Fully three-dimensional defocus-gradient corrected backprojection in cryoelectron microscopy

Modelling and Simulation of Compton Scatter Image Formation in Positron Emission Tomography

We present the comparative study of the analytical forward model and the statistical simulation of the Compton single scatter in the Positron Emission Tomography. The formula of the forward model has been obtained using the Single Scatter Simulation approximation under simplified assumptions and therefore we calculate scatter projections using independent Monte Carlo simulation mimicking the scatter physics. The numerical comparative study has been performed using a digital cylindrical phantom filled in with water and containing spherical sources of emission activity located at the central and several displaced positions. Good fits of the formula-based and statistically generated profiles of scatter projections are observed in the presented numerical results.

Subtomogram averaging from cryo-electron tomograms

Cryo-electron tomography (cryo-ET) allows three-dimensional (3D) visualization of frozen-hydrated biological samples, such as protein complexes and cell organelles, in near-native environments at nanometer scale. Protein complexes that are present in multiple copies in a set of tomograms can be extracted, mutually aligned, and averaged to yield a signal-enhanced 3D structure up to sub-nanometer or even near-atomic resolution. This technique, called subtomogram averaging (StA), is powered by improvements in EM hardware and image processing software. Importantly, StA provides unique biological insights into the structure and function of cellular machinery in close-to-native contexts. In this chapter, we describe the principles and key steps of StA. We briefly cover sample preparation and data collection with an emphasis on image processing procedures related to tomographic reconstruction, subtomogram alignment, averaging, and classification. We conclude by summarizing current limitations and future directions of this technique with a focus on high-resolution StA.

PSF correction in soft X-ray tomography

In this article, we introduce a linear approximation of the forward model of soft X-ray tomography, such that the reconstruction is solvable by standard iterative schemes. This linear model takes into account the three-dimensional point spread function (PSF) of the optical system, which consequently enhances the reconstruction of data. The feasibility of the model is demonstrated on both simulated and experimental data, based on theoretically estimated and experimentally measured PSFs.

Pushing the resolution limit by correcting the Ewald sphere effect in single-particle Cryo-EM reconstructions

The Ewald sphere effect is generally neglected when using the Central Projection Theorem for cryo electron microscopy single-particle reconstructions. This can reduce the resolution of a reconstruction. Here we estimate the attainable resolution and report a “block-based” reconstruction method for extending the resolution limit. We find the Ewald sphere effect limits the resolution of large objects, especially large viruses. After processing two real datasets of large viruses, we show that our procedure can extend the resolution for both datasets and can accommodate the flexibility associated with large protein complexes.

Conventional reconstruction methods used in cryo-EM single particle analysis do not take the depth of field effect into account. Here the authors present a block-based reconstruction method to deal with the depth of field effect and show that this approach can improve the resolution of cryo-EM virus structures.

Ewald sphere correction using a single side-band image processing algorithm

Curvature of the Ewald sphere limits the resolution at which Fourier components in an image can be approximated as corresponding to a projection of the object. Since the radius of the Ewald sphere is inversely proportional to the wavelength of the imaging electrons, this normally imposes a limit on the thickness of specimen for which images can be easily interpreted to a particular resolution. Here we present a computational method for precisely correcting for the curvature of the Ewald sphere using defocused images that delocalise the high-resolution Fourier components from the primary image. By correcting for each approximately Friedel-symmetry-related sideband separately using two distinct complex transforms that effectively move the displaced Fourier components back to where they belong in the structure, we can determine the amplitude and phase of each of the Fourier components separately. This precisely accounts for the effect of Ewald sphere curvature over a bandwidth defined by the defocus and the size of the particle being imaged. We demonstrate this processing algorithm using: 1. simulated images of a particle with only a single, high-resolution Fourier component, and 2. experimental images of gold nanoparticles embedded in ice. Processing micrographs with this algorithm will allow higher resolution imaging of thicker specimens at lower energies without any image degradation or blurring due to errors made by the assumption of a flat Ewald sphere. Although the procedure will work best on images recorded with higher defocus settings than used normally, it should still improve 3D single-particle structure determination using images recorded at any defocus and any electron energy. Finally, since the Ewald sphere curvature is in a known direction in the third dimension which is parallel to the direction of view, this algorithm automatically determines the absolute hand of the specimen without the need for pairs of images with a known tilt angle difference.

