Twinning (another verbose reply)

[Randy Read]

In article <PHIL.95Feb14103055 at xtreme1.mskcc.org> phil at xtreme1.mskcc.org (Phil  
Jeffrey) writes:
>
> (most of the post deleted)
> 
> We were, however, assisted by the fact that the non-xtallographic symmetry
> and the twinning operation were similar operators in reciprocal space (they
> might act in different ways mathematically, but they both work to increase
> the correlation between I(hkl) and I(h'k'l'), so I actually ignored the
> twinning in the case where alpha=0.11 for most of the refinement). The 
> twinning-ignored maps were often better than the twin-corrected maps,
> presumably because the data was more complete in the former case.
> 
This brings up a point that has to be considered in dealing with data from  
twinned crystals.  Most often exact twinning seems to arise *as a result* of  
parallel non-crystallographic symmetry.  So most of the time, there would be  
some correlation between "twin-related" reflections even in data from a  
completely untwinned crystal.  Almost all of the methods that have been  
proposed for estimating the twinning fraction assume that the reflections  
superimposed by the twinning operation are uncorrelated, or independent.  If  
the intrinsic correlation is not too high (because the interplay between  
crystallographic and non-crystallographic symmetry causes the contributions of  
the NCS related molecules to add up with different relative phases in the  
twin-related reflections), these methods probably work reasonably well.  But  
you can't count on it, and you should be certain of what is going on.

The particular case I'm thinking of is human GAPDH, the high resolution  
refinement of which we took over from Herman Watson while I was working with  
Wim Hol.  Herman's group published a nice paper about the twinning (JMB  
104:277-283, 1976), which you can read to get the background.  It turns out  
that, in this case, NCS-related contributions have more-or-less the same  
relative phase in the twin-related reflections, so reflections related by the  
twinning operation would have very similar intensities in an untwinned crystal.   
(It's close enough to exact that the methods to estimate twinning fraction by  
assuming independence are completely invalidated, but far enough from exact  
that you can't get away with ignoring the twinning.)  We used an idea we got  
from Herman to work out the twinning fraction.  It turns out that only half of  
the reflections are twinned, while the other half interleave in the diffraction  
pattern.  If you assume that the average intensities should be the same in both  
halves of the untwinned data (and there's no translational pseudosymmetry to  
invalidate this), you can work out the twinning fraction simply by comparing  
the average intensities in the observed data.  The twinned reflections have a  
higher average intensity because they are the sum of diffraction from both twin  
components.  This seems to work quite well, but of course, it is a special  
solution to a very special case.

Randy Read
rndy at mycroft.mmid.ualberta.ca