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Real-time detection and single-pass minimization of TEM objective lens astigmatism
Abstract
Minimization of the astigmatism of the objective lens is a critical daily instrument alignment task essential for high resolution TEM imaging. Fast and sensitive detection of astigmatism is needed to provide real-time feedback and adjust the stigmators to efficiently reduce astigmatism. Currently the method used by many microscopists is to visually examine the roundness of a diffractogram (Thon rings) and iteratively adjust the stigmators to make the Thon rings circular. This subjective method is limited by poor sensitivity and potentially biased by the astigmatism of human eyes. In this study, an s² power spectra based method, s²stigmator, was developed to allow fast and sensitive detection of the astigmatism in TEM live images. The “radar”-style display provides real-time feedback to guide the adjustment of the objective lens stigmators. Such unique capability allowed us to discover the mapping of the two stigmators to the astigmatism amplitude and angle, which led us to develop a single-pass tuning strategy capable of significantly quicker minimization of the objective lens astigmatism.
... Other approaches use least-squares to fit the minima of the power spectrum to the zeros of the CTF model [2], or to identify a single ring of zero crossings as an indicator for astigmatism, and derive a closed-form solution of the CTF parameters [7]. Regardless of the CTF estimation method, estimating the background-subtracted power spectrum is necessary. ...
When using an electron microscope for imaging of particles embedded in vitreous ice, the objective lens will inevitably corrupt the projection images. This corruption manifests as a band-pass filter on the micrograph. In addition, it causes the phase of several frequency bands to be flipped and distorts frequency bands. As a precursor to compensating for this distortion, the corrupting point spread function, which is termed the contrast transfer function (CTF) in reciprocal space, must be estimated. In this paper, we will present a novel method for CTF estimation. Our method is based on the multi-taper method for power spectral density estimation, which aims to reduce the bias and variance of the estimator. Furthermore, we use known properties of the CTF and of the background of the power spectrum to increase the accuracy of our estimation. We will show that the resulting estimates capture the zero-crossings of the CTF in the low-mid frequency range.
... The magnification used was 18,000× in superresolution counting mode, which corresponded to a pixel size of 0.81 Å per pixel. The objective lens astigmatism was corrected with s2stigmator (35). Frames were recorded every 0.2 s for 8 s, with a dose rate of ∼8 electrons per physical pixel per second. ...
Flaviviruses assemble initially in an immature, noninfectious state and undergo extensive conformational rearrangements to generate mature virus. Previous cryo-electron microscopy (cryo-EM) structural studies of flaviviruses assumed icosahedral symmetry and showed the concentric organization of the external glycoprotein shell, the lipid membrane, and the internal nucleocapsid core. We show here that when icosahedral symmetry constraints were excluded in calculating the cryo-EM reconstruction of an immature flavivirus, the nucleocapsid core was positioned asymmetrically with respect to the glycoprotein shell. The core was positioned closer to the lipid membrane at the proximal pole, and at the distal pole, the outer glycoprotein spikes and inner membrane leaflet were either perturbed or missing. In contrast, in the asymmetric reconstruction of a mature flavivirus, the core was positioned concentric with the glycoprotein shell. The deviations from icosahedral symmetry demonstrated that the core and glycoproteins have varied interactions, which likely promotes viral assembly and budding.
... We have recently published a method, s 2 stigmator 9 , with a single-pass tuning strategy, that allows rapid and sensitive detection of astigmatism using TEM live images and can reliably and efficiently guide the user to manually adjust the two stigmators to correct astigmatism. This new method opens up possibilities to minimize astigmatism with real-time feedback at a wide range of imaging conditions that are not available by visual examination. ...
