Holographic Methods in X-ray Crystallography. V. Multiple IsomorphousReplacement, Multiple Anomalous Dispersion and Non-crystallographic Symmetry

The holographic method for the recovery of the electron density of macromolecules is based on the expansion of the electron density into Gaussian basis functions. The technique makes consistent use of real- and reciprocal-space information to produce electron-density maps. It enforces positivity of the recovered electron density and makes effective use of previously known information about the electron density, such as knowledge of a solvent region or knowledge of a partial structure. In this paper, we summarize the theory underlying the holographic method, and describe how we extend the range of information that can be used by the method to include information from multiple-isomorphous-replacement (MIR) data, multiple-anomalous-dispersion (MAD) data and knowledge of non-crystallographic symmetry. The convergence properties and the limiting accuracy of the method are discussed. Its power for synthetic problems is demonstrated and the method is applied to experimentally measured MIR data from kinesin, a motor protein domain that has recently been solved. Appendix A gives a detailed description of the algorithms and the equations used in EDEN, the computer program that implements the holographic method.