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. 11.4, pp. 290-291   | 1 | 2 |

Section Stabilization of scaling parameters based on prior knowledge

Z. Otwinowski,a* W. Minor,b D. Boreka and M. Cymborowskib

aUT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390–9038, USA, and bDepartment of Molecular Physiology and Biological Physics, University of Virginia, 1300 Jefferson Park Avenue, Charlottesville, VA 22908, USA
Correspondence e-mail: Stabilization of scaling parameters based on prior knowledge

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The unknown parameters in equation ([link] are estimated with various level of uncertainty depending on the multiplicity of observations and how symmetry-equivalent reflections are related to each other. Potentially, this may result in unreasonable values of scaling parameters due to insufficient information to determine the values of parameters. In SCALEPACK, the method to stabilize such ill-conditioned calculations is closely related to Tikhonov stabilization (Tikhonov & Arsenin, 1977[link]), where additional, a priori knowledge about the expected mag­nitude of the physical effect modelled is used to restrain the solutions, based on the same argument as in the case of restraints in the atomic refinement.

For example, logarithms of scale factors typically do not fluctuate by more than ws between frames, where expectation about ws is a function of the data-collection stability (beam stability, goniostat and/or crystal vibrations). This knowledge is described by adding a penalty term (scale restrain) to the functions being optimized:[(1/w_s^2)(p_{i_s}-p_{(i+1)_s})^2.\eqno(]A similar approach can be used in calculations of absorption coefficients. For smooth absorption with the expectation of decreasing magnitude of parameters for high orders of spherical harmonics [equation ([link]], a reasonable restraint term parameterized by wa results in[{l^2(\,p_{as,lm}^2+p_{ac, lm}^2)\over w_a^2}.\eqno(]If we do not want to penalize high-order terms more than the low-order ones, the following restraint can be used:[{(\,p_{as,lm}^2+p_{ac, lm}^2)\over w_a^2}.\eqno(]


Tikhonov, A. N. & Arsenin, V. I. A. (1977). Solutions of ill-posed problems. Washington, New York, Winston: Halsted Press.

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