title: Detection and treatment of twinning: an improvement and new results.
abstract: This work deals with two aspects of the twinning problem. Firstly, an improvement of a known statistical test aimed at detecting twinning is presented and, secondly, a new parametrization of twinning is described, as well as a new method to obtain an accurate estimate of the degree of twinning. During work on crystals of the dimerization-initiation site of the HIV-1 genomic RNA, perfectly twinned crystals were obtained which were not immediately recognized as such by use of a known statistical method. This method, reminiscent of Wilson tests for the detection of centrosymmetric space groups, relies on calculation of <F2>/<F>2 or, equivalently, of <I2>/<I>2. It is shown that overlooking experimental errors may lead to erroneously large values of this index and, in turn, to ambiguous or incorrect conclusions. An immediate solution to this problem is presented. Independently, an alternative parametrization which expresses both the effect of twinning on intensities and the operation of untwinning to recover the correct intensities is proposed. A new method for estimating the degree of twinning is also presented. It is based upon maximization of the cross-correlation coefficients between intensities of all available data sets, and yields a fully analytical solution. Tests made with experimental data are quite satisfactory. It is suggested that the latter results could be used efficiently within the MIR method by allowing refinement, through one additional parameter only, of the twinning ratios of all data sets considered for phasing. Finally, the new parametrization of twinning has striking consequences in this correlation-based twinning determination: very unexpectedly, it yields a novel estimate of the 'twinning ratio' of a potentially twinned crystal which is fully independent of the data set used for the comparison.
mesh_terms: D015497:HIV-1 / Q000235:genetics*; D009690:Nucleic Acid Conformation; D012367:RNA, Viral / Q000737:chemistry*