AN EXACT DESCRIPTION OF 5-MEMBERED RING CONFIGURATIONS .1. PARAMETERIZATION VIA AN AMPLITUDE-S, AN ANGLE-GAMMA, THE PSEUDOROTATION AMPLITUDE-Q AND PHASE ANGLE-P, AND THE BOND LENGTHS

A method is developed which exactly parameterizes the configurations o f five membered rings. In addition to the five bond lengths, {b(j)}, f our other parameters are needed to describe completely all configurati ons of a three-dimensional figure with five sides. Two of these parame ters are taken as the phase angle. P, and amplitude, q (angstrom), int roduced 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 s econd amplitude, S (angstrom), and an orientation angle, GAMMA (the up per case Greek letter ''gamma''), are introduced. These two new parame ters 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 ri ng is maximally ''spread'' or ''stretched,'' where the stretch is a le ast 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 sour ce codes MAKERING and BREAKRING for these transformations are availabl e. The set of four parameters aids in direct comparisons of Ting struc tures and in detailed numerical analyses of data. They also aid the sy stematic generation of all possible rings, useful for theoretical stud ies of ring conformation. To demonstrate the descriptive method. we ha ve generated distributions of q, P, S, and GAMMA for furanose rings in the structures of 665 nucleosides determined by X-ray crystallography .