Definition of Pseudorotation Angles

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• Previous message: Frances Bernstein: "_atom_sites.cartn_transf_matrix"

	   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

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