THE SAYRE EQUATION IN ELECTRON CRYSTALLOGRAPHY

The Sayre equation was evaluated as a technique for phase refinement i n electron crystallography. Atomic-resolution electron diffraction dat a from copper perchlorophthalocyanine were assigned phase values from the Fourier transforms of various experimental electron micrographs, i ncluding one at 2.3 Angstrom containing errors due to lens astigmatism . In each case, an atomic-resolution structure could be found after Fo urier refinement. In addition, it was possible to begin with a basis s et derived from symbolic addition for phase extension. Such a source o f phases was also found to be useful for extending zonal electron diff raction sets from six polymer crystals, even though there was consider able 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 ma cromolecular electron crystallography. Comparison to known values indi cated that the low-resolution range (e.g. to 6 Angstrom) was reasonabl y stable for phase extension from a 10-15 Angstrom resolution image. O nly when a minimum in average intensity was approached (near 5 Angstro m) did the direct extension encounter serious difficulties. If this mi nimum was treated as a ''phase node'' to generate two possible solutio ns, a model more similar to the true phase set was found. In general, this rather simple convolutional technique for phase extension seems t o be particularly suitable for a variety of electron crystallographic applications.