Finding heavy atoms with centrosymmetric projections

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 242-243 | 1 | 2 |

Section 2.3.3.2. Finding heavy atoms with centrosymmetric projections

M. G. Rossmann^a ^* and E. Arnold^b

^aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

2.3.3.2. Finding heavy atoms with centrosymmetric projections

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Phases in a centrosymmetric projection will be 0 or π if the origin is chosen at the centre of symmetry. Hence, the native structure factor, $[{\bf F}_{N}]$ , and the heavy-atom-derivative structure factor, $[{\bf F}_{NH}]$ , will be collinear. It follows that the structure amplitude, $[|{\bf F}_{H}|]$ , of the heavy atoms alone in the cell will be given by $[|{\bf F}_{H}| = |(|{\bf F}_{NH}| \pm |{\bf F}_{N}|)| + \varepsilon,]$ where ɛ is the error on the parenthetic sum or difference. Three different cases may arise (Fig. 2.3.3.1). Since the situation shown in Fig. 2.3.3.1(c) is rare, in general $[|{\bf F}_{H}|^{2} \simeq (|{\bf F}_{NH}| - |{\bf F}_{N}|)^{2}. \eqno(2.3.3.1)]$ Thus, a Patterson computed with the square of the differences between the native and derivative structure amplitudes of a centrosymmetric projection will approximate to a Patterson of the heavy atoms alone.

Figure 2.3.3.1 | top | pdf |

Three different cases which can occur in the relation of the native, $[{\bf F}_{N}]$ , and heavy-atom derivative, $[{\bf F}_{NH}]$ , structure factors for centrosymmetric reflections. $[{\bf F}_{N}]$ is assumed to have a phase of 0, although analogous diagrams could be drawn when $[{\bf F}_{N}]$ has a phase of π. The crossover situation in (c) is clearly rare if the heavy-atom substitution is small compared to the native molecule, and can in general be neglected.

The approximation (2.3.3.1) is valid if the heavy-atom substitution is small enough to make $[|{\bf F}_{H}| \ll |{\bf F}_{NH}|]$ for most reflections, but sufficiently large to make $[\varepsilon \ll (|{\bf F}_{NH}| - |{\bf F}_{N}|)^{2}]$ . It is also assumed that the native and heavy-atom-derivative data have been placed on the same relative scale. Hence, the relation (2.3.3.1) should be re-written as $[|{\bf F}_{H}|^{2} \simeq (|{\bf F}_{NH}| - k|{\bf F}_{N}|)^{2},]$ where k is an experimentally determined scale factor (see Section 2.3.3.7). Uncertainty in the determination of k will contribute further to ɛ, albeit in a systematic manner.

Centrosymmetric projections were used extensively for the determination of heavy-atom sites in early work on proteins such as haemoglobin (Green et al., 1954), myoglobin (Bluhm et al., 1958) and lysozyme (Poljak, 1963). However, with the advent of faster data-collecting techniques, low-resolution (e.g. a 5 Å limit) three-dimensional data are to be preferred for calculating difference Pattersons. For noncentrosymmetric reflections, the approximation (2.3.3.1) is still valid but less exact (Section 2.3.3.3). However, the larger number of three-dimensional differences compared to projection differences will enhance the signal of the real Patterson peaks relative to the noise. If there are N terms in the Patterson synthesis, then the peak-to-noise ratio will be proportionally $[\sqrt{N}]$ and 1/ɛ. With the subscripts 2 and 3 representing two- and three-dimensional syntheses, respectively, the latter will be more powerful than the former whenever $[{\sqrt{N_{3}} \over \varepsilon_{3}} \gt {\sqrt{N_{2}} \over \varepsilon_{2}}.]$ Now, as $[\varepsilon_{3} \simeq \sqrt{2} \varepsilon_{2}]$ , it follows that $[N_{3}]$ must be greater than $[2N_{2}]$ if the three-dimensional noncentrosymmetric computation is to be more powerful. This condition must almost invariably be true.

References

Bluhm, M. M., Bodo, G., Dintzis, H. M. & Kendrew, J. C. (1958). The crystal structure of myoglobin. IV. A Fourier projection of sperm-whale myoglobin by the method of isomorphous replacement. Proc. R. Soc. London Ser. A, 246, 369–389.Google Scholar

Green, D. W., Ingram, V. M. & Perutz, M. F. (1954). The structure of haemoglobin. IV. Sign determination by the isomorphous replacement method. Proc. R. Soc. London Ser. A, 225, 287–307.Google Scholar

Poljak, R. J. (1963). Heavy-atom attachment to crystalline lysozyme. J. Mol. Biol. 6, 244–246.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 242-243