PhaseT3M: 3D Imaging at 1.6 Å Resolution via Electron Cryo-Tomography with Nonlinear Phase Retrieval

TL;DR

PhaseT3M introduces a 3D phase-retrieval framework for cryo-electron tomography that explicitly models multiple scattering via a multislice forward model and optimizes alignment with Bayesian methods. By enforcing a positivity constraint, it recovers missing-wedge information and achieves near-atomic $1.6~\AA$ resolution for a $\,Co_3O_4$ nanoparticle on a carbon support, using standard HRTEM equipment. Simulations and biological tests (HIV-1 EMPIAR-10164) demonstrate broader applicability and superior resolution/artifact suppression relative to conventional tomography, including significant improvements in carbon-support visibility and particle features from single tilt-series data. Although computationally intensive, PhaseT3M establishes a practical pathway to high-fidelity 3D imaging of heterogeneous, radiation-sensitive materials across materials, biology, and related fields.

Abstract

Electron cryo-tomography (cryo-ET) enables 3D imaging of complex, radiation-sensitive structures with molecular detail. However, image contrast from the interference of scattered electrons is nonlinear with atomic density and multiple scattering further complicates interpretation. These effects degrade resolution, particularly in conventional reconstruction algorithms, which assume linearity. Particle averaging can reduce such issues but is unsuitable for heterogeneous or dynamic samples ubiquitous in biology, chemistry, and materials sciences. Here, we develop a phase retrieval-based cryo-ET method, PhaseT3M. We experimentally demonstrate its application to a ~7 nm Co3O4 nanoparticle on ~30 nm carbon substrate, achieving a maximum resolution of 1.6 Å, surpassing conventional limits using standard cryo-TEM equipment. PhaseT3M uses a multislice model for multiple scattering and Bayesian optimization for alignment and computational aberration correction, with a positivity constraint to recover 'missing wedge' information. Applied directly to biological particles, it enhances resolution and reduces artifacts, establishing a new standard for routine 3D imaging of complex, radiation-sensitive materials.

Figures (13)

