Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 84-85   | 1 | 2 |

Section Solvent flattening

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Solvent flattening

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Crystals of proteins and nucleic acids contain large amounts of mother liquor, often in excess of 50% of the unit-cell volume, occupying connected channels. The well ordered electron density [\rho\llap{$-\!$}_{M}({\bf x})] corresponding to the macromolecule thus occupies only a periodic subregion [{\scr U}] of the crystal. Thus [\rho\llap{$-\!$}_{M} = \chi_{\scr U} \times \rho\llap{$-\!$}_{M},] implying the convolution identity between structure factors (Main & Woolfson, 1963[link]): [F_{M}({\bf h}) = \sum\limits_{{\bf k}} \bar{{\scr F}} \left[{1 \over {\scr U}} \chi_{{\scr U}}\right] ({\bf h} - {\bf k}) F_{M} ({\bf k})] which is a form of the Shannon interpolation formula (Sections[link],[link]; Bricogne, 1974[link]; Colman, 1974[link]).

It is often possible to obtain an approximate `molecular envelope' [{\scr U}] from a poor electron-density map [\rho\llap{$-\!$}], either interactively by computer graphics (Bricogne, 1976[link]) or automatically by calculating a moving average of the electron density within a small sphere S. The latter procedure can be implemented in real space (Wang, 1985[link]). However, as it is a convolution of [\rho\llap{$-\!$}] with [\chi_{S}], it can be speeded up considerably (Leslie, 1987[link]) by computing the moving average [\rho\llap{$-\!$}_{\rm mav}] as [\rho\llap{$-\!$}_{\rm mav}({\bf x}) = {\scr F}[\bar{{\scr F}}[\rho\llap{$-\!$}] \times \bar{{\scr F}}[\chi_{S}]]({\bf x}).]

This remark is identical in substance to Booth's method of computation of `bounded projections' (Booth, 1945a[link]) described in Section[link], except that the summation is kept three-dimensional.

The iterative use of the estimated envelope [{\scr U}] for the purpose of phase improvement (Wang, 1985[link]) is a submethod of the previously developed method of molecular averaging, which is described below. Sampling rules for the Fourier analysis of envelope-truncated maps will be given there.


Booth, A. D. (1945a). Two new modifications of the Fourier method of X-ray structure analysis. Trans. Faraday Soc. 41, 434–438.
Bricogne, G. (1974). Geometric sources of redundancy in intensity data and their use for phase determination. Acta Cryst. A30, 395–405.
Bricogne, G. (1976). Methods and programs for direct-space exploitation of geometric redundancies. Acta Cryst. A32, 832–847.
Colman, P. M. (1974). Non-crystallographic symmetry and the sampling theorem. Z. Kristallogr. 140, 344–349.
Leslie, A. G. W. (1987). A reciprocal-space method for calculating a molecular envelope using the algorithm of B. C. Wang. Acta Cryst. A43, 134–136.
Main, P. & Woolfson, M. M. (1963). Direct determination of phases by the use of linear equations between structure factors. Acta Cryst. 16, 1046–1051.
Wang, B. C. (1985). Resolution of phase ambiguity in macromolecular crystallography. In Diffraction methods for biological macromolecules (Methods in enzymology, Vol. 115), edited by H. Wyckoff, C. H. W. Hirs & S. N. Timasheff, pp. 90–112. New York: Academic Press.

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