III.E.1. Planar specimens (two-dimensional sheets)

Planar specimens such as membranes, cell walls, and thin crystals, make excellent subjects for Fourier image analysis in both 2D and 3D. Optical, digital, and photographic superposition methods of filtering provide clear projected images of thin specimens (<50 nm), i.e. those which contain only one or a few units cells in the direction normal to the plane of the specimen.

Several papers describe the methods and applications of 2D image processing of planar specimens (see Table 3 and Table 2.III.A.2a and B.10a,b; Baker, 1981). Included among those planar specimens that have been studied by 3D reconstruction techniques are purple membrane, cytochrome oxidase vesicles, membrane-bound ribosomes, actin filament bundles and tubulin sheets (See Table 3 of Baker, 1981 for references).

The basic steps in the process consist of the following:

1. Select best micrograph(s) on basis of quality of optical diffraction pattern. Highly coherent crystalline areas give strong, sharp Bragg reflections. Look for minimal radiation damage, astigmatism and specimen drift or vibration and for 'best' defocus (i.e. giving the desired CTF characteristics).

2. Digfitize the micrograph at a sampling interval fine enough not to limit image resolution but not too fine or the digitized image will be too large and will therefore needlessly slow down computations.

3. Box (window) out the desired region of interest, making sure to exclude, if possible, as much extraneous portions of the digitized imaghe as practical. This is easy to do with 'perfect' specimens like catalase crystals that grow large enough (several µm²) to fill the entire field of view at 30,000 magnification or higher.

4. Float the boxed image by subtracting from every pixel within the image the average value of the pixels that form the perimeter of the box.

5. Fourier transform the image.

6. Display and index the Fourier transform. This indexing could equally well be done with an optical diffraction pattern, however, the digital transform allows one to check other properties of the specimen such as the presence of certain plane group symmetries.

7. Perform either pseudo-optical filtering or Fourier averaging of the data. In pseudo-optical filtering, filter masks can be generated in the computer that are then multiplied by the Fourier transform and back-transformed to generate the filtered image. A number of variations of pseudo-optical filtering can be employed. For example, low and/or high-pass as well as lattice filter masks can be generated to perform different types of filtering of the data. To compute a Fourier-averaged reconstruction, in which all unit cells within the boxed area are averaged together, the data are reduced to single structure factors at each of the Bragg reflections. Again, a number of variations of this process can be implemented to produce Fourier averages.

8. Assess and apply additional symmetry (other than simple translational [p1] symmetry) if eveidence exists that the specimen indeed possesses such symmetry. It is very easy to apply any symmetry you want by means of computer processing methods, but one must always use caution when forcing additional symmetry onto the specimen because the end product will of course show whatever symmetry has been applied.

Fig. 4.1 (a) Bright-field image of uranyl acetate-stained purple membrane from Halobacterium halobium together with its optical transform (c). (b) shows the enhanced image, printed at the same magnification as the original (a), obtained by retaining only image information that is consistent with a hexagonal lattice (unit cell, 6.2nm x 6.2nm). The 'protein' is shown as white in (b). (a, b) image bar = 20nm, (c) transform bar = 0.2nm1 and the highest order diffraction spot corresponds to 0.5nm-1 or a resoltuion of 2nm. (M.V. Nermut, unpublished.) (From Misell, 4.1, p.127)

Fig. 4.9 (a) An area of negatively-stained hexagonal S-layer from Clostridium thermohydro-sulfuricum (image bar = 100 nm). In this and subsequent figures the stain is dark, so that by implication protein appears light. (b) The optical transform is taken from the boxed area (diffraction bar = 0.2 nm-1). (c) Shows an enlarged and rotated view of the boxed area in (a) (image bar = 50 nm) and (d) is the lattice image of the area shown in (c), produced by computer filtering using only the lowest order peaks in the transform and large apertures in the filter mask. Notice that the place where the lattice image becomes very weak in the top right-hand corner corresponds to an area of S-layer in which the pattern of morphological units disappears. (From Misell, 4.9, p.150; adapted from Crowther and Sleytr, 1977)

