AUTOMATED CONFORMATIONAL-ANALYSIS FROM CRYSTALLOGRAPHIC DATA .4. STATISTICAL DESCRIPTORS FOR A DISTRIBUTION OF TORSION ANGLES

The methods of linear and circular statistics are used to derive summary descriptors for a unimodal distribution of torsion angles (tau-i). The arithmetic methods of normal linear statistics are complicated by the phase restriction -180 < tau-i less-than-or-equal-to 180-degrees. Phase adjustments must be made to generate a correct arithmetic mean (tauBAR-a) which spans +/- 180-degrees in a Newman projection of the tau-i. A general single-pass algorithm is described for the calculation of this mean, its standard error sigma(tauBAR-a) and the sample standard deviation sigma(tau-a). The single-pass technique is restricted to distributions in the range r less-than-or-equal-to 180-degrees; a preliminary pass is required for broader distributions. The tau-i are, however, properly represented as a circular distribution and the established formalisms of circular statistics should be applied. Here a circular mean or preferred direction (pi-BAR(c)) may be derived in a single pass for a distribution of any range. The variance of the distribution may be assessed in terms of the concentration, RBAR, of data points around the mean tau-BAR(c). A circular standard error sigma(tau-BAR(c)) and a circular sample standard deviation sigma(tau-BAR(c)) may then be derived. It is shown that the arithmetic and circular descriptors are numerically similar, except for broad distributions. The circular method has computational advantages in minimizing phase-shift operations and the results are more realistic and reliable when used in further statistical tests.