• Figure 1: Steps for Experimental Phase Retrieval-based Electron Cryo-Tomography, PhaseT$\boldsymbol{\exists}$M.a-c, Workflow for collecting tomographic tilt-series data at multiple defocus values. a, Photograph of a commercial Cryo-TEM setup for measuring HRTEM tomographic data. b, Schematic illustration of tomographic tilt-series acquisition at different defocus levels. c, Acquisition of HRTEM tilt series images at multiple defocus settings using a standard Cryo-TEM instrument. d-f, Preprocessing steps for 3D reconstruction. d, Plot illustrating the estimated defocus values at each tilt angle, showing tilt-dependent variations. e, Alignment procedure, where the optimal alignment with minimal reconstruction error is identified through a grid search for 2D shifts. f, Estimation of microscope parameters via Bayesian optimization to minimize reconstruction errors. g-h, Workflow for experimental tomography reconstruction. g, Forward and inverse multislice models for solving the scattering problem. At each tilt angle, the multislice forward model calculates simulated HRTEM projections, and the inverse model reconstructs the 3D volume. h, Schematic illustration of phase contrast tomography setup. The phase contrast tomography directly reconstructs the 3D potential using multiple tilted focal series of HRTEM images. The 3D volume within the black box represents the reconstructed potential of a Co$_3$O$_4$ nanoparticle embedded in a thick carbon support. Note that, to enhance visualization, distinct color scales are applied to differentiate the Co$_3$O$_4$ nanoparticle from the carbon support. The total 3D reconstruction achieves a voxel size of 2.08 Å, after applying a binning factor of 4.
• Figure 2: Experimental 3D Reconstruction with Phase Retrieval (PhaseT$\boldsymbol{\exists}$M) and Resolution Analysis.a, Half-sectioned 3D density map showing the internal cross-section of the full 3D reconstruction of a Co$_3$O$_4$ nanoparticle partially embedded in a carbon support. b, 3D density map of the Co$_3$O$_4$ nanoparticle after applying a 3D mask to remove the carbon support. Scale bar: 1 nm. c, Top panel: 2-Å thick central slices of the 3D reconstruction in (b), with frame colors corresponding to the slice positions in the 3D volume of (b). Bottom panel: 2D Fourier transform of the central slice shown in the top panel. Scale bars: Top panel, 2 nm; Bottom panel, 0.4 Å$^{-1}$. d, 3D Fourier transform of the reconstruction in (b), displaying diffraction peaks at up to 1.6 Å resolution. e, 3D Fourier transform of the reconstruction in (b) from a different view, highlighting the missing wedge region. The image shows restored diffraction peaks within the missing wedge and weak diffraction peaks at 1.6 Å resolution. Red arrows indicate the 1.6 Å resolution peaks. f, Power spectrum of the 3D reconstruction shown in (b), indicating a resolution limit of 2.0 Å.
• Figure 3: Simulated 3D Reconstruction with Phase Retrieval (PhaseT$\boldsymbol{\exists}$M) and Resolution Analysis.a, 3D atomic models of the Co$_3$O$_4$ nanoparticle and the carbon support. The 3D atomic position of the Co$_3$O$_4$ nanoparticle is based on a perfect crystal, whereas the 3D atomic position of the amorphous carbon support is generated randomly. b, 3D density map of the Co$_3$O$_4$ nanoparticle after applying a 3D mask to remove the carbon support. Scale bar: 1 nm. c, Top panel: 2-Å thick central slices of the 3D reconstruction in (b), with each color frame corresponding to the slice color in the 3D reconstruction. Bottom panel: 2D Fourier transform of the central slice shown in the top panel. Scale bars: Top panel, 2 nm; Bottom panel, 0.4 Å$^{-1}$. d, 3D Fourier transform of the reconstruction in (b), displaying diffraction peaks at 1.4 Å resolution. e, 3D Fourier transform of the reconstruction in (b) from a different view, highlighting the missing wedge region. The image reveals restored diffraction peaks within the missing wedge and weak diffraction peaks at 1.4 Å resolution. The red arrows represent the 1.4 Å resolution limit. f, Power spectrum of the 3D reconstruction in (b), indicating a resolution of 1.6 Å.
• Figure 4: Comparative Analysis of Phase Retrieval (PhaseT$\boldsymbol{\exists}$M) and SIRT Reconstructions for the Co$_3$O$_4$ nanoparticle.a, e, 3D density maps of the Co$_3$O$_4$ nanoparticle and the carbon support reconstructed using phase retrieval (PhaseT$\exists$M) (a) and SIRT (e). The 3D volumes were reconstructed with a voxel size of 2.08 Å. Scale bar: 5 nm. b, f, 3D density maps of the Co$_3$O$_4$ nanoparticle reconstructed using phase retrieval (PhaseT$\exists$M) (b) and SIRT (f), after applying a 3D mask to remove the carbon support intensity. The 3D volumes were reconstructed with a voxel size of 0.52 Å. Scale bar: 1 nm. c, g, 1 Å-thick slices extracted from the 3D reconstructions in (b) and (f), respectively. Scale bar: 2 nm. d, h, 2D Fourier transforms of the slices shown in (c) and (g), respectively. Red circles indicate diffraction peaks present in (d) but absent in (h). Scale bar: 0.4 Å$^{-1}$. The slice numbers in (c–d) and (g–h) correspond to the slice positions indicated in the 3D reconstructions in (b) and (f), respectively.
• Figure 5: Comparative Analysis of Phase Retrieval (PhaseT$\boldsymbol{\exists}$M) and SIRT Reconstructions of HIV-1 Particles (EMPIAR-10164).a-b, Half-sectioned 3D density maps showing the internal cross-sections of the full 3D reconstructions of HIV-1 particles using our phase retrieval method (PhaseT$\exists$M) (a) and the SIRT algorithm (without positivity) (b). Faint, ring-shaped features correspond to HIV-1 particles, while bright, spherical intensities represent gold fiducial markers. c, Fourier shell correlation (FSC) plots comparing the resolution of 3D reconstructions obtained from our phase retrieval method (a) and the SIRT algorithm (b). The FSC resolutions, determined using the 0.143 criterion, are 9.0 nm for phase contrast tomography and 13.1 nm for SIRT, respectively. d, e, 24 nm-thick slices extracted from the reconstructions using our phase-contrast tomography (d) and the SIRT (e), corresponding to the 3D volumes shown in (a) and (b), respectively. The frame colors indicate their positions within the 3D volumes shown in (a) and (b). Scale bars: 50 nm. e. f, Enlarged views (3$\times$) of the boxed regions in the phase retrieval slice (d) and the SIRT reconstruction slice (g), respectively. Pink dotted rings highlight regions where our phase-retrieval-based tomography shows improved resolution compared to the SIRT reconstruction. Scale bars: 150 nm.