The papers you are looking for are:
Crick FHC & Magdoff BS (1956)."The theory of the method of isomorphous
replacement for protein crystals. I", Acta Crystallogr. 9, 901-908.
http://dx.doi.org/10.1107/S0365110X56002552
Magdoff BS & Crick FHC (1955)."Ribonuclease II. Accuracy of measurement
and shrinkage", Acta Crystallogr. 8, 461-468.
http://dx.doi.org/10.1107/S0365110X55001461
The 1956 one is where everyone gets the "1% rule" for non-isomorphism,
and it is based not on observations but rather a calculation that by
modern standards I would consider a bit flawed. The 1955 paper contains
all the experimental observations, and nobody ever cites it.
The "flaw" I think is that they just changed the unit cell without
changing the atomic coordinates and then re-calculated the Fs. This is a
rather non-physical thing to do, since crystal packing interfaces are
being stretched and compressed a huge amount and intra-molecular bonds
remain rigid. In reality, the molecule is a lot more squishy than
that. To put it another way, if instead of changing the unit cell in
the "PDB file" you change it in the map header before doing the FFT,
then there is 0% change in the structure factors. That is, for any
given HKL index, the value of "F" will be identical, just at a different
d-spacing. This is because the value at each map grid point has not
changed, just the unit cell. This is equivalent to a perfectly elastic
stretch, where the fractional coordinates of all the atoms remain
unchanged. You can do the same thing by scaling all the coordinates and
all the unit cell dimensions up by 10%, and you will generally get less
than 1% change in the resulting Fs.
So, it is possible to have a 10% change in all three unit cell
dimensions and ~1% non-isomorphism, but it is also possible to have 0%
change in unit cell dimensions and 67% non-isomorphism. The latter
situation was also explored by Crick and Magdoff by tilting the molecule
a few degrees and keeping the unit cell fixed. They were not trying to
go for a realistic estimate of the non-isomorphism in reality, but
rather trying to come up with triage strategies, where the "worst case
scenario" for a given unit cell change could be considered before
spending the next few years collecting data. These days, we tend to try
to balance errors, and therefore need more accurate estimates.
A perfectly realistic example of non-isomorphism is actually tetragonal
lysozyme. Have a look at 3aw6 vs 3aw7. These are from the same
crystal, and the unit cell dimensions are <0.8% different in all three
directions, but the structure factors are 44% different. This is due to
dehydration. Nevertheless, if you drop the model from 3aw7 into the
data for 3aw6 and do rigid-body refinement you rapidly converge to a
decent R/Rfree. Same goes for the converse.
Now you can ask questions about the phase shift. The problem with
phases is that the phase of weak reflections doesn't really matter
because they don't contribute to the map. Also, poorly-measured phases
get low FOM weights in map calculations as well. For this reason, the
phase shift in rms degrees is not really all that useful. Better to
consider the correlation coefficient of the two electron density maps.
In the case of 3aw6 vs 3aw7 you get CC= 0.7. Not too bad really.
Unfortunately, 3aw6 and 3aw7 don't include anomalous differences, but if
you calculate them from the model and use the phases from that same
model to do a phased anomalous difference Fourier to ~3A resolution you
get sulfur peaks as high as 13-14 sigmas, and if you "swap the phases",
using phases from 3aw7 to make a phased anomalous difference Fourier for
Fs from 3aw6 or vise versa, you also get peak heights around 13 sigmas.
Theoretically, this is more than enough to get phases (Bunkoczi et al.
2014, dx.doi.org/10.1038/nmeth.3212). In practice, if you try just
pasting in DANO from one dataset up against the F column from another
dataset then maximum likelihood phasing programs will get very cross
with you. Still trying to figure out why...
-James Holton
MAD Scientist
On 1/27/2015 12:28 AM, Frank von Delft wrote:
Hi all - anybody know the answer, or can tell me where to look:
Regarding that oft-stated rule-of-thumb from the 60s (Blow? Crick?),
that a 1% change in cell parameters causes a 3% change in
intensities: is there an equivalent statement to be made about how
phases change with increasing non-isomorphism - and related to that,
at what point do maps become unrecognisable?
Or another way to ask this: how isomorphous must two datasets be for
a phase transplant from one to the other to be valid?
Cheers
Frank
