2. Cryo-EM

### 2.1 Macromolecules and their structure

A central fact of molecular biology is that large polymeric molecules (proteins, RNA, and DNA) and their complexes are vital to the functioning of a cell [11]. These macromolecules not only make biochemical reactions possible in the cell, but they also maintain cell structure, cause cell motion, and sense and respond to signals in the environment. Macromolecules are able to do all this because of their threedimensional structure. Reconstructing the 3D structure and explaining the function of the molecule using the 3D structure is one of the goals of cryo-EM. Biological macromolecules and their assemblies are generically referred to as particles in cryo-EM. Most of the particles that we are interested in are proteins or protein complexes.

In very simple terms, the cryo-EM method is the following: first, several biochemical steps are carried out to isolate and purify many copies of the macromolecule from real cells. This is sample preparation. Then, the copies of the particle are frozen in a layer of vitreous (noncrystallized) ice, and a single image of the preparation is created using a transmission electron microscope (Figure 1). This image, called a micrograph, contains tomographic projection images of the particle at random orientations; the orientation is determined by how the particle is frozen in ice. After a micrograph is obtained, each tomographic projection of the particle

#### Figure 1.

A simplified schematic of a cryo-EM experiment. Particles are embedded in vitreous ice and exposed to the electron beam.

### A Gentle Introduction to Cryo-EM Single-Particle Reconstruction Algorithms DOI: http://dx.doi.org/10.5772/intechopen.90099

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 particle using images obtained from one or more micrographs.

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 this interaction are important:


Figure 2. Contrast transfer function.

Several cryo-EM reconstruction packages are freely available (e.g., SPIDER [5],

A central fact of molecular biology is that large polymeric molecules (proteins, RNA, and DNA) and their complexes are vital to the functioning of a cell [11]. These macromolecules not only make biochemical reactions possible in the cell, but they also maintain cell structure, cause cell motion, and sense and respond to signals in the environment. Macromolecules are able to do all this because of their threedimensional structure. Reconstructing the 3D structure and explaining the function of the molecule using the 3D structure is one of the goals of cryo-EM. Biological macromolecules and their assemblies are generically referred to as particles in cryo-EM. Most of the particles that we are interested in are proteins or protein

In very simple terms, the cryo-EM method is the following: first, several biochemical steps are carried out to isolate and purify many copies of the macromolecule from real cells. This is sample preparation. Then, the copies of the particle are frozen in a layer of vitreous (noncrystallized) ice, and a single image of the preparation is created using a transmission electron microscope (Figure 1). This image, called a micrograph, contains tomographic projection images of the particle at random orientations; the orientation is determined by how the particle is frozen in ice. After a micrograph is obtained, each tomographic projection of the particle

A simplified schematic of a cryo-EM experiment. Particles are embedded in vitreous ice and exposed to the

EMAN [6], FREALIGN [7], RELION [8, 9], and cryoSPARC [10]), and where

relevant, I will point out which algorithms are used in these packages.

Technology, Science and Culture - A Global Vision, Volume II

2. Cryo-EM

complexes.

Figure 1.

44

electron beam.

2.1 Macromolecules and their structure

A final fact to consider is that cryo-EM images are noisy. Noise is introduced into the image by the camera and potentially also by the beam. A simple model for noise—one that is used in most reconstruction algorithms—is the Gaussian.

The effect of noise is compounded by the fact that long exposures to the electron beam damage the particles (they are bombarded by high-energy electrons), thereby altering their structure. To limit this damage, exposures are typically short, which in turn limit the amount of "signal" in the images.

The result is that the signal-to-noise ratio (SNR) in cryo-EM images is quite low. In engineering terms, it can be less than �10 db.

> At this point, it is useful to introduce the adjoints of the projection operator P<sup>n</sup> and the CTF operator C. The adjoint of the projection operator is the back-projection

> duces a 3D function S as follows: S xð Þ¼ fð Þ Πnð Þ x , where x is a point in threedimensional space and Πnð Þ x is the orthogonal projection of x onto the image plane

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

"transpose") of Pn, and not its inverse. The CTF operator C is self-adjoint, so its

Ii ¼ Rθi,ti

particle is usually located somewhere close to the center of the image).

R<sup>θ</sup>i,ti

CiPnið Þþ S ni, or R<sup>θ</sup>i,ti

S, the set of projection directions N ¼ f g n1, … , n<sup>N</sup> , the set of 2D rotations and translations T ¼ fðθ1,t1Þ, … ,ðθN,tNÞg, and the set of scalars ρ ¼ ρ<sup>1</sup> f g , … , ρ<sup>N</sup> (if using the model of Eq. (3)). Of these variables, we are only interested in S; the rest

the image rather than to the projected structure:

Ii ¼ ρiR<sup>θ</sup>i,ti

In this version, scalar ρ<sup>i</sup> is also unknown.

the variable ice thickness:

are nuisance variables.

47

known (because the defocus at which the image micrograph was obtained is known). The challenge in cryo-EM reconstruction is to estimate S given that ni, θi, ti, and noise are also unknown. Note that the set of all possible values of n<sup>i</sup> is identical to the set of points on the surface of the unit sphere in 3D. The set of all possible values of θ<sup>i</sup> is the interval [0; 2π). And the set of all possible values of ti is identical to the set of points in some square in the plane (it is not the entire plane because the

Every image picked from a micrograph has its own projection direction n, CTF, and in-plane rotation and translation. Suppose N images Ii, i ¼ 1, … , N are picked, then

where n<sup>i</sup> is the projection direction of Ii, Ci is the CTF operator of image Ii, θ<sup>i</sup> and ti are the in-plane rotations and translations of image Ii, and n<sup>i</sup> is the noise in image Ii. Of all the terms that appear in the right-hand side of Eq. (1), only the CTF Ci is

Although Eq. (1) is commonly used, there are two variations of the equation that are worth noting. The first simply applies the in-plane rotations and translations to

The second introduces a positive scalar ρi, which models contrast change due to

The unknown variables in the cryo-EM reconstruction problem are the structure

Cryo-EM reconstruction algorithms can be classified according to how they treat the nuisance variables. On the one hand, there are algorithms that simultaneously

<sup>n</sup> . The back-projection operator takes an image f and pro-

<sup>n</sup> is the adjoint (loosely speaking, the

CiPnið Þþ S ni, (1)

Ii ¼ CiPnið Þþ S ni: (2)

Ii ¼ ρiCiPnið Þþ S ni: (3)

operator denoted by P<sup>∗</sup>

Signal flow for cryo-EM.

adjoint is itself.

Figure 4.

(which is perpendicular to n). Note that P<sup>∗</sup>

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