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. Author manuscript; available in PMC: 2017 Aug 1.
Published in final edited form as: J Struct Biol. 2016 Jun 4;195(2):207–215. doi: 10.1016/j.jsb.2016.06.003

An Algorithm for Estimation and Correction of Anisotropic Magnification Distortion of Cryo-EM Images without Need of Pre-calibration

Guimei Yu 1, Kunpeng Li 1, Yue Liu 1, Zhenguo Chen 1, Zhiqing Wang 1, Rui Yan 1, Thomas Klose 1, Liang Tang 2, Wen Jiang 1,#
PMCID: PMC4935934  NIHMSID: NIHMS795583  PMID: 27270241

Abstract

Anisotropic magnification distortion of TEM images (mainly the elliptic distortion) has been recently found as a potential resolution-limiting factor in single particle 3-D reconstruction. Elliptic distortions of ~1–3% have been reported for multiple microscopes under low magnification settings (e.g., 18,000x), which significantly limited the achievable resolution of single particle 3-D reconstruction, especially for large particles. Here we report a generic algorithm that formulates the distortion correction problem as a generalized 2-D alignment task and estimates the distortion parameters directly from the particle images. Unlike the present pre-calibration methods, our computational method is applicable to all datasets collected at a broad range of magnifications using any microscope without need of additional experimental measurements. Moreover, the per-micrograph and/or per-particle level elliptic distortion estimation in our method could resolve potential distortion variations within a cryo-EM dataset, and further improve the 3-D reconstructions relative to constant-value correction by the pre-calibration methods. With successful applications to multiple datasets and cross-validation with the pre-calibration method, we have demonstrated the validity and robustness of our algorithm in estimating the distortion; Correction of the elliptic distortion significantly improved the achievable resolutions by ~1–3 folds and enabled 3-D reconstructions of multiple viral structures at 2.4–2.6 Å resolutions. The resolution limits with elliptic distortion and the amounts of resolution improvements with distortion correction were found to strongly correlate with the product of the particle size and the amount of distortion, which can help assess if elliptic distortion is a major resolution limiting factor for single particle cryo-EM projects.

Keywords: anisotropic magnification distortion, elliptic distortion, jspr, generalized 2-D alignment, single particle 3-D reconstruction

Introduction

Development of direct electron detectors (DDDs) with superior electron detective quantum efficiency allows fast speed recording of movies to correct beam-induced motions of particles. This has significantly boosted the achievable resolutions of single particle cryo-Electron Microscopy (cryo-EM) of a wide range of structures including small protein complexes, to near-atomic resolutions (Bai et al., 2015; Banerjee et al., 2016; Bartesaghi et al., 2015; Clemens et al., 2015; Jiang et al., 2015; Liao et al., 2013). Application of DDDs with small physical pixels sizes (e.g., 5 μm for Gatan K2 Summit) makes it possible to use lower nominal magnifications (e.g., 18,000x) of transmission electron microscopes (TEMs) than those (e.g., 50,000x or higher) previously used for photographic film or CCD for near-atomic resolution cryo-EM data collection (Guo et al., 2014; Liu et al., 2010; Liu et al., 2016; Yu et al., 2008; Zhang et al., 2010; Zhang et al., 2008). However, it has been observed recently that under these lower magnification settings of many microscopes the anisotropic magnification distortion caused by imperfect projector lenses could be a significant issue for high-resolution single particle cryo-EM work (Grant and Grigorieff, 2015c). With the presence of anisotropic magnification, the object will be magnified differently along different directions, thereby causing distortion of the projection images of target macromolecules (Capitani et al., 2006). The dominant distortion has been approximated as elliptic distortion (Capitani et al., 2006) by which a circle is transformed into an ellipse. The deteriorating effects of elliptic distortion are proportional to the size of the imaged objects. Therefore, it is a more severe problem for high-resolution cryo-EM of large macromolecules (e.g., viruses), which is probably one reason why viral samples have lost their leading role in high-resolution single particle cryo-EM in recent years (Banerjee et al., 2016; Bartesaghi et al., 2015; Campbell et al., 2015; Liao et al., 2013).

Current elliptic distortion estimation approaches all require additional experimental calibration for individual microscopes at each magnification using polycrystalline standards (Capitani et al., 2006; Grant and Grigorieff, 2015c; Hou, 2008; Mugnaioli et al., 2009; Zhao et al., 2015). The calibrations need to be remeasured after major services of the microscope or the camera, and even performed periodically to ensure the validity of the calibrations. In contrast, here we present a computational method to directly discover the elliptic distortion information from cryo-EM datasets. Our algorithm can computationally determine the amount and angle of the distortion without need of experimental calibrations. Moreover, as a data-adaptive method, our algorithm can be applied to any single particle cryo-EM dataset (old or newly collected), regardless of the facilities and settings used for data collection.

