A fast cross-validation method for alignment of electron tomography images based on Beer-Lambert law

Abstract

In electron tomography, accurate alignment of tilt series is an essential step in attaining high-resolution 3D reconstructions. Nevertheless, quantitative assessment of alignment quality has remained a challenging issue, even though many alignment methods have been reported. Here, we report a fast and accurate method, tomoAlignEval, based on the Beer-Lambert law, for the evaluation of alignment quality. Our method is able to globally estimate the alignment accuracy by measuring the goodness of log-linear relationship of the beam intensity attenuations at different tilt angles. Extensive tests with experimental data demonstrated its robust performance with stained and cryo samples. Our method is not only significantly faster but also more sensitive than measurements of tomogram resolution using Fourier shell correlation method (FSCe/o). From these tests, we also conclude that while current alignment methods are sufficiently accurate for stained samples, inaccurate alignments remain a major limitation for high resolution cryo-electron tomography.

Keywords: Alignment accuracy; Beer–Lambert law; Cross validation; Electron tomography; Inelastic scattering; Least square fitting.

Fig. 1. Zero tilt image of an aligned tilt series of Sindbis virus infected BHK cell section

The infected cells were plastic embedded, sectioned, and then stained before being imaged. The tilt axis is vertical. The 8 red squares indicate regions selected and tracked through the tilt series. The region size is 100×100 pixels.

Fig. 2. Relationship between image intensity and tilt angles

(A) Profile of image intensity and tilt angle (θ) for the red square region marked with symbol * shown in Fig. 1. The blue squares and red circles indicate left and right branches, corresponding to negative and positive tilts, respectively. The images were taken in order of negative to positive tilt angles. (B) Replot of (A) using log of intensity (Y-axis) and 1/cosθ (X-axis). Colors are the same as in (A). The two dash lines indicate independent linear regression to points of two branches. (C, D) Expansion of plots (A) and (B) to include multiple regions. (E) Plots in (D) are rescaled and averaged using a common reference line with median slope. The errors are too small to make the error bars visible.

Fig. 3. Quantitative tests with computationally added alignment errors

A series of random errors, including shifts (A), in-plane rotations (B), and combination of shifts and in-plane rotations (C), were added to the aligned stack of stained cell section (same dataset shown in Figs. 1 and 2). The X-axis values represent a maximum pixel error up to the value (Joyeux and Penczek, 2002). The points and error bars indicate mean and standard deviation of residuals, respectively. Note that the mean residual of aligned stack is set as 1 here and all other mean and standard deviation are represented as ratios. A different random error was used for each image of the aligned tilt series.

Fig. 4. Quantitative comparison of alignment quality for raw stack, prealigned stack and aligned stack of stained sample

The linear patterns of log intensity as a function of 1/cosθ are shown for the raw stack (A), prealigned stack (B) and aligned stack (C). (D) Sorted residual plot of the three stacks. Note that the mean residual of raw stack is set at 1 and all residuals are represented as ratios. (E) The bar graph illustrates the statistical analysis (mean ± standard deviation) of residuals from 10 selected regions. The column denotes mean and the error bar denotes standard deviation. The region size is 100×100 pixels. (F) The plot depicts resolution comparisons for tomographic reconstructions of raw stack, prealigned stack and aligned stack on the basis of FSC_e/o. The small dip at around 0.03 Å⁻¹ in the curve of aligned stack (red curve) reflects the first CTF zero frequency of the mean defocus (~5.8 µm) for this dataset.

Fig. 5. Quantitative comparison of alignment quality for raw stack, prealigned stack and aligned stack of cryo dataset

The linear patterns of log intensity as a function of 1/cosθ are shown for the raw stack (A), prealigned stack (B) and aligned stack (C). (D) Sorted residual plot of the three stacks. Note that the mean residual of raw stack is set as 1 and all residuals are represented as ratios. (E) The bar graph illustrates the statistical analysis (mean ± standard deviation) of residuals from 16 selected regions. The region size is 200×200 pixels. (F) The plot depicts resolution comparisons for tomographic reconstructions of raw stack, prealigned stack and aligned stack on the basis of FSC_e/o.

Fig. 6. Evaluations of alignment qualities of additional datasets

Shown are sorted residual plots of tilt series of additional stained (A-D) and cryo (E-H) samples. (A-C) Sindbis virus infected BHK cell sections embedded in resin and stained. (D) Flock House virus infected Drosophila S2 cell sections embedded in resin and stained. (E-G) Frozen-hydrated, purified Sindbis virus. (H) Frozen-hydrated Borreliaburgdorferi cells.

Fig. 7. Performances with varying defocuses

(A-C) The three panels represent three different sets of regions with different amounts of defocus gradient: minimum (A), intermediate (B) and large defocus gradient (C). The dash lines represent the tilt axis. (D-F) The three panels compare the residuals when multiple regions were selected as shown in (A-C) from one stained sample tilt series. (G-I) The three panels compare the residuals when multiple regions were selected as shown in (A-C) from one cryo sample tilt series.