Bayesian perspective for orientation determination in cryo-EM with application to structural heterogeneity analysis

Xu, S.; Balanov, A.; Singer, A.; Bendory, T.

doi:10.1107/S2059798326001415

Figure 1
Comparison between maximum cross-correlation ( $[{\hat g}_{\rm {MLE}}]$ ) and Bayesian ( $[{\hat g}_{\rm {MMSE}}]$ ) orientation estimators in cryo-electron microscopy (cryo-EM) and cryo-electron tomography (cryo-ET) applications. The figure illustrates the general workflow in cryo-EM and cryo-ET techniques, highlighting the role of orientation estimation in each technique. (a) illustrates the model with 2D projections (single-particle cryo-EM model; equation 1

), while (b) shows the model of subtomogram averaging in cryo-ET (equation 2

). (a) Cryo-EM involves imaging macromolecules embedded in a thin layer of vitreous ice using an electron beam in a transmission electron microscope (TEM). The process generates 2D projection images (micrographs) of particles in unknown 3D orientations. These 2D particles are then identified and extracted from the micrographs, forming the basis for subsequent steps of the macromolecule's 3D structure reconstruction. (b) Cryo-ET involves imaging a sample from multiple known tilt angles (typically from −60° to + 60°) to create 2D projections, which are then combined computationally to reconstruct 3D subtomograms. In this context, a subtomogram refers to a small volume containing an individual 3D particle. The subtomograms are extracted by a particle-picker algorithm for further analysis. In both (a) and (b) the rotation-estimation problem involves determining the relative orientation of a noisy 2D particle (in cryo-EM) or a noisy 3D subtomogram (in cryo-ET) relative to a reference volume V. The reference volume structure used in both setups is identical and corresponds to the 80S ribosome (Wong et al., 2014

). Under high SNR conditions, both rotation estimators closely approximate the true relative rotation. However, as the SNR decreases the estimation accuracy deteriorates. Importantly, across all SNR levels, the geodesic angular distance between the MMSE orientation estimator and the true rotation consistently remains lower than that of the MLE orientation estimator. For (a) the estimation was conducted using a grid size of L = 3000 samples of the rotation group $[{\sf{SO}}(3)]$ , while for (b) a grid size of L = 300 was used. Each point in the two curve plots represents the average error computed over 3000 trials.

STRUCTURAL
BIOLOGY

ISSN: 2059-7983

Volume 82| Part 4| April 2026| Pages 274-294

https://doi.org/10.1107/S2059798326001415

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