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 16.1.7. Resolution enhancement: the `free lunch' algorithm

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:

16.1.7. Resolution enhancement: the `free lunch' algorithm

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Direct methods take a set of phases, refine them and also deter­mine new ones. There is no reason, however, why they cannot be used to predict new amplitudes as well. If density modification of a real-space map is performed, then any process of real-space density modification will, following a Fourier transformation, give structure-factor amplitudes for reflections that were not used to generate it, and these can be outside the resolution limit. If direct methods require atomic resolution data, can we use these techniques to extrapolate structure factors (i.e. predict not only phases, but also amplitudes for missing data) and extend data resolution? The idea is not new, but it has been quite extensively studied in recent years. Sheldrick has termed such algorithms `free lunch', with reference to the saying: `There is no such thing as a free lunch'! In one example (Usón et al., 2007[link]), weak SIRAS starting phase information followed by density modification led to an |Fo| weighted mean phase error (MPE) of 54° at 1.98 Å resolution, but when the density modification was performed with amplitude extrapolation to 1.0 Å, the MPE fell to 17°. Caliandro et al. (2005a[link],b[link]) used Patterson or direct methods to obtain trial phases that are submitted to various density-modification methods. Following this, extrapolated phases were generated. This too transformed uninterpretable maps into a solution amenable to automatic tracing. Palatinus et al. (2007[link]) used maximum entropy (ME) methods for amplitude extrapolation. In some ways these should be ideal for this purpose, and it is worth noting that ME maps have, de facto, optimal resolution enhancement built in, although they can be difficult to generate for large structures.

Why does this work, and why is it sometimes so spectacular? The answer probably lies with the fact that maps are much more sensitive to phases than amplitudes and, if the model bias of predicting new amplitudes is not too great, then using a nonzero value is better than zero, which is the default. Fourier-truncation errors may also be reduced, resulting in less spurious map detail.


Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2005a). Phasing at resolution higher than the experimental resolution. Acta Cryst. D61, 556–565.
Caliandro, R., Carrozzini, B., Cascarano, G. L., De Caro, L., Giacovazzo, C. & Siliqi, D. (2005b). Ab initio phasing at resolution higher than experimental resolution. Acta Cryst. D61, 1080–1087.
Palatinus, L., Steurer, W. & Chapuis, G. (2007). Extending the charge-flipping method towards structure solution from incomplete data sets. J. Appl. Cryst. 40, 456–462.
Usón, I., Stevenson, C. E. M., Lawson, D. M. & Sheldrick, G. M. (2007). Structure determination of the O-methyltransferase NovP using the `free lunch algorithm' as implemented in SHELXE. Acta Cryst. D63, 1069–1074.

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