EIGENSYSTEM ANALYSIS OF THE REFINEMENT NORMAL MATRIX

K. D. Cowtan¹, L. F. Ten Eyck²

¹ Department of Chemistry, University of York, York, YO1 5DD, United Kingdom.
² Science Department, San Diego Supercomputer Center, San Diego, CA 92186-5608, U.S.A.

Keywords: REFINMENT, VALIDATION, ERROR-ESTIMATION

The normal matrix for the full-matrix least-squares problem contains a great deal of information about the quality of a model [1]. In particular the eigenvalues and eigenvectors contain the precision of the model parameters, and correlations between those parameters. They also allow the isolation of those parameters or combinations of parameters which are not determined by the available data. Since a protein refinement is usually underdetermined without the application of geometric restraints, such indicators of the reliability of a model offer an important contribution to structural knowledge.

Eigensystem analysis is applied to the normal matrices for the refinement of a small metalloprotein using two datasets and models determined at different resolutions. The eigenvalue spectra reveals considerable information about the conditioning of the problem as the resolution varies. In the case of a restrained refinement, it also provides information about the impact of various restraints on the refinement. Initial results support conclusions drawn from the Free-R factor.

Examination of the eigenvectors provides information about which regions of the model are poorly determined. In the case of a restrained refinement, it is also possible to isolate places where X-ray and geometric restraints are in disagreement, usually indicating a problem in the model. The methods described here are also applicable to maximum-likelihood refinement.

1. L. F. Ten Eyck, Crystallographic Computing 7, Eds. Philip E. Bourne and Keith Watenpaugh. (1996) Oxford University Press