lamellae, increase the likelihood of imaging rare cellular events, and facilitate visual proteomics.     

Radiation dose reduction and image enhancement in biological imaging through equally-sloped tomography

Electron tomography is currently the highest resolution imaging modality available to study the 3D structures of pleomorphic macromolecular assemblies, viruses, organelles and cells. Unfortunately, the resolution is currently limited to 3-5nm by several factors including the dose tolerance of biological specimens and the inaccessibility of certain tilt angles. Here we report the first experimental demonstration of equally-sloped tomography (EST) to alleviate these problems. As a proof of principle, we applied EST to reconstructing frozen-hydrated keyhole limpet hemocyanin molecules from a tilt-series taken with constant slope increments. In comparison with weighted back-projection (WBP), the algebraic reconstruction technique (ART) and the simultaneous algebraic reconstruction technique (SART), EST reconstructions exhibited higher contrast, less peripheral noise, more easily detectable molecular boundaries and reduced missing wedge effects. More importantly, EST reconstructions including only two-thirds the original images appeared to have the same resolution as full WBP reconstructions, suggesting that EST can either reduce the dose required to reach a given resolution or allow higher resolutions to be achieved with a given dose. EST was also applied to reconstructing a frozen-hydrated bacterial cell from a tilt-series taken with constant angular increments. The results confirmed similar benefits when standard tilts are utilized.     

Evaluation of denoising algorithms for biological electron tomography

Tomograms of biological specimens derived using transmission electron microscopy can be intrinsically noisy due to the use of low electron doses, the presence of a "missing wedge" in most data collection schemes, and inaccuracies arising during 3D volume reconstruction. Before tomograms can be interpreted reliably, for example, by 3D segmentation, it is essential that the data be suitably denoised using procedures that can be individually optimized for specific data sets. Here, we implement a systematic procedure to compare various nonlinear denoising techniques on tomograms recorded at room temperature and at cryogenic temperatures, and establish quantitative criteria to select a denoising approach that is most relevant for a given tomogram. We demonstrate that using an appropriate denoising algorithm facilitates robust segmentation of tomograms of HIV-infected macrophages and Bdellovibrio bacteria obtained from specimens at room and cryogenic temperatures, respectively. We validate this strategy of automated segmentation of optimally denoised tomograms by comparing its performance with manual extraction of key features from the same tomograms.     

Enhancement of tomogram interpretability using the locked self-rotation function

The interpretation of cryo-electron tomograms of macromolecular complexes can be difficult because of the large amount of noise and because of the missing wedge effect. Here it is shown how the presence of rotational symmetry in a sample can be utilized to enhance the quality of a tomographic analysis. The orientation of symmetry axes in a sub-tomogram can be determined using a locked self-rotation function. Given this knowledge, the sub-tomogram density can then be averaged to improve its interpretability. Sub-tomograms of the icosahedral bacteriophage phiX174 are used to demonstrate the procedure.     

Compensation of Missing Wedge Effects with Sequential Statistical Reconstruction in Electron Tomography

Electron tomography (ET) of biological samples is used to study the organization and the structure of the whole cell and subcellular complexes in great detail. However, projections cannot be acquired over full tilt angle range with biological samples in electron microscopy. ET image reconstruction can be considered an ill-posed problem because of this missing information. This results in artifacts, seen as the loss of three-dimensional (3D) resolution in the reconstructed images. The goal of this study was to achieve isotropic resolution with a statistical reconstruction method, sequential maximum a posteriori expectation maximization (sMAP-EM), using no prior morphological knowledge about the specimen. The missing wedge effects on sMAP-EM were examined with a synthetic cell phantom to assess the effects of noise. An experimental dataset of a multivesicular body was evaluated with a number of gold particles. An ellipsoid fitting based method was developed to realize the quantitative measures elongation and contrast in an automated, objective, and reliable way. The method statistically evaluates the sub-volumes containing gold particles randomly located in various parts of the whole volume, thus giving information about the robustness of the volume reconstruction. The quantitative results were also compared with reconstructions made with widely-used weighted backprojection and simultaneous iterative reconstruction technique methods. The results showed that the proposed sMAP-EM method significantly suppresses the effects of the missing information producing isotropic resolution. Furthermore, this method improves the contrast ratio, enhancing the applicability of further automatic and semi-automatic analysis. These improvements in ET reconstruction by sMAP-EM enable analysis of subcellular structures with higher three-dimensional resolution and contrast than conventional methods.     

A "flip-flop" rotation stage for routine dual-axis electron cryotomography

Electron cryotomography can be used to solve the three-dimensional structures of individual large macromolecules, assemblies, and even small intact cells to medium (approximately 4-8 nm) resolution in a near-native state, but restrictions in the range of accessible views are a major limitation. Here we report on the design, characterization, and demonstration of a new "flip-flop" rotation stage that allows facile and routine collection of two orthogonal tilt-series of cryosamples. Single- and dual-axis tomograms of a variety of samples are compared to illustrate qualitatively the improvement produced by inclusion of the second tilt-series. Exact quantitative expressions are derived for the volume of the remaining "missing pyramid" in reciprocal space. When orthogonal tilt-series are recorded to +/-65 degrees in each direction, as this new cryostage permits, only 11% of reciprocal space is left unmeasured. The tomograms suggest that further improvement could be realized, however, through better software to align and merge dual-axis tilt-series of cryosamples.     

High resolution electron tomography and segmentation‐by‐modeling interpretation in Bsoft

Bsoft offers many tools for the processing of tomographic tilt series and the interpretation of tomograms. Since I introduced tomography into Bsoft almost two decades ago, the field has advanced significantly, requiring refinement of old algorithms and development of new ones. The current direct detectors allow us to collect data more efficiently and with better quality, progressing towards automation. The goal is then to also automate alignment of tilt series and reconstruction. I added an estimation of the specimen thickness as well as fiducialless alignment, to augment the existing fiducial‐based alignment. High‐resolution work requires correction for the contrast transfer function, in tomography complicated by the tilted specimen. For this, I developed a method to generate a power spectrum using the whole micrograph, compensating for tilting. This is followed by routine determination of the contrast transfer function, and correction for it during reconstruction. The next steps involve interpretation of the tomogram, either by subtomogram averaging where possible, or by segmentation and modeling otherwise. Such interpretation actually constitutes the main time‐consuming part of tomography and is less amenable to automation compared to the initial reconstruction.     

A procedure to deposit fiducial markers on vitreous cryo-sections for cellular tomography

We describe a novel approach for the accurate alignment of images in electron tomography of vitreous cryo-sections. Quantum dots, suspended in organic solvents at cryo-temperatures, are applied directly onto the sections and are subsequently used as fiducial markers to align the tilt series. Data collection can be performed from different regions of the vitreous sections, even when the sections touch the grid only at a few places. We present high-resolution tomograms of some organelles in cryo-sections of human skin cells using this method. The average error in image alignment was about 1nm and the resolution was estimated to be 5-7nm. Thus, the use of section-attached quantum dots as fiducial markers in electron tomography of vitreous cryo-sections facilitates high-resolution in situ 3D imaging of organelles and macromolecular complexes in their native hydrated state.