Methods

1. Quantification of anisotropic magnification distortion

The elliptic distortion of images can be represented as a general form of scaling transform of the image (Equation 1):

[x′y′]=[Sx00Sy][xy]=Savg[11+Δ2001+Δ2][xy] (1)

in which Sx and Sy represent the magnifications along X and Y directions, respectively. Sx and Sy can be described by an average magnification (Savg) modulated by differential magnifications (Δ > 0) and rewritten as Sx = Savg/(1 + Δ/2), Sy = Savg (1 + Δ/2). This general form of scaling is reduced to isotropic magnification when there is no distortion (i.e. Sx = Sy = Savg when Δ= 0) (Fig. 1A). Here we have adopted the formulation [11+Δ2001+Δ2] to ensure that the transform is area preserving ( Sx·Sy≡Savg2 for arbitrary Δ) and therefore does not change the mean magnification and sampling rate of images. In this definition, the Y-axis with a larger magnification (i.e. more elongated) is defined as the major axis of distortion. Equation 1 represents the simplified case when the major axis of distortion is along Y and the minor axis is along X-axis (Fig. 1B).

Figure 1. Illustration of elliptic anisotropic magnification distortion.

Figure 1

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(A) Isotropic magnification in the absence of elliptic distortion. (B) Elliptic distortion with the major axis along Y-axis and minor axis along X-axis. (C) A generalized form of elliptic distortion when the minor axis is at an arbitrary angle θ. (D) Uneven sampling of Δ and θ in a polar coordinate system with Δ along radial direction and θ along angular direction. Sampling of the space is finer at smaller Δ, while much coarser at larger Δ. (E) Uniform sampling of the search space of elliptic distortion parameters by converting into Cartesian coordinate system with Δx=Δ cosθ, Δy=Δ·sinθ as the search variables. Recursive decrease of the step size was performed to find the final solution (Δ) at high accuracy.

If the minor axis is at an arbitrary angle θ (0° ≤ θ < 180° when the 2-fold symmetry of an ellipse is considered) (Figure 1C), Equation 1 needs to be extended to account for the angular offset of the distortion. Mathematically, elliptic distortion along an arbitrary angle θ is equivalent to a three-step process: rotate the image by − θ to align the minor axis along X axis; perform anisotropic scaling as shown in Equation 1; then rotate the image by θ to bring back the image to its original angular view. Since the 2-D in-plane rotation matrix is R=[cosθ-sinθsinθcosθ], Equation 1 is thus extended to

[x′y′]=Savg[cosθ-sinθsinθcosθ][11+Δ2001+Δ2][cosθsinθ-sinθcosθ][xy] (2)

Three parameters are thus needed to fully quantify elliptic magnifications, including the average magnification (Savg), the amount of distortion (Δ), and the angle of minor axis (θ). Savg is a known constant provided by user based on the calibrated instrument magnification, or optionally could be refined separately (i.e., refineScale in jspr) (Guo and Jiang, 2014). We thus will not further discuss the mean magnification here and will focus on the other two parameters, Δ and θ, for the problem of anisotropic magnification parameter determination.

2. A generalized 2-D alignment method using projection matching for determination of elliptic distortion parameters

We formulated the task of determining the anisotropic magnification parameters as a generalized 2-D alignment problem based on matching projections of 3-D reference model to experimental images. The 3-D reconstructions can be considered isotropic in magnification even when images have not been corrected for anisotropic distortion. This is a good approximation as the large number of particles in a wide range of views, including in-plane rotation angles, can effectively average out the anisotropic distortion in the 3-D model. Further, the 3-D model will become even more isotropic after imposing the symmetry, if existing. Nevertheless, the resolution of the 3-D reconstruction is inevitably reduced if the elliptic distortions of images are not properly corrected. Prior to search of the elliptic distortion parameters, other particle parameters including Euler angles, center positions, CTF parameters and mean magnifications, need to be first determined. Therefore, in our workflow, we first perform image processing tasks, including particle selection, CTF fitting, orientation/center search, 3-D reconstructions, etc., as usual without determination/correction of elliptic distortion (Guo and Jiang, 2014). The 3-D models are then used to generate 2-D projections, which are computationally distorted to different combinations of Δ and θ parameters. Correlations of the distorted projection images with the experimental images are computed and the Δ-θ pair resulting in the best correlation will be accepted as the distortion parameters of the experimental images. These parameters will be used to correct anisotropic distortion in computer memory when reconstructing the 3-D models. Therefore, in our workflow, there is no need to store extra sets of micrographs or particle images with elliptic distortions corrected.

There are two parameters, Δ and θ, to be searched. The Δ and θ parameters naturally form the basis of a polar coordinate system with Δ along radial direction and θ along angular direction (Figure 1D). We converted these two parameters into Cartesian coordinate system (Δx = Δ · cosθ, Δy = Δ · sinθ), and then used Δx and Δy as direct search variables to allow a more uniform sampling of the searching space (Figure 1E). In contrast to Δ and θ, which are different in scale and require different step sizes for the search, Δx and Δy have the same scale and can share a common step size of searching. Recursive reduction of the search step sizes around the best search results (Figure 1E) makes the searching efficient by starting with larger step sizes (e.g. 0.01) to cover a large range (e.g., 0.05) but ending with smaller step sizes (e.g., 0.001) to reach high accuracy.

