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A new multisolution phasing method based on entropy maximization and likelihood ranking, proposed for the specific purpose of extending probabilistic direct methods to the field of macromolecules, has been implemented in two different computer programs and applied to a wide variety of problems. The latter comprise the determination of small crystal structures from X-ray diffraction data obtained from single crystals or from powders, and from electron diffraction data partially phased by image processing of electron micrographs, the ab initio generation and ranking of phase sets for small proteins; and the improvement of poor quality phases for a larger protein at medium resolution under constraint of solvent flatness. These applications show that the primary goal of this new method - namely increasing the accuracy and sensitivity of probabilistic phase indications compared with conventional direct methods - has been achieved. The main components of the method are (1) a tree-directed search through a space of trial phase sets; (2) the saddle-point method for calculating joint probabilities of structure factors, using entropy maximization; (3) likelihood-based scores to rank trial phase sets and prune the search tree; (4) efficient schemes, based on error-correcting codes, for sampling trial phase sets; (5) a statistical analysis of the scores for automatically selecting reliable phase indications. They have been implemented to varying degrees of completeness in a computer program (BUSTER) and tested on two small structures as well as on the small protein crambin. The main obstructions to successful ab initio phasing in the latter case seem to reside in the accumulation of phase sampling errors and in the lack of a properly defined molecular envelope, both of which can be remedied within the methods proposed. A review of the Bayesian statistical theory encompassing all phasing procedures, proposed earlier as an extension of the initial theory, shows that the techniques now available in BUSTER bring closer a number of major enhancements of standard macromolecular phasing techniques, namely isomorphous replacement, molecular replacement, solvent flattening and non-crystallographic symmetry averaging. The gradual implementation of the successive stages of this `Bayesian programme' should lead to an increasingly integrated, effective and dependable phasing procedure for macromolecular structure determination.
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