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Re: [ccp4bb]: NCS analysis

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Trixie Wagner, Ph.D. wrote:
> > I am working on the structure of a tetramer with non-crystallographic 222
> > symmetry. It is not difficult to determine the three two-fold axes
> > separately by superimposing dimers, but
> > a) they are not necessarily perpendicular to each other and/or might not
> >    intersect in one point and
> > b) from the direction cosines of the axes you cannot conclude where
> >    exactly to put the two-folds with respect to the tetramer (i.e.
> >    where the origin of the 222 system is located).
> >
For a), I don't know of such a program. It should be a fairly simple
exercize in geometry to write a program that simultaneously minimizes
rmsd for the three operators as a function of 8 parameters: x,y,z at the
origin of the 2,2,2 system, x,y,z of a vector parallel to the first axis,
and x,y of a vector parallel to the second axis (z being fixed by the
requirement for perpendicularity, as is the third vector)

b) if your derived operators were perfect 2-fold operators,
points on each axis can be found as the midpoint between a point and
the point generated from that point by the operator: 0,0,0 -> t1,t2,t3
and the point t1/2, t2/2, t3/2 is on the axis. Operating on another
point and likewise taking the midpoint gives a second point on the
axis, from which the equation for the line defining the axis can
be derived. Repeating this for the other two axes and solving for the 
intersection of the three lines gives the origin of the system. 

Because the operators will not be perfect proper 2-folds, the point
located thus will not be on the axis, but will be much closer than the
starting point. Operate on it again. If it were on the axis, it would be
unmoved or only moved along the axis by the screw component of the operator.
If it is slightly off the axis, the midpoint between it and the the point 
generated from it by the operator will be much closer to the axis. This
can be repeated until convergence (distance moved in each step is constant 
and in constant direction- along the axis) to give a point on the axis.
Applying your direction cosines, or finding another point on the axis,
then gives the equation of the line along the axis.

Someone may have an equation for the least-squares closest point between
three non-intersecting lines, or be able to derive one.

To get back to CCP4, the points should be atoms and you can preform
the successive operations with 

#!/bin/csh -f
pdbset xyzin point1.pdb xyzout point2.pdb <<eof
transform (ODB) lsq1.o

To get the most accurate operators lsqi.pdb, determine them
for the whole tetramer, not dimers. Find the operator to take
dimer AB to CD as the operator to take tetramer ABCD to
tetramer CDAB. This should give a very nearly proper 2-fold
with no screw, since operating twice has to bring everything
back where it started.