Daily alignment of the microscope is a prerequisite to reaching optimal lens conditions for high resolution imaging in cryo-EM. In this study, we have investigated how image astigmatism varies with the imaging conditions (e.g. defocus, magnification). We have found that the large change of defocus/magnification between visual correction of astigmatism and subsequent data collection tasks, or during data collection, will inevitably result in undesirable astigmatism in the final images. The dependence of astigmatism on the imaging conditions varies significantly from time to time, so that it cannot be reliably compensated by pre-calibration of the microscope. Based on these findings, we recommend that the same magnification and the median defocus of the intended defocus range for final data collection are used in the objective lens astigmatism correction task during microscope alignment and in the focus mode of the iterative low-dose imaging. It is also desirable to develop a fast, accurate method that can perform dynamic correction of the astigmatism for different intended defocuses during automated imaging. Our findings also suggest that the slope of astigmatism changes caused by varying defocuses can be used as a convenient measurement of objective lens rotation symmetry and potentially an acceptance test of new electron microscopes.
Illuminating a specimen with a parallel electron beam is critical for many experiments in transmission electron microscopy as deviations from this condition cause considerable deterioration of image quality. Carefully establishing parallel illumination is particularly important on two-condenser lens transmission electron microscopes (TEMs) as the parallel illumination condition is limited to a single beam intensity value on these instruments. It was recently shown that a Thermo Fisher Scientific Talos Arctica, a two-condenser lens TEM operating at 200 kV, equipped with a Gatan K2 Summit direct electron detector is capable of resolving frozen-hydrated macromolecules of various sizes and internal symmetries to better than 3 Å resolution using single particle methodologies. A critical aspect of the success of these findings was the careful alignment of the electron microscope to ensure the specimen was illuminated with a parallel electron beam. Here, this chapter describes how to establish parallel illumination conditions in a Talos Arctica TEM for high-resolution cryogenic data collection for structure determination.
When using an electron microscope for imaging of particles embedded in vitreous ice, the recorded image, or micrograph, is a significantly degraded version of the tomographic projection of the sample. Apart from noise, the image is affected by the optical configuration of the microscope. This transformation is typically modeled as a convolution with a point spread function. The Fourier transform of this function, known as the contrast transfer function (CTF), is oscillatory, attenuating and amplifying different frequency bands, and sometimes flipping their signs. High-resolution reconstruction requires this CTF to be accounted for, but as its form depends on experimental parameters, it must first be estimated from the micrograph. We present a new method for CTF estimation based on multitaper techniques that reduce bias and variance in the estimate. We also use known properties of the CTF and the background power spectrum to further reduce the variance through background subtraction and steerable basis projection. We show that the resulting power spectrum estimates better capture the zero-crossings of the CTF and yield accurate CTF estimates on several experimental micrographs.
Three-dimensional electron microscopy (3DEM) of ice-embedded samples allows the structural analysis of large biological macromolecules close to their native state. Different techniques have been developed during the last forty years to process cryo-electron microscopy (cryo-EM) data. Not surprisingly, success in analysis and interpretation is highly correlated with the continuous development of image processing packages. The field has matured to the point where further progress in data and methods sharing depends on an agreement between the packages on how to describe common image processing tasks. Such standardization will facilitate the use of software as well as seamless collaboration, allowing the sharing of rich information between different platforms. Our aim here is to describe the electron microscopy exchange (EMX) initiative, launched at the 2012 Instruct Image Processing Center Developer Workshop, with the intention of developing a first set of standard conventions for the interchange of information for single-particle analysis (EMX version 1.0). These conventions cover the specification of the metadata for micrograph and particle images, including contrast transfer function (CTF) parameters and particle orientations. EMX v1.0 has already been implemented in the Bsoft, EMAN, Xmipp and Scipion image processing packages. It has been and will be used in the CTF and EMDataBank Validation Challenges respectively. It is also being used in EMPIAR, the Electron Microscopy Pilot Image Archive, which stores raw image data related to the 3DEM reconstructions in EMDB.
