1. Introduction

In cryogenic electron microscopy (cryo-EM), single-particle reconstruction is a method for reconstructing the three-dimensional (3D) structures of biological macromolecules. This paper provides an overview of the algorithms used for these reconstructions (to avoid cumbersome terminology, I will use the term "cryo-EM" as a simplified form of "cryo-EM single-particle reconstruction").

Most histories of cryo-EM trace its origin to the late 1960s and early 1970s [1, 2]. Circa 2012, a minirevolution occurred in cryo-EM with the introduction of direct detector cameras; these cameras pushed cryo-EM reconstructions to near-atomic resolutions. In 2017, the Nobel Prize in Chemistry was awarded to Jacques Dubochet, Joachim Frank, and Richard Henderson for "The Development of Cryo-electron Microscopy." The essay accompanying the announcement of the prize explains clearly that image processing algorithms were critical to the development of cryo-EM [3].

#### 1.1 Organization of the paper

This paper is quite informal; I have stayed away from detailed mathematical calculations and proofs. My goal is to describe the reconstruction problem at a level that students in engineering and computer science might find accessible. I have also simplified the problem somewhat, hoping to retain the essential core while eliminating distracting details. I do provide pointers to the literature where the curious reader can glean additional information. For more general background information, the reader may consult [4].

I begin in Section 2 by briefly describing the imaging process in cryo-EM. Following this, Section 3 contains the signal model used in reconstruction. Section 4 describes the best-alignment class of reconstruction algorithms. Section 5 describes algorithms that are based on the expectation-maximization (EM) approach. Section 6 contains a discussion of postprocessing. Finally, Section 7 concludes the paper.

Several cryo-EM reconstruction packages are freely available (e.g., SPIDER [5], EMAN [6], FREALIGN [7], RELION [8, 9], and cryoSPARC [10]), and where relevant, I will point out which algorithms are used in these packages.

in the micrograph is isolated by a bounding box. This is called particle picking and is usually a semiautomatic process. The content of each bounding box is an image. The single-particle reconstruction (SPR) problem is to estimate the 3D structure of

The above description of cryo-EM image formation as a tomographic projection is highly simplified; a more realistic description takes into account the details of how the electron beam interacts with the ice-embedded particle. Three effects of

1.The wave nature of the electron causes the image produced by the microscope to be a tomographic projection of the particle followed by convolution with a filter. The spectral response of the filter is called the contrast transfer function (CTF). The CTF depends on the microscope defocus. Figure 2 shows a CTF in the Fourier domain. The CTF is real-valued and circularly symmetric and takes positive and negative values. The CTF has zeros in the Fourier domain, and information about the particle is lost at these spatial frequencies. However, the CTF and its zeros can be changed by changing the microscope defocus. To take advantage of this, micrographs of the same particle are obtained at different defocus values. If the CTF zeros at these defoci do not coincide, then

2.The contrast in the image depends on the ice thickness. The thicker the ice, the

reconstruction. When the beam is turned on in the microscope, there is a drift of the ice and the ice-embedded particles. If the micrograph is formed as an

information about the particles is lost due to the drift. Direct detector cameras are designed to overcome this problem. They do not average over the exposure time; instead, they create a multi-frame movie. Judicially discarding some movie frames and then aligning the remaining frames, followed by averaging, compensate for the drift and create a micrograph that retains high-frequency

3.The third effect is more subtle but important for high-resolution

average image over this exposure time, then high-frequency spatial

the particle using images obtained from one or more micrographs.

A Gentle Introduction to Cryo-EM Single-Particle Reconstruction Algorithms

information is available at every frequency.

this interaction are important:

DOI: http://dx.doi.org/10.5772/intechopen.90099

lower the contrast.

information.

Figure 2.

45

Contrast transfer function.
