Automated conformational analysis from crystallographic data. 4. Statistical descriptors for a distribution of torsion angles

Abstract

The methods of linear and circular statistics are used to derive summary descriptors for a unimodal distribution of torsion angles (r;). The arithmetic methods of normal linear statistics are complicated by the phase restriction - 180 < 7 i ~--- 180 °. Phase adjustments must be made to generate a correct arithmetic mean (-L,) which spans _180 ° in a Newman projection of the re. A general single-pass algorithm is described for the calculation of this mean, its standard error ~r(-L,) and the sample standard deviation o'(r,,). The single-pass technique is restricted to distributions in the range r_< 180°; a preliminary pass is required for broader distributions. The r; 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 (-L.) 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, R, of data points around the mean -~c. A circular standard error o-(~c) and a circular sample standard deviation ~r(rc) 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.