Having explored the use of Sayre’s squaring relation in phase refinement during year 1, we encountered an impasse arising from the fact that the objective function defined by Sayre’s equations is multimodal, and has minima that do not correspond to the correct phase set. An important limitation identified in this work is the difficulty of incorporating different atom types into the equations. We have therefore reformulated Sayre’s squaring relation in real space and evaluated it locally by forming a local squaring function, whose value depends on the extent to which the electron density near a point approximates that of a particular atom type. Local squaring functions can be defined over the entire unit cell for multiple atom types and can then be combined to form a probabilistic filter that can be used for phase improvement by generalized density modification. The overall procedure provides a method for extending the power of solvent flattening into the molecular region of a map, thus affording the full power of density modification.

We implemented a two-stage phase refinement algorithm using local squaring functions first for phase refinement via density modification, and next by using the local squaring function mimima as the basis for automatic map interpretation, leading to parametrization into atomic models. It works very well on several different kinds of model systems, reducing phase errors by 20-30 degrees. This phase refinement procedure vastly outperforms the least squares phase refinements envisioned in our initial publication. Nevertheless, it stops short of our goal, which is a fully convergent, model-independent phase determination. The most important reason for the failure to converge appears to be that the spherical atomic templates are not appropriate for many of the local features of the electron density. A key example is the backbone carbonyl group, which appears to violate sphericity at most contour levels even for very high resolution Fourier terms if the isotropic temperature factor exceeds around 12Å2. This surprising result places a quite fundamental limitation on the use of spherical atom types in phasing protein structures. We have therefore begun to investigate the use of non-spherical templates ranging from the carbonyl group to the entire peptide group. As results with the local squaring functions based on spherical templates gave significant improvement to experimental phase sets obtained at 1.7Å resolution for Tryptophanyl-tRNA synthetase, it is quite possible that the use of non-spherical templates will extend the usefulness of local squaring functions into even lower resolution regimes. This provides the basis for the work we plan to do in the coming year 3.