Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, p. 317   | 1 | 2 |

Section Discretization

B. K. Vainshteinc Discretization

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In direct methods of reconstruction as well as in Fourier methods the space is represented as a discrete set of points [\varphi ({\bf x}_{jk})] on a two-dimensional net or [\varphi ({\bf r}_{jkl})] on a three-dimensional lattice. It is sometimes expedient to use cylindrical or spherical coordinates. In two-dimensional reconstruction the one-dimensional projections are represented as a set of discrete values [L^{i}], at a certain spacing in [x_{\psi}]. The reconstruction ([link] is carried out over the discrete net with [m^{2}] nodes [\varphi_{jk}]. The net side A should exceed the diameter of an object [D,\ A \gt D]; the spacing [a = A/m]. Then ([link] transforms into the sum [L^{i} = {\textstyle\sum\limits_{k}} \varphi_{jk}. \eqno(] For oblique projections the above sum is taken over all the points within the strips of width a along the axis [{\boldtau}_{\psi_{i}}] (Fig.[link]).


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Discretization and oblique projection.

The resolution δ of the reconstructed function depends on the number h of the available projections. At approximately uniform angular distribution of projections, and diameter equal to D, the resolution at reconstruction is estimated as [\delta \simeq 2D / h. \eqno(] The reconstruction resolution δ should be equal to or somewhat better than the instrumental resolution d of electron micrographs [(\delta \;\lt\; d)], the real resolution of the reconstructed structure being d. If the number of projections h is not sufficient, i.e. [\delta \;\gt\; d], then the resolution of the reconstructed structure is δ (Crowther, DeRosier & Klug, 1970[link]; Vainshtein, 1978[link]).

In electron microscopy the typical instrumental resolution d of biological macromolecules for stained specimens is about 20 Å; at the object with diameter [D \simeq 200] Å the sufficient number h of projections is about 20. If the projections are not uniformly distributed in projection angles, the resolution decreases towards [{\bf x} \perp {\boldtau}] for such τ in which the number of projections is small.

Properties of projections of symmetric objects . If the object has an N-fold axis of rotation, its projection has the same symmetry. At orthoaxial projection perpendicular to the N-fold axis the projections which differ in angle at [j(2 \pi / N)] are identical: [\varphi_{2} ({\bf x}_{\psi}) = \varphi_{2} [{\bf x}_{\psi + j(2 \pi / N)}] \quad (j = 1, 2, \ldots, N). \eqno(] This means that one of its projections is equivalent to N projections. If we have h independent projections of such a structure, the real number of projections is hN (Vainshtein, 1978[link]). For a structure with cylindrical symmetry [(N = \infty)] one of its projections fully determines the three-dimensional structure.

Many biological objects possess helical symmetry – they transform into themselves by the screw displacement operation [s_{p/q}], where p is the number of packing units in the helical structure per q turns of the continuous helix. In addition, the helical structures may also have the axis of symmetry N defining the pitch of the helix. In this case, a single projection is equivalent to [h = pN] projections (Cochran et al., 1952[link]).

Individual protein molecules are described by point groups of symmetry of type N or [N/2]. Spherical viruses have icosahedral symmetry 532 with two-, three- and fivefold axes of symmetry. The relationship between vectors τ of projections is determined by the transformation matrix of the corresponding point group (Crowther, Amos et al., 1970[link]).


First citation Cochran, W., Crick, F. H. C. & Vand, V. (1952). The structure of synthetic polypeptides. 1. The transform of atoms on a helix. Acta Cryst. 5, 581–586.Google Scholar
First citation Crowther, R. A., Amos, L. A., Finch, J. T., DeRosier, D. J. & Klug, A. (1970). Three dimensional reconstruction of spherical viruses by Fourier synthesis from electron micrographs. Nature (London), 226, 421–425.Google Scholar
First citation 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.Google Scholar
First citation Vainshtein, B. K. (1978). Electron microscopical analysis of the three-dimensional structure of biological macromolecules. In Advances in optical and electron microscopy, Vol. 7, edited by V. E. Cosslett & R. Barer, pp. 281–377. London: Academic Press.Google Scholar

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