IMPROVED STRUCTURE REFINEMENT THROUGH MAXIMUM LIKELIHOOD. Randy J. Read and Navraj S. Pannu, Departments of Medical Microbiology & Immunology, and Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2H7, Canada.

The least-squares target is not theoretically justified for crystal structure refinement, so it is preferable to use a maximum likelihood target instead. With a maximum likelihood treatment, the need for ad hoc weighting schemes and resolution cutoffs is eliminated, observational errors are used appropriately and, above all, the refinement is more successful.

When crystal structures of proteins or small molecules are used to address questions of scientific relevance, the accuracy and precision of the atomic coordinates are crucial. Accordingly, the atomic model is generally improved by refining it to improve agreement with the observed diffraction data. The use of least-squares methods would only be justified (by the principle of maximum likelihood) if the probability distribution relating the observed and calculated diffraction measurements were Gaussian. As the relationship is not Gaussian, the least-squares target is inappropriate.

We have implemented two maximum likelihood targets in the program XPLOR: 1) an amplitude-based Gaussian approximation assuming Gaussian errors in the observed amplitudes; and 2) an intensity-based likelihood function assuming Gaussian errors in the observed amplitudes squared. The amplitude-based target can be implemented easily in any least-squares refinement program, while the intensity-based target has a number of advantages including the ability to use negative observed intensities.

Preliminary tests with protein structures give dramatic results. Compared to least-squares refinement, maximum likelihood refinement can achieve more than twice the improvement in average phase error. The resulting electron density maps are correspondingly clearer and suffer less from model bias.

The authors were supported in these investigations by AHFMR, MRC (Canada), NSERC (Canada) and an International Research Scholars award to RJR from HHMI.