Bayesian Perspective for Orientation Determination in Cryo-EM with Application to Structural Heterogeneity Analysis

TL;DR

A Bayesian framework for more accurate and flexible orientation estimation is proposed, with the minimum mean square error (MMSE) estimator serving as a key example, and it is demonstrated that orientation estimation accuracy has a decisive effect on downstream structural heterogeneity analysis.

Abstract

Accurate orientation estimation is a crucial component of 3D molecular structure reconstruction, both in single-particle cryo-electron microscopy (cryo-EM) and in the increasingly popular field of cryo-electron tomography (cryo-ET). The dominant approach, which involves searching for the orientation that maximizes cross-correlation relative to given templates, is sub-optimal, particularly under low signal-to-noise conditions. In this work, we propose a Bayesian framework for more accurate and flexible orientation estimation, with the minimum mean square error (MMSE) estimator serving as a key example. Through simulations, we demonstrate that the MMSE estimator consistently outperforms the cross-correlation-based method, especially in challenging low signal-to-noise scenarios, and we provide a theoretical framework that supports these improvements. When incorporated into iterative refinement algorithms in the 3D reconstruction pipeline, the MMSE estimator markedly improves reconstruction accuracy, reduces model bias, and enhances robustness to the ``Einstein from Noise'' artifact. Crucially, we demonstrate that orientation estimation accuracy has a decisive effect on downstream structural heterogeneity analysis. In particular, integrating the MMSE-based pose estimator into frameworks for continuous heterogeneity recovery yields accuracy improvements approaching those obtained with ground-truth poses, establishing MMSE-based pose estimation as a key enabler of high-fidelity conformational landscape reconstruction. These findings indicate that the proposed Bayesian framework could substantially advance cryo-EM and cryo-ET by enhancing the accuracy, robustness, and reliability of 3D molecular structure reconstruction, thereby facilitating deeper insights into complex biological systems.

Figures (6)

