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Reconstructing continuous distributions of 3D protein structure from cryo-EM images
Ellen D. Zhong, Tristan Bepler, Joseph H. Davis, Bonnie Berger
TL;DR
Cryo-EM reconstruction must recover 3D structure from noisy projections even when particles vary in conformation and orientation. CryoDRGN learns continuous structural variation from unlabeled images by combining a neural-network generative model with exact pose inference and variational inference. The paper reports automatic identification of structural states in real data and continuous representation of simulated structural variation.
Problem
Structural heterogeneity complicates cryo-EM reconstruction, while discrete multiclass methods struggle when conformations are complex or lie along a continuum.
Source: §1 · p. 1, §2.2 · p. 3Method
CryoDRGN uses an image encoder and coordinate-based volume decoder to model continuous structural variation from unlabeled images, with exact pose inference and variational inference for heterogeneity.
Source: §1 · p. 2, §3 · p. 3, §3.2 · p. 5Results
CryoDRGN automatically identifies major states in a real ribosome dataset and learns a continuous representation along the true reaction coordinate in simulated heterogeneity.
Source: §6 · p. 9Takeaways & Limitations
The authors suggest the techniques may also apply to image and volume generative modeling in computer vision and 3D shape reconstruction.
Source: §6 · p. 9
Abstract
from arXiv · showhide
Cryo-electron microscopy (cryo-EM) is a powerful technique for determining the structure of proteins and other macromolecular complexes at near-atomic resolution. In single particle cryo-EM, the central problem is to reconstruct the three-dimensional structure of a macromolecule from $10^{4-7}$ noisy and randomly oriented two-dimensional projections. However, the imaged protein complexes may exhibit structural variability, which complicates reconstruction and is typically addressed using discrete clustering approaches that fail to capture the full range of protein dynamics. Here, we introduce a novel method for cryo-EM reconstruction that extends naturally to modeling continuous generative factors of structural heterogeneity. This method encodes structures in Fourier space using coordinate-based deep neural networks, and trains these networks from unlabeled 2D cryo-EM images by combining exact inference over image orientation with variational inference for structural heterogeneity. We demonstrate that the proposed method, termed cryoDRGN, can perform ab initio reconstruction of 3D protein complexes from simulated and real 2D cryo-EM image data. To our knowledge, cryoDRGN is the first neural network-based approach for cryo-EM reconstruction and the first end-to-end method for directly reconstructing continuous ensembles of protein structures from cryo-EM images.
1 INTRODUCTION
Cryo-EM reconstruction must infer 3D structure from noisy projections with unknown orientations, and structural variability makes the task harder. The paper introduces a neural-network method for learning continuous conformational variation from unlabeled images.
1 INTRODUCTION
- Reconstruction must infer a 3D electron-density volume and unknown image orientations from noisy projections.The images also involve unknown in-plane translations, low signal-to-noise ratios, and other measurement challenges.Source: §1 · p. 1
- Protein flexibility creates structural heterogeneity that discrete clusters may poorly approximate when conformations lie along a continuum.Each cryo-EM image contains a separate molecular instance, offering the potential to resolve this heterogeneity.Source: §1 · pp. 1–2
- CryoDRGN learns a continuous low-dimensional conformational manifold from unlabeled 2D images using an image encoder and volume decoder.Its coordinate-based decoder and Fourier-space inference model image formation, while exact pose inference is combined with variational inference for structural heterogeneity.Source: §1 · p. 2
2 BACKGROUND AND NOTATION
The background describes cryo-EM reconstruction as inferring a 3D volume and particle poses from noisy 2D projections. It then outlines Fourier-space image formation and the limits of traditional methods for complex or continuous heterogeneity.
