International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 |
International Tables for Crystallography (2012). Vol. F, ch. 11.4, pp. 290-291
Section 11.4.8.2. Stabilization of scaling parameters based on prior knowledge^{a}UT Southwestern Medical Center at Dallas, 5323 Harry Hines Boulevard, Dallas, TX 75390–9038, USA, and ^{b}Department of Molecular Physiology and Biological Physics, University of Virginia, 1300 Jefferson Park Avenue, Charlottesville, VA 22908, USA |
The unknown parameters in equation (11.4.8.4) 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), 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 w_{s} between frames, where expectation about w_{s} 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: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 (11.4.8.7)], a reasonable restraint term parameterized by w_{a} results inIf we do not want to penalize high-order terms more than the low-order ones, the following restraint can be used:
References
Tikhonov, A. N. & Arsenin, V. I. A. (1977). Solutions of ill-posed problems. Washington, New York, Winston: Halsted Press.