Pseudorotation is a measure of the nature of the pucker of a non-planar ring. Just as the conformation about a single bond is named gauche or eclipsed but measured with an angle, the pucker of a deoxyribose ring is described as 2'endo or 3'endo (amongst other names) but measured with an angle called the pseudorotation angle. The messy part of pseudorotation angles is that there are two competing definitions of the quantity. There is the definition used by nucleic acid people to describe the pucker of the ribose sugar and there is the definition used by everyone else in the world (including polysaccharides people). Since there are two competing definitions and other implementation issues for mmCIF this letter will be a long and boring discussion of the details of pseudorotation. At its base, the pseudorotation of a ring is calculated by choosing one atom as the origin and Fourier transforming a measure of the out-of-plane-ness of each atom. The zero and first Fourier coefficients must be equal to zero for various reasons and are ignored. For a five membered ring the only independent coefficient is number 2 and this coefficient is complex. The magnitude of the coefficient is a measure of the amplitude of the pucker and the coefficient's phase angle is the pseudorotation angle. The amplitude is correlated with the bond lengths and angles and is usually ignored. This leaves us with the pseudorotation angle. The angle varies from 0 to 360 degrees. A mirroring of the ring through the best-fit plane results in a change in the pseudorotation of 180 degrees. Just as with torsion angles a given standard dictionary can have multiple ideal pseudorotation angles. A mmCIF definition for the pseudorotation of five membered rings can be constructed just like the torsion angle definition except that five atoms are involved instead of four. Now we come to the first hard part. The definition of pseudorotation above says it is the Fourier transform of a measure of "out-of-plane-ness". Here is the problem. There are two different measures being used. The measure used by nucleic acid people is defined in Altona, C., and Sundaralingam, M., J. Am. Chem. Soc, (1972) 94, 8205-8212. They use the five internal torsion angles as their measure. Their equation is (theta2 + theta4) - (theta1 + theta3) P = ATAN(-------------------------------------) 2 theta0 (sin 36 + sin 72) If theta0 < 0 then P = P + 180, where the theta's are the five torsion angles. Because the bond lengths in the ring differ, relabeling the atoms can result in differing values for P. The authors recommended using all five origins and averaging the results (after compensating for the phase change). Whether or not anyone does this I don't know. In TNT I do not. The alternative form is much more general and does not have the problem of origin choice. It is more complicated too. The paper is Cremer, D., and Pople, J.A., J. Am. Chem. Soc, (1975) 97, 1354-1358. This formulation uses the out-of-plane distance and therefore requires the definition of a plane. The origin of the plane is defined as the mean location of the atoms in the plane. The normal of the plane is defined by (R' cross R'') N = -------------- |R' cross R''| where -- R' = \ / Rj sin (2 Pi (j-1)/n) -- j -- R'' = \ / Rj cos (2 Pi (j-1)/n) -- j and Rj is the position of the jth atom, and n is the number of atoms in the ring. If you look at this definition in a Fourier sense it is simply a convention to ensure that the 1th coefficient is zero. Putting the plane through the mean position results in the 0th coefficient being set to zero. Now we calculate the out-of-plane distance as zj = Rj dot N. The pseudorotation parameters are calculated with 1/2 -- qm exp phim = /2\ \ --- / zj exp (2 Pi m(j-1)/n) \n/ -- j where m = 2,3, ..., (n-1)/2. For a five membered ring m can only be 2 giving a single pucker amplitude, q2, and pseudorotation angle phi2. (Rings with an even number of atoms have an additional parameter q(n/2). There is no corresponding phi(n/2) which means q(n/2) can be either positive or negative. Whew... That was painful. The bottom line is that this definition is general to all size rings and gives the equivalent answers regardless of origin. To the best of my knowledge this is the form used in most chemistry and certainly is the form used for 6 membered rings and larger. As to mmCIF implementation, if you want to consider only 5 membered rings and ignore the amplitude you can implement either form in the same fashion as torsion angles, except for the presence of five atoms instead of four. To implement Cremer and Pople in general, some means must be developed to handle the varying number of atoms in the ring (n) and the corresponding n-3 numbers (some amplitudes and other pseudorotations). If this cannot be done in general I would suggest that special cases be developed for at least 4, 5, 6, and 7 membered rings. All of these appear in macromolecular models (aren't inhibitors wonderful things?). Dale Tronrud