2.1 IMAGE FORMATION MODEL
- Each noisy cryo-EM image is a projection of a molecular instance, so reconstruction must infer the 3D volume and each image’s unknown pose.The pose includes a 3D orientation and an in-plane translation.Source: Figure 1 · p. 2, §2.1 · p. 2
- The Fourier slice theorem represents a 2D projection as a central 2D slice of the 3D volume’s Fourier transform.The Fourier-space image model applies the microscope CTF and translation phase shift to the rotated slice, with frequency-dependent noise.Source: §2.1 · pp. 2–3
2.2 TRADITIONAL CRYO-EM RECONSTRUCTION
- Traditional reconstruction alternates between aligning images to the current volume and backprojecting them to update it, but the optimization is sensitive to initialization.The text describes expectation maximization and coordinate-ascent variants for estimating the volume while marginalizing over poses.Source: §2.2 · p. 3
- Multiclass refinement models heterogeneous samples as images generated from K independent volumes, but struggles with complex or continuous conformational variation.It often requires hierarchical refinement with manual choices of class counts and initial models; the underlying states are unknown.Source: §2.2 · p. 3
3 METHODS
CryoDRGN models a continuous distribution of 3D volumes from unlabeled 2D images, separating structural heterogeneity from imaging pose. Its coordinate-based decoder and distinct inference strategies connect image pixels to volumes and estimate latent variation and pose.
3.1 GENERATIVE MODEL
- A coordinate-based MLP represents a continuous manifold of Fourier-domain 3D electron densities, with coordinates enforcing consistency between 2D views through the Fourier slice theorem.The decoder models a distribution over volume density at each Fourier coordinate, conditioned on latent variable z.Source: §3.1 · p. 4
- For each image, a rotated fixed x-y lattice supplies the 3D Fourier coordinates used to compute its likelihood under pose R, translation t, and latent variable z.The image is centered for translation, and positional encodings of coordinates are supplied to the decoder.Source: §3.1 · p. 4
- For noisy datasets, the positional encoding omits its highest-frequency 10 percentile, which performed better than the noiseless-data encoding.The noiseless encoding spans wavelengths from 1 to the image-data Nyquist limit.Source: §3.1 · p. 4
3.2 INFERENCE
- A variational encoder estimates latent structural heterogeneity, while branch-and-bound globally searches image pose and frequency marching progressively expands the search to higher frequencies.The pose search is performed over SO(3) × R2 and uses low-frequency components first.Source: §3.2 · pp. 4–5
- Training compares many 2D views to learn the volume, and the authors describe this process as denoising; a separate network is trained for each experimental dataset.The stated optimization objective is a variational lower bound estimated using one Monte Carlo sample.Source: §3.2 · p. 5
4 RELATED WORK
Prior work spans homogeneous reconstruction pipelines, discrete and continuous approaches to structural heterogeneity, and neural methods for 3D reconstruction in computer vision. CryoDRGN extends coordinate-based modeling from 2D images to 3D cryo-EM reconstruction in Fourier space.
Homogeneous cryo-EM reconstruction
- Homogeneous cryo-EM reconstruction commonly starts with a low-resolution model, then iteratively alternates projection matching and structure refinement.Existing methods include Bayesian expectation-maximization refinement and voxel-based variational inference with known poses.Source: §4 · pp. 5–6
Heterogeneous cryo-EM reconstruction
- Multiclass refinement models heterogeneity as discrete independent volumes, often requiring expert-guided hierarchical clustering and manual choices of cluster count.Rigid-body models impose assumptions on heterogeneity; another continuous-manifold approach requires pose supervision and later clustering.Source: §4 · p. 6
Neural network 3D reconstruction in computer vision
- Computer-vision reconstruction methods often use known viewing directions, and their 2D viewpoints are not projections as in cryo-EM.Some methods explicitly encode viewpoint projection operations or geometric consistency.Source: §4 · p. 6
Coordinate-based neural networks in computer vision
- Coordinate-based image models have been used for 2D image modeling, and cryoDRGN extends these ideas to 3D cryo-EM reconstruction in the Fourier domain.Spatial-VAE uses coordinate-based modeling to enforce consistency between rotated 2D images.Source: §4 · p. 6
5 RESULTS
The results show cryoDRGN matching established tools on homogeneous reconstruction, identifying major LSU states, and recovering continuous structural variation that discrete reconstruction does not represent as a continuum.
