Single-Particle Cryo-Electron Microscopy: Mathematical Theory, Computational Challenges, and Opportunities

Abstract

In recent years, an abundance of new molecular structures have been elucidated using cryo-electron microscopy (cryo-EM), largely due to advances in hardware technology and data processing techniques. Owing to these new exciting developments, cryo-EM was selected by Nature Methods as Method of the Year 2015, and the Nobel Prize in Chemistry 2017 was awarded to three pioneers in the field.The main goal of this article is to introduce the challenging and exciting computational tasks involved in reconstructing 3-D molecular structures by cryo-EM. Determining molecular structures requires a wide range of computational tools in a variety of fields, including signal processing, estimation and detection theory, high-dimensional statistics, convex and non-convex optimization, spectral algorithms, dimensionality reduction, and machine learning. The tools from these fields must be adapted to work under exceptionally challenging conditions, including extreme noise levels, the presence of missing data, and massively large datasets as large as several Terabytes.In addition, we present two statistical models: multi-reference alignment and multi-target detection, that abstract away much of the intricacies of cryo-EM, while retaining some of its essential features. Based on these abstractions, we discuss some recent intriguing results in the mathematical theory of cryo-EM, and delineate relations with group theory, invariant theory, and information theory.

Multi-Target Detection With an Arbitrary Spacing Distribution

“…Our study of the MTD model is primarily motivated by the goal to reconstruct the structures of small biomolecules using cryo-EM [11], [21]. In a cryo-EM experiment, individual copies of the target biomolecule are dispersed at unknown 2D locations and 3D orientations in a thin layer of vitreous ice, from which 2D tomographic projection images are produced by an electron microscope.…”

CryoGAN: A New Reconstruction Paradigm for Single-Particle Cryo-EMviaDeep Adversarial Learning

“…Alternatively, the problem can be minimized using stochastic gradient descent (e.g., during the ab initio phase of [11]); here, the challenge is that the involved gradients require computations over all poses. For a more in-depth discussion, see [14,15,38]. Likelihood-free methods for cryo-EM reconstruction are relatively few.…”

“…Hence, the outcome of the global process is predicated on the quality of the initial reconstruction [12,13]. Moreover, one still often relies on the input of an expert user for appropriate processing decisions and parameter tuning [14]. Even for more automated methods, the risk of outputting incorrect and misleading 3D reconstructions is ever-present.…”

Image Recovery from Rotational And Translational Invariants

“…The measurements in cryo-EM consist of 2-D tomographic projections of identical biomolecules of unknown 3-D orientations embedded in a large noisy image, called a micrograph. In the current analysis workflow of cryo-EM data [2], [7], [8], [17], [19], [20], the 2-D projections are first detected and extracted from the micrograph, and later rotationally and translationally aligned to reconstruct the 3-D molecular structure. This approach is problematic for small molecules, which are difficult to detect due to their lower SNR.…”