New ways of looking at experimental phasing

In the original work by Blow and Crick, experimental phasing was formulated as a least-squares problem. For good data on good derivatives this approach works reasonably well, but we now attempt to extract more information from poorer data than in the past. As in many other crystallographic problems, the assumptions underlying the use of least squares for phasing are not satisfied, particularly for poor derivatives. The introduction of maximum likelihood (and more powerful computers) has led to substantial improvements. For computational convenience, these new methods still make many assumptions about the independence of different measurements and sources of error. A more general formulation for the probability distributions underlying likelihood-based methods for both experimental phasing and molecular-replacement phasing is reviewed. In the new formulation, all the structure factors associated with a particular hkl are considered to be related by a complex multivariate normal distribution. When it is assumed that certain errors are independent, the general formulation reduces to current likelihood targets. However, the new formulation makes the necessary assumptions more explicit and points the way to improving phasing using both isomorphous and anomalous differences.