III.E.4. Helical Specimens
The 3D structure of helical particles is usually reconstructed because features revealed by two-dimensional filtering techniques are often uninterpretable. Thus, optical and digital filtering have limited value in studying the structures of helical particles. The high symmetry of most helical aggregates makes these structures ideal subjects for image processing since a single view often suffices for determining 3D structure to 2-3 nm resolution (Table 1.III.C.1.a, Baker, 1981).
Particles chosen for 3D processing are selected on the basis of the quality of optical diffraction patterns and the helical parameters determined from indexing the diffraction patterns. These criteria indicate how well the symmetry and structure of the particle is preserved. Indexing helical diffraction patterns can often be non-trivial since the diffraction spots tend to be weak and diffuse in comparison with the intense, sharp spots characteristic of extended, well-ordered crystalline arrays. In addition, helical diffraction patterns often do not display perfect mirror symmetry owing to distortion or uneven staining of the particles. Computed transforms aid indexing because they reveal symmetry or anti-symmetry in the phases of diffraction spots, arising from the top (side closest to the electron beam) and bottom sides of the particle, related by mirror symmetry across the meridional (parallel to the helix axis) line of the diffraction pattern (Table 1.III.B.4, Baker, 1981). These phase relationships identify if the pair of spots arises from an even or odd numbered set of helices (helical family). DeRosier and Moore (1970) give an excellent and comprehensive description of how 3D reconstruction methods are applied to the study of helically symmetric particles. They also discuss how to analyze data from particles supported on the microscope grid with their axes tilted relative to the electron beam.
Independent reconstructions from different particles are often compared and averaged together to reduce bias in the results. Averaging helps reduce artifacts due to staining or distortions which might incorrectly influence the reconstruction obtained for a particular helical particle. Computer averaging also allows similarities and differences between the separate reconstructions to be quantitatively assessed. Several examples of averaged reconstructions are cited in Table 1.III.C.3.i of Baker (1981).
The absolute handedness of the arrangement of subunits that make up a helical particle may be determined by one of several methods. Helical particles (e.g. TMV) often have a serrated edge, the appearance of which changes in a predictable manner upon tilting the particle about an axis normal to the helix axis (see e.g. Finch (1972) J. Mol. Biol. 66:291-294; Linck & Amos (1974) J. Cell Sci. 14:551-559; Nonomura & Kohama (1974) J. Mol. Biol. 86:621-626). If the changes are too difficult to see directly, optical diffraction patterns obtained from the right and left halves of the tilted particle can be used to establish a consistent choice of hand (Chasey (1974) Nature 248:611-612; Finch (1972) J. Mol. Biol. 66:291-294; Nonomura & Kohama (1974) J. Mol. Biol. 86:621-626). When tilted about the helix axis, phase changes in the computed transform are monitored to determine the relative hand of the various helical families (Bloomer, et al. (1976) J. Mol. Biol. 105:453-457). Model building studies can also aid in deciding the correct hand (Finch (1972) J. Mol. Biol. 66:291-294; Linck & Amos (1974) J. Cell Sci. 14:551-559). Metal shadowing, in which only one side of the specimen is revealed, can provide unequivocal evidence in the determination of handedness (Weiss, et al. (1976) Sixth Europ. Reg. Conf. Elec. Microsc. (Jerusalem) 1:20-23).
The sheath of the T4 bacteriophage tail was the first helical particle, and the first biological particle of any kind, to be studied by the 3D Fourier reconstruction method developed by DeRosier and Klug (1968). This method is now routinely used to study a wide range of helical structures including actin, tobacco mosaic virus rods and stacked discs, helical aggregates of ribosomes, bacterial flagella, flagellar microtubules, sickle-cell hemoglobin fibers, alfalfa mosaic virus, catalase tubes, glutamine synthetase cables, haemocyanin, etc. (see Table 3, Baker, 1981 for references).
