We remark that calculating averages and standard deviations of angular variables is not trivial, unless they cover restricted angular ranges. There is also no simple definition of maximal and minimal values. This problem occurs in many branches of science with broadly distributed angular variables, for example, in analysing wind directions (30 ). While angular helical variables generally lie within limited ranges, backbone dihedrals can easily span the full range of 360°. In this case, maximal and minimal values in the Curves+ analysis are replaced with the parameter ‘range’ and angular averages and standard deviations are calculated using a vectorial approach. Range is defined as the number of 1° bins visited by a given variable in the interval 0–360°. This gives a good idea of the angular spread of variables. Note that when analysing molecular dynamics trajectories, this value may increase with sampling, giving an indication that more sampling probably needs to be done. However, the details of the angular distribution can be checked using the histogram output option of the supplementary program Canal (see below). For averages, angles are added as vectors in 2D space (with an angle θ having components x = Cos θ and y = Sin θ). The result is converted to a unit vector, whose X and Y components yield the average. Other approaches require assuming that the angles obey a presupposed type of distribution. We have checked our values against one such model (31 ), and found negligible differences for standard deviations up to roughly 20°. Larger values differ more significantly (5–10°), but in these cases it is the qualitative result that the variables in question fluctuate very strongly that is the most important.
The sugar ring is usefully described using pseudorotation parameters. Although strictly speaking there are four pseudorotation parameters for a five-membered ring (32 (link)), only two of these, the so-called phase (Pha) and amplitude (Amp), are generally useful. While the amplitude describes the degree of ring puckering, the phase describes which atoms are most displaced from the mean ring plane. We calculate these parameters using the formulae given below (33 ), which have the advantage of treating the ring dihedrals ν1 (C1′-C2′-C3′-C4′) to ν5 (O4′-C1′-C2′-C3′) in an equivalent manner. In this approach:
where and b =−0.4 note, if then .
Conventionally, sugar ring puckers are divided into 10 families described by the atom which is most displaced from the mean ring plane (C1′, C2′, C3′, C4′ or O4′) and the direction of this displacement (endo for displacements on the side of the C5′ atom and exo for displacements on the other side). These pucker families can be easily calculated from the phase angle and are also output by the Curves+ program.
In order to deal with non-standard nucleic acids the backbone parameters are not hard-wired into the program, but are contained in a data file (standard_s.lib) which can be modified or extended by this user. This makes it easy to analyse chemically modified backbones such as those, for example, in PNA (34 (link)).