Contrast Transfer Function Correction Research Articles

CryoEM single particle reconstruction with a complex-valued particle stack

Single particle reconstruction (SPR) in cryoEM is an image processing task with an elaborate hierarchy that starts with many very noisy multi-frame images. Efficient representation of the intermediary image structures is critical for keeping the calculations manageable. One such intermediary structure is called a particle stack and contains cut-out images of particles in square boxes of predefined size. The micrograph that is the source of the boxed images is usually corrected for motion between frames prior to particle stack creation. However, the contrast transfer function (CTF) or its Fourier Transform point spread function (PSF) are not considered at this step. Historically, the particle stack was intended for large particles and for a tighter PSF, which is characteristic of lower resolution data. The field now performs analyses of smaller particles and to higher resolution, and these conditions result in a broader PSF that requires larger padding and slower calculations to integrate information for each particle. Consequently, the approach to handling structures such as the particle stack should be reexamined to optimize data processing.Here we propose to use as a source image for the particle stack a complex-valued image, in which CTF correction is implicitly applied as a real component of the image. We can achieve it by applying an initial CTF correction to the entire micrograph first and perform box cutouts as a subsequent step. The final CTF correction that we refine and apply later has a very narrow PSF, and so cutting out particles from micrographs that were approximately corrected for CTF does not require extended buffering, i.e. the boxes during the analysis only have to be large enough to encompass the particle. The Fourier Transform of an exit-wave reconstruction creates an image that has complex values. This is a complex value image considered in real space, opposed to standard SPR data processing where complex numbers appear only in Fourier space. This extension of the micrograph concept provides multiple advantages because the particle box size can be small and calculations crucial for high resolution reconstruction such as Ewald sphere correction, aberration refinement, and particle-specific defocus refinement can be performed on the small box data.

Unified fast reconstruction algorithm for conventional, phase-contrast, and diffraction tomography.

A unified method for three-dimensional reconstruction of objects from transmission images collected at multiple illumination directions is described. The method may be applicable to experimental conditions relevant to absorption-based, phase-contrast, or diffraction imaging using x rays, electrons, and other forms of penetrating radiation or matter waves. Both the phase retrieval (also known as contrast transfer function correction) and the effect of Ewald sphere curvature (in the cases with a shallow depth of field and significant in-object diffraction) are incorporated in the proposed algorithm and can be taken into account. Multiple scattering is not treated explicitly but can be mitigated as a result of angular averaging that constitutes an essential feature of the method. The corresponding numerical algorithm is based on three-dimensional gridding which allows for fast computational implementation, including a straightforward parallelization. The algorithm can be used with any scanning geometry involving plane-wave illumination. A software code implementing the proposed algorithm has been developed, tested on simulated and experimental image data, and made publicly available.

A Method for High-Resolution Three-Dimensional Reconstruction with Ewald Sphere Curvature Correction from Transmission Electron Images.

A method for three-dimensional reconstruction of objects from defocused images collected at multiple illumination directions in high-resolution transmission electron microscopy is presented. The method effectively corrects for the Ewald sphere curvature by taking into account the in-particle propagation of the electron beam. Numerical simulations demonstrate that the proposed method is capable of accurately reconstructing biological molecules or nanoparticles from high-resolution defocused images under conditions achievable in single-particle electron cryo-microscopy or electron tomography with realistic radiation doses, non-trivial aberrations, multiple scattering, and other experimentally relevant factors. The physics of the method is based on the well-known Diffraction Tomography formalism, but with the phase-retrieval step modified to include a conjugation of the phase (i.e., multiplication of the phase by a negative constant). At each illumination direction, numerically backpropagating the beam with the conjugated phase produces maximum contrast at the location of individual atoms in the molecule or nanoparticle. The resultant algorithm, Conjugated Holographic Reconstruction, can potentially be incorporated into established software tools for single-particle analysis, such as, for example, RELION or FREALIGN, in place of the conventional contrast transfer function correction procedure, in order to account for the Ewald sphere curvature and improve the spatial resolution of the three-dimensional reconstruction.

