The Sayre equation in electron crystallography

The Sayre equation was evaluated as a technique for phase refinement in electron crystallography. Atomic-resolution electron diffraction data from copper perchlorophthalocyanine were assigned phase values from the Fourier transforms of various experimental electron micrographs, including one at 2.3 A, containing errors due to lens astigmatism. In each case, an atomic-resolution structure could be found after Fourier refinement. In addition, it was possible to begin with a basis set derived from symbolic addition for phase extension. Such a source of phases was also found to be useful for extending zonal electron diffraction sets from six polymer crystals, even though there was considerable overlap of atomic positions in the projection down the chain axes. Other tests of the Sayre equation were made with zonal protein data sets (bacteriorhodopsin, halorhodopsin) to evaluate what difficulties are to be expected when direct phasing techniques are to be used in macromolecular electron crystallography. Comparison to known values indicated that the low-resolution range (e.g. to 6 A) was reasonably stable for phase extension from a 10-15 A resolution image. Only when a minimum in average intensity was approached (near 5 A) did the direct extension encounter serious difficulties. If this minimum was treated as a "phase node" to generate two possible solutions, a model more similar to the true phase set was found. In general, this rather simple convolutional technique for phase extension seems to be particularly suitable for a variety of electron crystallographic applications.