3. Implementation

The above search algorithm was implemented in C++ as a 2-D aligner anisoscale in our jspr software (Guo and Jiang, 2014) that is built based on EMAN2 library (Tang et al., 2007). It can be used alone or together with other existing 2-D aligners such as refineOrientationCenter and refineMicrographDefocus in jspr. The determined distortion parameters can be used to correct the distortion using the new jspr processor, xform.anisoscale, which is also implemented in C++. We typically use quadratic interpolation although both bilinear and quadratic interpolations are supported. The xform.anisoscale processor is internally used in both refinements and 3-D reconstruction to correct the distortion. Following the EMAN2 design, both anisoscale aligner and xform.anisoscale processor are also exposed to Python scripting. The jspr software and user document are available on the author’s Web site (http://jiang.bio.purdue.edu/jspr).

4. Single particle 3-D reconstructions with jspr

The overall process of single particle 3-D reconstructions with jspr has been described before (Guo and Jiang, 2014). After installation of K2 Summit camera, we have further extended it to support processing of movies. For cryo-EM data collected using DDDs, motion among raw frames was first determined using a driver script motionCorrect.py that runs the dosefgpu_driftcorr method (Li et al., 2013) in batch mode. The dosefgpu_driftcorr program was modified to support running average of neighboring frames (i.e., the –nrw option) to make the alignment trajectory smoother (supplementary Fig. 1). Both the motionCorrect.py script and the modified dosefgpu_driftcorr program are available on GitHub (https://github.com/jianglab/motioncorr). The resultant movie sums, which are motion-corrected but without dose-weighting, were used for particle picking using either e2boxer.py (Tang et al., 2007) or RELION particle picking module (Scheres, 2012). The generated coordinates files, together with the frame shift parameters and electron dose, were used to extract particles from raw movie frames with jspr program batchboxer.py while both motion correction and dose weighting were performed. Two different dose weighting strategies, mode 1 (Wang et al., 2014) and mode 2 (Grant and Grigorieff, 2015b), are incorporated into batchboxer.py. Similar results were obtained for these two dose-weighting modes, and the mode 1 strategy has been used for all the datasets processed in this work. The contrast transfer function (CTF) parameters were determined with jspr program fitctf2.py (Jiang et al., 2012) using the extracted particles. The whole dataset was then split into two halves, even and odd subsets, and processed independently with jspr as described before (Guo and Jiang, 2014; Liu et al., 2016). After reaching the convergence of Euler/center parameters, the anisoscale aligner was included into the iterative refinement to determine and correct the anisotropic magnification distortion, which together with Euler/center aligners (e.g., refineOrientationCenter) also improved the accuracy of Euler/center parameters. More aligners could be added either before or after anisoscale refinement, including refineMicrographDefocus to refine the initial defocus determined by fitctf2.py and also to determine per-particle defoucs, refineAstigmatism to determine the astigmatism aberration, refineScale to compensate the relative scale variance among all the 2-D images, and refineBeamTilt to correct potential phase errors from off-axis coma (Guo and Jiang, 2014; Liu et al., 2016). After all the refinements, the even and odd halves were pooled to generate the final 3-D reconstruction. The 0.143 cutoff of Fourier Shell Correlation (FSC) curves of the independently refined even and odd models was used to estimate the resolution (Rosenthal and Henderson, 2003).

Results and Discussion

1. The amount and angle of elliptic distortion can be reliably estimated from particle images

Multiple datasets collected using the Titan Krios TEM at different nominal magnifications were tested with the anisoscale aligner of jspr to estimate the elliptic distortion parameters. As shown in Figure 2, the amount and angle of elliptic distortion estimated using jspr (Δjspr, θjspr) exhibited a pattern where a central peak was surrounded with some scattered points, when the distortion angle θjspr and the distortion amplitude Δjspr are plotted in a polar coordinate system. The peak positions, representing the dominant angle and amount of the magnification distortion, were consistent among datasets that were collected within a year using the same nominal magnification of the same instrument (Fig. 2A–D for 22.5K; Fig. 2E and supplementary Fig. 4I for 18K).

Figure 2. Elliptic distortion parameters (Δjspr, θjspr) determined using anisoscale aligner in jspr.

Figure 2

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Multiple datasets collected at different magnifications using the Titan Krios at Purdue recorded with K2 electron detector were tested. (A–D) Four datasets, Tulane virus (TV), enterovirus D68 (EV-D68), phage ST1 and phage W2, were imaged at the same nominal magnification of 22,500×. (E) The phage T4 dataset was imaged at 18,000× nominal magnification. (F) The human rhinovirus (HRV) dataset was imaged at 14,000× nominal magnification. Δjspr, and θjspr were plotted in a polar coordinate system with Δjspr along radial direction and θjspr along angular direction. The peak positions indicating the dominant elliptic distortion parameters were labeled accordingly.