Defocus and twofold astigmatism are the key parameters governing the contrast transfer function (CTF) in transmission electron microscopy (TEM) of weak phase objects. We present a new algorithm to estimate these aberrations and the associated uncertainties. Tests show very good agreement between simulated and estimated defocus and astigmatism. We evaluate the reproducibility of the algorithm on experimental data by repeating measurements of an amorphous sample under identical imaging conditions and by analyzing the linearity of the stigmator response. By using a new Thon ring averaging method, the modulation depth of the rings in a 1D averaged power spectrum density (PSD) can be enhanced compared to elliptical averaging. This facilitates a better contrast transfer assessment in the presence of spherical aberration. Our algorithm for defocus and astigmatism estimation inverts the contrast of the Thon rings and suppresses the background in the PSD using an adaptive filtering strategy. Template matching with kernels of various ellipticities is applied to the filtered PSD after transformation into polar coordinates. Maxima in the resulting 3D parameter space provide multiple estimates of the long axis orientation, frequencies and apparent ellipticities of the rings. The frequencies of the detected rings, together with outlier rejection and assignment of an order to the CTF zeros, are used to estimate the defocus and its uncertainty. From estimations of defocus and ellipticity, we derive astigmatism and its uncertainty. A two-pass approach refines the astigmatism and defocus estimate by taking into account the influence of the known spherical aberration on the shape and frequencies of the rings. The implementation of the presented algorithm is freely available for non-commercial use.
Image formation in bright field electron microscopy can be described with the help of the contrast transfer function (CTF). In this work the authors describe the "CTF Estimation Challenge", called by the Madrid Instruct Image Processing Center (I2PC) in collaboration with the National Center for Macromolecular Imaging (NCMI) at Houston. Correcting for the effects of the CTF requires accurate knowledge of the CTF parameters, but these have often been difficult to determine. In this challenge, researchers have had the opportunity to test their ability in estimating some of the key parameters of the electron microscope CTF on a large micrograph data set produced by well-known laboratories on a wide set of experimental conditions. This work presents the first analysis of the results of the CTF Estimation Challenge, including an assessment of the performance of the different software packages under different conditions, so as to identify those areas of research where further developments would be desirable in order to achieve high-resolution structural information.
Copyright © 2015 Elsevier Inc. All rights reserved.
In recent work with large high-symmetry viruses, single-particle electron cryomicroscopy (cryo-EM) has achieved the determination of near-atomic-resolution structures by allowing direct fitting of atomic models into experimental density maps. However, achieving this goal with smaller particles of lower symmetry remains challenging. Using a newly developed single electron-counting detector, we confirmed that electron beam-induced motion substantially degrades resolution, and we showed that the combination of rapid readout and nearly noiseless electron counting allow image blurring to be corrected to subpixel accuracy, restoring intrinsic image information to high resolution (Thon rings visible to ∼3 Å). Using this approach, we determined a 3.3-Å-resolution structure of an ∼700-kDa protein with D7 symmetry, the Thermoplasma acidophilum 20S proteasome, showing clear side-chain density. Our method greatly enhances image quality and data acquisition efficiency-key bottlenecks in applying near-atomic-resolution cryo-EM to a broad range of protein samples.
In this work we approach the task of estimating the contrast transfer function (CTF) of a transmission electron microscope by applying mathematical tools extracted from the field of spectral estimation in multidimensional signal processing, such as periodogram averaging and autoregressive modelling. We prove that the clarity and precision by which the CTF can be detected using these approaches is far better than by any conventional method based on the Fourier transform amplitude alone. We also present a unified signal processing-based framework in which recent development in CTF detection approach can be studied, helping to understand their respective benefits and problems.
The three-dimensional reconstruction of macromolecules from two-dimensional single-particle electron images requires determination and correction of the contrast transfer function (CTF) and envelope function. A computational algorithm based on constrained non-linear optimization is developed to estimate the essential parameters in the CTF and envelope function model simultaneously and automatically. The application of this estimation method is demonstrated with focal series images of amorphous carbon film as well as images of ice-embedded icosahedral virus particles suspended across holes.
