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

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'

On account of the five-member 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, in the range of 0° to 360°).

One set of widely used formula to convert the five torsion angles to the pseudorotation parameters is due to Altona & Sundaralingam (1972): “Conformational Analysis of the Sugar Ring in Nucleosides and Nucleotides. A New Description Using the Concept of Pseudorotation” [J. Am. Chem. Soc., 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), with Matlab/Octave code:

# xyz coordinates of the sugar ring: G4 (chain A), 1bna
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, following the NewHelix program of Dr. Dickerson. The Curves+ program, on the other hand, uses another (newer) set of formula due to Westhof & Sundaralingam (1983): “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., 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 (τ_m and P).

Since 3DNA and Curves+ are currently two of the most widely used programs for conformational analysis of nucleic acid structures, the subtle differences in pseudorotation parameters may cause confusions for users who use (or are familiar with) both programs. Over the past few years, I have indeed received such questions via email.

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)

# 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 of G4 (chain A, 1bna), the pseudorotation parameters as calculated by 3DNA per Altona & Sundaralingam (1972) and Curves+ per Westhof & Sundaralingam (1983) are as follows:

           amplitude        phase angle
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.

For consistency with the 3DNA output, DSSR (by default) also follows the Altona & Sundaralingam (1972) definitions of sugar pseudorotation. Nevertheless, DSSR also contains an undocumented option, --sugar-pucker=westhof83, to output τ_m and P according to the Westhof & Sundaralingam (1983) definitions.

Each sugar is assigned into one of the following ten puckering modes, by dividing the phase angle (P, in the range of 0° to 360°) into 36° ranges reach.

C3'-endo, C4'-exo,   O4'-endo, C1'-exo,  C2'-endo,
C3'-exo,  C4'-endo,  O4'-exo,  C1'-endo, C2'-exo

For sugars in nucleic acid structures, C3’-endo [0°, 36°) and C2’-endo [144°, 180°) are predominant. The former corresponds to sugars in ‘canonical’ RNA or A-form DNA, and the latter in sugars of standard B-form DNA. In reality, RNA structures as deposited in the PDB could also contain C2′-endo sugars. One significant example is the GpU dinucleotide platforms, where the 5′-ribose sugar (G) is in the C2′-endo form and the 3′-sugar (U) in the C3′-endo form — see my blog post, titled ‘Is the O2′(G)…O2P H-bond in GpU platforms real?’.

Notes:

• This post is based on my 2011-06-11 blog post with the same title.
• While visiting Lyon in July 2014, I had the opportunity to hear Dr. Lavery’s opinion on adopting the Westhof & Sundaralingam (1983) sugar-pucker definitions in Curves+. I learned that the new formula are more robust in rare, extreme cases of sugar conformation than the 1972 variants. After all, Dr. Sundaralingam is a co-author on both papers. It is possible that in future releases of DSSR, the new 1983 formula for sugar pucker would become the default.