International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 18.4, p. 397
Section 18.4.4.2. Restraints and/or constraints on coordinates and ADPs
a
National Cancer Institute, Brookhaven National Laboratory, Building 725A-X9, Upton, NY 11973, USA,bStructural Biology Laboratory, Department of Chemistry, University of York, York YO10 5DD, England, and CLRC, Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, England, and cStructural Biology Laboratory, Department of Chemistry, University of York, York YO10 5DD, England |
Even for small-molecule structures, disordered regions of the unit cell require the imposition of stereochemical restraints or constraints if the chemical integrity is to be preserved and the ADPs are to be realistic. The restraints are comparable to those used for proteins at lower resolution and this makes sense, since the poorly ordered regions with high ADPs in effect do not contribute to the high-angle diffraction terms, and as a result their parameters are only defined by the lower-angle amplitudes.
Thus, even for a macromolecule for which the crystals diffract to atomic resolution, there will be regions possessing substantial thermal or static disorder, and restraints on the positional parameters and ADPs are essential for these parts. Their effect on the ordered regions will be minimal, as the X-ray terms will dominate the refinement, provided the relative weighting of X-ray and geometric contributions is appropriate.
Another justification for use of restraints is that refinement can be considered a Bayesian estimation. From this point of view, all available and usable prior knowledge should be exploited, as it should not harm the parameter estimation during refinement. Bayesian estimation shows asymptotic behaviour (Box & Tiao, 1973), i.e., when the number of observations becomes large, the experimental data override the prior knowledge. In this sense, the purpose of the experiment is to enhance our knowledge about the molecule, and the procedure should be cumulative, i.e., the result of the old experiment should serve as prior knowledge for the design and treatment of new experiments (Box & Tiao, 1973; Stuart et al., 1999; O'Hagan, 1994). However, there are problems in using restraints. For example, the probability distribution reflecting the degree of belief in the restraints is not good enough. Use of a Gaussian approximation to distributions of distances, angles and other geometric properties has not been justified. Firstly, the distribution of geometric parameters depends strongly on ADPs, and secondly, different geometric parameters are correlated. This problem should be the subject of further investigation.
References
Box, G. E. P. & Tiao, G. C. (1973). Bayesian inference in statistical analysis. Reading, Massachusetts/California/London: Addison-Wesley.Google ScholarO'Hagan, A. (1994). Kendal's advanced theory of statistics; Bayesian inference, Vol. 2B. Cambridge: Arnold, Hodder Headline and Cambridge University Press.Google Scholar
Stuart, A., Ord, K. J. & Arnold, S. (1999). Kendall's advanced theory of statistics; classical inference and linear model, Vol. 2A. London/Sydney/Auckland: Arnold, Hodder Headline.Google Scholar