Structural Variability from Noisy Tomographic Projections

Abstract

In cryo-electron microscopy, the three-dimensional (3D) electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy two-dimensional images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For n images of size N-by-N pixels, we propose an algorithm for calculating this covariance estimator with computational complexity $O(n{N}^4+\sqrt{\kappa }{N}^{6}logN)$ , where the condition number κ is empirically in the range 10-200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in six dimensions. This is then solved by the conjugate gradient method with an appropriate circulant preconditioner. The result is the first computationally efficient algorithm for consistent estimation of the 3D covariance from noisy projections. It also compares favorably in runtime with respect to previously proposed nonconsistent estimators. Motivated by the recent success of eigenvalue shrinkage procedures for high-dimensional covariance matrix estimation, we incorporate a shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We evaluate our methods on simulated datasets and achieve classification results comparable to state-of-the-art methods in shorter running time. We also present results on clustering volumes in an experimental dataset, illustrating the power of the proposed algorithm for practical determination of structural variability.

Keywords: 44A12; 62G05; 62H30; 62J07; 62J10; 65R32; 68U10; 92C55; Toeplitz matrices; conjugate gradient; cryo-electron microscopy; deconvolution; heterogeneity; principal component analysis; shift invariance; single-particle reconstruction.

Figure 1

Two sample cryo-EM images from a 10000-image dataset depicting the 70S ribosome complex in E. Coli [46]. Each image measures 130-by-130 with a pixel size of 2.82 Å. The images depict two similar molecular structures projected in approximately the same viewing direction, but the high noise level makes it difficult to distinguish the difference in structure.

Figure 2

The eigenvalue distribution of the sample covariance matrix for (a) p = 256, n = 512 and (b) p = 128, n = 1024 with σ = 1 and ℓ = 3 in both regimes. For (a), we have γ = 1/2 and the spiked covariance model predicts a maximum noise eigenvalue at ${(1+\sqrt{1/2})}^2\approx 2.91$ and a signal eigenvalue at λ(3, 1/2) ≈ 4.67, while for (b), γ = 1/8 gives ${(1+\sqrt{1/8})}^2\approx 1.83$ and λ(3, 1/8) ≈ 4.17.

Figure 3

Effect of shrinkage on the relative error of (a) B_n and (b) Σ_n for simulated data with N = 16, C = 2, and 7 distinct CTFs. The signal-to-noise ratio (see (91)) is 0.001.

Figure 4

The square magnitudes |ℱh(ω)|² of two sample CTFs. Their sum forms a bandpass filter, worsening the conditioning of the least-squares estimators.

Figure 5

The simulation ground truth at N = 130 (top) and N = 16 (bottom). (a), (b) Two conformations of the 70S ribosome. (c) Their mean volume (red) and difference map (positive in blue, negative in green).

Figure 6

Covariance estimation results for different simulations with N = 16, C = 2, seven distinct CTFs, and unless otherwise noted, n = 4096 and uniform distribution of orientations. (a) The relative error in Σ_n as a function of SNR_h for n = 1024, n = 4096, and n = 16384. (b) The correlation of the top eigenvector of Σ_n with that of Cov[x]. (c) Top eigenvector correlations for Σ_n and ${\mathrm{\sum }}_n^{(\mathrm{s})}$. (d) Top eigenvector correlations for Σ_n with orientations estimated using the ASPIRE toolbox. (e) Top eigenvector correlations for Σ_n with different orientation distributions over SO(3) described by (92). (f) The cosine of the maximum principal angle between the top three eigenvectors of Σ_n and those of Cov[x] for a simulation with C = 4 classes.

Figure 7

Clustering results for discrete variability with C = 2 classes imaged using n = 4096 images with resolution N = 16 for uniform distribution of viewing angles and seven distinct CTFs. (a) The top 32 eigenvalues of Σ_n obtained at SNR_h = 0.01. (b) A histogram of the coordinates ${\widehat{\alpha }}_s^{(1)}$ corresponding to the images y_s for s = 1, … , n subject to the same SNR_h. (c) The fraction of images classified correctly as a function of SNR_h. (d) The normalized root mean squared error (NRMSE) of the reconstructed volumes.

Figure 8

Manifold learning results for continuous variability (a) Volumes are generated by independently rotating two parts (green and blue) by angles θ₁ and θ₂ while keeping the remainder (red) fixed. (b) The cosine of the maximum principal angle between the top four population eigenvectors and those of Σ_n. (c) 3D diffusion map embedding coordinates of the volume coordinates {ᾱ₁, … , ᾱ_n}, colored according to the first and second rotation angles. (d) The NRMSE of each volume estimate as a function of its diffusion map coordinate. (e) The NRMSE of the reconstructed volumes as a function of SNR_h.

Figure 9

(a) The relative residuals for each iteration of CG applied to A_nμ_n = b_n, denoted by ${\mu }_n^{(t)}$, with no CTF and uniform distribution of viewing angles. (b)–(d) The relative residuals for each CG iterate ${\mathrm{\sum }}_n^{(t)}$ of L_n(Σ_n) = B_n with (b) no CTF and uniform distribution of viewing angles, (c) three distinct CTFs and uniform distribution of viewing angles, and (d) three distinct CTFs and nonuniform distribution of viewing angles. For all plots, the residuals of the standard (nonpreconditoned) CG method are compared with using a circulant preconditioner. All methods were applied to n = 16384 images with size N = 16 and σ² = 1.

Figure 10

Running times for the whole covariance estimation algorithm applied to a dataset of size n = 16384 with varying image size N. Three scenarios are considered: no CTF with uniform distribution of viewing angles, three distinct CTFs with uniform distribution of viewing angles, and three CTFs with nonuniform distribution of viewing angles.

Figure 11

Covariance estimation on the 70S ribosome dataset. (a) Largest eigenvalues of the estimated covariance matrix Σ_n. (b) The estimated mean volume (red), together with the positive (blue) and negative (green) components of the top eigenvector. (c) Histogram of coordinates {ᾱ₁, … , ᾱ_n} from the Wiener filter estimator. (d) 2D histogram of coordinates { $({\widehat{\alpha }}_1^{(1)},{\widehat{\alpha }}_1^{(2)}),\dots ,({\widehat{\alpha }}_n^{(1)},{\widehat{\alpha }}_n^{(2)})$} from the Wiener filter estimator. (e), (f) Full-resolution reconstructions obtained using RELION applied to the clusters identified in (c).