5 RESULTS
- Across three evaluation settings, results validate homogeneous reconstruction, automatic learning of a pose-supervised latent manifold, and fully unsupervised reconstruction of continuous structure distributions.The paper reports the last capability as unavailable in existing tools.Source: §5 · p. 6
5.1 UNSUPERVISED HOMOGENEOUS RECONSTRUCTION
- On the real 80S ribosome dataset, cryoDRGN's inferred poses and volumes matched those from state-of-the-art tools.The dataset contained 105,247 images; no ground truth was available.Source: §5.1 · p. 7
5.2 HETEROGENEOUS RECONSTRUCTION WITH POSE SUPERVISION
- With poses fixed from alignment to an LSU template, cryoDRGN automatically identified all four major states and its latent encoding aligned with cryoSPARC clusters.It also separated impurity images into distinct clusters, but did not recover their structures because template-based poses were incorrect.Source: §5.2 · p. 7, Figure 3 · p. 7
5.3 UNSUPERVISED HETEROGENEOUS RECONSTRUCTION
- For a simulated circular 1D motion, cryoDRGN reconstructed a continuous sequence of volumes matching ground truth, while cryoSPARC's three-class model produced blurring artifacts.The decoder can reconstruct the full continuum; only 10 sampled structures were visualized.Source: §5.3 · p. 8, Figure 4 · p. 8
- CryoDRGN achieved higher FSC-resolution accuracy than cryoSPARC multiclass reconstruction on simulated heterogeneous datasets, with further improvement from tilt-series pairs.Performance was averaged across per-image comparisons between each MAP volume and its ground-truth volume.Source: §5.3 · p. 8
6 CONCLUSIONS
The conclusion presents cryoDRGN as a neural-network method for learning continuous protein-structure variation, with evidence from ribosome assembly intermediates and simulated continuous heterogeneity.
6 CONCLUSIONS
- CryoDRGN learns continuous variation in protein structure for single-particle cryo-EM reconstruction.The paper applies it to highly heterogeneous ribosome assembly intermediates and reports automatic partitioning of structural states.Source: §6 · p. 9
- In simulated continuous heterogeneity, cryoDRGN learns a continuous representation along the true reaction coordinate while disentangling imaging orientation from structural heterogeneity.Source: §6 · p. 9
- The described techniques may also apply to image and volume generative modeling in computer vision and 3D shape reconstruction.Source: §6 · p. 9
A.1 BRANCH AND BOUND IMPLEMENTATION DETAILS
CryoDRGN searches image poses with a multiresolution branch-and-bound procedure and frequency marching, alongside training options for homogeneous or heterogeneous reconstruction. The implementation uses a simplified image representation and noise model.
A.1 BRANCH AND BOUND IMPLEMENTATION DETAILS
- Pose search combines a uniform SO(3) × R2 grid, branch-and-bound pruning, and progressively higher-frequency objectives to find each image’s maximum-likelihood pose.The SO(3) grid starts with 576 orientations and is subdivided five times; frequency marching begins with low-frequency components.Source: §A.1 · p. 12
- At a sufficiently fine base resolution, pose inference is tractable in hours on one GPU, but pruning can discard a pose whose finer-grid neighbor has lower loss.Incorporating neighboring-pose loss into the lower bound would require assumptions about loss smoothness and is left for future work.Source: §A.1 · p. 12
A.2 TRAINING DETAILS
- CryoDRGN supports homogeneous reconstruction with pose inference, heterogeneous reconstruction with known poses, and unsupervised heterogeneous reconstruction with joint pose and latent-variable inference.Source: §A.2 · p. 13
- Images use a real-valued Hartley representation, while the simplified white-noise model makes the reconstruction loss a mean squared error.A colored-noise model is left for future work.Source: §A.2 · p. 13
B.1 DATASET PREPARATION
The experiments use simulated projections and a prepared real cryo-EM dataset, then train and refine a volume decoder using inferred poses.
B.1 DATASET PREPARATION
- Each simulated dataset contains 50,000 projections of a 128 × 128 volume, with uniform rotations, translations, and optional noise at SNR 0.1.SNR is defined as signal variance divided by noise variance.Source: §B.1 · p. 13
- The real homogeneous-reconstruction dataset comes from EMPIAR-10028, downsampled fourfold and phase-flipped using its contrast transfer function.Source: §B.1 · p. 13
B.2 TRAINING
- Training uses an initial epoch with random orientations, four epochs of branch-and-bound pose inference, then 15 refinement epochs with a larger decoder and fixed poses.Reported times are for 50,000-image, 128 × 128 datasets on a Titan V GPU.Source: §B.2 · p. 13
B.3 SUPPLEMENTARY RESULTS
The supplementary results report homogeneous reconstructions matching state-of-the-art and describe preparation and training for a pose-supervised heterogeneous ribosome reconstruction.