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Fig. XXX. Diffraction by helices: amplitude variations of the Bessel function Jn(X) for |
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| Fig. 29 Cylindrical coordinates. (From Holmes and Blow, 29, p. 210) |
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| Fig. 30 The Bessel function Jp(x) as functions of p and x. Note that Jo is large at the origin but that all other orders are zero at the origin. At large values of p the first maximum value of the Bessel function Jp(x) occurs at a value of x close to p. After the first maximum all of the Bessel functions oscillate like an attenuated sine wave. Reproduced by permission from Tables of Functions with Formulae and Curves by Jahnke and Emde (112). (From Holmes and Blow, 30, p. 211) |
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| Fig. 34 (a) An n-l plot of the helix given in Figure 32a. (b) An n-l plot of the helix given in Figure 32b. (c) An n-l plot of the helix given in Figure 32c. (From Holmes and Blow, 34, p. 220) |
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| Fig. 35 Two helices of different radii but the same pitch are shown. A point on the inner helix becomes a point on the outer helix if projected along a radius. Imagine the outer helix to be drawn on a cylinder. If a cut is made in the cylinder (broken line) it may be opened out to give a flat sheet. On this sheet the helix becomes a straight line. The flat sheet is called a radial projection (120). (From Holmes and Blow, 35, p. 221) |
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| Fig. 36 A helical projection: The point A projected down a helix of pitch P becomes a point A1 on the horizontal plane, and the point B becomes B1. Alternatively the projection may be made onto a vertical plane giving the points A2 and B2. (From Holmes and Blow, 36, p. 223) |
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| Fig. 32 (a) Optical transform of a continuous helix. (b) Optical transform of a helix with 10 pints per turn. (c) Optical transform of a helix with 5 points per turn. (From Holmes and Blow, 32, p. 215) |
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| From Holmes and Blow, 32, p. 215. |
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Diagrammatic representation of some of the Bessel functions contributing to the diffraction pattern from (a) a continuous helix of pitch P, (b) a discontinuous helix with 5 units per turn, (c) a discontinuous helix with 5-1/3 units per turn, (d) a discontinuous helix with 2-1/3 units per turn. The dark regions on the left hand side of the diagrams correspond approximatley to the first maxima of the Bessel functions. (From Wilson, 2.25, pp.42-43.) |
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| Fig. XXX. Diffraction by helices: (a) a continuous helix, its projection and the definition of polar coordinates (r,f). |
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| Fig. XXX. Diffraction by helices: amplitude variations of the Bessel function Jn(X) for n = 0 to 3. The progressive displacement of the first maxima explains the cross shape of the diffraction pattern in Fig. 3.25b. (From Misell, 3.26, p.98) |
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| Fig. XXX. Diffraction by helices: (a) a discontinuous helix, with subunits spaced in z by p and the helix pitch P, is equal to the multiplication of a continuous helix by a lattice with a repeat distance p, (b) the diffraction pattern, with layer lines separated by 1/P and successive meridional diffractiion orders separated by 1/p. (From Misell, 3.27, p.99) |
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| Fig. XXX. (a) Optical, (b) computer transforms of a negatively-stained T4 phage tail (uncontracted) together with indexation of the layer lines. The spacing from the equator (l = 0) to the first meridional (l = ± 7) = 1/4.1nm-1 and the layer line spacing = 1/(7 x 4.1) = 1/28.7nm-1. Image bar = 20nm, diffraction bar = 0.2nm-1. (From Misell, 3.28, p.101) | |
 Fig. XXX. Diffraction by helices: the separation of the top and bottom of a helix. (a) the diffraction pattern of a helix with 5 subunits/helix turn. (b) the lattice drawn through meridional points (full line = top, dashed line = bottom). (c) the 'one-sided' diffraction pattern. Generally complete separation is not possible because of the size of the diffraction spots and the overlap of different diffraction orders. Z (or integer I) and R define the axial and radial coordinates of the diffraction pattern and the integer m denotes the branch of the helical diffraction pattern. p is the separation between successive subunits on the helix, and x is the effective angle between the top and bottom of the helix. (From Misell, 3.29, p.105)
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 Fig. XXX. Diffraction by helices: (a) (n, l) plot for a basic helix with 7 units in two turns, c, = 2P, c~= 7p;; selection rule 1 = -2n + 7m, (b) six such helices related by a 6-fold axis. The spacing of the layer lines in unaltered, but Bessel functions whose order are a multiple of six appear (Jo, J6, J12 etc). Ringed spots correspond to those that appear in the original (a). Note the change in the n scale. (From Misell, 3.44, p.122)
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