All, something for you to think about over the Study Weekend - sorry I
can't be there with you this time :-( :
I was recently asked whether a formal derivation of the expression we
use for the 'minimally biased' difference Fourier coefficient, i.e.
delta-Fm = mFo-DFc, is published anywhere, and I was forced to admit
that I didn't know of any such publication. It's certainly mentioned in
program documentation (SIGMAA, REFMAC, etc), and also in online course
material, but I could find no proof of the formula. Then I got thinking
about how one would go about deriving it, and that opened up a whole new
can of worms!
First, if we define the coefficient Fm for the 'minimally biased'
Fourier (as derived by Read, AC 1986, A42, 140) then:
Fm = 2mFo - DFc for acentrics,
= mFo for centrics.
I'm reasonably happy with Randy's proof of this. My first question is
why isn't the difference map coefficient given by:
delta-Fm = Fm - DFc
= 2(mFo - DFc) for acentrics,
= mFo - DFc for centrics.
This would reflect the fact that provided the amount of missing
structure is small, peaks in a centrosymmetric difference Fourier show
up around the full expected height, whereas those in a
non-centrosymmetric one show up at about half-height, so you need to
double the Fo-Fc difference. Indeed this is the whole rationale for the
formulae Fo = Fc + (Fo-Fc) for centrics and 2Fo-Fc = Fc + 2(Fo-Fc) for
acentrics.
AFAIK all programs use the same expression delta-Fm = mFo-DFc for both
acentrics and centrics, and looking at the SIGMAA code it certainly does
(code snippet edited for clarity):
IF (IC.EQ.1) THEN
C
C CENTRIC DATA: EITHER M*FO OR M*FO-D*FC
C
FOUT = W*FO
FOUTD = FOUT -DLUZ*FC
ELSE
C
C NON-CENTRIC DATA: EITHER 2*M*FO-D*FC OR M*FO-D*FC
C
FOUT = 2.0*W*FO - DLUZ*FC
FOUTD = W*FO - DLUZ*FC
ENDIF
Incidentally there's the following code snippet in the centric part of
the IF..ELSE..ENDIF block above:
cv C. Vonrhein Jul 5 1999
c
c if we don't have FP set it to -D*Fc
c
IF (LOGMSS(IFO)) FOUT = -DLUZ*FC
Shouldn't this be FOUT = DLUZ*FC (so that FOUTD = 0), if not then why
the minus sign? And shouldn't the same test be applied to acentrics?
Then I got thinking even more (always dangerous!), and so my second
question is why isn't the difference map coefficient given instead by:
delta-Fm = Fm - Fc
= 2mFo - (1+D)Fc for acentrics,
= mFo - Fc for centrics.
My rationale for this is that the difference map is supposed to tell you
what changes you need to make to the current model (represented by Fc)
in order to obtain the minimally biased model (represented by Fm). The
point is that DFc does not represent the current model: as I understand
it, it's a partial structure factor representing the part of the current
model that's correct, the remaining part being random error.
I modified SIGMAA and looked at some difference maps computed using the
new expressions, and I have to admit they're not strikingly different,
though there are some differences from the maps computed using the
standard version. That could be because my D's are close to 1, in which
case there would be no difference: maybe there would be a bigger effect
if my D's were significantly less than 1 (D of course can't be < 0 or >
1). Even so it's always nice to use expressions that can be justified
theoretically. A formal derivation of the mFo-DFc expression would
settle this, and I would be happy to accept whatever result such a
derivation gives, assuming that one exists.
Then just for fun I looked at the FWT & DELFWT columns in an MTZ file
output by Refmac, and this is where the worms really get out of the can!
I picked an acentric and a centric reflection at random (SG P21):
hkl = (0,13,10) FP=383.5 FC=182.9 FWT= 501.6 DELFWT= 159.4
FOM=0.89
hkl = (-5,0,11) FP= 82.4 FC=410.9 FWT=-271.6 DELFWT=-341.2
FOM=0.85
Note that the -ve values of FWT & DELFWT arise because the phases differ
by 180 from phi(calc).
First I checked for consistency of the FOM values since for acentric:
m = ((2mFo-DFc)-(mFo-DFc))/Fo = mFo/Fo = (501.6-159.4)/383.5 = 0.89 so
OK.
For centric case m = mFo/Fo = -271.6/82.4 = -3.3 so not OK!
However if I use the acentric formula for the centric case: m =
(-271.6-(-341.2))/82.4 = 0.845 . So Refmac is either not detecting
centrics correctly or is using the wrong formula.
Then I checked for consistency of the D values since for acentrics:
D = ((2mFo-DFc)-2(mFo-DFc))/Fc = DFc/Fc
For the acentric case D = (501.6-2x159.4)/182.9 = 1.000 .
I've already established it's using the acentric formulae for centrics,
so for the centric case:
D = (-271.6-2(-341.2))/410.9 = 1.000 .
In fact D computes to exactly 1 for every reflection, so something is
not right; either D isn't being computed correctly, or a possible
explanation is that the FC column contains DFc instead of Fc. This
would be unfortunate since we use Refmac to compute structure factors
containing the solvent background contribution (which SFALL doesn't of
course).
Happy New Year!
Cheers
-- Ian
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