My apologies, a correction to my last posting:
On my final point concerning the calculation of D and the content of the
FC column in the MTZ file output by Refmac: I actually used the FC_ALL
column for all calculations, not the FC column as I implied. I use the
FC_ALL, PHIC_ALL columns for SF calculations because I believe that
these contain Fcalc including the bulk solvent background, whereas I
have been assuming that the FC, PHIC columns contain Fcalc omitting this
correction. An mtzdump of the file for the (0,13,10) reflection shows:
H K L FOM F SIGF FreeR_flag FC PHIC
0 13 10 0.89 383.48 7.05 14.00 98.96 99.76
FC_ALL PHIC_ALL FWT PHWT DELFWT PHDELWT
182.87 129.48 501.58 129.48 159.36 129.48
>From the phase values PHIC, PHIC_ALL, PHWT & PHDELWT above it is clear
that FWT and DELFWT have been computed using FC_ALL not FC, so that's
why I actually used the FC_ALL value (182.9) in my example.
Cheers
-- Ian
> -----Original Message-----
> From: owner-ccp...@jiscmail.ac.uk
> [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of Ian Tickle
> Sent: 30 December 2008 23:24
> To: CCP4BB@jiscmail.ac.uk
> Subject: Published derivation of mFo-DFc formula?
>
> 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).
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