• Figure 1: Comparison between maximum cross-correlation ($\hat{g}_{\text{MLE}}$) and Bayesian ($\hat{g}_{\text{MMSE}}$) 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. Panel (a) illustrates the model with 2D projections (single-particle cryo-EM model, \ref{['eq:cryoEM']}), while panel (b) shows the model of the subtomogram averaging in cryo-ET, \ref{['eq:cryoET']}). (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 $-60^\circ$ to $+60^\circ$) 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. (a+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 wong2014cryo. 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 $\mathsf{SO}(3)$, 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.
• Figure 2: Impact of incorporating the prior rotation distribution on estimation accuracy. Simulations are performed for the cryo-ET model \ref{['eq:cryoET']}, excluding the projection step. The true rotation distribution is modeled as an isotropic Gaussian $\mathcal{IG}_{\mathsf{SO}(3)}(\eta=0.1)$. Estimation performance is measured using the geodesic distance defined in \ref{['eq:geo_dist']}. Here, $g$ denotes the true rotation, $\hat{g}_{\mathrm{MLE}}$ is the maximum-likelihood estimator from \ref{['eqn:g_MAP_discrete']}, $\hat{g}_{\mathrm{MAP}}$ is the maximum a posteriori estimator from \ref{['eq:mapEstimator']}, and $\hat{g}_{\mathrm{MMSE}}$ denotes the Bayesian minimum mean square error estimator from \ref{['eqn:g_MMSE_numerical']}. The MMSE estimators are computed assuming isotropic Gaussian priors on $\mathsf{SO}(3)$ with different concentration parameters $\eta \in \{0.5, 0.1\}$ (see Appendix \ref{['apx:isotropicGaussianDef']}). As $\eta$ decreases, the prior becomes more concentrated and closer to the true underlying distribution, leading to improved accuracy of both the MAP and MMSE estimators. Each data point in the plot is averaged over 3000 Monte Carlo trials using a rotation grid of size $L=2976$. Bottom $2\times 2$ panel. The four images compare denoised 3D volumes obtained using different rotation estimators at two representative noise regimes marked on the curve plot. Rows correspond to the estimator: top row uses $\hat{g}_{\mathrm{MLE}}$, bottom row uses $\hat{g}_{\mathrm{MMSE}}$ with the true prior $\mathcal{IG}_{\mathsf{SO}(3)}(\eta=0.1)$. Columns correspond to SNR: left column is a low-SNR example, and right column is a high-SNR example (where the true rotation is correctly classified).
• Figure 3: Impact of the sampling grid size of $\mathsf{SO}(3)$ ($L$) and the signal-to-noise ratio (SNR) on rotation estimation accuracy. This figure shows the accuracy of rotation estimation under varying sampling grid sizes $L$ of the rotation group $\mathsf{SO}(3)$ and different SNR levels of the observed data $y$ in the model \ref{['eq:model']}. Simulations are performed for the cryo-ET model \ref{['eq:cryoET']}, excluding the projection step. The metric used for comparison is the geodesic distance, as defined in \ref{['eq:geo_dist']}. Here, $g$ denotes the true rotation, $\hat{g}_{\mathrm{MLE}}$ represents the MLE estimator from \ref{['eqn:g_MAP_discrete']}, and $\hat{g}_{\mathrm{MMSE}}$ denotes the Bayesian MMSE estimator from \ref{['eqn:g_MMSE_numerical']}. In the high SNR regime ($\sigma \to 0$), the MLE and MMSE estimators converge, and the geodesic distance scales empirically as $\propto L^{1/3}$. This scaling reflects the three-parameter nature of $\mathsf{SO}(3)$ rotations, where the resolution of the sampling grid improves as $L$ increases. The results shown are based on Monte Carlo simulations with 3000 trials per data point.
• Figure 4: Comparison of 2D image recovery using the MMSE and the MLE rotation estimators. Iterative image recovery procedures with the MMSE estimator ($\hat{g}_{\text{MMSE}}$) and MLE estimator ($\hat{g}_{\text{MLE}}$) are defined in \ref{['eqn:V_t_as_MMSE_est']} and \ref{['eqn:V_t_as_MAP_est']}, respectively. The experiment employs a template image of Einstein and a ground truth image of Newton, both rotated in 2D over a uniform polar grid with $L = 30$ samples. Each image is of size $100\times 100$ pixels, and the radial direction is discretized using $R=300$ points. The reconstructed images within the dark-orange rectangle (right panel) show superior performance with the MMSE rotation estimator, with Pearson cross-correlation (PCC) provided for each reconstructed image. The MLE and MMSE reconstructions are nearly identical at high SNR ($\sigma \to 0$), as predicted by Proposition \ref{['thm:mapAndMmseEstimatorsConicideness']}. The SNR values used for the panels (from right to left) are $10^{-2}$, $4 \times 10^{-3}$, $2 \times 10^{-3}$, $7 \times 10^{-4}$, and $2 \times 10^{-4}$. At very low SNR ($\sigma \to \infty$), the "Einstein from Noise" effect appears, where the estimator resembles the template image of Einstein rather than the underlying truth of Newton. In intermediate SNR ranges, using the MMSE estimator in the iterative step clearly outperforms the MLE estimator.
• Figure 5: Comparison of 3D structure reconstruction in cryo-ET subtomograms averaging using the MMSE and MLE rotation estimators. Iterative structure reconstruction procedures with the MMSE estimator ($\hat{g}_{\text{MMSE}}$) and MLE estimator ($\hat{g}_{\text{MLE}}$) are defined in \ref{['eqn:V_t_as_MMSE_est']} and \ref{['eqn:V_t_as_MAP_est']}, respectively. At high SNR levels, a low-resolution structure emerges due to finite grid sampling of the rotation group $\mathsf{SO}(3)$, effectively acting as a low-pass filter. The 3D reconstruction using the MMSE estimator consistently outperforms the reconstruction using the MLE estimator. For high SNR conditions (i.e., $\sigma \to 0$), both estimators yield similar 3D structures, as expected from Proposition \ref{['thm:mapAndMmseEstimatorsConicideness']}. The SNR values used for the panels (from right to left) are $10^{-2}$, $2 \times 10^{-3}$, $7 \times 10^{-4}$, and $2 \times 10^{-4}$, with a volume size of $32 \times 32 \times 32$. The boxes highlighted in purple resemble the true input structure (Ribosome-S80 wong2014cryo), while the orange-highlighted boxes are more similar to the initial template (beta-galactosidase bartesaghi2014structure), illustrating the "Einstein from Noise" phenomenon. Notably, at very low SNRs, the "Einstein from Noise" effect is evident with the MLE estimator but not with the MMSE estimator.

Theorems & Definitions (5)

• Proposition 2.1
• Proposition 2.2
• Proposition 4.1: MMSE operator form of the EM M-step
• Remark 4.2
• Theorem E.1