The conformation of the five-membered sugar ring in DNA/RNA structures can be characterized by the five consecutive endocyclic torsion angles (see Figure below), i.e.,
ν0: C4′-O4′-C1′-C2′ ν1: O4′-C1′-C2′-C3′ ν2: C1′-C2′-C3′-C4′ ν3: C2′-C3′-C4′-O4′ ν4: 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 (τm) 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 (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]. The concept is easily illustrated with an example — here with the G4 sugar ring on chain A of the Dickerson dodecamer (1bna), using Octave/Matlab code:
# xyz coordinates of the G4 sugar ring on chain A of 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. The Curves+ program, on the other hand, uses another 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 — Altona & Sundaralingam (1972) and Westhof & Sundaralingam (1983) — give slightly different numerical values for the two pseudorotation parameters (see below).
Since Curves+ and 3DNA are currently the most commonly used programs for conformational analysis of nucleic acid structures, the subtle differences in these two pseudorotation parameters may cause confusions for users who use both programs. With the same G4 (on chain A of 1bna) sugar ring, here is the Octave/Matlab script showing how Curve+ calculates the pseudorotation parameters:
# xyz coordinates of the G4 sugar ring on chain A of 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 G4 sugar ring on chain A of 1bna, the pseudorotation parameters as calculated by 3DNA following Altona & Sundaralingam (1972) and Curves+ following Westhof & Sundaralingam (1983) are as follows:
|amplitude (τm)||phase angle (P)|
Needless to say, for the majority of cases like the one shown here, the differences are subtle; very few people would notice them or be bothered at all. For those who do care about such little details, however, this post shows where the discrepancies really come from.