Discretization

Vainshtein, B. K.

doi:10.1107/97809553602060000558

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 2.5.6.3. Discretization

B. K. Vainshtein^c ^‡

2.5.6.3. 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 (2.5.6.9) 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 (2.5.6.8) transforms into the sum $[L^{i} = {\textstyle\sum\limits_{k}} \varphi_{jk}. \eqno(2.5.6.10)]$ 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. 2.5.6.4).

Figure 2.5.6.4| top | pdf |

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 $[\delta \simeq 2D / h. \eqno(2.5.6.11)]$ 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; Vainshtein, 1978).

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: $[\varphi_{2} ({\bf x}_{\psi}) = \varphi_{2} [{\bf x}_{\psi + j(2 \pi / N)}] \quad (j = 1, 2, \ldots, N). \eqno(2.5.6.12)]$ 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). 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).

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).

References

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

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

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

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

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