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International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 19.6, pp. 459-461
Section 19.6.4.6. Image processing and 3D reconstruction
a
Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907-1392, USA, and bMedical Research Council, Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England |
Although the general concepts of signal averaging, together with combining different views to reconstruct the 3D structure, are common to the different computer-based procedures which have been implemented, it is important to emphasize one or two preliminary points. First, a homogeneous set of particles must be selected for inclusion in the 3D reconstruction. This selection may be made by eye to eliminate obviously damaged particles or impurities, or by the use of multivariate statistical analysis (van Heel & Frank, 1981![[link]](/graphics/greenarr.gif)
) or some other classification scheme. This allows a subset of the particle images to be used to determine the structure of a better-defined entity. All image-processing procedures require the determination of the same parameters that are needed to specify unambiguously how to combine the information from each micrograph or particle. These parameters are: the magnification, defocus, astigmatism and, at high resolution, the beam tilt for each micrograph; the electron wavelength used (i.e. accelerating voltage of the microscope); the spherical aberration coefficient, ![[C_{s}]](/teximages/abch10o1/abch10o1fi30.svg)
, of the objective lens; and the orientation and phase origin for each particle or unit cell of the 1D, 2D or 3D crystal. There are 13 parameters for each particle, eight of which may be common to each micrograph and two or three (, accelerating voltage, magnification) to each microscope. The different general approaches that have been used in practice to determine the 3D structure of different classes of macromolecular assemblies from one or more electron micrographs are listed in Table 19.6.4.2.
†Electron tomography is the subject of an entire issue of Journal of Structural Biology [(1997), 120, pp. 207–395] and a book edited by Frank (1992 ).
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![[C_{1}]](/teximages/abch4o3/abch4o3fi298.svg)
)![[C_{n}, D_{n}\hbox{;}\ n\gt 1]](/teximages/fach19o6/fach19o6fi29.svg)
)The precise way in which each general approach codes and determines the particle or unit-cell parameters varies greatly and is not described in detail. Much of the computer software used in image-reconstruction studies is relatively specialized compared with that used in the more mature field of macromolecular X-ray crystallography. In part, this may be attributed to the large diversity of specimen types amenable to cryo EM and reconstruction methods. As a consequence, image-reconstruction software is evolving quite rapidly, and references to software packages cited in Table 19.6.4.2 are likely to become quickly outdated. Extensive discussion of algorithms and software packages in use at this time may be found in a number of recent special issues of Journal of Structural Biology (Vol. 116, No. 1; Vol. 120, No. 3; Vol. 121, No. 2; Vol. 125, Nos. 2–3).
In practice, attempts to determine or refine some parameters may be affected by the inability to determine accurately one of the other parameters. The solution of the structure is therefore an iterative procedure in which reliable knowledge of the parameters that describe each image is gradually built up to produce a more and more accurate structure until no more information can be squeezed out of the micrographs. At this point, if any of the origins or orientations are wrongly assigned, there will be a loss of detail and a decrease in signal-to-noise ratio in the map. If a better-determined or higher-resolution structure is required, it would then be necessary to record images on a better microscope or to prepare new specimens and record better pictures.
The reliability and resolution of the final reconstruction can be measured using a variety of indices. For example, the differential phase residual (DPR) (Frank et al., 1981), the Fourier shell correlation (FSC) (van Heel, 1987b) and the Q factor (van Heel & Hollenberg, 1980) are three such measures. The DPR is the mean phase difference, as a function of resolution, between the structure factors from two independent reconstructions, often calculated by splitting the image data into two halves. The FSC is a similar calculation of the mean correlation coefficient between the complex structure factors of the two halves of the data as a function of resolution. The Q factor is the mean ratio of the vector sum of the individual structure factors from each image divided by the sum of their moduli, again calculated as a function of resolution. Perfectly accurate measurements would have values of the DPR, FSC and Q factor of 0°, 1.0 and 1.0, respectively, whereas random data containing no information would have values of 90°, 0.0 and 0.0. The spectral signal-to-noise ratio (SSNR) criterion has been advocated as the best of all (Unser et al., 1989): it effectively measures, as a function of resolution, the overall signal-to-noise ratio (squared) of the whole of the image data. It is calculated by taking into consideration how well all the contributing image data agree internally.