To further validate these elliptic distortion parameters (Δjspr, θjspr), experimental measurements of the magnification distortion for the Titan Krios TEM at Purdue University were then performed with a polycrystalline gold sample (Grant and Grigorieff, 2015c). As listed in table 1, there was ~0.5–3% elliptic distortion at nominal magnifications ranging from 14,000x to 59,000x. Comparison of the experimental magnification distortion parameters (Δprecalib , θprecalib) with the peak values of (Δjspr, θjspr) revealed excellent agreement at all the tested magnifications (Fig. 2A–F; table 1). The consistent elliptic distortion parameters among multiple datasets at the same magnification of the same instrument (Fig. 2A–D; Fig. 2E and supplementary Fig. 4I), together with their agreement with the calibration measurements (table 1) indicate the reliability and robustness of the anisoscale aligner of jspr in estimating elliptic magnification distortion directly from cryo-EM particle images. This data-adaptive method of elliptic distortion estimation will make it possible to revisit and improve 3-D reconstructions of some years-old datasets without elliptic distortion calibration data.

Table 1. Experimental measurement of anisotropic magnification distortion of the Titan Kiros microscope at Purdue University.

Images of a polycrystalline gold sample were taken with a Gatan K2 Summit direct electron detector under different nominal magnifications, and the anisotropic magnification distortion parameters were determined using mag_distortion_estimate program (Grant and Grigorieff, 2015c).

nominal mag α (degree) minor axis scale factor major axis scale factor %distortion θpreclib* Δpreclib*
14,000 31.3 0.986 1.014 2.87 121.3 0.028
18,000 28.1 0.985 1.015 3.02 118.1 0.030
22,500 28.2 0.987 1.013 2.72 118.2 0.026
29,000 24.8 0.987 1.013 2.57 114.8 0.026
37,000 24.9 0.989 1.011 2.16 114.9 0.022
47,000 34.8 0.992 1.008 1.61 124.8 0.016
59,000 60.2 0.998 1.002 0.55 150.2 0.004
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*

The determined parameters were converted to corresponding parameters in jspr as described in Methods section, named Δpreclib and θpreclib.

2. Variations of the elliptic distortion resolved by anisoscale refinements

Furthermore, to compare the quality of 3-D reconstructions corrected using (Δprecalib, θprecalib) or (Δjspr, θjspr) distortion parameters, new 3-D maps were reconstructed after replacing the (Δjspr, θjspr) parameters of individual particles with (Δprecalib, θprecalib) at corresponding magnifications (table 1); FSC curves were then recalculated and compared with the original FSC curves. As shown in Figure 3, similar resolutions were achieved for 3-D reconstructions of relatively small particles (i.e., human rhinovirus of ~30 nm, and Tulane virus of ~40 nm) (Fig. 3A–B), while for particles of relatively large sizes (i.e., phage Sf6 of ~62 nm, phage W2 of ~64 nm, and T4 isometric head of ~90 nm), applying the (Δjspr, θjspr) distortion parameters usually resulted in 3-D reconstructions of slightly better resolutions (Fig. 3C–E). Consistently, replacing the refined elliptic distortion parameters of individual particles with the peak values as shown in Figure 2, also resulted in 3-D reconstructions with similarly reduced quality for the larger particles (supplementary Fig. 2A–E). Unlike the calibration measurement method that assumes a constant set of distortion parameters for the entire cryo-EM dataset (Capitani et al., 2006; Grant and Grigorieff, 2015c; Zhao et al., 2015), elliptic distortion parameters of individual particles were determined with the anisoscale algorithm for the tested datasets. The observation of resolution reduction for 3-D reconstructions corrected using a constant value from either pre-calibration or the jspr computed peak value, indicates potential variations of anisotropic distortions among particles within a dataset, which has been resolved by our algorithm.

Figure 3. Resolution comparison of 3-D reconstructions with elliptic distortion corrected based on (Δprecalib, θprecalib) and (Δjspr, θjspr).

Figure 3

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FSC curves from five datasets of (A) human rhinovirus (HRV), (B) Tulane virus (TV), (C) phage Sf6, (D) phage W2 and (E) phage T4, covering particles of different sizes (~30–90nm), were plotted. The blue curves represent FSCs calculated based on maps with elliptic distortion corrected with (Δjspr, θjspr), and the green ones are from maps with elliptic distortion corrected with (Δprecalib, θprecalib). Corresponding resolutions at 0.143 cutoff of FSC were labeled. The data collection session of phage W2 was interrupted for re-alignments of the microscope due to beam instability; The large discrepancy of the two FSC curves in (D) likely also reflects the extra variations of instrument conditions.

To further evaluate the spread of (Δjspr, θjspr) among particles within a dataset as shown in Figure 2, different amounts of (Δjspr, θjspr) “outliers” of a testing dataset of Tulane virus recorded on photographic film (see table 2) were replaced with the peak value prior to 3-D reconstruction. Such replacement of (Δjspr, θjspr) “outliers” with the peak value slightly reduced the quality of 3-D reconstruction, and the reduction of quality became more obvious when more “outliers” were replaced (supplementary Fig. 3A–B). To better understand the variations of elliptic distortion, we have investigated the particle location dependence of (Δjspr, θjspr); Tulane virus particles located at the bottom and top right corners on films were found having larger Δjspr, while no significant location dependence for θjspr was observed (supplementary Fig. 3C). Such location dependence of Δjspr was likely due to scanner defects during digitization (Henderson et al., 2007) and it was not observed for datasets recorded with the K2 detector. All these tests suggested that the spread of the elliptic distortion parameters (Δjspr, θjspr) among particles revealed by the anisoscale aligner indeed reflected true variable magnification distortion within a dataset to a large extent.

