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Mixture of neural fields for heterogeneous reconstruction in cryo-EM
Axel Levy, Rishwanth Raghu, David Shustin, Adele Rui-Yang Peng, Huan Li, Oliver Biggs Clarke, Gordon Wetzstein, Ellen D. Zhong
TL;DR
Cryo-EM reconstruction lacks methods that jointly recover compositional and conformational heterogeneity from mixtures with unknown poses. Hydra models structures with K neural fields and jointly optimizes poses, classes, and conformations using a likelihood-based objective. It improves over previous neural-based methods on compositionally heterogeneous datasets and handles simultaneous compositional and conformational heterogeneity in synthetic and experimental data.
Problem
Existing cryo-EM methods do not adequately model samples with mixed compositional and conformational heterogeneity, which requires joint inference of structure, pose, class, and conformation.
Method
Hydra represents each compositional state with one of K neural fields and jointly estimates structures, orientations, conformations, and class assignments with a likelihood-based objective.
Results
Hydra improves over previous neural-based methods on compositionally heterogeneous datasets and simultaneously handles compositional and conformational heterogeneity in synthetic and experimental datasets.
Takeaways & Limitations
Hydra expands neural-based heterogeneous reconstruction to mixtures of multiple protein species with strong compositional and conformational variability.
Takeaways & Limitations
Hydra requires specifying the number of classes K before reconstruction, and selecting K by repeated runs can be time- and energy-consuming.
Abstract
from arXiv · showhide
Cryo-electron microscopy (cryo-EM) is an experimental technique for protein structure determination that images an ensemble of macromolecules in near-physiological contexts. While recent advances enable the reconstruction of dynamic conformations of a single biomolecular complex, current methods do not adequately model samples with mixed conformational and compositional heterogeneity. In particular, datasets containing mixtures of multiple proteins require the joint inference of structure, pose, compositional class, and conformational states for 3D reconstruction. Here, we present Hydra, an approach that models both conformational and compositional heterogeneity fully ab initio by parameterizing structures as arising from one of K neural fields. We employ a new likelihood-based loss function and demonstrate the effectiveness of our approach on synthetic datasets composed of mixtures of proteins with large degrees of conformational variability. We additionally demonstrate Hydra on an experimental dataset of a cellular lysate containing a mixture of different protein complexes. Hydra expands the expressivity of heterogeneous reconstruction methods and thus broadens the scope of cryo-EM to increasingly complex samples.
1 Introduction
Cryo-EM can reveal dynamic structures in near-native states, but existing ab initio methods do not jointly recover compositional and conformational heterogeneity. Hydra addresses this gap with a mixture of neural fields optimized for simultaneous pose, class, and conformation inference.
- Cryo-EM reveals dynamic information about large macromolecular complexes in near-native states, supporting structural biology and rational drug design.
- Each single-particle image is a noisy, randomly oriented projection of an individual particle with unknown identity, composition, and state.
- No existing approach simultaneously reveals discrete compositional and continuous conformational heterogeneity, while sequential strategies fail because inaccurate consensus poses undermine class-first processing.
- Hydra models heterogeneous reconstructions with a mixture of K neural fields and a likelihood-based objective that jointly optimizes orientations, conformations, and class assignments.
- Hydra handles strong compositional heterogeneity and simultaneously reconstructs compositional and conformational heterogeneity, including multiple protein complexes from an unpurified experimental sample.
2 Related Work
Prior cryo-EM methods model discrete or continuous variability but struggle with complex nonlinear motion, strong compositional mixtures, or unknown poses. Hydra combines an ensemble of neural fields with joint estimation of compositional states and orientations to address these limitations.
- Discrete Variability: Classical methods such as RELION and cryoSPARC model heterogeneous structures, but their sequential treatment of discrete and continuous variability can produce inaccurate poses for large motions.
- Non-Linear Methods for Heterogeneous Reconstruction: Nonlinear approaches capture continuous motion using manifolds, Laplacian methods, normal modes, or Zernike representations, but have limited applicability or rely on structural priors.
