On the application of phase relationships to complex structures. XXVII. Phase extension for small proteins

Proteins and direct methods

Conventional direct methods, which work so well for small structures, are less successful for macromolecules. Where it has been demonstrated that a solution might be found using direct methods it is then found that the usual figures of merit are unable to distinguish the few good sets of phases from the large number of sets generated. The reasons for the difficulties with very large structures are considered from a first-principles approach taking into account both the factors of having a large number of atoms and low resolution data. A proposal is made for trying to recognize good phase sets by taking a large structure as a sum of a number of smaller structures for each of which a conventional figure of merit can be applied.

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.