Averaging of electron subtomograms and random conical tilt reconstructions through likelihood optimization

Abstract

The reference-free averaging of three-dimensional electron microscopy (3D-EM) reconstructions with empty regions in Fourier space represents a pressing problem in electron tomography and single-particle analysis. We present a maximum likelihood algorithm for the simultaneous alignment and classification of subtomograms or random conical tilt (RCT) reconstructions, where the Fourier components in the missing data regions are treated as hidden variables. The behavior of this algorithm was explored using tests on simulated data, while application to experimental data was shown to yield unsupervised class averages for subtomograms of groEL/groES complexes and RCT reconstructions of p53. The latter application served to obtain a reliable de novo structure for p53 that may resolve uncertainties about its quaternary structure.

Figure 1. Model Calculations

(A) Class purities (see below) for the ML (black) and CC (gray) approach to unsupervised multireference refinement of simulated data sets with varying amounts of noise. Averages and standard deviations (error bars) are calculated over five runs that were started from different random class and orientation assignments. (B) As in A, for simulated data with SNR = 0.007, comparing the ML approach with a modified ML approach (MLσ = 0) where the standard deviation of the noise was fixed at a value 100 times lower than the actual noise level. (C) As in B, comparing the ML approach with a modified ML approach (MLwa) where the marginalization over the missing data regions was replaced by a weighted-averaging approach. (D) As in C, but for data with a more severe preferred orientation of the particles. (E) Class purities for the ML multireference refinements (as in A, black) and for the ML refinements with perfectly aligned particles (gray). (F) Errors in the angular assignments of the groEL₁₄ complexes in the ML multireference refinements (black) and the ML refinements with structurally homogenous subsets (gray). To calculate class purities and angular errors, class and angular assignments were based on the maximum of the probability distribution of each particle. Then, as given on p. 549 of (Tan et al., 2006), class purity was defined as $$\frac{1}{N}\sum _{k=1}^2\underset{t}{max}{m}_{\mathit{kt}},$$with m_kt being the number of particles from true structure t assigned to class k. Angular errors were calculated as the average angular distance between the three vectors of the coordinate systems defined by the perfect Euler angles and the angles obtained in the ML refinements. All 14 equivalent orientations due to the D7 symmetry of the groEL₁₄ complex were taken into account. Note that, as a consequence of this high symmetry the angular errors at low SNR (almost 30°) correspond to near-random angular assignments.

Figure 2. GroEL/GroES Subtomogram Averaging

(A) Central slices through XY and YZ of the three class averages obtained for the data set of subtomograms of groEL and groEL/groES complexes. (B) Symmetrized average subtomogram for class 2 with fitted atomic model of groEL₁₄. (C) Symmetrized average subtomogram for class 3 with fitted atomic model of groEL₁₄groES₇.

Figure 3. p53-DNA RCT Reconstruction Averaging

(A) RCT average with imposed C2 symmetry. (B) Refinement of the RCT average against the untilted data set, imposing C2 symmetry. (C) Refinement of a common-lines generated model against the untilted data set, imposing C2 symmetry. (D) RCT average without imposing any symmetry. (E) Refinement of the RCT average against the untilted data set, without imposing any symmetry. (F) Comparison of the refined RCT models with (orange mesh) or without (green) imposing C2 symmetry shows a departure of the top domain from the symmetry axis (arrows) and a gain in linker density with one of the core dimers (asterisk) in the asymmetric model. All C2-symmetrized maps are shown in orange; all C1 maps are shown in green. Maps are thresholded at 50% of the expected mass in green and orange, and at 100% of the expected mass in (mesh) gray. The side views (top rows) are related to the top views (bottom rows) by a 90-degree rotation, and the putative two-fold symmetry axis is perpendicular to the plane of the top views.