A Survey of the Use of Iterative Reconstruction Algorithms in Electron Microscopy

One of the key steps in Electron Microscopy is the tomographic reconstruction of a three-dimensional (3D) map of the specimen being studied from a set of two-dimensional (2D) projections acquired at the microscope. This tomographic reconstruction may be performed with different reconstruction algorithms that can be grouped into several large families: direct Fourier inversion methods, back-projection methods, Radon methods, or iterative algorithms. In this review, we focus on the latter family of algorithms, explaining the mathematical rationale behind the different algorithms in this family as they have been introduced in the field of Electron Microscopy. We cover their use in Single Particle Analysis (SPA) as well as in Electron Tomography (ET).

Efficient 3D-CTF correction for cryo-electron tomography using NovaCTF improves subtomogram averaging resolution to 3.4Å

Cryo-electron tomography (cryo-ET) allows cellular ultrastructures and macromolecular complexes to be imaged in three-dimensions in their native environments. Cryo-electron tomograms are reconstructed from projection images taken at defined tilt-angles. In order to recover high-resolution information from cryo-electron tomograms, it is necessary to measure and correct for the contrast transfer function (CTF) of the microscope. Most commonly, this is performed using protocols that approximate the sample as a two-dimensional (2D) plane. This approximation accounts for differences in defocus and therefore CTF across the tilted sample. It does not account for differences in defocus of objects at different heights within the sample; instead, a 3D approach is required. Currently available approaches for 3D-CTF correction are computationally expensive and have not been widely implemented. Here we simulate the benefits of 3D-CTF correction for high-resolution subtomogram averaging, and present a user-friendly, computationally-efficient 3D-CTF correction tool, NovaCTF, that is compatible with standard tomogram reconstruction workflows in IMOD. We validate the approach on synthetic data and test it using subtomogram averaging of real data. Consistent with our simulations, we find that 3D-CTF correction allows high-resolution structures to be obtained with much smaller subtomogram averaging datasets than are required using 2D-CTF. We also show that using equivalent dataset sizes, 3D-CTF correction can be used to obtain higher-resolution structures. We present a 3.4 Å resolution structure determined by subtomogram averaging.

The advent of structural biology in situ by single particle cryo-electron tomography

Single particle tomography (SPT), also known as subtomogram averaging, is a powerful technique uniquely poised to address questions in structural biology that are not amenable to more traditional approaches like X-ray crystallography, nuclear magnetic resonance, and conventional cryoEM single particle analysis. Owing to its potential for in situ structural biology at subnanometer resolution, SPT has been gaining enormous momentum in the last five years and is becoming a prominent, widely used technique. This method can be applied to unambiguously determine the structures of macromolecular complexes that exhibit compositional and conformational heterogeneity, both in vitro and in situ. Here we review the development of SPT, highlighting its applications and identifying areas of ongoing development.

Electron tomography simulator with realistic 3D phantom for evaluation of acquisition, alignment and reconstruction methods

Because of the significance of electron microscope tomography in the investigation of biological structure at nanometer scales, ongoing improvement efforts have been continuous over recent years. This is particularly true in the case of software developments. Nevertheless, verification of improvements delivered by new algorithms and software remains difficult. Current analysis tools do not provide adaptable and consistent methods for quality assessment. This is particularly true with images of biological samples, due to image complexity, variability, low contrast and noise. We report an electron tomography (ET) simulator with accurate ray optics modeling of image formation that includes curvilinear trajectories through the sample, warping of the sample and noise. As a demonstration of the utility of our approach, we have concentrated on providing verification of the class of reconstruction methods applicable to wide field images of stained plastic-embedded samples. Accordingly, we have also constructed digital phantoms derived from serial block face scanning electron microscope images. These phantoms are also easily modified to include alignment features to test alignment algorithms. The combination of more realistic phantoms with more faithful simulations facilitates objective comparison of acquisition parameters, alignment and reconstruction algorithms and their range of applicability. With proper phantoms, this approach can also be modified to include more complex optical models, including distance-dependent blurring and phase contrast functions, such as may occur in cryotomography.

