The Translation Problem in Molecular Replacement Techniques. II. The Estimate of Quartet Invariants

The method of joint probability distribution functions of structure factors has been used to estimate quartet invariants when prior information on the orientation of molecular fragments is available. The mathematical approach makes use of the Gram–Charlier expansion of the characteristic function, as described by Giacovazzo [Acta Cryst. (1976), A32, 91–99] for deriving quartet estimates in the absence of prior information. The conclusive formula is a von Mizes distribution: the expected value of the quartet phase may lie anywhere between 0 and 2π. The reliability parameter may be large even for proteins, provided the fractionary scattering power of the molecular fragments with known orientation is sufficiently large. The first practical applications prove the correctness of the probabilistic approach and suggest the usefulness of the quartet information even in molecular replacement methods when a model molecule has been oriented by some rotation function and needs to be translated into a proper position.