What may be counter-intuitive when looked at in one way may be
perfectly expected from another point of view.  I look at these maps as
the result of a single cycle of steepest descent refinement where the
parameters are the density values of the map sampled on the grid.  If
you start with a map calculated from the coefficients

(|Fcalc|, PhiCalc)

One cycle of steepest descent gives the shift

2(|Fobs|-|Fcalc|, PhiCalc)

giving a new and improved map with the coefficients

(|Fcalc|, PhiCalc) + 2(|Fobs|-|Fcalc|, PhiCalc)
   = (|Fcalc| + 2|Fobs| -2|Fcalc|, PhiCalc)
   = (2|Fobs| - |Fcalc|, PhiCalc)

or the classic 2Fo-Fc map.

   If you start with (|Fobs|, PhiObs) then your shift will be zero
because the R value is already perfect.  You cannot improve an
experimental map unless you refine against other criteria.

   On the other hand, if you start with (|Fcalc|, PhiObs) you have to
question your sanity a bit because |Fobs| is so much better, in fact
perfect.  If you decide to press ahead anyway you find that the
coefficients of the updated map are (2|Fobs| - |Fcalc|, PhiObs).  These
are better than (|Fcalc|, PhiObs) but still not as good as (|Fobs|, PhiObs).

   There really is no justification for simply attaching the observed
phases to the calculated amplitudes.  The reason we are doing atomic
model refinement (instead of density map refinement as described above)
is to impose a lot of external knowledge such as atomic shape, solvent
flatness, and various relationships between atoms.  All of that
information gets encoded in the Fcalc's (complex numbers) so their
amplitudes and phases are tightly coupled.  It is no surprise that just
ripping out half and replacing it with something else would lower the
quality of the contained information.

   If you want a map to help evaluate your model when you have
experimental phase information you should run one cycle of steepest
descent optimization on the map with both amplitudes and phases
restrained.  If I ignore the complication of much larger uncertainty of
the phase relative to the amplitude, I believe the single cycle shift is
a map calculated from the complex coefficients 2(Fobs-Fcalc) and this is
your "difference map".  The 2Fo-Fc equivalent would have the
coefficients 2Fobs-Fcalc.  Remember they are all complex numbers with
their proper phases.

   I did this derivation in Least-Squares formalism so I can't be
confident of the m's and D's.  I also assumed that Fridel's Law holds,
but that assumption was made with the traditional maps as well.

Dale Tronrud


On 12/6/2018 11:01 AM, James Holton wrote:
> Sorry for the confusion, I was going for brevity.
> 
> Any time you do a thought experiment you make a fake-data data set, the
> "true" phases and "true" amplitudes become the ones you put into the
> simulation process.  This is by definition.  Is there potential for
> circular reasoning?  Of course!  But you can do controls:
> 
>   If you start with an ordinary single-conformer coordinate model and
> flat bulk solvent from refmac to make your Ftrue, then what you will
> find is that even after adding all plausible experimental errors to the
> data the final Rwork/Rfree invariably drop to small-molecule levels of
> 3-4%.  This is true even if you prune the structure back, shake it, and
> rebuild it in various ways.  The difference features always guide you
> back to Rwork/Rfree = 3/4%. However, if you refine with phenix.refine,
> you will find Rwork/Rfree stall at around 10-11%.  This is because Ftrue
> came from refmac and refmac and phenix.refine have somewhat different
> bulk solvent models.  If Ftrue comes from phenix and you refine with
> refmac you get similar "high" R values.  High for a small molecule
> anyway. And, of course, if you get Ftrue from phenix and refine with
> phenix you also get final Rwork/Rfree = 3/4%. If you do more things that
> automated building doesn't do, like multi-headed side chains, or get the
> bulk solvent from an MD simulation, then you can get "realistic"
> Rwork/Rfree in the 20%s.  All of this is the main conclusion from this
> paper: https://dx.doi.org/10.1111/febs.12922
> 
> But, in all these situations with various types of "systematic error"
> thrown in, because you know Ftrue and PHItrue you can compare different
> kinds of maps to this ground "truth" and see which is closest when you
> compare electron density. In my experience, this is the 2mFo-DFc map,
> phased with PHIcalc from the model. You might think that replacing
> PHIcalc with PHItrue would make the map even better because PHItrue is a
> "better" phase than PHIcalc, but it turns out this actually make things
> worse!  That's what is counter-intuitive: 2mFo-DFc amplitudes are
> "designed" to be used with the slightly-wrong phase of PHIcalc, not
> PHItrue.
> 
> That's what I was trying to say.
> 
> -James Holton
> MAD Scientist
> 
> 
> On 12/5/2018 7:36 PM, Keller, Jacob wrote:
>>>> That said, model phases are not so bad.  In fact, in all my
>>>> experiments with fake data the model-phased 2mFo-DFc map always has
>>>> the best correlation to the "true" map.  If you substitute the
>>>> "true" phases and use the 2mFo-DFc coefficients you actually make
>>>> things worse. Counter-intuitive, but true.
>> I don't understand what you mean by true and fake here--can you
>> clarify? How are the true map and phases generated (from an original
>> true model, I assume?), and how are the fake data generated? (Also
>> from the true model?) I am wondering whether there is some circular
>> reasoning?
>>
>> JPK
> 
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