- Ab Initio Heterogeneous Reconstruction: Ab initio methods address unknown poses through pose search, distribution matching, or neural architectures, but the related approaches do not provide Hydra's stated joint compositional and conformational treatment.
- Heterogeneous Reconstruction in Cryo-EM: Hydra represents accessible structures with K neural networks, each specialized to one compositional state, while jointly estimating structures and orientations.
- Dynamic Scenes and Large-Scale Scenes: Hydra uses modulation for continuous protein motion and an ensemble of neural fields for capacity, while a variational approach estimates each image's compositional state during pose and network optimization.
- Adaptive Mixtures of Experts: As an adaptive mixture of experts, Hydra uses a Gaussian-mixture negative log-probability objective, while its noisy-image gating relies on autodecoding rather than image content.
3 Methods
Hydra models cryo-EM density maps as a finite union of low-dimensional neural-field manifolds, jointly representing compositional classes and within-class conformations. Its latent-variable likelihood model and hybrid optimization infer poses, conformations, class assignments, and shared neural-field parameters from observed images.
- Image formation model: Each cryo-EM image is modeled from a randomly oriented projection of a density map, modified by its contrast transfer function and isotropic Gaussian noise.The pose comprises a rotation and in-plane translation, while the projection operator maps the 3D density field to a 2D image.
- Heterogeneity model: Hydra represents each density map as belonging to one of K compositional states, with each state’s conformational variation described by d degrees of freedom.The model assumes all density maps lie on a finite union of low-dimensional manifolds parameterized by neural fields.
- Latent-variable model: Hydra treats pose, conformation, and class identity as latent variables, while the K neural-field parameters are shared model parameters.Class probabilities are parameterized with a softmax over per-image scores.
- Objective: The negative log-likelihood combines image-reconstruction error under each class with the corresponding class probability.This likelihood-based objective is minimized over poses, conformations, class scores, and neural-field parameters.
- Implementation: Each low-dimensional manifold is implemented with a residual multilayer perceptron, and each image’s final class is the highest-probability class.The class is selected from the largest entry of the per-image softmax score vector.
- Optimization: Hydra alternates stochastic gradient descent for conformations, class scores, and neural fields with hierarchical pose search to avoid pose-related local minima.After a computationally expensive initial search, pose optimization switches to stochastic gradient descent once poses are near their global optima.
4 Experiments
Hydra is evaluated across synthetic and experimental datasets to reconstruct compositional heterogeneity, including mixtures with simultaneous conformational variation. It matches or exceeds competing methods on classification and reconstruction metrics while operating ab initio.
- 4 Experiments: The experiments cover synthetic static mixtures, an experimental lysate mixture, and synthetic data combining compositional and conformational heterogeneity.The evaluation is organized around expressiveness, experimental compositional separation, and simultaneous reconstruction of both heterogeneity types.
- 4.1 Ab initio reconstruction of compositional heterogeneity: With K = 3, Hydra recovers all three tomotwin3 density maps with perfect classification accuracy, whereas K = 1 fails and K = 5 leaves two classes empty.The K = 1 model is DRGN-AI with d = 8, while Hydra uses d = 2 in the K = 3 and K = 5 settings.
- 4.1 Ab initio reconstruction of compositional heterogeneity: Hydra outperforms all methods on per-image FSC while matching cryoSPARC’s classification quality on the tomotwin3 mixture.The comparison uses ARI for classification and mean area under the FSC curve for reconstruction quality.
- 4.2 Ab initio reconstruction of an experimental cryo-EM mixture dataset: On the experimental mixture, Hydra with K = 4 separates three protein complexes plus a junk class, while DRGN-AI fails to learn distinct non-RyR structures.DRGN-AI also fails to reveal the CIII class when trained from homogeneous-reconstruction poses.
- 4.3 Ab initio reconstruction of conformational and compositional heterogeneity: On ribosplike data, Hydra with K = 3 nearly perfectly separates the three classes and captures conformational changes within each neural field.It achieves superior ARI and per-image FSC and the lowest pose error among the evaluated ab initio methods.
