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

International Tables for Crystallography (2006). Vol. F, ch. 11.3, pp. 222-223   | 1 | 2 |

Section 11.3.4. Scaling

W. Kabscha*

aMax-Planck-Institut für medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, 69120 Heidelberg, Germany
Correspondence e-mail:

11.3.4. Scaling

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Usually, many statistically independent observations of symmetry-related reflections are recorded in the rotation images taken from one or several similar crystals of the same compound. The squared structure-factor amplitudes of equivalent reflections should be equal and the idea of scaling is to exploit this a priori knowledge to determine a correction factor for each observed intensity. These correction factors compensate to some extent for effects such as radiation damage, absorption, and variations in detector sensitivity and exposure times, as well as variations in size and disorder between different crystals.

The usual methods of scaling split the data into batches of roughly the same size, each covering one or more adjacent rotation images, and then determine a single scaling factor for all reflections in each batch. Neighbouring reflections may then receive quite different corrections if they are assigned to different batches. Since the selection of batch boundaries is to some extent arbitrary, a more continuous correction function would be preferable. This function could be modelled analytically (for example by using spherical harmonics) or empirically, as implemented in XSCALE and described below.

For each reflection, observational equations are defined as [\psi_{hl\alpha} = (I_{hl} - g_{\alpha} I_{h})/\sigma_{hl}.] The subscript h represents the unique reflection indices and l enumerates all symmetry-related reflections to h. By definition, the unique reflection indices have the largest h, then k, then l value occuring in the set of all indices related by symmetry to the original indices, including Friedel mates. Thus, two reflections are symmetry-related if and only if their unique indices are identical. [I_{h}] is the unknown `true' intensity and [I_{hl}], [\sigma_{hl}] are symmetry-related observed intensities and their standard deviations, respectively. The subscript α denotes the coordinates at which the scaling function [g_{\alpha}] should be evaluated. As implemented in XDS and XSCALE, [\alpha = 1,\ldots,9] denotes nine positions uniformly distributed in the detector plane at the beginning of data collection, [\alpha = 10,\ldots,18] the same positions on the detector but after the crystal has been rotated by, say, 5°, and so on. The scaling factors [g_{\alpha}] and the estimated intensities [I_{h}] are found at the minimum of the function [\Psi = {\textstyle\sum\limits_{hl\alpha}}w_{hl\alpha}\Psi_{hl\alpha}^2.]

The main difference from the method of Fox & Holmes (1966[link]) is the introduction of the weights [w_{hl\alpha}]. These weights depend upon the distance between each reflection hl and the positions α. They are monotonically decreasing functions of this distance, implemented as Gaussians in XDS and XSCALE. This results in a smoothing of the scaling factors since each reflection contributes to the observational equations in proportion to the weights [w_{hl\alpha}].

Minimization of Ψ is done iteratively. After each step, the [g_{\alpha}] are replaced by [\Delta g_{\alpha}+g_{\alpha}] and rescaled to a mean value of 1. The corrections [\Delta g_{\alpha}] are determined from the normal equations [{\textstyle\sum\limits_{\beta}}\ A_{\alpha\beta}\Delta g_{\beta}=b_{\alpha},] where [\eqalign{A_{\alpha\beta} &= {\textstyle\sum\limits_h}[\delta_{\alpha\beta}I_h^2 u_{h\alpha} + (r_{h\alpha}v_{h\beta}+v_{h\alpha}r_{h\beta}-v_{h\alpha}v_{h\beta})/u_h]\cr b_{\alpha} &= {\textstyle\sum\limits_{h}} I_hr_{h\alpha}\cr I_{h} &= v_h/u_h\cr r_{h\alpha} &= v_{h\alpha}-g_{\alpha}u_{h\alpha}I_h\cr u_{h\alpha} &= {\textstyle\sum\limits_l} w_{hl\alpha}/\sigma_{hl}^2\cr v_{h\alpha} &= {\textstyle\sum\limits_l} w_{hl\alpha}I_{hl}/\sigma_{hl}^2\cr u_h &= {\textstyle\sum\limits_\alpha} g_{\alpha}^2u_{h\alpha}\cr v_h &= {\textstyle\sum\limits_\alpha} g_{\alpha}v_{h\alpha}.}]

In case a `true' intensity [I_{h}] is available from a reference data set, the non-diagonal elements are omitted from the sum over h in the normal matrix [A_{\alpha\beta}]. The corrections [\Delta g_{\alpha}] are expanded in terms of the eigenvectors of the normal matrix, thereby avoiding shifts along eigenvectors with very small eigenvalues (Diamond, 1966[link]). This filtering method is essential since the normal matrix has zero determinant if no reference data set is available.


Diamond, R. (1966). A mathematical model-building procedure for proteins. Acta Cryst. 21, 253–266.
Fox, G. C. & Holmes, K. C. (1966). An alternative method of solving the layer scaling equations of Hamilton, Rollett and Sparks. Acta Cryst. 20, 886–891.

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