Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 381-382

Section Initial determination of 3D structure using tilt experiments

P. A. Penczekg Initial determination of 3D structure using tilt experiments

| top | pdf |

The 2D analysis of projection images provides insight into the behaviour of the protein on the grid in terms of the structural consistency and the number and shape of projection images. In order to obtain 3D information, it is necessary to find geometrical relations between different observed 2D images. The most robust and historically the earliest approach is based on tilt experiments. By tilting the stage in the microscope and acquiring additional pictures of the same area of the grid it is possible to collect projection images of the same molecule with some of the required Eulerian angles determined accurately by the setting of the goniometer of the microscope.

In random conical tilt (RCT) reconstruction (Radermacher et al., 1987[link]), two micrographs of the same specimen area are collected: the first one is recorded at a tilt angle of ~50° while the second one is recorded at 0° (Fig.[link]). If particles have preferred orientation on the support carbon film (or within the amorphous ice layer, if no carbon support is used), the projections of particles in the tilted micrographs form a conical tilt series. Since in-plane rotations of particles are random, the azimuthal angles of the projections of tilted particles are also randomly distributed; hence the name of the method. The untilted image is required for two reasons: (i) the particle projections from the untilted image are classified, thus a subset corresponding to possibly identical images can be selected ensuring that the projections originated from similar and similarly oriented structures; and (ii) the in-plane rotation angle found during alignment corresponds to the azimuthal angles in three dimensions (one of the three Eulerian angles needed). The second Eulerian angle, the tilt, is either taken from the microscope setting of the goniometer or calculated based on geometrical relations between tilted and untilted micrographs. The third Eulerian angle corresponds to the angle of the tilt axis of the microscope stage and is also calculated using the geometrical relations between two micrographs. In addition, it is necessary to centre the particle projections selected from tilted micrographs; although various correlation-based schemes have been proposed, the problem is difficult as the tilt data tend to be very noisy and have very low contrast.


Figure | top | pdf |

Principle of random conical tilt reconstruction. A tilt pair of images of the same grid area is collected. By aligning the particle images in the untilted micrograph (left), the Eulerian angles of their counterparts in the tilted micrograph (right) are established. The particle images from the tilted micrograph are used for 3D reconstruction of the molecule (bottom). The set of projections form a cone in Fourier space; information within the cone remains undetermined.

Given three Eulerian angles and centred tilted projections, a 3D reconstruction is calculated. There are numerous advantages of the RCT method. (i) Assuming the sign of the tilt angle is read correctly (it can be confirmed by analysing the defocus gradient in the tilted micrographs), the method yields a correct hand of the structure. (ii) With the exception of the in-plane rotation of untilted projections, which can be found relatively easily using alignment procedures, the remaining parameters are determined by the experimental settings. Even if they are not extremely accurate, the possibility of a gross error is eliminated, which positively distinguishes the method from the ab initio computational approaches that use only untilted data. (iii) The computational analysis is entirely done using the untilted data, which have high contrast. (iv) The RCT method is often the only method of obtaining 3D information if the molecule has strongly preferential orientation and only one view is observed in untilted micrographs. The main disadvantage is that the conical projection series leaves a significant portion of the Fourier space undetermined. This follows from the central section theorem [equation ([link] of Section 2.5.6[link]]: as the tilt angle is less than 90°, the undetermined region can be thought to form a cone in three dimensions and is referred to as the missing cone. The problem can be overcome if the molecule has more than one preferred orientation. Subsets of particles that have similar untilted appearance (as determined by clustering) are processed independently and for each a separate 3D structure is calculated. If the preferred orientations are sufficiently different, i.e., the orientations of the original particles in three dimensions are sufficiently different in terms of their angles with respect to the z axis, the 3D structures can be aligned and merged, all but eliminating the problem of the missing cone and yielding a robust, if resolution-limited, initial model of the molecule (Penczek et al., 1994[link]). It should be noted that RCT by itself almost never results in a high-resolution 3D model of the molecule. This is due to a variety of reasons, the main ones being the already mentioned poor quality of high-tilt data and difficulties with the collection of large numbers of high-quality tilted micrographs (they are often marred by drift).

In cases when the molecule does not have well defined preferred orientations, it is possible to use electron tomography to obtain the initial model. In this method, a single-axis tilt series of projection images of the same specimen area is collected using an angular step of ~2° and a maximum tilt angle not exceeding 60° (Crowther, DeRosier & Klug, 1970[link]). The single-axis tilt data-collection geometry yields worse coverage of the Fourier space than the RCT method, leaving missing wedges uncovered (Penczek & Frank, 2006[link]). This results in severe artifacts in real space, which make smaller objects virtually unrecognizable. The situation can be largely rectified using so-called double-axis tomography, in which a second single-axis tilt series of data are collected after rotating the specimen grid in-plane by 90° (Penczek et al., 1995[link]). This reduces the undetermined region to a missing pyramid and makes the resolution almost isotropic in the xy plane.

The tomographic projection data have to be aligned. This is done using either correlation techniques that enforce pairwise alignment of images (Frank & Mcewen, 1992[link]) or by taking advantage of fiducial markers and enforcing their consistency with respect to a 3D model (Lawrence, 1992[link]; Penczek et al., 1995[link]). In the application to single-particle work, it is possible to use locations of protein in the micrographs as markers. After the 3D reconstruction is calculated, regions collecting individual molecules are windowed from the volume and all molecules are aligned in three dimensions (Walz et al., 1997[link]). While generally robust, the procedure is labour- and computer-intensive. Unlike RCT, where only two exposures of the same field are required, electron tomography may require over one hundred images, raising serious concerns about radiation damage. Moreover, most of the data have to be collected at high tilt angle, thus are of lower quality. Particularly troublesome is alignment of 3D molecules deteriorated by the missing wedge/pyramid artifacts, with the directions of artifacts different for each object. However, when successful, electron tomography yields a very good initial model of the molecule, free from missing-Fourier-space-related artifacts and with defined handedness.


Crowther, R. A., DeRosier, D. J. & Klug, A. (1970). The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. Proc. R. Soc. London Ser. A, 317, 319–340.
Frank, J. & Mcewen, B. (1992). Alignment by crosscorrelation. In Electron Tomography, edited by J. Frank, pp. 205–214. New York: Plenum.
Lawrence, M. C. (1992). Least-squares method of alignment using markers. In Electron Tomography, edited by J. Frank, pp. 197–204. New York: Plenum Press.
Penczek, P., Marko, M., Buttle, K. & Frank, J. (1995). Double-tilt electron tomography. Ultramicroscopy, 60, 393–410.
Penczek, P. A. & Frank, J. (2006). Resolution in electron tomography. In Electron Tomography: Methods for Three-Dimensional Visualization of Structures in the Cell, 2nd ed., edited by J. Frank, pp. 307–330. Berlin: Springer.
Penczek, P. A., Grassucci, R. A. & Frank, J. (1994). The ribosome at improved resolution: new techniques for merging and orientation refinement in 3D cryo-electron microscopy of biological particles. Ultramicroscopy, 53, 251–270.
Radermacher, M., Wagenknecht, T., Verschoor, A. & Frank, J. (1987). Three-dimensional reconstruction from a single-exposure, random conical tilt series applied to the 50S ribosomal subunit of Escherichia coli. J. Microsc. 146, 113–136.
Walz, J., Typke, D., Nitsch, M., Koster, A. J., Hegerl, R. & Baumeister, W. (1997). Electron tomography of single ice-embedded macromolecules: three-dimensional alignment and classification. J. Struct. Biol. 120, 387–395.

to end of page
to top of page