An algorithm for estimation and correction of anisotropic magnification distortion of cryo-EM images without need of pre-calibration

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., 18,000×), 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 2.4-2.6Å 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; Generalized 2-D alignment; Single particle 3-D reconstruction; jspr.

Figure 1. Illustration of elliptic anisotropic magnification distortion

(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 Δ_x=Δ cosθ, Δ_y=Δ·sinθ as the search variables. Recursive decrease of the step size was performed to find the final solution (Δ) at high accuracy.

Figure 2. Elliptic distortion parameters (Δ_jspr, θ_jspr) determined using anisoscale aligner in jspr

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. Δ_jspr, and θ_jspr were plotted in a polar coordinate system with Δ_jspr along radial direction and θ_jspr along angular direction. The peak positions indicating the dominant elliptic distortion parameters were labeled accordingly.

Figure 3. Resolution comparison of 3-D reconstructions with elliptic distortion corrected based on (Δ_precalib, θ_precalib) and (Δ_jspr, θ_jspr)

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 (Δ_jspr, θ_jspr), and the green ones are from maps with elliptic distortion corrected with (Δ_precalib, θ_precalib). 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.

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

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.

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

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).