TomoAlign: A novel approach to correcting sample motion and 3D CTF in CryoET

TomoAlign is a software package that integrates tools to mitigate two important resolution limiting factors in cryoET, namely the beam-induced sample motion and the contrast transfer function (CTF) of the microscope. The package is especially focused on cryoET of thick specimens where fiducial markers are required for accurate tilt-series alignment and sample motion estimation. TomoAlign models the beam-induced sample motion undergone during the tilt-series acquisition. The motion models are used to produce motion-corrected subtilt-series centered on the particles of interest. In addition, the defocus of each particle at each tilt image is determined and can be corrected, resulting in motion-corrected and CTF-corrected subtilt-series from which the subtomograms can be computed. Alternatively, the CTF information can be passed on so that CTF correction can be carried out entirely within external packages like Relion. TomoAlign serves as a versatile tool that can streamline the cryoET workflow from initial alignment of tilt-series to final subtomogram averaging during in situ structure determination.

TEM bright field imaging of thick specimens: nodes in Thon ring patterns

The thickness of an object will, at some point, exceed the depth of field of a transmission electron microscope; the value at which this occurs, depends on the resolution and the wavelength considered. An image is then no longer a true projection of the 3D structure. This effect will be expressed in the power spectrum. Here, we first demonstrate this phenomenon experimentally, using carbon foils of different thicknesses and working at 40, 60, 80 and 300 kV. Since we determined the thicknesses of the foils by tomography, we are also able to confirm experimentally that in the case of a thick object, the Thon ring pattern can be described as the sum of the power spectra originating from thin, independently scattering slices. Thus, a sinc function envelope is observed that attenuates the Thon rings' amplitudes, yielding "nodes" in the pattern at which the amplitudes are zero. These nodes move to lower spatial frequencies with decreasing acceleration voltages and increasing thicknesses. Conversely, the object thickness can be directly derived from node positions at a particular acceleration voltage. We validate our approach by applying it to frozen-hydrated bacteria with experimentally determined thicknesses. Our model will contribute to more reliably determining the defocus to be used with contrast transfer function correction for thicker objects and at lower acceleration voltages.

Z-Contrast Enhancement for Small Protein Cryo-EM Structure Determination

Single-particle cryo-EM structure determination of small proteins has been challenging, which is mainly due to the poor signal-to-noise ratio (SNR) dictated by the low-dose imaging conditions required to avoid severe radiation damage. With the advent of direct electron detectors and phase plates, protein molecules with molecular weights (MWs) as low as 64 (hemoglobin [1]) and 52 (streptavidin [2]) kDa have recently been imaged and reconstructed to near-atomic resolution. These achievements indicate the potential of single-particle cryo-EM in structure determination of even smaller proteins. However, given the current state of phase-plate hardware and the associated control software development, optimal phase-plate utilization requires frequent conditioning and re-focusing during data collection, which is not yet suitable for routine, fully-automated large dataset acquisition. The electron scattering cross-section of an element depends on its atomic number (Z) and therefore differs substantially for heavier metal ions compared to the light atoms that are abundant in proteins (namely C, N, O). This physical property naturally endorses enhanced amplitude contrast in cryo-EM imaging of metalloproteins. Here we demonstrate that a copper storage protein [3] with a MW less than 50 kDa can be visualized by single-particle cryo-EM without the assistance of phase-plate. The electron density map has been reconstructed to 3.5 Å resolution upon multiset contrast transfer function correction. Beyond this work, the use of this Z-contrast enhancement approach has broader applications in single-particle EM structural biology.

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.

Alignment algorithms and per-particle CTF correction for single particle cryo-electron tomography.

