Ian's analysis is, as usual, much more sophisticated then my little
back-of-the-envelope calculation.  I would like to make one philosophical
note:  The non-isomorphism between the two crystals is not "noise" in the
Fo-Fo map, it is the whole point.  Sometimes the non-isomorphism is
dominated by the direct effects of the presence of a compound binding to
the protein (for instance) but other times by the large scale changes
in orientation of the domains of the protein in response to that compound.

   If your protein undergoes large scale motion upon binding and you are
interested in those motions then the Fo-Fo map will show them to you,
even though the small changes directly caused by the compound might
be difficult to discern.  If the protein doesn't have domain motions
then you are likely to see very interesting, specific, motions of
side chains and solvent around the binding site.  If you are not
interested by the major non-isomorphism of your pair of crystals you
will have to find some way to filter out the boring non-isomorphism
from the interesting non-isomorphism.  This will require a model
and the potential for model bias.  Very often this cannot be avoided.

   The variability of cell constants after freezing is a property of
the protein, not intrinsic to the freezing process itself.  With my
favorite protein the cell constants are very consistent after freezing
and I can calculate Fo-Fo maps that show great detail.

   Perhaps Fo-Fo maps would be more popular if they weren't so cumbersome
to calculate in the CCP4 world.

Dale Tronrud

On 05/04/10 04:48, Ian Tickle wrote:
> Dale,
> 
> On Tue, May 4, 2010 at 12:19 AM, Dale Tronrud <[log in to unmask]> wrote:
>>  The greater the difference in cell constants the greater the "noise"
>> in the map.  I think the high resolution cutoff for the maps should
>> be
>>
>>    2 A delta/(A+delta)
>>
>> where A is the cell edge with the largest change, and delta is the
>> amount of change (in Angstrom).  Basically a 1A change for a 100A
>> edge would require a 2A resolution limit.  A 5A change would imply
>> a 10A cutoff and a very boring map.
> 
> I think this estimate may be a little over-optimistic.  Crick &
> Magdoff (1956, Acta Cryst.9, 901–908) calculated that for a 100Å cubic
> unit cell a 0.5% change in all the cell dimensions, or a 0.5° rotation
> of the molecule within the unit cell, would produce an average 15%
> change in intensity, equivalent to a 7.5% average change in F, for
> data to 3Å resolution.  So your 1% change in cell  at 2Å would give ~
> 22.5% average change in F, just from the non-isomorphism effects.  For
> an Fo-Fc map, given that measurement errors are typically ~ 4 or 5% in
> F, you need to have the average difference due the ligand or heavy
> atom at least 3 times the noise level at the bare minimum, say at
> least 12%, and indeed we see average differences (i.e. partial Fc) due
> to a soaked ligand in the range 10-20%.  For an Fo-Fo map the noise
> level is multiplied by sqrt(2) since you have 2 lots of measurement
> errors, so you have 6-8% average experimental error in Fo-Fo.  Now
> clearly if you have experimental errors of ~ 7% plus non-isomorphism
> errors of 22.5% and you're looking for a signal of 15% due to the
> ligand then you have a problem!  It implies you need to reduce the
> non-isomorphism errors by a factor of at least 3, i.e. to <= 0.3%
> change in cell at 2Å.  Such a stringent requirement, which is
> virtually impossible to achieve for frozen/soaked crystals (freezing &
> soaking can easily induce > 1% and sometimes > 5% change), is surely
> the reason that Fo-Fo maps never caught on!
> 
> One objection to this that I've seen is that Fo-Fc differences include
> errors due to the model, say ~ 20% (assuming that's the R factor),
> which Fo-Fo differences clearly do not.  However the point is that
> this 20% contribution from the model error is spread throughout the
> cell, not just at the ligand site, since it arises from co-ordinate &
> B factor errors, disorder etc in all atoms, of which the ligand makes
> up a very small fraction, so the contribution to this from errors in
> the ligand parameters are likely to be negligible (adding the ligand
> has a negligible effect on the R factor!).  In contrast the ~ 5%
> experimental error in Fo-Fc, or ~ 7% + non-isomorphism error for
> Fo-Fo, contribute roughly equally to the error in the density over the
> whole cell.
> 
> Cheers
> 
> -- Ian