Updating direct methods V. Phasing when triplet invariants are estimated via the Patterson map

In previous articles in this series, a novel probabilistic method was described which is capable of estimating triplet invariants using the Patterson map as prior information. The first experimental tests demonstrated the superiority of the new method compared with the traditional Cochran estimate. The advantages were so significant that the ab initio solution of macromolecular structures was considered to be feasible even when the data resolution is worse than 2 Å. However, several questions remained unanswered. For example: (i) which and how many Patterson peaks should be used to optimize a direct-methods phasing procedure applied to experimental data up to 2.2 Å resolution?, (ii) is the presence of heavy atoms a necessary ingredient for the validity of the method?, (iii) which and how many reflections must be used in the triplet search?, (iv) is the information contained in the Patterson map able to identify negative cosine triplets? and (v) may a computer program be made that routinary solves macromolecular structures with data resolution up to 2.2 Å? This article recalls these five unresolved questions and answers them. In particular, criteria have been defined to determine both the number of Patterson peaks to be actively used for triplet estimation and the number of reflections to be used in the triplet search. It has also been shown that the presence of heavy atoms is a necessary ingredient for success of the theory. In particular, the theory is unable to accurately identify triplet invariants with a negative cosine, but rather can identify enantiomorph-sensitive triplets. A paradox of the theory is discussed and resolved. Finally, a computer program is presented that is capable of automatically, with a few directives, solving some of the test structures at non-atomic resolution (proteins and nucleic acids) with data resolution up to 2.2 Å, but not in a straightforward way. The limitations of the computer program and its prospects are discussed.