An example of a strategy for determination of the 3D structure of a new and unknown molecule without any symmetry and which does not crystallize might be as follows:
For large single particles with no symmetry, particles with higher symmetry or crystalline arrays, it is usually possible to miss out the negative-staining steps and go straight to alignment of particle images from ice embedding, because the particle or crystal tilt angles can be determined internally from comparison of phases along common lines in reciprocal space or from the lattice or helix parameters from a 2D or 1D crystal.
The following discussion briefly outlines for a few specific classes of macromolecule the general strategy for carrying out image processing and 3D reconstruction.
For 2D crystals, the general 3D reconstruction approach consists of the following steps. First, a series of micrographs of single 2D crystals are recorded at different tilt angles, with random azimuthal orientations. Each crystal is then unbent using cross-correlation techniques to identify the precise position of each unit cell (Henderson et al., 1986), and amplitudes and phases of the Fourier components of the average of that particular view of the structure are obtained for the transform of the unbent crystal. The reference image used in the cross-correlation calculation can either be a part of the whole image masked off after a preliminary round of averaging by reciprocal-space filtering of the regions surrounding the diffraction spots in the transform, or it can be a reference image calculated from a previously determined 3D model. The amplitudes and phases from each image are then corrected for the CTF and beam tilt (Henderson et al., 1986, 1990; Havelka et al., 1995) and merged with data from many other crystals by scaling and origin refinement, taking into account the proper symmetry of the 2D space group of the crystal. Finally, the whole data set is fitted by least squares to constrained amplitudes and phases along the lattice lines (Agard, 1983) prior to calculating a map of the structure. The initial determination of the 2D space group can be carried out by a statistical test of the phase relationships in one or two images of untilted specimens (Valpuesta et al., 1994). The absolute hand of the structure is automatically correct since the 3D structure is calculated from images whose tilt axes and tilt angle are known. Nevertheless, care must be taken not to make any of a number of trivial mistakes that would invert the hand.
The basic steps involved in processing and 3D reconstruction of helical specimens include: Record a series of micrographs of vitrified particles suspended over holes in a perforated carbon support film. The micrographs are digitized and Fourier transformed to determine image quality (astigmatism, drift, defocus, presence and quality of layer lines, etc.). Individual particle images are boxed, floated, and apodized within a rectangular mask. The parameters of helical symmetry (number of subunits per turn and pitch) must be determined by indexing the computed diffraction patterns. If necessary, simple spline-fitting procedures may be employed to `straighten' images of curved particles (Egelman, 1986), and the image data may be reinterpolated (Owen et al., 1996) to provide more precise sampling of the layer-line data in the computed transform. Once a preliminary 3D structure is available, a much more sophisticated refinement of all the helical parameters can be used to unbend the helices on to a predetermined average helix so that the contributions of all parts of the image are correctly treated (Beroukhim & Unwin, 1997). The layer-line data are extracted from each particle transform and two phase origin corrections are made: one to shift the phase origin to the helix axis (at the centre of the particle image) and the other to correct for effects caused by having the helix axis tilted out of the plane normal to the electron beam in the electron microscope. The layer-line data are separated out into near- and far-side data, corresponding to contributions from the near and far sides of each particle imaged. The relative rotations and translations needed to align the different transforms are determined so the data may be merged and a 3D reconstruction computed by Fourier–Bessel inversion procedures (DeRosier & Moore, 1970).
The typical strategy for processing and 3D reconstruction of icosahedral particles consists of the following steps: First, a series of micrographs of a monodisperse distribution of particles, normally suspended over holes in a perforated carbon support film, is recorded. After digitization of the micrographs, individual particle images are boxed and floated with a circular mask. The astigmatism and defocus of each micrograph is measured from the sum of intensities of the Fourier transforms of all particle images (Zhou et al., 1996). Auto-correlation techniques are then used to estimate the particle phase origins, which coincide with the centre of each particle where all rotational symmetry axes intersect (Olson & Baker, 1989). The view orientation of each particle, defined by three Eulerian angles, is determined either by means of common and cross-common lines techniques or with the aid of model-based procedures (e.g. Crowther, 1971; Fuller et al., 1996; Baker et al., 1999). Once a set of self-consistent particle images is available, an initial low-resolution 3D reconstruction is computed by merging these data with Fourier–Bessel methods (Crowther, 1971). This reconstruction then serves as a reference for refining the orientation, origin and CTF parameters of each of the included particle images, for rejecting `bad' images, and for increasing the size of the data set by including new particle images from additional micrographs taken at different defocus levels. A new reconstruction, computed from the latest set of images, serves as a new reference and the above refinement procedure is repeated until no further improvements as measured by the reliability criteria mentioned above are made.
