Mathematics for Cryo-Electron Microscopy

Abstract

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryoelectron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. A IntroductionThe field of structural biology is currently undergoing a transformative change [1,2]. Structures of many biomolecular targets previously insurmountable by X-ray crystallography are now being obtained using single particle cryo-EM to resolutions beyond 4Å on a regular basis [3][4][5]. This leap in cryo-EM technology, as recognized by the 2017 Nobel Prize in Chemistry, is mainly due to hardware advancements including the invention of the direct electron detector and the methodological development of algorithms for data processing. Cryo-EM is a very general and powerful technique because it does not require the formation of crystalline arrays of macromolecules. In addition, unlike X-ray crystallography and nuclear magnetic resonance (NMR) that measure ensembles of particles, single particle cryo-EM produces images of individual particles. Cryo-EM therefore has the potential to analyze conformational changes and energy landscapes associated with structures of complexes in different functional states.The main purpose of this brief review paper is to expose mathematicians to the exciting field of cryo-EM. As there exist many excellent review articles and textbooks on single particle cryo-EM [6-12], we choose to solely focus here on the mathematical foundations of this technique. Topics of great importance to practitioners, such as the physics and optics of the electron microscope, sample preparation, and data acquisition are not treated here.In cryo-EM, biological macromolecules are imaged in an electron microscope. The molecules are rapidly frozen in a thin layer of vitreous ice, trapping them in a nearly-physiological state. The molecules are randomly oriented and positioned within the ice layer. The electron microscope produces a two-dimensional tomographic projection image (called a micrograph) of the molecules embedded in the ice layer. More specifically, what is being measured by the detector is the integral in the direction of the beaming electrons of the electrostatic potential of the individual molecules.

Multireference Alignment Is Easier With an Aperiodic Translation Distribution

“…The MRA model (I.1) can be thought of as a simplified model for the cryo-EM problem, where cyclic translations replace actions of elements of the group SO(3) [14]. The tomographic projection does not appear in (I.1).…”

Multi-Reference Alignment in High Dimensions: Sample Complexity and Phase Transition

“…If T i = \pi S for all i, we get the projected MRA model (2.2). In cryo-EM---the main motivation of this work---G is the group of 3D rotations SO(3), \BbbX is the space of 3D``band-limited"" functions (that is, functions that can be expanded by finitely many basis functions), and T i encodes the (fixed) tomographic projection, as well as other linear effects, such as the microscope's point spread function (which varies across images) and sampling [10,41].…”

The numerics of phase retrieval

“…Some of the most challenging problems related to phase retrieval arise in blind ptychography, in imaging proteins at the single molecule level (Longchamp et al 2017), and in non-crystallographic 'single-shot' X-ray imaging (Chapman et al 2007, Loh et al 2010. In the latter problem, in addition to the phase retrieval problem, we face the major task of tomographic three-dimensional reconstruction of the object from diffraction images with unknown rotation angles -a challenge that we also encounter in cryo-EM (Singer 2019). The review article by Shechtman et al (2015) contains a detailed discussion of current bottlenecks and future challenges, such as taking the CDI techniques to the regime of attosecond science.…”