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. 15.1, p. 398   | 1 | 2 |

Section 15.1.6. Statistical density-modification methods

K. Y. J. Zhang,a K. D. Cowtanb* and P. Mainc

aDivision of Basic Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N., Seattle, WA 90109, USA,bDepartment of Chemistry, University of York, York YO1 5DD, England, and cDepartment of Physics, University of York, York YO1 5DD, England
Correspondence e-mail:

15.1.6. Statistical density-modification methods

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Statistical density-modification methods arise from a reinterpretation of the problem of phase improvement in statistical terms, and as a result reduce the problems of bias associated with the classical density-modification methods described above.

This is achieved by expressing all information about probable electron-density values in the map in terms of probability distributions, with a probability distribution of electron-density values assigned to each point in the electron-density map. The probable electron-density values at any point in the map may depend, for example, on whether that point lies in the solvent or protein region, and on any NCS relationships with other regions of the map. These distributions are then used to infer corresponding phase probability distributions in reciprocal space, which may then be combined with any existing phase information. This avoids working with a single map representing a single sample from the phase probability distributions.

This formulation in turn weakens the link between the newly introduced information and the initial phase probability distributions, thus reducing the bias introduced in a single cycle of phase improvement. Since the current `best' map is not used as a starting map to be modified, the phase probability distributions from which the `best' map is derived are not directly included in any new phase information. The only way in which the current phases are used is in the classification of the asymmetric unit into regions of different density types, e.g. solvent and protein.

The result of these changes is that statistical density-modification techniques lead to reduced phase bias and more realistic estimates of the figures of merit. The resulting method has been implemented in the RESOLVE software (Terwilliger, 1999[link]). In addition to its application to conventional density-modification problems, it has been particularly effective in removing bias from maps phased from an atomic model through the `prime-and-switch' approach (Terwilliger, 2004[link]). The statistical approach to density modification requires substantially more computation that the simpler classical methods.


Terwilliger, T. C. (1999). Reciprocal-space solvent flattening. Acta Cryst. D55, 1863–1871.
Terwilliger, T. C. (2004). Using prime-and-switch phasing to reduce model bias in molecular replacement. Acta Cryst. D60, 2144–2149.

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