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. 2.2, pp. 70-71

Section 2.2.7. Quality indicators for molecular replacement

H. M. Einspahra* and M. S. Weissb

aPO Box 6483, Lawrenceville, NJ 08648–0483, United States, and bHelmholtz-Zentrum Berlin für Materialien und Energie, Macromolecular Crystallography (HZB-MX), Albert-Einstein-Str. 15, D-12489 Berlin, Germany
Correspondence e-mail:

2.2.7. Quality indicators for molecular replacement

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In a case where a known structure is assumed to be similar to that of a target molecule (structural similarity is typically inferred by the degree of sequence similarity), the known structure, also termed the search model, can be used to determine the structure of the target molecule. The approach, termed molecular replacement, was first described in 1962 (Rossmann & Blow, 1962[link]). Nowadays, about two thirds of all newly determined structures are determined by molecular replacement (Long et al., 2008[link]).

Rotation function, RF. The rotation function RF is a measure of the overlap or the agreement of the stationary Patterson function P2 calculated from the observed data and the rotated Patterson function P1 from the search model.[{\rm RF}=\textstyle \int \limits_{r}P2\, \underline{R}\,P1\,{\rm d}r.\eqno(]In the equation for RF, R is the rotation operator. The integration is performed between a minimum value and a maximum value for the radius r. These values are chosen according to the size of the search model, with the aim of including as many intramolecular Patterson peaks (self vectors) as possible and to exclude as many intermolecular Patterson peaks (cross vectors) as possible. The ratio of the height of a peak in the RF to the background level is used as an indicator of how likely it is that this peak describes the orientation of a molecule in the target structure.

Translation function, TF. There are numerous ways of defining a translation function TF, making it impractical to discuss quality indicators here. For a thorough treatment of translation-function applications, the reader is referred to Chapter 2.3[link] of International Tables for Crystallography Volume B and Chapter 13.3[link] of the present volume.

Log-likelihood gain, LLG. In likelihood-based molecular replacement (McCoy et al., 2007[link]), potential molecular-replacement solutions are evaluated using likelihood, which is defined as the probability P that the observed diffraction data would have been measured if the orientation (and, usually, position) of the model were correct (Read, 2001[link]). The score is reported in terms of the log-likelihood gain (LLG), which is defined as the logarithm of the likelihood score for the model p(Fobs; model) minus the logarithm of the likelihood score for a random-atom Wilson distribution pWilson(Fobs). The LLG measures how much better the data can be predicted from the molecular-replacement model than from a collection of random atoms.[{\rm LLG}=\textstyle \sum \limits_{hkl}\ln[p(F_{\rm obs}\semi{\rm model})]-\textstyle \sum \limits_{hkl}\ln[p_{\rm Wilson}(F_{\rm obs})].\eqno(]

LLG-Z score. It is important to note that the LLG depends on the quality of the model and the number of reflections, so the absolute values cannot be compared between different molecular-replacement applications. Instead, the quality of a molecular-replacement solution can be judged by the LLG-Z score, which is defined as the number of standard deviations a score is above the mean score in a particular rotation or translation search.[\hbox{LLG-Z}={\rm LLG} -\langle{\rm LLG}\rangle/[({\rm LLG} -\langle{\rm LLG}\rangle)^2]^{1/2}.\eqno(]The translation function Z score (TFZ) for the last component placed in a molecular-replacement search is often a good indicator of the confidence that can be placed in the solution. If TFZ is greater than 8 and there is no translational pseudo-symmetry, the solution is almost always correct.

Detailed descriptions of the background and proper application of molecular-replacement approaches are presented in Chapter 2.3[link] of International Tables for Crystallography Volume B and Chapters 13.2[link] and 13.3[link] of the present volume.


Long, F., Vagin, A. A., Young, P. & Murshudov, G. N. (2008). BALBES: a molecular-replacement pipeline. Acta Cryst. D64, 125–132.
McCoy, A. J., Grosse-Kunstleve, R. W., Adams, P. D., Winn, M. D., Storoni, L. C. & Read, R. J. (2007). Phaser crystallographic software. J. Appl. Cryst. 40, 658–674.
Read, R. J. (2001). Pushing the boundaries of molecular replacement with maximum likelihood. Acta Cryst. D57, 1373–1382.
Rossmann, M. G. & Blow, D. M. (1962). The detection of sub-units within the crystallographic asymmetric unit. Acta Cryst. 15, 24–31.

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