Table 2. Resolution improvements of 3-D reconstructions by anisotropic magnification distortion estimation and correction with the anisoscale aligner of jspr.

Multiple datasets collected using Titan Krios microscope and K2 camera at Purdue with a dose rate of 8 electrons/physical pixel/second and frame rate of 5 frames/second were tested. The datasets were Tulane virus (TV), enterovirus D68 (EV-D68), phage ST1, Sf6, W2 and T4, and human rhinovirus (HRV). Three previously published datasets1–3 were also tested.

mag detector sample diameter /nm total mass/MDa res/Å (−anisoscale) res/Å (+anisoscale) improvement4 #ptcls
14,000 K2 HRV 30 5.8 4.14 3.16 1.3 3,739
18,000 K2 phage T45 90 42.9 ~10 3.27 3.1 17,954
18,000 K2 phage Sf6 62 19.5 6.05 2.89 2.1 68,237
22,500 K2 EV-D68 32 5.0 3.20 2.42 1.3 11,344
22,500 K2 phage ST1 30 6.0 3.20 2.44 1.3 32,120
22,500 K2 TV 40 10.4 3.44 2.62 1.3 14,154
22,500 K2 phage W2 64 20.5 8.23 3.59 2.3 8,663
37,000 film TV1 40 10.4 6.13 4.17 1.5 4,278
59,000 film PCV22 19 1.7 ~3 2.92 1.03 55,178
22,500 K2 20S proteasome3 18 0.8 2.65 2.65 1 49,954
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1

Dataset reported in (Yu et al., 2013).

2

Dataset reported in (Liu et al., 2016).

3

Dataset reported in (Campbell et al., 2015) that was imaged using the Titan Krios at The Scripps Research Institute.

4

The “improvement” is computed as res(−anisoscale)/res(+anisoscale).

5

A T4 mutant with icosahedral head.

Since the spread was inevitably contaminated by noise caused errors given the intrinsic low signal-to-noise ratio of cryo-EM images, we also further examined the results of per-micrograph and per-particle level refinements, which can be easily specified with a “batchsize” option of our anisoscale aligner. Refinement with the anisoscale aligner typically started from the per-micrograph level that was followed by per-particle level refinement. Extending refinement from per-micrograph to per-particle level retained or marginally improved the reconstruction in our test cases (supplementary Fig. 4A, D and G), and per-particle level estimation and correction of elliptic distortion have been performed for all the test cases in this work. Depending on the size of particles and/or the amounts of particles in individual micrographs, per-particle level and per-micrograph level refinement and correction of the elliptic distortion may include more computing errors relative to the peak-values. Therefore, peak-value, per-micrograph and per-particle levels elliptic distortion correction can be performed separately in practice and compared for the gain or loss of 3-D reconstruction quality.

We speculated that the hysteresis of electromagnetic lenses of microscopes was possibly related to the variation of distortions among different micrographs, while potential local charging of specimens and/or other potential distortions such as scanner distortion for data collected with photographic films (Henderson et al., 2007) might further contribute to the variation of distortions among particles. However, more cases and more systematic studies will be required to fully understand the factors contributing to elliptic distortion and the variations.

3. Computational estimation and correction of the elliptic distortion significantly improves the achievable resolutions of single particle 3-D reconstructions

To evaluate the benefits of the anisoscale distortion estimation and correction method implemented in jspr for single particle 3-D reconstructions, we have tested the method with multiple datasets that led to structures of subnanometer to near-atomic resolutions without correction of the elliptic distortion. As expected, iterative estimation and correction of elliptic distortion using anisoscale aligner improved both the accuracy of 2-D alignments and 3-D reconstructions, and resulted in significant resolution improvements for all the datasets containing unignorable amounts of elliptic distortions as shown in Figure 4 and (Sirohi et al., 2016).

Figure 4. Improved quality of 3-D reconstructions after anisoscale estimation and correction of elliptic distortion.

Figure 4

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Results from four datasets covering different resolutions ranging from subnanometer to near-atomic resolution are displayed. Shown are four datasets: (A, B) Tulane virus (TV) (K2 detector), (C, D) human rhinovirus (HRV), (E, F) TV (previously published dataset imaged on film (Yu et al., 2013)), and (G, H) phage T4. FSC curves from 3-D reconstructions with (blue)/without (green) including anisoscale aligner in iterative refinement were shown (A, C, and E). (B, D, and F) illustrate the map quality improvement; Panels in the left show the maps from refinements without anisoscale aligner, and those in the right are from refinements with the anisoscale aligner. Arrows in (B) point to two Ile residues with different side chain rotamer conformations. (G and H) show the FSC curves, and the maps of phage T4 final reconstruction with/without performing the elliptic distortion correction, respectively.