Single particle cryo-electron tomography (cryoSPT) extracts features from cryo-electron tomograms, followed by 3D classification, alignment and averaging to generate improved 3D density maps of such features. Robust methods to correct for the contrast transfer function (CTF) of the electron microscope are necessary for cryoSPT to reach its resolution potential. Many factors can make CTF correction for cryoSPT challenging, such as lack of eucentricity of the specimen stage, inherent low dose per image, specimen charging, beam-induced specimen motions, and defocus gradients resulting both from specimen tilting and from unpredictable ice thickness variations. Current CTF correction methods for cryoET make at least one of the following assumptions: that the defocus at the center of the image is the same across the images of a tiltseries, that the particles all lie at the same Z-height in the embedding ice, and/or that the specimen, the cryo-electron microscopy (cryoEM) grid and/or the carbon support are flat. These experimental conditions are not always met. We have developed a CTF correction algorithm for cryoSPT without making any of the aforementioned assumptions. We also introduce speed and accuracy improvements and a higher degree of automation to the subtomogram averaging algorithms available in EMAN2. Using motion-corrected images of isolated virus particles as a benchmark specimen, recorded with a DE20 direct detection camera, we show that our CTF correction and subtomogram alignment routines can yield subtomogram averages close to 4/5 Nyquist frequency of the detector under our experimental conditions.

Computational Tools to Improve Visualization by Cryo-Electron Tomography

Cryo-electron tomography (CryoET) can directly visualize frozen, hydrated macromolecules inside cells without staining artifacts or fluorescent tags at nanometer resolution. However, imaging tilted specimens is limited to ±60° tilt, causing a “missing wedge” of data that induces smearing artifacts in tomograms. Furthermore, each image is subject to a defocusing artifact known as the contrast transfer function (CTF) of the electron microscope. Lastly, high noise and low contrast make feature identification difficult and error prone. Although contrast is typically improved by denoising (filtering), the tomographic reconstruction process introduces additional artifacts that can be accentuated when filters are applied to tomograms in 3-D.We developed tools for CTF correction in cryoET tiltseries and to fill in the missing wedge. We implemented them in EMAN2, a widely used software package for cryo elctron microscopy data processing. Using our first too, we applied CTF correction on cryoET tiltseries of viral particles of known structure recorded with a direct electron detector (DE20). Images collected in “movie-mode” were first corrected for drift. We achieve a resolution improvement from ∼25A to ∼13A. Our second tool “fills in” the missing wedge with reprojections of tomographic data from angular directions not accessible experimentally in the electron microscope. Results on viral particle data show decreased missing wedge artifacts (smearing). We also applied pre-reconstruction denoising filters directly to 2-D images in tiltseries, which improved the accuracy of automated particle picking (measured against manual particle picking, considered ground truth).Lastly, we applied pre-reconstruction denoising filters to tiltseries of mammalian cells to minimize artifacts and enhance the reliability of feature identification. For example, in platelets, microtubules and the cell membrane become clearer with pre-filtration, compared to the standard post-reconstruction filtering approach, and mitochondria become more easily distinguishable from granules.

Gctf: Real-time CTF determination and correction.

Accurate estimation of the contrast transfer function (CTF) is critical for a near-atomic resolution cryo electron microscopy (cryoEM) reconstruction. Here, a GPU-accelerated computer program, Gctf, for accurate and robust, real-time CTF determination is presented. The main target of Gctf is to maximize the cross-correlation of a simulated CTF with the logarithmic amplitude spectra (LAS) of observed micrographs after background subtraction. Novel approaches in Gctf improve both speed and accuracy. In addition to GPU acceleration (e.g. 10–50×), a fast ‘1-dimensional search plus 2-dimensional refinement (1S2R)’ procedure further speeds up Gctf. Based on the global CTF determination, the local defocus for each particle and for single frames of movies is accurately refined, which improves CTF parameters of all particles for subsequent image processing. Novel diagnosis method using equiphase averaging (EPA) and self-consistency verification procedures have also been implemented in the program for practical use, especially for aims of near-atomic reconstruction. Gctf is an independent program and the outputs can be easily imported into other cryoEM software such as Relion (Scheres, 2012) and Frealign (Grigorieff, 2007). The results from several representative datasets are shown and discussed in this paper.

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.

