Figure 1. Diagram of an Euler sphere indicating how θ and φ are used to describe any point on the surface of the sphere.
Simple Electron Microscopy Primer
In short, the general purpose of single particle reconstruction is to generate a three-dimensional model from many two-dimensional images. This, of course, is a general goal, as a lot of information can be gathered from the various intermediate steps. Regardless, for the purpose of this tutorial, we will follow the steps necessary to generate a 3D model.
In order to most clearly understand the various steps that will be discussed, it is important to know what the goal is and what our starting conditions are. The data typically consists of micrographs containing projections of proteins or protein complexes generated in a transmission electron microscope. From these experimental projections (particles), we will generate a 3D model. Later on we will discuss how the actual model is generated, but for now we just need to know that the process requires the assignment of Euler angles to each experimental projection. Once these angles are assigned to each experimental projection, the model can be generated.
Euler angles are directional angles used to indicate a position and orientation in space around a common center. Their definitions are as follows:
Defines the elevation above or below the equator.
Defines the rotation (azimuth) around the equator.
Defines the rotation around the center of the position defined by θ and φ.
If one imagines standing in the center of a large hollow sphere, in order to define a point on the surface of the sphere, one needs to know two pieces of information: what the elevation of the spot is relative to the equator of the sphere (θ) and what the rotation is around the vertical axis of the sphere (φ - azimuth). To generate a reconstruction, we will need to determine θ and φ for each experimental particle in order define its position on the Euler sphere. The center of the Euler sphere represents where our 3D model is. Assignment of the Euler angles is most commonly done by the method known as random conical tilt.
The process itself is very iterative (Figure 2). Experimental projections are first selected from electron micrographs. These particles are then centered, aligned, and classified. From the centered/aligned/classified particles, a preliminary model is generated. To refine this solution, the model is used to better align and classify the original particles. A new model is generated from the refined data, and so on until a satisfactory solution is reached.