Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 16.1, p. 419   | 1 | 2 |

Section 16.1.6. Fourier refinement

G. M. Sheldrick,a C. J. Gilmore,b H. A. Hauptman,c C. M. Weeks,c* R. Millerc and I. Usónd

aLehrstuhl für Strukturchemie, Georg-August-Universität Göttingen, Tammannstrasse 4, D-37077 Göttingen, Germany,bDepartment of Chemistry, University of Glasgow, Glasgow G12 8QQ, UK,cHauptman–Woodward Medical Research Institute, Inc., 700 Ellicott Street, Buffalo, NY 14203–1102, USA, and dInstitució Catalana de Recerca i Estudis Avançats at IBMB-CSIC, Barcelona Science Park. Baldiri Reixach 15, 08028 Barcelona, Spain
Correspondence e-mail:

16.1.6. Fourier refinement

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E-map recycling, but without phase refinement (Sheldrick, 1982[link], 1990[link]; Kinneging & de Graaff, 1984[link]), has been frequently used in conventional direct-methods programs to improve the completeness of the solutions after phase refinement. It is important to apply Fourier refinement to Shake-and-Bake solutions also because such processing significantly increases the number of resolved atoms, thereby making the job of map interpretation much easier. Since phase refinement via either the tangent formula or the minimal function requires relatively accurate invariants that can only be generated using the larger E magnitudes, a limited number of reflections are phased during the actual dual-space cycles. Working with a limited amount of data has the added advantage that less CPU time is required. However, if the current trial structure is the `best' so far based on a figure of merit (either the minimal function or a real-space criterion), then it makes sense to subject this structure to Fourier refinement using additional data, thereby reducing series-termination errors. The correlation coefficient [\eqalignno{\hbox{CC} &= \left[\left(\textstyle\sum wE_{o}^{2} E_{c}^{2} \textstyle\sum w\right) - \left(\textstyle\sum wE_{o}^{2} \textstyle\sum wE_{c}^{2}\right)\right]\cr &\quad \times \left\{\left[\left(\textstyle\sum wE_{o}^{4} \textstyle\sum w\right) - \left(\textstyle\sum wE_{o}^{2}\right)^{2}\right]\right.\cr&\quad\times\left.\left[\left(\textstyle\sum wE_{c}^{4} \textstyle\sum w\right) - \left(\textstyle\sum wE_{c}^{2}\right)^{2}\right]\right\}^{-1/2} &(\cr}](Fujinaga & Read, 1987[link]), where weights [w = 1/[0.04 + \sigma^{2} (E_{o})]], has been found to be an especially effective figure of merit when used with all the data and is, therefore, suited for identifying the most promising trial structure at the end of Fourier refinement. Either simple peak picking or iterative peaklist optimization can be employed during the Fourier-refinement cycles in conjunction with weighted E maps (Sim, 1959[link]). The final model can be further improved by isotropic displacement parameter [(B_{\rm iso})] refinement for the individual atoms (Usón et al., 1999[link]) followed by calculation of the Sim (1959)[link] or sigma-A (Read, 1986[link]) weighted map. This is particularly useful when the requirement of atomic resolution is barely fulfilled, and it makes it easier to interpret the resulting maps by classical macromolecular methods.


Fujinaga, M. & Read, R. J. (1987). Experiences with a new translation-function program. J. Appl. Cryst. 20, 517–521.
Kinneging, A. J. & de Graaf, R. A. G. (1984). On the automatic extension of incomplete models by iterative Fourier calculation. J. Appl. Cryst. 17, 364–366.
Read, R. J. (1986). Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Cryst. A42, 140–149.
Sheldrick, G. M. (1982). Crystallographic algorithms for mini- and maxi-computers. In Computational Crystallography, edited by D. Sayre, pp. 506–514. Oxford: Clarendon Press.
Sheldrick, G. M. (1990). Phase annealing in SHELX-90: direct methods for larger structures. Acta Cryst. A46, 467–473.
Sim, G. A. (1959). The distribution of phase angles for structures containing heavy atoms. II. A modification of the normal heavy-atom method for non-centrosymmetical structures. Acta Cryst. 12, 813–815.
Usón, I., Sheldrick, G. M., de La Fortelle, E., Bricogne, G., di Marco, S., Priestle, J. P., Grütter, M. G. & Mittl, P. R. E. (1999). The 1.2 Å crystal structure of hirustasin reveals the intrinsic flexibility of a family of highly disulphide bridged inhibitors. Structure, 7, 55–63.

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