Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 91-92   | 1 | 2 |

Section Other nonlinear operations

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Other nonlinear operations

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A category of phase improvement procedures known as `density modification' is based on the pointwise application of various quadratic or cubic `filters' to electron-density maps after removal of negative regions (Hoppe & Gassmann, 1968[link]; Hoppe et al., 1970[link]; Barrett & Zwick, 1971[link]; Gassmann & Zechmeister, 1972[link]; Collins, 1975[link]; Collins et al., 1976[link]; Gassmann, 1976[link]). These operations are claimed to be equivalent to reciprocal-space phase-refinement techniques such as those based on the tangent formula. Indeed the replacement of[\rho\llap{$-\!$} ({\bf x}) = {\textstyle\sum\limits_{{\bf h}}} \,F_{{\bf h}} \exp (-2 \pi i {\bf h} \cdot {\bf x})]by [P[\rho\llap{$-\!$} ({\bf x})]], where P is a polynomial[P(\rho\llap{$-\!$}) = a_{0} + a_{1} \rho\llap{$-\!$} + a_{2} \rho\llap{$-\!$}^{2} + a_{3} \rho\llap{$-\!$}^{3} + \ldots]yields[\eqalign{ P[\rho\llap{$-\!$}({\bf x})] =\ &a_{0} + {\textstyle\sum\limits_{{\bf h}}} \left[a_{1} F_{{\bf h}} + a_{2} {\textstyle\sum\limits_{{\bf k}}} \,F_{{\bf k}} F_{{\bf h} - {\bf k}}\right.\cr &\left.+\ a_{3} {\textstyle\sum\limits_{{\bf k}}} {\textstyle\sum\limits_{\bf l}} \,F_{{\bf k}} F_{\bf l} F_{{\bf h} - {\bf k} - {\bf l}} + \ldots \right] \exp (- 2 \pi i{\bf h} \cdot {\bf x})}]and hence gives rise to the convolution-like families of terms encountered in direct methods. This equivalence, however, has been shown to be rather superficial (Bricogne, 1982[link]) because the `uncertainty principle' embodied in Heisenberg's inequality (Section[link]) imposes severe limitations on the effectiveness of any procedure which operates pointwise in both real and reciprocal space.

In applying such methods, sampling considerations must be given close attention. If the spectrum of [\rho\llap{$-\!$}] extends to resolution Δ and if the pointwise nonlinear filter involves a polynomial P of degree n, then P([\rho\llap{$-\!$}]) should be sampled at intervals of at most [\Delta/2n] to accommodate the full bandwidth of its spectrum.


Barrett, A. N. & Zwick, M. (1971). A method for the extension and refinement of crystallographic protein phases utilizing the fast Fourier transform. Acta Cryst. A27, 6–11.
Bricogne, G. (1982). Generalised density modification methods. In Computational Crystallography, edited by D. Sayre, pp. 258–264. New York: Oxford University Press.
Collins, D. M. (1975). Efficiency in Fourier phase refinement for protein crystal structures. Acta Cryst. A31, 388–389.
Collins, D. M., Brice, M. D., Lacour, T. F. M. & Legg, M. J. (1976). Fourier phase refinement and extension by modification of electron-density maps. In Crystallographic Computing Techniques, edited by F. R. Ahmed, pp. 330–335. Copenhagen: Munksgaard.
Gassmann, J. (1976). Improvement and extension of approximate phase sets in structure determination. In Crystallographic Computing Techniques, edited by F. R. Ahmed, pp. 144–154. Copenhagen: Munksgaard.
Gassmann, J. & Zechmeister, K. (1972). Limits of phase expansion in direct methods. Acta Cryst. A28, 270–280.
Hoppe, W. & Gassmann, J. (1968). Phase correction, a new method to solve partially known structures. Acta Cryst. B24, 97–107.
Hoppe, W., Gassmann, J. & Zechmeister, K. (1970). Some automatic procedures for the solution of crystal structures with direct methods and phase correction. In Crystallographic Computing, edited by F. R. Ahmed, S. R. Hall & C. P. Huber, pp. 26–36. Copenhagen: Munksgaard.

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