Characterization of transfer function, resolution and depth of field of a soft X-ray microscope applied to tomography enhancement by Wiener deconvolution

Full field soft X-ray microscopy is becoming a powerful imaging technique to analyze whole cells preserved under cryo conditions. Images obtained in these X-ray microscopes can be combined by tomographic reconstruction to quantitatively estimate the three-dimensional (3D) distribution of absorption coefficients inside the cell. The impulse response of an imaging system is one of the factors that limits the quality of the X-ray microscope reconstructions. The main goal of this work is to experimentally measure the 3D impulse response and to assess the optical resolution and depth of field of the Mistral microscope at ALBA synchrotron (Barcelona, Spain). To this end we measure the microscope apparent transfer function (ATF) and we use it to design a deblurring Wiener filter, obtaining an increase in the image quality when applied to experimental datasets collected at ALBA.

Three-dimensional CTF correction improves the resolution of electron tomograms

Correction of the contrast transfer function (CTF) of the microscope is a necessary step, in order to achieve high resolution from averaged electron microscopic images. Thereby, the CTF is first estimated and subsequently the electron micrograph is corrected, so that the negative oscillations of the CTF are equalized. Typically, the CTF correction is performed in 2D and the tilt-induced focus gradient is taken into account. Most often, the sample-thickness-induced focus gradient is ignored. Theoretical considerations, as well as implementation suggestions, for a 3D CTF correction that considers both gradients have been proposed before, although an implementation achieving a resolution improvement has been lacking, primarily due to computational reasons. Here, we present a comprehensive solution for a 3D CTF correction based on the Jensen-Kornberg scheme, which performs a slice-by-slice correction of the CTF within the tomographic reconstruction. We show that the computational requirements are comparable to those of 2D CTF correction. Using the examples of mitochondrial ribosomes and tobacco mosaic virus we demonstrate the improvement of the reconstruction quality with the 3D CTF correction, and the resolution gain on sub-tomogram averaging. More interestingly, for tomographic applications, the quality of the individual sub-tomograms before averaging increases significantly. We find that 3D CTF correction always produces equal or better results than 2D CTF correction.

Quantifying resolution limiting factors in subtomogram averaged cryo-electron tomography using simulations

Cryo-electron tomography (CET) is the only available technique capable of characterizing the structure of biological macromolecules in conditions close to the native state. With the advent of subtomogram averaging, as a post-processing step to CET, resolutions in the (sub-) nanometer range have become within reach. In addition to advances in instrumentation and experiments, the reconstruction scheme has improved by inclusion of more accurate contrast transfer function (CTF) correction methods, better defocus estimation, and better alignments of the tilt-series and subtomograms. To quantify the importance of each contribution, we have split the full process from data collection to reconstruction into different steps. For the purpose of evaluation we have acquired tilt-series of ribosomes in such a way that we could precisely determine the defocus of each macromolecule. Then, we simulated tilt-series using the InSilicoTEM package and applied tomogram reconstruction and subtomogram averaging. Through large scale simulations under different conditions and parameter settings we find that tilt-series alignment is the resolution limiting factor for our experimental data. Using simulations, we find that when this alignment inaccuracy is alleviated, tilted CTF correction improves the final resolution, or equivalently, the same resolution can be achieved using less particles. Furthermore, we predict from which resolution onwards better CTF correction and defocus estimation methods are required. We obtain a final average using 3198 ribosomes with a resolution of 2.2nm on the experimental data. Our simulations suggest that with the same number of particles a resolution of 1.2nm could be achieved by improving the tilt-series alignment.

A fast algorithm for computing and correcting the CTF for tilted, thick specimens in TEM

Today, the resolution in phase-contrast cryo-electron tomography is for a significant part limited by the contrast transfer function (CTF) of the microscope. The CTF is a function of defocus and thus varies spatially as a result of the tilting of the specimen and the finite specimen thickness. Models that include spatial dependencies have not been adopted in daily practice because of their high computational complexity. Here we present an algorithm which reduces the processing time for computing the 'tilted' CTF by more than a factor 100. Our implementation of the full 3D CTF has a processing time on the order of a Radon transform of a full tilt-series. We derive and validate an expression for the damping envelope function describing the loss of resolution due to specimen thickness. Using simulations we quantify the effects of specimen thickness on the accuracy of various forward models. We study the influence of spatially varying CTF correction and subsequent tomographic reconstruction by simulation and present a new approach for space-variant phase-flipping. We show that our CTF correction strategies are successful in increasing the resolution after tomographic reconstruction.