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SHREC: A Spectral Embedding-Based Approach for Ab-Initio Reconstruction of Helical Molecules

Guy Shapira, Yoel Shkolnisky

Abstract

Cryo-electron microscopy (cryo-EM) has emerged as a powerful technique for determining the three-dimensional structures of biological molecules at near-atomic resolution. However, reconstructing helical assemblies presents unique challenges due to their inherent symmetry and the need to determine unknown helical symmetry parameters. Traditional approaches require an accurate initial estimation of these parameters, which is often obtained through trial and error or prior knowledge. These requirements can lead to incorrect reconstructions, limiting the reliability of ab initio helical reconstruction. In this work, we present SHREC (Spectral Helical REConstruction), an algorithm that directly recovers the projection angles of helical segments from their two-dimensional cryo-EM images, without requiring prior knowledge of helical symmetry parameters. Our approach leverages the insight that projections of helical segments form a one-dimensional manifold, which can be recovered using spectral embedding techniques. Experimental validation on publicly available datasets demonstrates that SHREC achieves high resolution reconstructions while accurately recovering helical parameters, requiring only knowledge of the specimen's axial symmetry group. By eliminating the need for initial symmetry estimates, SHREC offers a more robust and automated pathway for determining helical structures in cryo-EM.

SHREC: A Spectral Embedding-Based Approach for Ab-Initio Reconstruction of Helical Molecules

Abstract

Cryo-electron microscopy (cryo-EM) has emerged as a powerful technique for determining the three-dimensional structures of biological molecules at near-atomic resolution. However, reconstructing helical assemblies presents unique challenges due to their inherent symmetry and the need to determine unknown helical symmetry parameters. Traditional approaches require an accurate initial estimation of these parameters, which is often obtained through trial and error or prior knowledge. These requirements can lead to incorrect reconstructions, limiting the reliability of ab initio helical reconstruction. In this work, we present SHREC (Spectral Helical REConstruction), an algorithm that directly recovers the projection angles of helical segments from their two-dimensional cryo-EM images, without requiring prior knowledge of helical symmetry parameters. Our approach leverages the insight that projections of helical segments form a one-dimensional manifold, which can be recovered using spectral embedding techniques. Experimental validation on publicly available datasets demonstrates that SHREC achieves high resolution reconstructions while accurately recovering helical parameters, requiring only knowledge of the specimen's axial symmetry group. By eliminating the need for initial symmetry estimates, SHREC offers a more robust and automated pathway for determining helical structures in cryo-EM.
Paper Structure (22 sections, 9 theorems, 117 equations, 17 figures, 1 algorithm)

This paper contains 22 sections, 9 theorems, 117 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Prior work
  4. Spectral angle recovery
  5. Spectral embedding
  6. Projection angle recovery
  7. Discrete helices
  8. Implementation details
  9. Preprocessing
  10. Denoising
  11. Spectral angle recovery
  12. 3D reconstruction and refinement
  13. Initial model generation
  14. Helical parameters estimation
  15. 3D classification and refinement
  16. ...and 7 more sections

Key Result

Lemma 1.1

Let $\psi: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a function such that for all $x \in \mathbb{R}$, Let $R \in \mathrm{SO}(3)$ be a rotation matrix, and let $M: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ denote reflection across the $xy$-plane. In coordinates, $M$ is given by or equivalently, by the matrix $M = \mathrm{diag}(1, 1, -1)$. Define the mirror of $\psi$, denoted $\psi^M: \mathbb{R}^3 \rig

Figures (17)

  • Figure 1: Two helical segments extracted from different positions are related by rotation around the screw axis. Therefore, a 2D projection of one segment corresponds to a projection of the other from a different angle.
  • Figure 2: Power spectra estimation for projections from the EMPIAR-10019 dataset. (a) The (log) power spectrum of the projections. (b) The estimated (log) power spectrum of the noise. (c) The estimated (log) power spectrum of the signal.
  • Figure 3: The effect of the denoising process on a projection from the EMPIAR-10019 dataset. (a) The projection before denoising. (b) The same projection after denoising.
  • Figure 4: 2D class averages with their estimated resolution. The average of the chosen class is marked with a red frame.
  • Figure 5: The 2D embedding of the selected 3,023 helical segments from EMPIAR-10022. The circular structure corresponds to the varying projection angles. These embedding coordinates are used to derive initial angle estimates for 3D reconstruction.
  • ...and 12 more figures

Theorems & Definitions (24)

  • Lemma 1.1
  • Definition 1.2: Helix
  • Definition 1.3: Helical segment
  • Lemma 1.4: Translation-rotation correspondence
  • proof
  • Definition 1.5: Segment's angle
  • Lemma 1.6: Existence of a segment's angle
  • proof
  • Definition 1.7: Segment's projection
  • Definition 1.8: Projection angle
  • ...and 14 more