5 Discussion
Hydra expands neural-field reconstruction to mixed protein species and continuous conformational variation, but its current workflow requires advance specification of the class count. The method is positioned for complex in situ mixtures, with cryo-ET extension left for future work.
- 5 Discussion: Hydra models each particle as arising from one of K neural fields, enabling fully ab initio reconstruction of compositional and conformational heterogeneity.It improves over previous neural-based methods on compositionally heterogeneous datasets and is demonstrated on synthetic and experimental data.
- 5 Discussion: Hydra requires the number of classes K to be specified before reconstruction, often requiring multiple runs over candidate values.The authors note that each reconstruction can take up to 4 GPU-days, making hyperparameter sweeps costly.
- 5 Discussion: The current implementation also assigns the same latent dimension d to every class.The authors suggest dictionary-based representations as a possible way to relax this constraint.
- 5 Discussion: Hydra is especially suited to mixtures of dynamic complexes in situ, but extending it to subtomogram averaging for cryo-ET remains future work.Cryo-ET is described as collecting data while progressively tilting the sample.
A Architectural Details
Hydra represents each density-map manifold with a neural network conditioned on conformation and uses hierarchical pose search before switching to stochastic optimization. Fourier features and residual layers parameterize the neural fields.
- A Architectural Details: Each neural field maps a 3D frequency coordinate and conformation to the Hartley transform of a particle’s electron-scattering potential.The frequency coordinate is expanded using randomly sampled Fourier features, and the network contains three residual hidden layers of size 128.
- A Architectural Details: Hierarchical pose search evaluates a predefined SO(3) × R2 grid, retains the eight lowest-error poses, and refines them locally.The grid contains 4,608 rotations and 49 translations, with bandlimited images and an increasing frequency cutoff.
- A Architectural Details: After at least two pose-search epochs, Hydra initializes stochastic-gradient pose optimization from the latest hierarchical-search poses.This switches the poses from a fixed grid to continuous optimization.
C Conformation Estimation
Hydra independently optimizes conformations and class scores with stochastic gradient descent. Conformations begin from a low-variance Gaussian initialization, while scores become class probabilities through softmax.
- C Conformation Estimation: Conformations are independently optimized by SGD after random initialization from a d-dimensional Gaussian with standard deviation 0.1.Unless otherwise stated, the conformational dimension is d = 2.
- C Conformation Estimation: Class scores are optimized by SGD and converted into K-dimensional probability vectors with a softmax operator.This provides the class-assignment probabilities used by the mixture model.
E Optimization Parameters
Hydra training uses separate learning rates for scores, conformations, poses, and neural-network weights, with Adam and no weight decay.
- Adam optimization uses learning rates of 0.1 for scores, 0.01 for conformations, 0.001 for poses, and 0.0001 for neural-network weights.Weight decay is not used.
F Synthetic Datasets
The synthetic evaluations cover compositional heterogeneity alone and combined compositional–conformational heterogeneity, using controlled image-generation and training protocols across Hydra and baseline methods.
- Compositional heterogeneity: The tomotwin3 experiment evaluates compositional heterogeneity using 3,000 synthetic images from three protein structures.Hydra is tested with both the correct class count and an overparameterized configuration.
- Compositional heterogeneity: Hydra is compared with CryoDRGN2, DRGN-AI, and cryoSPARC under specified latent-space, training-duration, and class-count settings.The baselines use an 8-dimensional latent or conformational space where specified, while cryoSPARC uses three classes.
- Compositional heterogeneity: Volume metrics compare reconstructions with downsampled ground truth maps, and tomotwin3 experiments run on one NVIDIA A100 GPU.Hydra requires 4h00min for K = 3 and 6h20min for K = 5, compared with 1h20min for DRGN-AI and 1h50min for cryoDRGN2.
