Efficient high-resolution refinement in cryo-EM with stochastic gradient descent.

Electron cryomicroscopy (cryo-EM) is an imaging technique widely used in structural biology to determine the three-dimensional structure of biological molecules from noisy two-dimensional projections with unknown orientations. As the typical pipeline involves processing large amounts of data, efficient algorithms are crucial for fast and reliable results. The stochastic gradient descent (SGD) algorithm has been used to improve the speed of ab initio reconstruction, which results in a first, low-resolution estimation of the volume representing the molecule of interest, but has yet to be applied successfully in the high-resolution regime, where expectation-maximization algorithms achieve state-of-the-art results, at a high computational cost. In this article, we investigate the conditioning of the optimization problem and show that the large condition number prevents the successful application of gradient descent-based methods at high resolution. Our results include a theoretical analysis of the condition number of the optimization problem in a simplified setting where the individual projection directions are known, an algorithm based on computing a diagonal preconditioner using Hutchinson's diagonal estimator, and numerical experiments showing the improvement in the convergence speed when using the estimated preconditioner with SGD. The preconditioned SGD approach can potentially enable a simple and unified approach to ab initio reconstruction and high-resolution refinement with faster convergence speed and higher flexibility, and our results are a promising step in this direction.

Methods for Cryo-EM Single Particle Reconstruction of Macromolecules Having Continuous Heterogeneity.

Macromolecules change their shape (conformation) in the process of carrying out their functions. The imaging by cryo-electron microscopy of rapidly-frozen, individual copies of macromolecules (single particles) is a powerful and general approach to understanding the motions and energy landscapes of macromolecules. Widely-used computational methods already allow the recovery of a few distinct conformations from heterogeneous single-particle samples, but the treatment of complex forms of heterogeneity such as the continuum of possible transitory states and flexible regions remains largely an open problem. In recent years there has been a surge of new approaches for treating the more general problem of continuous heterogeneity. This paper surveys the current state of the art in this area.

Integrating Molecular Models Into CryoEM Heterogeneity Analysis Using Scalable High-resolution Deep Gaussian Mixture Models.

Resolving the structural variability of proteins is often key to understanding the structure-function relationship of those macromolecular machines. Single particle analysis using Cryogenic electron microscopy (CryoEM), combined with machine learning algorithms, provides a way to reveal the dynamics within the protein system from noisy micrographs. Here, we introduce an improved computational method that uses Gaussian mixture models for protein structure representation and deep neural networks for conformation space embedding. By integrating information from molecular models into the heterogeneity analysis, we can analyze continuous protein conformational changes using structural information at the frequency of 1/3 Å-1, and present the results in a more interpretable form.

On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena.

Many techniques in machine learning attempt explicitly or implicitly to infer a low-dimensional manifold structure of an underlying physical phenomenon from measurements without an explicit model of the phenomenon or the measurement apparatus. This paper presents a cautionary tale regarding the discrepancy between the geometry of measurements and the geometry of the underlying phenomenon in a benign setting. The deformation in the metric illustrated in this paper is mathematically straightforward and unavoidable in the general case, and it is only one of several similar effects. While this is not always problematic, we provide an example of an arguably standard and harmless data processing procedure where this effect leads to an incorrect answer to a seemingly simple question. Although we focus on manifold learning, these issues apply broadly to dimensionality reduction and unsupervised learning.

Image Reconstruction in Light-Sheet Microscopy: Spatially Varying Deconvolution and Mixed Noise.

We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. We introduce a model of the image formation process that incorporates this interaction and we formulate a variational model that accounts for the combination of Poisson and Gaussian noise through a data fidelity term consisting of the infimal convolution of the single noise fidelities, first introduced in L. Calatroni et al. (SIAM J Imaging Sci 10(3):1196-1233, 2017). We establish convergence rates and a discrepancy principle for the infimal convolution fidelity and the inverse problem is solved by applying the primal-dual hybrid gradient (PDHG) algorithm in a novel way. Numerical experiments performed on simulated and real data show superior reconstruction results in comparison with other methods.