References
Agard, D. A. (1983). A least-squares method for determining structure factors in three-dimensional tilted-view reconstructions. J. Mol. Biol. 167, 849–852.Google Scholar
Baker, T. S., Olson, N. H. & Fuller, S. D. (1999). Adding the third dimension to virus life cycles: three-dimensional reconstruction of icosahedral viruses from cryo-electron micrographs. Microbiol. Mol. Biol. Rev. 63, 862–922.Google Scholar
Beroukhim, R. & Unwin, N. (1997). Distortion correction of tubular crystals: improvements in the acetylcholine receptor structure. Ultramicroscopy, 70, 57–81.Google Scholar
Crowther, R. A. (1971). Procedures for three-dimensional reconstruction of spherical viruses by Fourier synthesis from electron micrographs. Philos. Trans. R. Soc. London, 261, 221–230.Google Scholar
DeRosier, D. J. & Moore, P. B. (1970). Reconstruction of three-dimensional images from electron micrographs of structures with helical symmetry. J. Mol. Biol. 52, 355–369.Google Scholar
Egelman, E. H. (1986). An algorithm for straightening images of curved filamentous structures. Ultramicroscopy, 19, 367–374.Google Scholar
Frank, J., Verschoor, A. & Boublik, M. (1981). Computer averaging of electron micrographs of 40S ribosomal subunits. Science, 214, 1353–1355.Google Scholar
Fuller, S. D., Butcher, S. J., Cheng, R. H. & Baker, T. S. (1996). Three-dimensional reconstruction of icosahedral particles – the uncommon line. J. Struct. Biol. 116, 48–55.Google Scholar
Havelka, W. A., Henderson, R. & Oesterhelt, D. (1995). Three-dimensional structure of halorhodopsin at 7 Å resolution. J. Mol. Biol. 247, 726–738.Google Scholar
Heel, M. van (1987b). Similarity measures between images. Ultramicroscopy, 21, 95–100.Google Scholar
Heel, M. van & Frank, J. (1981). Use of multivariate statistics in analyzing the images of biological macromolecules. Ultramicroscopy, 6, 187–194.Google Scholar
Heel, M. van & Hollenberg, J.(1980). On the stretching of distorted images of two-dimensional crystals. In Electron microscopy at molecular dimensions, edited by W. Baumeister & W. Vogell, pp. 256–260. Berlin: Springer-Verlag.Google Scholar
Henderson, R., Baldwin, J. M., Ceska, T. A., Zemlin, F., Beckmann, E. & Downing, K. H. (1990). Model for the structure of bacteriorhodopsin based on high-resolution electron cryo-microscopy. J. Mol. Biol. 213, 899–929.Google Scholar
Henderson, R., Baldwin, J. M., Downing, K. H., Lepault, J. & Zemlin, F. (1986). Structure of purple membrane from halobacterium: recording, measurement, and evaluation of electron micrographs at 3.5 Å resolution. Ultramicroscopy, 19, 147–178.Google Scholar
Olson, N. H. & Baker, T. S. (1989). Magnification calibration and the determination of spherical virus diameters using cryo-microscopy. Ultramicroscopy, 30, 281–298.Google Scholar
Owen, C. H., Morgan, D. G. & DeRosier, D. J. (1996). Image analysis of helical objects: the Brandeis helical package. J. Struct. Biol. 116, 167–175.Google Scholar
Radermacher, M. (1992). Weighted back-projection methods. In Electron tomography, edited by J. Frank, pp. 91–115. New York: Plenum Press. Google Scholar
Unser, M., Trus, B. L., Frank, J. & Steven, A. C. (1989). The spectral signal-to-noise ratio resolution criterion: computational efficiency and statistical precision. Ultramicroscopy, 30, 429–434.Google Scholar
Valpuesta, J. M., Carrascosa, J. L. & Henderson, R. (1994). Analysis of electron microscope images and electron diffraction patterns of thin crystals of Φ29 connectors in ice. J. Mol. Biol. 240, 281–287.Google Scholar
Zhou, Z. H., Hardt, S., Wang, B., Sherman, M. B., Jakana, J. & Chiu, W. (1996). CTF determination of images of ice-embedded single particles using a graphics interface. J. Struct. Biol. 116, 216–222.Google Scholar
