Self Fourier shell correlation: properties and application to cryo-ET.

The Fourier shell correlation (FSC) is a measure of the similarity between two signals computed over corresponding shells in the frequency domain and has broad applications in microscopy. In structural biology, the FSC is ubiquitous in methods for validation, resolution determination, and signal enhancement. Computing the FSC usually requires two independent measurements of the same underlying signal, which can be limiting for some applications. Here, we analyze and extend on an approach to estimate the FSC from a single measurement. In particular, we derive the necessary conditions required to estimate the FSC from downsampled versions of a single noisy measurement. These conditions reveal additional corrections which we implement to increase the applicability of the method. We then illustrate two applications of our approach, first as an estimate of the global resolution from a single 3-D structure and second as a data-driven method for denoising tomographic reconstructions in electron cryo-tomography. These results provide general guidelines for computing the FSC from a single measurement and suggest new applications of the FSC in microscopy.

Moment-based metrics for molecules computable from cryogenic electron microscopy images.

Single-particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution three-dimensional (3D) structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3D reconstruction, known as Kam's method, relies on the moments of the two-dimensional (2D) images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate the uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

A Bayesian Framework for Cryo-EM Heterogeneity Analysis using Regularized Covariance Estimation.

Proteins and the complexes they form are central to nearly all cellular processes. Their flexibility, expressed through a continuum of states, provides a window into their biological functions. Cryogenic-electron microscopy (cryo-EM) is an ideal tool to study these dynamic states as it captures specimens in non-crystalline conditions and enables high-resolution reconstructions. However, analyzing the heterogeneous distribution of conformations from cryo-EM data is challenging. Current methods face issues such as a lack of explainability, overfitting caused by lack of regularization, and a large number of parameters to tune; problems exacerbated by the lack of proper metrics to evaluate or compare heterogeneous reconstructions. To address these challenges, we present RECOVAR, a white-box method based on principal component analysis (PCA) computed via regularized covariance estimation that can resolve intricate heterogeneity with similar expressive power to neural networks with significantly lower computational demands. We extend the ubiquitous Bayesian framework used in homogeneous reconstruction to automatically regularize principal components, overcoming overfitting concerns and removing the need for most parameters. We further exploit the conservation of density and distances endowed by the embedding in PCA space, opening the door to reliable free energy computation. We leverage the predictable uncertainty of image labels to generate high-resolution reconstructions and identify high-density trajectories in latent space. We make the code freely available at https://github.com/ma-gilles/recovar.

Self Fourier shell correlation: properties and application to cryo-ET.

The Fourier shell correlation (FSC) is a measure of the similarity between two signals computed over corresponding shells in the frequency domain and has broad applications in microscopy. In structural biology, the FSC is ubiquitous in methods for validation, resolution determination, and signal enhancement. Computing the FSC usually requires two independent measurements of the same underlying signal, which can be limiting for some applications. Here, we analyze and extend on an approach proposed by Koho et al. [1] to estimate the FSC from a single measurement. In particular, we derive the necessary conditions required to estimate the FSC from downsampled versions of a single noisy measurement. These conditions reveal additional corrections which we implement to increase the applicability of the method. We then illustrate two applications of our approach, first as an estimate of the global resolution from a single 3-D structure and second as a data-driven method for denoising tomographic reconstructions in electron cryo-tomography. These results provide general guidelines for computing the FSC from a single measurement and suggest new applications of the FSC in microscopy.

A molecular prior distribution for Bayesian inference based on Wilson statistics.

BACKGROUND AND OBJECTIVE: Wilson statistics describe well the power spectrum of proteins at high frequencies. Therefore, it has found several applications in structural biology, e.g., it is the basis for sharpening steps used in cryogenic electron microscopy (cryo-EM). A recent paper gave the first rigorous proof of Wilson statistics based on a formalism of Wilson's original argument. This new analysis also leads to statistical estimates of the scattering potential of proteins that reveal a correlation between neighboring Fourier coefficients. Here we exploit these estimates to craft a novel prior that can be used for Bayesian inference of molecular structures. METHODS: We describe the properties of the prior and the computation of its hyperparameters. We then evaluate the prior on two synthetic linear inverse problems, and compare against a popular prior in cryo-EM reconstruction at a range of SNRs. RESULTS: We show that the new prior effectively suppresses noise and fills-in low SNR regions in the spectral domain. Furthermore, it improves the resolution of estimates on the problems considered for a wide range of SNR and produces Fourier Shell Correlation curves that are insensitive to masking effects. CONCLUSIONS: We analyze the assumptions in the model, discuss relations to other regularization strategies, and postulate on potential implications for structure determination in cryo-EM.