Cryo-EM data of Tulane virus (TV, ~40 nm in diameter) (Yu et al., 2013) collected using Titan Krios at 22,500x nominal magnification on K2 detector is one case where the estimation and correction of elliptic magnification distortion using jspr significantly boosted the achievable resolution. As shown in Figure 4A, without including the anisoscale aligner, the reconstruction of TV reached only a resolution of 3.44 Å with ~14,000 particle images, while including the anisoscale estimation and correction improved the resolution to 2.62 Å, essentially at the Nyquist resolution (2.6 Å) of the K2 physical pixel at this magnification. At this improved resolution of ~2.6 Å, better side chain densities of amino acids were obtained for more accurate atomic model interpretation (e.g., unambiguous assignment of side chain rotamers). For example, two rotameric conformations of Isoleucine residues were recognizable in the portion of density from the 2.62 Å map shown in Figure 4B.

Another case is shown in Figure 4C, for which a significant improvement of resolution from ~4.14 Å to ~3.16 Å was obtained by performing the anisotropic scale estimation and correction using jspr. This cryo-EM dataset contains ~3,700 images of human rhinovirus (HRV) particles (~30 nm in diameter) (Bochkov et al., 2011) collected using the Titan Krios at 14,000x nominal magnification on the K2 Summit DDD. Such improvement in resolution was confirmed also by the obviously improved map quality. As shown in Figure 4D, compared to the original map without anisotropic magnification distortion correction, more side chain densities were resolved in the improved map at ~3.2 Å.

Since the anisoscale method in jspr does not require pre-calibration, we also revisited some of our published datasets. We first re-processed a dataset of TV collected using Titan Krios at 37,000x nominal magnification recorded on photographic films and digitized using Nikon Coolscan scanner in 2012 (Yu et al., 2013), considering the ~2.2% magnification distortion now found for the 37,000x nominal magnification of the Titan Krios at Purdue (table 1). A peak of ~2.9% elliptic distortion at ~37° was estimated by the anisoscale aligner (supplementary Fig. 3A), which is different from the current pre-calibration values (~2.2% at ~115°, table 1). The differences, especially the amount, are probably primarily due to the inherent variation of the microscope over time and/or after many services performed to the microscope, and also potential distortion by nonflat photographic films and/or scanner defects during digitization of the photographic films (Henderson et al., 2007). The large change of the distortion angle should be mostly due to the different orientations of the photographic films and the K2 detector. Both our previous refinement and new refinement without considering the elliptic distortion using jspr led to a 3-D reconstruction of ~6.1–6.3 Å (Yu et al., 2013), while further refinement using the anisoscale aligner improved the reconstruction markedly to ~4.2 Å resolution (Fig. 4E), at which individual strands of β sheet became resolved (Fig. 4F). Introduction of the anisoscale aligner to the single particle 3-D reconstruction of a dataset of Porcine circovirus type 2 (PCV2, ~19 nm in diameter) collected using Titan Krios at 59,000x nominal magnification on photographic films in 2011, however, resulted in much smaller improvement in resolutions (from ~3.0 to 2.9 Å) (Liu et al., 2016), presumably due to the small amount of distortion at this high magnification of 59,000x (table 1) and the small size of the PCV2 particle.

One more case on particles of much larger size is shown in Figure 4G/H. Refinement of a dataset of a bacteriophage T4 mutant with an isometric head (~90 nm in diameter) (Olson et al., 2001) was limited to ~10 Å with ~18,000 particle images collected using Titan Krios at 18,000x nominal magnification on K2 detector. The inclusion of anisoscale aligner in jspr refinement to estimate and correct the elliptic distortion drastically improved the attainable resolution of this bacteriophage T4 dataset from subnanometer to near-atomic resolution (3.27 Å, Fig. 4G/H). A 3-D reconstruction using the finally refined parameters without elliptic distortion correction reached a resolution of ~7.16 Å (Fig. 4G) that was also notably better than original refinement without considering the elliptic distortion (~10 Å). This indicated that the correction of elliptic distortion also improved the accuracy of Euler/Center parameters, which was consistent among our tested cases.

In addition to our own datasets collected at Purdue, we also tested our method with a public dataset for the 2.8 Å structure of Thermoplasma acidophilum 20S proteasome (~18 nm in the long dimension) (Campbell et al., 2015) deposited in the Electron Microscopy Pilot Image Archive of EMDataBank (EMPIAR-10025). Although this dataset of 20S proteasome was collected using the same nominal magnification (22,500x) of Titan Krios on K2 detector as some of our datasets collected at Purdue University, little anisotropic magnification distortion was observed as revealed by the per-micrograph level refinement with anisoscale (supplementary Fig. 5A), suggesting that the Titan Krios at The Scripps Research Institute did not have significant elliptic lens distortion at 22,500x nominal magnification; 3-D reconstructions of similar resolutions (~2.65 Å) were obtained with/without this refinement (supplementary Fig. 5C). Further refinement with anisoscale at per-particle level introduced a larger spread of (Δjspr, θjspr) (supplementary Fig. 5B) potentially due to increased randomness of elliptic distortion among individual particles and more computing errors of anisoscale with individual images of low SNRs, while resulting in a 3-D reconstruction at a similar resolution (supplementary Fig. 5C). This indicated that the potential gain from resolving elliptic distortion variations among particles (e.g., due to local charging of specimens) by anisoscale was balanced by loss due to computing errors for the small 20S proteasome particles, and there was no net gain or loss of reconstruction quality.