Critical importance of the correction of contrast transfer function for transmission electron microscopy-mediated structural biology

Abstracts Background Transmission electron microscopy (TEM) is an excellent tool for studying detailed biological structures. High-resolution structure determination is now routinely performed using advanced sample preparation techniques and image processing software. In particular, correction for contrast transfer function (CTF) is crucial for extracting high-resolution information from TEM image that is convoluted by imperfect imaging condition. Accurate determination of defocus, one of the major elements constituting the CTF, is mandatory for CTF correction. Findings To investigate the effect of correct estimation of image defocus and subsequent CTF correction, we tested arbitrary CTF imposition onto the images of two-dimensional crystals of Rous sarcoma virus capsid protein. The morphology of the crystal in calculated projection maps from incorrect CTF imposition was utterly distorted in comparison to an appropriately CTF-corrected image. Conclusion This result demonstrates critical importance of CTF correction for producing true representation of the specimen at high resolution.

Fast and accurate reference-free alignment of subtomograms

In cryoelectron tomography alignment and averaging of subtomograms, each dnepicting the same macromolecule, improves the resolution compared to the individual subtomogram. Major challenges of subtomogram alignment are noise enhancement due to overfitting, the bias of an initial reference in the iterative alignment process, and the computational cost of processing increasingly large amounts of data. Here, we propose an efficient and accurate alignment algorithm via a generalized convolution theorem, which allows computation of a constrained correlation function using spherical harmonics. This formulation increases computational speed of rotational matching dramatically compared to rotation search in Cartesian space without sacrificing accuracy in contrast to other spherical harmonic based approaches. Using this sampling method, a reference-free alignment procedure is proposed to tackle reference bias and overfitting, which also includes contrast transfer function correction by Wiener filtering. Application of the method to simulated data allowed us to obtain resolutions near the ground truth. For two experimental datasets, ribosomes from yeast lysate and purified 20S proteasomes, we achieved reconstructions of approximately 20Å and 16Å, respectively. The software is ready-to-use and made public to the community.

Unraveling the structure of membrane proteins in situ by transfer function corrected cryo-electron tomography

Cryo-electron tomography in combination with subtomogram averaging allows to investigate the structure of protein assemblies in their natural environment in a close to live state. To make full use of the structural information contained in tomograms it is necessary to analyze the contrast transfer function (CTF) of projections and to restore the phases of higher spatial frequencies. CTF correction is however hampered by the difficulty of determining the actual defocus values from tilt series data, which is due to the low signal-to-noise ratio of electron micrographs. In this study, an extended acquisition scheme is introduced that enables an independent CTF determination. Two high-dose images are recorded along the tilt axis on both sides of each projection, which allow an accurate determination of the defocus values of these images. These values are used to calculate the CTF for each image of the tilt series. We applied this scheme to the mycobacterial outer membrane protein MspA reconstituted in lipid vesicles and tested several variants of CTF estimation in combination with subtomogram averaging and correction of the modulation transfer function (MTF). The 3D electron density map of MspA was compared with a structure previously determined by X-ray crystallography. We were able to demonstrate that structural information up to a resolution of 16.8Å can be recovered using our CTF correction approach, whereas the uncorrected 3D map had a resolution of only 26.2Å.

Fast, spatially varying CTF correction in TEM

We have developed new methods for contrast transfer function (CTF) correction of tilted and/or thick specimens. In order to achieve higher resolutions in cryo-electron tomography (CryoET), it is necessary to account for the defocus gradient on a tilted specimen and possibly the defocus gradient within a thick specimen. CTF correction methods which account for these defocus differences have recently gained interest. However, there is no global CTF correction method available to this date (to process the entire field-of-view at once) which can use different inverse filters, e.g. phase-flipping or Wiener filter, and which can do so within a reasonable time for realistic image sizes. We show that the CTF correction methods presented in this paper correctly account for the spatially varying defocus, can employ different inverse filters and are significantly faster (>50×) than existing methods. We provide proof-of-principle implementations of all the presented CTF correction methods online.

Structure projection reconstruction from through-focus series of high-resolution transmission electron microscopy images

A structure projection reconstruction method based on contrast transfer function correction of through-focus series of high-resolution transmission electron microscopy images is presented. In this method, defocus values are determined by evaluating phase similarities of the pixels on the Fourier transforms of the images after correction using trial defocus values. Two-fold astigmatism is also determined, by measuring focus variation along different directions. Each image in the series is corrected for the effects of contrast transfer function and then combined into a structure projection image. The method works for both crystalline and non-crystalline objects. Test results with experimental images are presented. Influences of experimental parameters for imaging and effects of crystal thickness on reconstruction are discussed.

