The conformation of the five-membered sugar ring in DNA/RNA structure can be characterized using the five corresponding endocyclic torsion angles (see Figure below).

i.e.,

v0: C4'-O4'-C1'-C2'
v1: O4'-C1'-C2'-C3'
v2: C1'-C2'-C3'-C4'
v3: C2'-C3'-C4'-O4'
v4: C3'-C4'-O4'-C1'

Due to the ring constraint, the conformation can be characterized approximately by

`5 - 3 = 2` parameters. Using the concept of pseudorotation of the sugar ring, the two parameters are the amplitude (

**τ**_{m}) and phase angle (

**P**).

One set of widely used formula to convert the five torsion angles to the pseudorotation parameters is due to Altona & Sundaralingam: "Conformational Analysis of the Sugar Ring in Nucleosides and Nucleotides. A New Description Using the Concept of Pseudorotation" [

*J. Am. Chem. Soc.*,

__1972__,

**94**(23), pp 8205–8212]. As always, the concept is best illustrated with an example. Here I use the sugar ring of G4 (chain A) of the Dickerson-Drew dodecamer (

1bna/

bdl001), with Matlab/Octave code:

# xyz coordinates of the sugar ring: G4 (chain A), 1bna/bdl001
ATOM 63 C4' DG A 4 21.393 16.960 18.505 1.00 53.00
ATOM 64 O4' DG A 4 20.353 17.952 18.496 1.00 38.79
ATOM 65 C3' DG A 4 21.264 16.229 17.176 1.00 56.72
ATOM 67 C2' DG A 4 20.793 17.368 16.288 1.00 40.81
ATOM 68 C1' DG A 4 19.716 17.901 17.218 1.00 30.52
# endocyclic torsion angles:
v0 = -26.7; v1 = 46.3; v2 = -47.1; v3 = 33.4; v4 = -4.4
Pconst = sin(pi/5) + sin(pi/2.5) # 1.5388
P0 = atan2(v4 + v1 - v3 - v0, 2.0 * v2 * Pconst); # 2.9034
tm = v2 / cos(P0); # amplitude: 48.469
P = 180/pi * P0; # phase angle: 166.35 [P + 360 if P0 < 0]

The Altona & Sundaralingam (1972) pseudorotation parameters are what have been adopted in 3DNA. The

Curves+ program, however, uses another set of formula due to Westhof & Sundaralingam: "A Method for the Analysis of Puckering Disorder in Five-Membered Rings: The Relative Mobilities of Furanose and Proline Rings and Their Effects on Polynucleotide and Polypeptide Backbone Flexibility" [

*J. Am. Chem. Soc.*,

__1983__,

**105**(4), pp 970–976]. The two sets of formula by Altona & Sundaralingam (1972) and Westhof & Sundaralingam (1983) give slightly different numerical values for the two pseudorotation parameters (amplitude

**τ**_{m }and phase angle

**P**).

Since

3DNA and Curves+ are currently the most commonly used programs for conformational analysis of nucleic acid structures, the subtle differences in pseudorotation parameters may cause confusions for users who use both programs. With the same G4 (chain A, 1bna) sugar ring, here is the Matlab/Octave script showing how Curve+ calculates the pseudorotation parameters:

# xyz coordinates of sugar ring G4 (chain A, 1bna/bdl001)
# endocyclic torsion angles, same as above
v0 = -26.7; v1 = 46.3; v2 = -47.1; v3 = 33.4; v4 = -4.4
v = [v2, v3, v4, v0, v1]; # reorder them into vector v[]
A = 0; B = 0;
for i = 1:5
t = 0.8 * pi * (i - 1);
A += v(i) * cos(t);
B += v(i) * sin(t);
end
A *= 0.4; # -48.476
B *= -0.4; # 11.516
tm = sqrt(A * A + B * B); # 49.825
c = A/tm; s = B/tm;
P = atan2(s, c) * 180 / pi; # 166.64

For this specific example, i.e., the sugar ring G4 (chain A, 1bna/bdl001), the pseudorotation parameters as calculated by 3DNA following Altona & Sundaralingam (1972) and Curves+ following Westhof & Sundaralingam (1983) are as follows:

amplitude (**τ**_{m}) phase angle (**P**)
3DNA 48.469 166.35
Curves+ 49.825 166.64

Needless to say, the differences are subtle, and few people will notice/bother at all. For those who do care about such little details, however, hopefully this post will help you understand where the differences actually come from.