B.3 SUPPLEMENTARY RESULTS
- CryoDRGN’s unsupervised homogeneous reconstructions match state-of-the-art on two simulated datasets and one real cryo-EM dataset.The accompanying FSC curves compare reconstructed volumes with ground truth for synthetic ribosome datasets.Source: Figure S5 · p. 14
C HETEROGENEOUS RIBOSOME RECONSTRUCTION WITH POSE SUPERVISION
- The EMPIAR-10076 dataset contains 131,899 E. coli large ribosomal subunit images, with poses aligned to a mature-subunit consensus reconstruction.Source: C HETEROGENEOUS RIBOSOME… · p. 14
- CryoSPARC multiclass refinement initialized from six published states reproduced image assignments, and structures other than low-population classes A and F correlated well with published volumes.The initial models were low-pass filtered to 25 Å.Source: C HETEROGENEOUS RIBOSOME… · p. 14, Figure S6 · p. 14
- Pose-supervised cryoDRGN training used a one-dimensional latent variable for 200 epochs with fixed image poses.Images were centered and phase-flipped before encoding.Source: C HETEROGENEOUS RIBOSOME… · p. 15
C.1 SUPPLEMENTARY RESULTS
The supplementary figures show latent encodings and sampled volumes for ribosome assembly states, and relate the encoding to cryoSPARC class assignments.
C.1 SUPPLEMENTARY RESULTS
- Figure S7 shows image latent encodings, sampled volumes along the latent space, and FSC curves comparing four structures with published classes B–E.Source: Figure S7 · p. 15
- Figure S8 compares the latent encoding with cryoSPARC multiclass assignments for images in classes D and E.Source: Figure S8 · p. 15
D.1 DATASET PREPARATION
The supplementary experiments construct simulated datasets spanning continuous and discrete structural variation, then describe tilt-pair imaging and cryoDRGN training settings.
D.1 DATASET PREPARATION
- Four simulated datasets cover linear 1D, linear 2D, circular 1D, and ten discrete states, each yielding 50,000 projection images.The datasets use uniformly sampled rotations and translations from −5 to 5 pixels, with 64×64 images.Source: §D.1 · pp. 15–16
D.2 TILT SERIES PAIRS
- Tilt-series pairs provide images of the same molecule before and after a known stage tilt; paired encodings encourage pose-invariant heterogeneity inference and can identify chirality.The encoder uses separate image-encoding and joint latent-mapping MLPs.Source: §D.2 · p. 16
D.3 TRAINING
- Training used 1-D latent variables for linear 1D motion and 10-D variables for the other datasets, with random angles initially and branch-and-bound pose inference afterward.Runs used 40 epochs without tilt pairs and 20 with them; pose inference took 120 versus 2 minutes per epoch with and without branch-and-bound, respectively.Source: §D.3 · p. 17
D.4 SUPPLEMENTARY RESULTS
Supplementary reconstructions show that cryoDRGN samples often track simulated structural variation, while manifold recovery varies by dataset and tilt-series use. The ten-state comparison reports 10 structures reconstructed with tilt series versus 8 of 10 learned by cryoSPARC.
D.4 SUPPLEMENTARY RESULTS
- For linear and circular 1D datasets, cryoDRGN sampled structures match ground truth; circular latent images also match the true manifold topology.CryoSPARC reconstructions show artifacts in both comparisons.Source: Figure S10 · p. 17, Figure S11 · p. 18
- For linear 2D motion, no-tilt samples roughly match structures but miss the manifold distribution; tilt-series results recover the variation and manifold topology.CryoSPARC volumes show artifacts at low isosurface.Source: Figure S12 · p. 19
- With tilt series, cryoDRGN reconstructs all 10 discrete structures and clusters images accordingly, while cryoSPARC correctly learns 8 of 10.Source: Figure S13 · p. 20
- Tables S3 and S4 relate cryoSPARC class count and cryoDRGN latent dimension to FSC=0.5 reconstruction resolution, averaged over 100 images; lower is better and 2 pixels is best.Source: Table S3 · p. 20, Table S4 · p. 20