X-ray crystallography at macromolecular resolution: A solution of the phase problem

The problem of crystal structure analysis at macromolecular resolution is tackled using notions borrowed from small‐angle solution‐scattering studies. The mathematical procedure ‐ essentially a pattern recognition approach ‐ consists of generating all the phase combinations compatible with the data and screening them: the critical steps are the choice of the “pattern” and of the criteria which preside the “recognition”. Two strategies are adopted. One is based upon use of the histogram of the electron density maps, a function more dependent on chemical composition than on physical structure: the histograms of samples of different structure and similar chemical composition are compared in search for similarity. The other strategy operates on the electron density maps and involves the principle of maximum smoothness. From the practical viewpoint, one single parameter, (Δp)4, is found to play an outstanding role in the two cases. Algorithms are worked out, implemented on the computer and tested using a variety of lipid‐water phases. The comparison of samples of different structure and similar chemical composition is found to provide a powerful screening criterion. The principle of maximum smoothness is also of great help, at least to the extent that the electron density distribution is a fairly smooth function, devoid of sharp peaks (associated with heavy atoms). The practical aspects of the algorithms are discussed, as well as their possible application to crystallographic problems of more general interest