Eigensystem analysis of the refinement of a small metalloprotein

The eigenvalues and eigenvectors of the least-squares normal matrix for the full-matrix refinement problem contain a great deal of information about the quality of a model; in particular 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 under-determined 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 data sets and models determined at different resolutions. The eigenvalue spectra reveal 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.