4. Correlation of resolution limit and resolution improvement with elliptic distortion

Additional datasets have also been tested, and the results of all tested cases were summarized in table 2. A linear correlation between the product of particle diameter and the dominant elliptic distortion amplitude (D*Δ) and the achievable resolutions without the distortion correction (resolution_limit) was observed based on these tested cases (Fig. 5A). As expected, cryo-EM datasets with larger D*Δ are more severely affected and limited to lower resolutions. A published rotavirus dataset with elliptic distortion (Grant and Grigorieff, 2015a) also followed this trend. This plot can be used as a guide to assess if elliptic distortion is a major resolution limiting factor for cryo-EM datasets. Resolution improvements (res(-anisoscale)/res(+anisoscale), table 2) ranging from ~1.3 to 3.1 folds were obtained for the tested datasets collected at nominal magnifications ranging from 14,000x to 37,000x bearing ~2–3% elliptic distortion. Such resolution improvements were also found strongly correlated with D*Δ (Fig. 5B). In general, datasets with larger D*Δ benefits more from the elliptic distortion correction. The aforementioned rotavirus dataset (Grant and Grigorieff, 2015a) also agreed well with the approximated trend line that can serve as a guideline to predict the potential benefits of elliptic distortion correction.

Figure 5. Resolution limits, improvements and their dependence on particle size and the amount of elliptic distortion.

Figure 5

Open in a new tab

Resolution limits (A) and improvements (B) as function of the product of particle size and the amount of elliptic distortion. The trend lines were fitted using the data points labeled with blue dots representing the datasets collected using Titan Krios at Purdue listed in table 2. Resolution_limit trend line: y =0.3485x+0.7832 (R2=0.8407); The improvements trend line: y=0.0951x+0.4728 (R2=0.9549). The data point represented by with diamond-shaped symbols is from the Rotavirus case reported in (Grant and Grigorieff, 2015a).

Conclusions

Elliptic distortion of TEM images is introduced mainly by projector lenses (e.g., the P2 lens for the Titan Krios at Purdue according to preliminary tests using cross-grating grids by previous facility manager Dr. Agustin Avila-Sakar, unpublished), and potentially further modulated by other issues in imaging such as sample charging. It can be a resolution limiting factor of single particle 3-D reconstructions in a large range of resolutions (Fig. 5). Correction of the elliptic distortion is essential for obtaining 3-D reconstructions of large macromolecules such as most viral particles at sub-3 Å and better resolutions, while for smaller protein complexes, it could become a problem when targeting at higher resolutions (e.g., beyond 2 Å). In this work, we have developed a generic algorithm for the elliptic anisotropic magnification distortion estimation and correction in single particle 3-D reconstructions. This method brings multiple benefits, including 1) it eliminates the need of pre-calibration using standard specimens; 2) it is data-adaptive and therefore is generally applicable to any single particle cryo-EM datasets irrespective of facility settings; 3) The per-micrograph and/or per-particle level determination of anisotropic magnification distortion in our algorithm alleviates the concern about the potentially variable anisotropic magnification distortions caused by charging of the samples and numerous changes of currents of all lenses during data collection spanning hours to days; 4) In our method, the raw images were left unmodified and the distortion correction is performed during the 3-D reconstruction step, and therefore there is no need for extra space to store distortion corrected micrographs.

Supplementary Material

1. Supplementary Figure 1. Motion trajectories of raw frames determined by motionCorrect.py with/without performing running average of neighboring frames.

Results of two movies were shown. (A) and (C) were aligned without performing running average, and (B) and (D) show the smoother motion trajectories determined after performing running average of 3 neighboring frames.

2. Supplementary Figure 2. Comparison of 3-D reconstructions with the elliptic distortion corrected using per-particle (Δjspr, θjspr) and the peak-values.

Blue FSC curves are for maps with elliptic distortion corrected with peak values of (Δjspr, θjspr), and the green ones are for maps with elliptic distortion corrected with (Δjspr, θjspr) values at per-particle level. Five cases of (A) TV recorded by K2 detector, (B) TV recorded on photographic films, (C) phage Sf6, (D) phage W2 and (E) phage T4 are shown. The amounts of elliptic distortion were labeled accordingly. Corresponding resolutions at 0.143 cutoff of FSC were labeled.

3. Supplementary Figure 3. Comparison of 3-D reconstructions of TV (recorded on photographic films) with different amounts of per-particle (Δjspr, θjspr) replaced with the peak value (0.029, 36.8°).

