Circular variance

The circular variance provides a measure of the spread of a set of dihedral angles. It is applied here to each residue's distribution of phi, psi, chi-1 and chi-2 angles across all the members of the NMR ensemble.

So, for example, it can provide a measure of how tightly or loosely a given residue's phi torsion angles cluster together across the entire ensemble of models.

It is defined as


where Rav = R / n, n being the number of members in the ensemble and R is given by the expression


The value of the circular variance varies from 0 to 1, with the lower the value the tighter the clustering of the values about a single mean value.

For two dimensional distributions, such as the distributions of the phi-psi values on the residue-by-residue Ramachandran plots, the expression for R squared above is modified to



Allen, F. H. & Johnson, O. (1991) Automated conformational analysis from crystallographic data. 4. Statistical descriptors for a distribution of torsion angles. Acta Cryst., B47, 62-67.

MacArthur M W & Thornton J M (1993). Conformational analysis of protein structures derived from NMR data. Proteins, 17, 232-251.