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 t he 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 re liability of a model offer an important contribution to structural knowledg e. Eigensystem analysis is applied to the normal matrices for the refinemen t of a small metalloprotein using two data sets and models determined at di fferent resolutions. The eigenvalue spectra reveal considerable information about the conditioning of the problem as the resolution varies. In the cas e of a restrained refinement, it also provides information about the impact of various restraints on the refinement. Initial results support conclusio ns drawn from the free R factor. Examination of the eigenvectors provides i nformation about which regions of the model are poorly determined. In the c ase of a restrained refinement, it is also possible to isolate places where X-ray and geometric restraints are in disagreement, usually indicating a p roblem in the model.