The Contrast Transfer Function (CTF) of an image, which modulates images taken from a Transmission Electron Microscope (TEM), is usually determined from the radial average of the power spectrum of the image (Frank, J., Three-dimensional Electron Microscopy of Macromolecular Assemblies, Oxford University Press, Oxford, 2006). The CTF is primarily defined by the defocus. If the defocus estimate is accurate enough then it is possible to demodulate the image, which is popularly known as the CTF correction. However, it is known that the radial average is somewhat attenuated if the image is astigmatic (see Fernando, K.V., Fuller, S.D., 2007. Determination of astigmatism in TEM images. Journal of Structural Biology 157, 189-200) but this distortion due to astigmatism has not been fully studied or understood up to now. We have discovered the exact mathematical relationship between the radial averages of TEM images with and without astigmatism. This relationship is determined by a zeroth order Bessel function of the first kind and hence we can exactly quantify this distortion in the radial averages of signal and power spectra of astigmatic images. The argument to this Bessel function is similar to an aberration function (without the spherical aberration term) except that the defocus parameter is replaced by the differences of the defoci in the major and minor axes of astigmatism. The ill effects due this Bessel function are twofold. Since the zeroth order Bessel function is a decaying oscillatory function, it introduces additional zeros to the radial average and it also attenuates the CTF signal in the radial averages. Using our analysis, it is possible to simulate the effects of astigmatism in radial averages by imposing Bessel functions on idealized radial averages of images which are not astigmatic. We validate our theory using astigmatic TEM images.
Cryoelectron microscopy provides the means of studying macromolecules in their native state. However, the contrast transfer function (CTF) makes the images and the three-dimensional (3D) maps derived from them difficult to interpret. We developed methods to determine the CTF from experimental data and to obtain a CTF-corrected 3D reconstruction. The CTF correction and 3D reconstruction accomplished in one step make it easy to combine different defocus data sets and decrease the error accumulation in the computation. These methods were applied to energy-filtered images of the 70S Escherichia coli ribosome, resulting in a distortion-free 3D map of the ribosome at 1/24.5 A-1 resolution, as determined by the differential phase residual resolution criterion.
Two-fold astigmatism, three-fold astigmatism and misalignment (coma) can be found from the orientations only of the diffractograms of a set of images with injected beam tilts--typically eight images with tilts of equal magnitude and azimuths 45 degrees apart. The spherical aberration must be determined independently, and the defocus is not measured; however, the measurement required is very much simpler than the measurement of image displacements or induced defocus/astigmatism values, and no calibration of the microscope magnification is needed.
Three-dimensional electron cryomicroscopy of randomly oriented single particles is a method that is suitable for the determination of three-dimensional structures of macromolecular complexes at molecular resolution. However, the electron-microscopical projection images are modulated by a contrast transfer function (CTF) that prevents the calculation of three-dimensional reconstructions of biological complexes at high resolution from uncorrected images. We describe here an automated method for the accurate determination and correction of the CTF parameters defocus, twofold astigmatism and amplitude-contrast proportion from single-particle images. At the same time, the method allows the frequency-dependent signal decrease (B factor) and the non-convoluted background signal to be estimated. The method involves the classification of the power spectra of single-particle images into groups with similar CTF parameters; this is done by multivariate statistical analysis (MSA) and hierarchically ascending classification (HAC). Averaging over several power spectra generates class averages with enhanced signal-to-noise ratios. The correct CTF parameters can be deduced from these class averages by applying an iterative correlation procedure with theoretical CTF functions; they are then used to correct the raw images. Furthermore, the method enables the tilt axis of the sample holder to be determined and allows the elimination of individual poor-quality images that show high drift or charging effects.
