An exact description of five-membered ring configurations. I. Parameterization via an amplitude S, an angle gamma,...

A method is developed which exactly parameterizes the configurations of five membered rings. In addition to the five bond lengths, (bj), four other parameters are needed to describe completely all configurations of a three-dimensional figure with five sides. Two of these parameters are taken as the phase angle, P, and amplitude, q (A), introduced by Cremer and Pople (J. Am. Chem. Soc. 97, 1354 (1975)) which give exactly the displacements of the atoms in any five-membered ring from a special plane (the CP plane). For the two other parameters, a second amplitude, S (A), and an orientation angle, gamma (the upper case Greek letter "gamma"), are introduced. These two new parameters describe the "distortion" of the projection of the ring in the CP plane. gamma is the angle in the CP plane at which the projected ring is maximally "spread" or "stretched," where the stretch is a least squares measure of position along a line through the ring center. S measures the difference between maximum stretch (in direction gamma) and minimum stretch (in direction gamma +/- 90 degrees) such that S = 0 denotes maximum symmetry. Transformations from Cartesian coordinates to the new internal coordinates, and from the new internal coordinates to the Cartesian set, are exact inverse transformations. Fortran source codes MAKERING and BREAKRING for these transformations are available. The set of four parameters aids in direct comparisons of ring structures and in detailed numerical analyses of data. They also aid the systematic generation of all possible rings, useful for theoretical studies of ring conformation. To demonstrate the descriptive method, we have generated distributions of q, P, S, and gamma for furanose rings in the structures of 665 nucleosides determined by X-ray crystallography.