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Jacob,
In the formula:
Kd=[P][L]/[PL] 
[P] and [L] are concentrations of UNBOUND protein and ligand, and [PL] is that in the complex.
Since the occupancy of the ligand in the crystal is 
[ PL]/[Po]= 1/(Kd/L+1),

varying [L] around Kd like from 0.1Kd to 10Kd will make the titration of occupancy. You can calculate from the provided formula which [L] will give 0.25, 0.5 and 0.75 occupancies.

Forget that the protein is crystallized. We assume that its behavior has not changed due to it. In reality, ligand affinity of conformationally flexible proteins can change by many orders of magnitude in both directions. This is why soaking does not work sometimes and you have to do co-crystallization.

If you decide to titrate a crystal with a ligand, you should collect data and refine the ligandless and fully-ocupied crystals first, then use the superimposition of their structures for refinement of all other cases.  Take care of waters that substitute for the partially bound ligand, they should have occupancies
 =1-Occ_of_ligand. 

Good luck.
Alex

On Jun 27, 2011, at 10:04 AM, Jacob Keller wrote:

Yes, I think you are right--the somewhat counterintuitive case I was
thinking of was, for example, when:

Kd = 20nM
[L] = 20uM
[Po in crystal] = 20mM

In this case, even though [L] = 20uM, since [L] is 1000 x Kd, the
occupancy should be ~100%, and [PL] at equilibrium should be about
20mM, so in the crystal, the total [L] should be ~20mM. This explains,
among other things, why bromophenol blue makes crystals bluer than the
surrounding solution--the Kd is probably significantly lower than the
BB concentration in the drop.

Thanks for your clarifications!

Jacob

The question would remain, then, whether there is any utility in
titrating ligands into crystals, and monitoring occupancies as a
readout for binding. Although crystallization conditions are horribly
non-physiological, perhaps there would be utility in the case where
there are multiple known binding sites of various affinities, and
other methods would have trouble resolving the binding events. One
could start with:

1. totally saturated conditions, set occ=1 for all sites, refine B's, then
2. fix B's at this value, and refine the occ's in a subsequent series
of dilutions.

All of this is not totally theoretical--I am considering a set of
experiments along these lines, where there really are multiple sites
of varying affinity.

*******************************************
Jacob Pearson Keller
Northwestern University
Medical Scientist Training Program
cel: 773.608.9185
email: [log in to unmask]
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