(A) Elliptic distortion parameters (Δjspr, θjspr) determined for TV at per-particle level. Δjspr, and θjspr were plotted in a polar coordinate system with Δjspr along radial direction and θjspr along angular direction. (B) The distance between individual (Δjspr, θjspr) and the peak position (0.029, 36.8°) was calculated. As labeled in (A), (Δjspr, θjspr) parameters within given distances were kept, while those beyond were replaced with the peak value prior to 3-D reconstruction. Gold-standard FSC curves were calculated for individual distance cutoffs. (C) Δjspr and θjspr were plotted against particle location in the micrographs, respectively. The magnitudes of Δjspr are represented by the sizes of the blue dots, and different distortion angles are labeled with different colors.

4. Supplementary Figure 4. Extending anisoscale refinements from per-micrograph level to per-particle level.

Three tested cases of (A–C) TV recorded on K2 detector, (B–F) TV recorded on photographic films and (G–I) phage Sf6 were shown. (A, D, G) FSC curves from iterative refinements at per-micrograph (blue) and per-particle (green) levels. Polar plots of (Δjspr, θjspr) at per-micrograph level were shown in (B, E, H), and those at per-particle level were displayed in (C, F, I).

5. Supplementary Figure 5. Single particle 3-D reconstruction of 20S proteasome using jspr.

The dataset was previously published (Campbell et al., 2015). Elliptic distortion parameters (Δjspr, θjspr) determined using anisoscale of jspr at per-micrograph (A) and per-particle (B) levels. (C) FSC curves of 3-D reconstructions without (red), and with per-micrograph level (green) or per-particle level (blue) anisoscale refinements for elliptic distortion estimation and correction.

Acknowledgments

This work was supported in part by NIH grant (1R01AI111095). Cryo-EM images were taken at the Purdue Cryo-EM Facility (http://cryoem.bio.purdue.edu).

Footnotes

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

1. Supplementary Figure 1. Motion trajectories of raw frames determined by motionCorrect.py with/without performing running average of neighboring frames.

Results of two movies were shown. (A) and (C) were aligned without performing running average, and (B) and (D) show the smoother motion trajectories determined after performing running average of 3 neighboring frames.

NIHMS795583-supplement-1.tif (3.9MB, tif)
2. Supplementary Figure 2. Comparison of 3-D reconstructions with the elliptic distortion corrected using per-particle (Δjspr, θjspr) and the peak-values.

Blue FSC curves are for maps with elliptic distortion corrected with peak values of (Δjspr, θjspr), and the green ones are for maps with elliptic distortion corrected with (Δjspr, θjspr) values at per-particle level. Five cases of (A) TV recorded by K2 detector, (B) TV recorded on photographic films, (C) phage Sf6, (D) phage W2 and (E) phage T4 are shown. The amounts of elliptic distortion were labeled accordingly. Corresponding resolutions at 0.143 cutoff of FSC were labeled.

NIHMS795583-supplement-2.tif (3.1MB, tif)
3. Supplementary Figure 3. Comparison of 3-D reconstructions of TV (recorded on photographic films) with different amounts of per-particle (Δjspr, θjspr) replaced with the peak value (0.029, 36.8°).

(A) Elliptic distortion parameters (Δjspr, θjspr) determined for TV at per-particle level. Δjspr, and θjspr were plotted in a polar coordinate system with Δjspr along radial direction and θjspr along angular direction. (B) The distance between individual (Δjspr, θjspr) and the peak position (0.029, 36.8°) was calculated. As labeled in (A), (Δjspr, θjspr) parameters within given distances were kept, while those beyond were replaced with the peak value prior to 3-D reconstruction. Gold-standard FSC curves were calculated for individual distance cutoffs. (C) Δjspr and θjspr were plotted against particle location in the micrographs, respectively. The magnitudes of Δjspr are represented by the sizes of the blue dots, and different distortion angles are labeled with different colors.

NIHMS795583-supplement-3.tif (15.6MB, tif)
4. Supplementary Figure 4. Extending anisoscale refinements from per-micrograph level to per-particle level.

Three tested cases of (A–C) TV recorded on K2 detector, (B–F) TV recorded on photographic films and (G–I) phage Sf6 were shown. (A, D, G) FSC curves from iterative refinements at per-micrograph (blue) and per-particle (green) levels. Polar plots of (Δjspr, θjspr) at per-micrograph level were shown in (B, E, H), and those at per-particle level were displayed in (C, F, I).

NIHMS795583-supplement-4.tif (3.9MB, tif)
5. Supplementary Figure 5. Single particle 3-D reconstruction of 20S proteasome using jspr.

The dataset was previously published (Campbell et al., 2015). Elliptic distortion parameters (Δjspr, θjspr) determined using anisoscale of jspr at per-micrograph (A) and per-particle (B) levels. (C) FSC curves of 3-D reconstructions without (red), and with per-micrograph level (green) or per-particle level (blue) anisoscale refinements for elliptic distortion estimation and correction.

NIHMS795583-supplement-5.tif (2.8MB, tif)

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