Subtomogram alignment by adaptive Fourier coefficient thresholding

Abstract

In the past few years, three-dimensional (3D) subtomogram alignment has become an important tool in cryo-electron tomography (CET). This technique allows one to produce higher resolution images of structures which can not be reconstructed using single-particle methods. Building on previous work, we present a new dissimilarity measure between subtomograms that works well for the noisy images that often occur in CET images. A technique that is more robust to noise provides the ability to analyze macromolecules in thicker samples such as whole cells or lower the defocus in thinner samples to push the first zero of the Contrast Transfer Function (CTF). Our method, Threshold Constrained Cross-Correlation (TCCC), uses statistics of the noise to automatically select only a small percentage of the Fourier coefficients to compute the cross-correlation, which has two main advantages: first, it reduces the influence of the noise by looking at only those peaks dominated by signal; and second, it avoids the missing wedge normalization problem since we consider the same number of coefficients for all possible pairs of subtomograms. We present results with synthetic and real data to compare our approach with other existing methods under different SNR and missing wedge conditions, and show that TCCC improves alignment results for datasets with SNR<0.1. We have made our source code freely available for the community.

Figure 1

Cumulative magnitude distribution of sorted Fourier coefficients for EMBD model 1581 shown in Fig. 4. Most of the energy is concentrated in a small percentage of the total number of coefficients.

Figure 2

Comparison between empirical cumulative distribution functions (cdf): first, obtained from 500 aligned subtomograms of C.c. whole cells for one voxel (blue continuous line); second, the analytical cdf of a Gaussian distribution (red dashed line) with μ and σ estimated from the empirical data. The agreement is almost perfect making hard to distinguish both curves.

Figure 3

Likelihood score as a function of C for different SNR. Left: likelihood estimated from 300 aligned subtomograms from the real data described in materials and methods where SNR is very low. C^* is very small. Right: likelihood estimated from 150 aligned subtomograms from the phantom described in section 4 with SNR = 10. C^* is analogous to selecting all the non-zero coefficients according to Fig. 1.

Figure 4

(A) Visualization of dynein’s microtubule-binding domain from EMDB entry 1581. (B) Phantom generated from (A) at SNR equal to 1 for different tilt ranges. (C) Phantom generated from (A) at SNR equal to 0.001 for different tilt ranges.

Figure 5

RMSE alignment comparison between three different metrics for synthetic data: TCCC (dashed blue), constrained cross-correlation from Forster et al. (2008) (dot-continuous red) and eq. (2) from Bartesaghi et al. (2008) (continuous green). Results are obtained at different SNR and tilt range levels.

Figure 6

Simulations of how CTF variations between tomographic projections affects subtomogram averaging resolution. We averaged B particles simulating a CTF effect with nominal defocus −10μm plus random uniform deviations in ±1.5μm range. Each particle was corrupted by Poisson noise to have SNR = 0.01. Dashed line is theoretical CTF at −10μm, continuous line has B = 1000, dotted line has B = 2000 and dot-dash line has B = 5000. Because we use synthetic tomogram, the decrease in resolution is due to noise and CTF variability, but not to misalignment.

Figure 7

FSC curve for tetramer S-layer (continuous line). Theoretical CTF for imaging conditions (dashed line) assuming 0.1 amplitude contrast and Cs = 3.2mm. Formulas for theoretical CTF from (Frank (2006)-Ch. 3).

Figure 8

XY, XZ, YZ projections of two aligned boxes from raw data for Caulobacter crescentus S-layer. Bar represents 20nm. (A) Same subtomogram as in (B) but with a low-pass filter to visualize main features. The hexagonal pattern is preserved (XY) but the outermembrane feature is almost blurred out. (B) Shows box were missing wedge blurring affects perpendicular to the cell wall. Noise masks all the features. (C) Same subtomogram as in (D) but with a low-pass filter to visualize main features. Outer membrane is visible but hexagonal pattern is almost blurred out. (D) Shows box were missing wedge blurring affects tangential to the cell wall. Again noise masks all the features.

Figure 9

(A) XZ plane of the C.c. average. Double layer of S-layer is very clear on top of the outer membrane. (B) YZ plane of the C.c. average. (C) Sequential view of slices in the direction perpendicular to the cell wall. Separation between slices is 12Å. Top slice is in the right top corner. Bars represent 30nm.

Figure 10

FSC curve for Caulobacter crescentus surface layer (continuous line). Theoretical CTF for the imaging conditions assuming amplitude contrast of 0.2 (dashed line) and Cs = 3.2mm. Formulas for theoretical CTF from (Frank (2006)-Ch. 3).