Objective comparison of protein structures: error-­scaled difference distance matrices

Understanding of macromolecular function in many cases relies on the comparison of related structural models. Commonly used least-squares superposition methods suffer from bias introduced into the comparison process by the subjective choice of atoms employed for the superposition. Difference distance matrices are a more objective means of comparing structures as they do not depend on a particular superposition scheme. However, they suffer from very high noise originating from coordinate errors. Modern refinement programs allow the rigorous estimation of standard uncertainties for individual atomic positions. These errors can be propagated through the calculation of a difference distance matrix allowing one to assess the significance level of structural differences. An algorithm is presented which produces an intuitive graphical representation of difference distance matrices after normalization to their error levels. Two examples where its application was revealing are described. Alternatives are suggested for cases where rigorous estimation of individual errors by the inversion of the full least-squares matrix is not feasible. The method offers an unbiased way to detect significant similarities and differences between related structures, as encountered in studies of complexes and mutants or when multiple models are obtained from experiments such as crystal structures involving non-crystallographic symmetry or different crystal modifications, or ensembles derived from NMR spectroscopy.