- Conformational and compositional heterogeneity: The mixed-heterogeneity simulation generates trajectories for spliceosome, ribosome, and spike-protein density maps before standardizing their sizes and applying soft masks.The maps are derived from trained cryoDRGN models and sampled along latent-space trajectories.
- Conformational and compositional heterogeneity: Synthetic projection images use uniformly sampled rotations and translations, experimental CTF values, and noise at SNR 0.1.Hydra is trained for 100,000 HPS images followed by 100 SGD epochs, while cryoDRGN2 is trained for 90 epochs.
G Real Dataset
The real-data evaluation uses a manually picked, downsampled ryanodine receptor dataset and compares Hydra with DRGN-AI and cryoSPARC processing workflows.
- Dataset details: The experimental dataset contains 85,656 manually picked particles downsampled to D = 150 pixels at 3.32 Å/pixel.The particles came from one round of 2D classification of a larger 148,596-particle dataset.
- DRGN-AI and Hydra: DRGN-AI and Hydra are trained on four 80GB A100 GPUs, with ab initio DRGN-AI using 500k images for HPS followed by 100 SGD epochs.Single-class DRGN-AI analyses also separate RyR and non-RyR particles using latent-space clustering.
- CryoSPARC processing: CryoSPARC ab initio is swept from K = 1 through K = 8, with K = 6 selected because lower values insufficiently separate junk from protein.The selected K = 6 run produces three different RyR density maps and is used for heterogeneous-refinement comparison with Hydra.
- Supplementary materials: The supplementary materials include three movies showing continuous motion and ten ribosome and spliceosome volumes in MRC format.
I Additional Figures
Additional figures document dataset examples, robustness to overestimating K, alternative baseline analyses, and Hydra’s experimental-data visualizations and comparisons.
- Dataset examples: Figure S1 provides 25 representative images for tomotwin3, the experimental ryanodine receptor dataset, and the ribosplike synthetic dataset.The datasets use different box sizes and pixel sizes.
- Tomotwin3: Hydra reconstructs the three tomotwin3 states with K = 7, while four excess classes remain empty.The figure includes reconstructed maps and a confusion matrix for 3,000 particles.
- Baseline analyses: The supplementary figures examine cryoSPARC ab initio uncertainty, DRGN-AI latent-space sampling and filtering, and baseline reconstructions on the ribosplike dataset.The DRGN-AI latent space shows main RyR and non-RyR clusters, while the ribosplike comparison includes CryoDRGN2, DRGN-AI, and cryoSPARC outputs.
- Experimental ryanodine receptor: Figure S6 shows Hydra K = 4 latent spaces, density-map views including transmembrane helices in CIII, and agreement with cryoSPARC K = 6 class assignments.Agreement bars are normalized by the total number of particles in each Hydra class.
J Additional Tables
The additional tables extend quantitative evaluation across datasets and Hydra configurations, covering reconstruction quality, classification, pose accuracy, and resolution.
- Table 3 reports tomowin3 classification accuracy by adjusted Rand index and reconstruction quality by mean area under the FSC curve, including ±1 standard deviation.Best and second-best results are visually marked in the table.
- Table 4 evaluates Hydra across varying K on the larger tomotwin3 dataset using pose errors and FSC-based resolution.The table notes good reconstruction quality despite high pose errors for structure 6up6, suggesting pose ambiguity.
- Hydra outperforms state-of-the-art methods on the riboslike dataset when jointly capturing conformational and compositional heterogeneity.The comparison includes ARI, rotational and translational errors, and per-image FSC area.
NeurIPS Paper Checklist
The checklist records that the paper presents an experimental neural reconstruction model, supplies reproducibility information, and reports replicated results. It also states that the work has no identified high-risk release or malicious-use concerns.
- The paper introduces a neural model for compositional and conformational heterogeneity, supported by synthetic and experimental results.
- The authors state that implementation details and experimental-setting information are provided in supplementary materials.
- The checklist states that open-access data and code are available through the Hydra project website.
- The paper reports results from three experimental replicas in Table 3 for statistical-significance assessment.