Fig. XXX. (a), (b), (c) show averaged images formed from part of the area shown in Fig. 4.9(c). In each case an area of 2x2 unit cells is shown, so that the edge of each figure corresponds to a length of 29.2 nm. In (a) the averaging is done without correcting the distortions, while in (b) the distortions were corrected. (c) is the 6-fold rotational average of the image in (b). (d) shows another area of negatively-stained hexagonal S-layer (image bar = 50 nm) with the corresponding (e) translationally and (f) rotationally averaged images. (From Misell, 4.10, p.151; adapted from Crowther and Sleytr, 1977)

Fig. XXX. How an undistorted unit cell (a) is produced from the distorted lattice (b). The dots represent the centre of the repeating motif, as determined from the lattice image, and the heavy lines represent the unit cell edges. Each cell is divided into a number of equal parts corresponding to the sampling chosen for the undistorted cell. By joining corresponding pairs of points on opposite cell edges a distorted sampling grid (light lines) is generated in each distorted unit cell. The density values at these grid points can now be interpolated from the underlying array of densitometer values and added into the average cell. (From Misell, 4.11, p.152; adapted from Crowther and Sleytr, 1977)

Fig. XXX. (a) Electron micrograph of periodic arrays of negatively-stained 'matrix' protein derived from spheroplasted E. coli by differential treatment in sodium dodecylsulphate (SDS). (b) The optical diffraction pattern of the windowed area is indexed on a simple hexagonal lattice (only a single layer) with diffraction maxima extending to 1/2.2 nm-1 (difffraction bar = 0.2 nm-1). (c) Digitally filtered result in which the 'protein' is white. (From Misell, 4.12, p.153; adapted from Steven et al., 1977)

Fig. XXX. Optical transform quadrants of unstained purple membrane micrographs taken in bright-field microscopy: (a) low dose image, (b) high dose image. Diffraction bar = 0.3 nm-1. (From Misell, 4.25, p.174; adapted from Unwin and Henderson, 1975)

Fig. XXX. Contour map of the projected structure of purple membrane at 0.7 nm resolution. Positive contours are shown by thicker lines; positive peaks are due to high concentrations of scattering material (protein). Low density regions indicated by thinner lines are due to lipid and glucose. Unit cell dimensions are 6.2 nm x 6.2 nm. (From Misell, 4.26, p.175; adapted from Unwin and Henderson, 1975)

Structural details are best understood only from a 3D reconstruction. This requires the recording and combination of images from several specimens viewed from different directions. Such images are usually obtained by systematically tilting the specimen at specified angles in the microscope. Unfortunately, the range of possible tilt angles in most microscopes is restricted to ±60°. This results in a missing cone of diffraction data which reduces resolution of structural features in the direction perpendicular to the plane of the specimen. In fortuitous cases, such as the purple membrane (Henderson and Unwin, 1975), in which diffraction is known to be very weak in the unobserved region, the absence of this data does not seriously affect the reconstruction results. For other specimens, however, the unobserved data in the missing cone may contain critical structural information, and strategies are needed to measure or generate this data. Thin sectioning of specimens at right angles to the plane provides one method to obtain the missing data (Amos & Baker, 1979b; Unwin, 1977). A constrained iterative Fourier refinement method (solvent flattening technique) provides another approach to the missing cone problem (Agard and Stroud, Biophys. J. (1982) 37:589-602).

Fig. XXX. The calculation of the number of projections, N, required to make a 3D reconstruction of an object, size D, at a resolution d. The numerical transform is sampled at approximately Æv = 1/D on each plane (central-section), and successive planes are rotated by f = p/N. The maximum spatial frequency vmax. determines the resolution of the 3D reconstruction, d = 1/vmax. (From Misell, 7.8, p.279)

Fig. XXX. Part of the 3D reciprocal lattice showing the geometry of the lattice lines in the hexagonal space group P3, a*, b* and c* are the reciprocal lattice vectors, a* and b* lie in, and c* is perpendicular to the plane of the membrane. A central-section, which is perpendicular to the incident electron beam, has been drawn through the lattice. The intersection of this central-section with the reciprocal lattice is determined by the angle of tilt and the axis about which the membrane is tilted. Individual diffraction patterns and micrographs provide the amplitudes and phases in the section at the points shown. z* represents the coordinate along the c* direction of one of these points. The accuracy of measurement of both the amplitudes and phases depended on having sharp lattice lines. It is therefore necessary to ensure that, on the microscope grid, the membranes remained coherently ordered and flat to within 1/5°. (From Misell, 7.9, p.282; adapted from Unwin and Henderson, 1975)