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.

Structure of 9.6nm Discoidal High-Density Lipoprotein Revealed by Individual-Particle Electron Tomography

Human high-density lipoprotein (HDL) conveys excess cholesterol from peripheral tissues to the liver and steroidogenic organs for clearance during reverse cholesterol transport (RCT). The majority of HDL's ability to reverse or prevent the onset of cardiovascular disease is derived from HDL's mediation of RCT. HDL particles in vivo vary in size, shape, components, and biological functions. 9.6nm discoidal HDL (140-240 kDa) is the disk-shaped precursor of mature spherical HDL and contains phospholipids and two apolipoprotein A-I (apoA-I) molecules. Discoidal HDL is a critical intermediate between lipid-poor apoA-I and mature spherical HDL during HDL assembly as preferential substrate for lecithin:cholesterol acyl transferase (LCAT), the enzyme that loads cholesterol ester onto HDL. However, HDL structure determination is frustrated by the dynamic nature and heterogeneity of HDL.Here we have studied the structure of 9.6nm discoidal HDL by individual-particle electron tomography (IPET), e.g. the combination of electron tomography (ET) and our newly developed local refinement reconstruction program (see details of the method in Lei Zhang and Gang Ren's abstract). Individual HDL particles were imaged by ET with tilt angles ranging from −70° to 70° in steps of 1°. By tracking each individual HDL particle image from the tilt series of micrographs after their contrast transfer function correction, we reconstructed three-dimensional (3D) density maps of each particle using our local-area refinement reconstruction program. By analysis of the 3D density maps, we propose a structural model for 9.6nm discoidal HDL that is in general agreement with the reported double-belt model. In our model, however, the two N-terminal domains are mainly alpha-helical, are oriented parallel to each other and fold back into the lipid-bilayer region near the center of disc. This research suggests the IPET is a novel method for the structure study of lipoproteins.

CTF determination and correction for low dose tomographic tilt series

The resolution of cryo-electron tomography can be limited by the first zero of the microscope's contrast transfer function (CTF). To achieve higher resolution, it is critical to determine the CTF and correct its phase inversions. However, the extremely low signal-to-noise ratio (SNR) and the defocus gradient in the projections of tilted specimens make this process challenging. Two programs, CTFPLOTTER and CTFPHASEFLIP, have been developed to address these issues. CTFPLOTTER obtains a 1D power spectrum by periodogram averaging and rotational averaging and it estimates the noise background with a novel approach, which uses images taken with no specimen. The background-subtracted 1D power spectra from image regions at different defocus values are then shifted to align their first zeros and averaged together. This averaging improves the SNR sufficiently that it becomes possible to determine the defocus for subsets of the tilt series rather than just the entire series. CTFPHASEFLIP corrects images line-by-line by inverting phases appropriately in thin strips of the image at nearly constant defocus. CTF correction by these methods is shown to improve the resolution of aligned, averaged particles extracted from tomograms. However, some restoration of Fourier amplitudes at high frequencies is important for seeing the benefits from CTF correction.

Contrast transfer function correction applied to cryo-electron tomography and sub-tomogram averaging

Cryo-electron tomography together with averaging of sub-tomograms containing identical particles can reveal the structure of proteins or protein complexes in their native environment. The resolution of this technique is limited by the contrast transfer function (CTF) of the microscope. The CTF is not routinely corrected in cryo-electron tomography because of difficulties including CTF detection, due to the low signal to noise ratio, and CTF correction, since images are characterised by a spatially variant CTF. Here we simulate the effects of the CTF on the resolution of the final reconstruction, before and after CTF correction, and consider the effect of errors and approximations in defocus determination. We show that errors in defocus determination are well tolerated when correcting a series of tomograms collected at a range of defocus values. We apply methods for determining the CTF parameters in low signal to noise images of tilted specimens, for monitoring defocus changes using observed magnification changes, and for correcting the CTF prior to reconstruction. Using bacteriophage PRD1 as a test sample, we demonstrate that this approach gives an improvement in the structure obtained by sub-tomogram averaging from cryo-electron tomograms.