Accurate knowledge of defocus and tilt parameters is essential for the determination of three-dimensional protein structures at high resolution using electron microscopy. We present two computer programs, CTFFIND3 and CTFTILT, which determine defocus parameters from images of untilted specimens, as well as defocus and tilt parameters from images of tilted specimens, respectively. Both programs use a simple algorithm that fits the amplitude modulations visible in a power spectrum with a calculated contrast transfer function (CTF). The background present in the power spectrum is calculated using a low-pass filter. The background is then subtracted from the original power spectrum, allowing the fitting of only the oscillatory component of the CTF. CTFTILT determines specimen tilt parameters by measuring the defocus at a series of locations on the image while constraining them to a single plane. We tested the algorithm on images of two-dimensional crystals by comparing the results with those obtained using crystallographic methods. The images also contained contrast from carbon support film that added to the visibility of the CTF oscillations. The tests suggest that the fitting procedure is able to determine the image defocus with an error of about 10nm, whereas tilt axis and tilt angle are determined with an error of about 2 degrees and 1 degrees, respectively. Further tests were performed on images of single protein particles embedded in ice that were recorded from untilted or slightly tilted specimens. The visibility of the CTF oscillations from these images was reduced due to the lack of a carbon support film. Nevertheless, the test results suggest that the fitting procedure is able to determine image defocus and tilt angle with errors of about 100 nm and 6 degrees, respectively.
The current theory of image formation in electron microscopy has been semi-quantitatively successful in describing data. The theory involves parameters due to the transfer function of the microscope (defocus, spherical aberration constant, and amplitude constant ratio) as well as parameters used to describe the background and attenuation of the signal. We present empirical evidence that at least one of the features of this model has not been well characterized. Namely the spectrum of the noise background is not accurately described by a Gaussian and associated "B-factor;" this becomes apparent when one studies high-quality far-from focus data. In order to have both our analysis and conclusions free from any innate bias, we have approached the questions by developing an automated fitting algorithm. The most important features of this routine, not currently found in the literature, are (i). a process for determining the cutoff for those frequencies below which observations and the currently adopted model are not in accord, (ii). a method for determining the resolution at which no more signal is expected to exist, and (iii). a parameter-with units of spatial frequency-that characterizes which frequencies mainly contribute to the signal. Whereas no general relation is seen to exist between either of these two quantities and the defocus, a simple empirical relationship approximately relates all three.
We present a completely automated algorithm for estimating the parameters of the contrast transfer function (CTF) of a transmission electron microscope. The primary contribution of this paper is the determination of the astigmatism prior to the estimation of the CTF parameters. The CTF parameter estimation is then reduced to a 1D problem using elliptical averaging. We have also implemented an automated method to calculate lower and upper cutoff frequencies to eliminate regions of the power spectrum which perturb the estimation of the CTF parameters. The algorithm comprises three optimization subproblems, two of which are proven to be convex. Results of the CTF estimation method are presented for images of carbon support films as well as for images of single particles embedded in ice and suspended over holes in the support film. A MATLAB implementation of the algorithm, called ACE, is freely available.
EMAN is a scientific image processing package with a particular focus on single particle reconstruction from transmission electron microscopy (TEM) images. It was first released in 1999, and new versions have been released typically 2-3 times each year since that time. EMAN2 has been under development for the last two years, with a completely refactored image processing library, and a wide range of features to make it much more flexible and extensible than EMAN1. The user-level programs are better documented, more straightforward to use, and written in the Python scripting language, so advanced users can modify the programs' behavior without any recompilation. A completely rewritten 3D transformation class simplifies translation between Euler angle standards and symmetry conventions. The core C++ library has over 500 functions for image processing and associated tasks, and it is modular with introspection capabilities, so programmers can add new algorithms with minimal effort and programs can incorporate new capabilities automatically. Finally, a flexible new parallelism system has been designed to address the shortcomings in the rigid system in EMAN1.
Transmission electron microscopy, as most imaging devices, introduces optical aberrations that in the case of thin specimens are usually modeled in Fourier space by the so-called contrast transfer function (CTF). Accurate determination of the CTF is crucial for its posterior correction. Furthermore, the CTF estimation must be fast and robust if high-throughput three-dimensional electron microscopy (3DEM) studies are to be carried out. In this paper we present a robust algorithm that fits a theoretical CTF model to the power spectrum density (PSD) measured on a specific micrograph or micrograph area. Our algorithm is capable of estimating the envelope of the CTF which is absolutely needed for the correction of the CTF amplitude changes.