Principles & Protocols

Principles - Single Particle Analysis

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Three-dimensional reconstruction from two-dimensional images

In a frozen-hydrated sample of homogeneous, identical particles, each particle is embedded within the vitreous ice layer in a different orientation. The image that is formed of such a layer of particles in a micrograph, recorded on photographic film or a charged-couple device (CCD), is a two-dimensional projection of each particle (Figure 1). The challenge is thus to reconstruct a three-dimensional image of the particle from different two-dimensional projections. The mathematical theory underlying three- dimensional reconstruction, which goes back to Radon (55), was first put into practice by DeRosier and Klug (20), who reconstructed a bacteriophage tail from electron micrographs. However, unlike the reconstruction of the highly ordered, helical bacteriophage tail, which only requires a single view, the reconstruction of a totally asymmetric particle requires that a large number of views are available, and that their relative orientation are known with high accuracy.

For a homogeneous sample, the spatial relationship of the particles in the vitreous ice layer can be described mathematically by a series of rigid-body translations and rotations of a single object. Thus, if the angular distribution of the particles is sufficiently uniform, i.e., if particles are present in orientations that sample a large part of the angular space, then a set of micrographs of the specimen, each showing projections of hundreds of particles, contains all the information necessary to reconstruct the particle in three dimensions.

The problem of determining the relative orientation among all projections is challenging because of the low SNR of the data. If no prior information about the structure of the particle is known, two ab initio methods can be used for orientation determination: (i) In the random-conical data collection method (69), which involves the use of an additional tilted view of the specimen, geometric relationships are established among a subset of particles that face the grid in the same orientation. (ii) In the method of common lines (34, 83), particle images are first classified, then class averages representing different views of the particle are related to one another following the common lines principle first formulated by Crowther (18). The latter method cannot establish the handedness, however, and requires an extra tilt for this purpose. From a set of projections whose angles have been determined by either one of these ab initio methods, a coarse reconstruction can then be computed, which is inaccurate yet contains important shape information allowing higher resolution and accuracy to be achieved through a succession of iterative steps, known as angular refinement.

In the refinement, the first, coarse reconstruction serves as an initial reference structure from which two-dimensional projections are computed, which are then compared to the experimental projections, yielding refined angles. With those angles, a new reconstruction is obtained, which is then used as reference in the next cycle, etc. Refinement continues until no improvement in the reconstructed three-dimensional model is observed, or the orientation angles have stabilized (see (27, 54, 84) for more detailed reviews on cryo-EM). Currently, resolutions in the range between 10 and 13 Å are routinely obtained, where detailed features of proteins and RNA are observed at the level of protein domains or RNA backbone. Resolutions in the range up to 7 Å, where structural motifs such as alpha-helices start to be discernable (see (8)), still require a large effort. Typically, more than 100,000 particle projections are required to meet this goal. Still larger numbers are needed an estimated 1,000,000 for the ribosome to approach 3 Å resolution. However, while the achievement of atomic resolution still awaits improvements in several instrumental and data processing aspects, existing moderate-resolution cryo-EM density maps of a molecular assembly can already be interpreted at the atomic level if X-ray or NMR structures of its components are known. The remainder of this chapter will elaborate on the way these hybrid methods of analysis can be put to use.

-- KakoliMitra - 11 Feb 2009