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

International Tables for Crystallography (2006). Vol. F, ch. 11.5, p. 236   | 1 | 2 |

Section 11.5.1. Introduction

C. G. van Beek,a R. Bolotovskya§ and M. G. Rossmanna*

aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA
Correspondence e-mail:

11.5.1. Introduction

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Recent advances in the use of frozen crystals of biological samples for X-ray diffraction data collection (Rodgers, 1994[link]) often result in data for which most of the observed reflections on each frame are partially observed. This might be avoided by increasing the oscillation ranges, but this would cause many reflections to overlap with their neighbours. Hence, it is necessary to develop scaling procedures that are independent of the exclusive use of fully recorded reflections.

A set of measured Bragg intensities is dependent on the properties of the crystal, radiation source and detector. Usually, these factors cannot be kept constant throughout the data collection. The crystal may decay, weakening the Bragg intensities, or even `die', which requires the use of several crystals for a full data set. The intensity and position of the primary X-ray beam may vary, especially at synchrotron-radiation sources. Finally, the detector response may change when, for example, different films or imaging plates are used during the data collection.

Most data sets can be divided into series of subsets, or frames, collected under more-or-less constant conditions. These frames need to be placed on a common arbitrary scale. The scaling can be performed by comparing the intensities of multiply measured reflections or symmetry-equivalent reflections on different frames.

A least-squares procedure frequently used for scaling frames of data is the Hamilton, Rollett and Sparks (HRS) method (Hamilton et al., 1965[link]). The target for the HRS least-squares minimization is [\psi = {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{i}} W_{hi} (I_{hi} - G_{m} I_{h})^{2}, \eqno(] where [I_{h}] is the best estimate of the intensity of a reflection with reduced Miller indices h, [I_{hi}] is the intensity of the ith measurement of reflection h, [W_{hi}] is a weight for reflection [h_{i}] and [G_{m}] is the inverse linear scale factor for frame m on which reflection [h_{i}] is recorded. The reduced Miller indices are those corresponding to an arbitrarily defined asymmetric unit of reciprocal space. The HRS expression ([link]) assumes that all reflections [h_{i}] are full, that is, their reciprocal-lattice points have completely passed through the Ewald sphere.

For all unique reflections h, the values of [I_{h}] must correspond to a minimum in ψ. Thus, [\partial \psi/\partial I_{h} = 0. \eqno(] Therefore, the best least-squares estimate of the intensity of a reflection is [I_{h} = {\textstyle\sum\limits_{i}} W_{hi} G_{m} I_{hi}\big/{\textstyle\sum\limits_{i}} W_{hi} G_{m}^{2}. \eqno(] Since ψ is not linear with respect to the scale factors [G_{m}], the values of the scale factors have to be determined by an iterative nonlinear least-squares procedure. As the scale factors are relative to each other, the HRS procedure requires that one of them be fixed.

Fox & Holmes (1966[link]) describe an improved method of solving the HRS normal equations. Their approach is based on the singular value decomposition of the normal equations matrix. The advantage of the Fox and Holmes method, apart from the accelerated convergence of the least-squares procedure, is that no ad hoc decision needs to be made as to which scale factor should be fixed. Furthermore, `troublesome' frames of data can be identified as causing negligibly small eigenvalues in the normal equations matrix.


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.
Hamilton, W. C., Rollett, J. S. & Sparks, R. A. (1965). On the relative scaling of X-ray photographs. Acta Cryst. 18, 129–130.
Rodgers, D. W. (1994). Cryocrystallography. Structure, 2, 1135–1140.

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