# [phenixbb] difference density map

Edward A. Berry BerryE at upstate.edu
Tue Jan 19 07:35:07 PST 2010

```Dale Tronrud wrote:
> Hi,
>
>     My favorite explanation for why differences show at full height
> in a 2Fo-Fc map is that the map is really a Fc + 2(Fo-Fc) map.  You
> a difference map to it to bring in the experimental data.  The half
> height features of the difference map are doubled to full high in
> the computation.
> . . .
Another explanation:
It is the difference between an 2F(o)Phi(c) map and an F(c)Phi(c) map.
Now consider three cases:
1. The feature (atom) exists in the real structure but not in the model:
Then the feature will make no contribution to Fc or Phi(c) since it was
not present in the model from which they were calculated. It will show
up in the FoPhi(c) map because it is represented in Fo and the Phi(c)
do have some validity, but it will show with reduced amplitude  because
it is not represented in the Phi(c). Roughly speaking, it will show up at
half height, so the factor of 2 gives full height. Subtracting the FcPhi(c)
map has no effect since this map just reproduces the model, from which
the feature is missing.
2. The feature exists in the real structure and also in the model, exactly correct.
Now the feature will show up at full height in both FoPhi(c) and FcPhi(c) maps,
in fact the contributions of the feature to Fo and Fc are identical.
Multiplying the Fo map by two gives double height, but subtracting out the
Fc map brings it back to full height.
3. The feature is present in the model but not in the real structure.
in both Fc and Phi(c). It will show up in the FoPhi(c) map because of the phase bias,
but at reduced height because it is missing from the Fo. Again lets say half height,
and the factor of 2 brings it to full height. But now it will be present at full height
in the FcPhi(c) map, so subtracting the Fc map cancels the contribution of the Fo map.
Result zero height.

Of course the assumption that a feature being represented in one of F or Phi but
not the other gives half height is an approximation- it is well known that correct
phases and random amplitudes gives a better map than correct amplitudes and random phses.
The Sigma-A theory, which I don't pretend to understand, allows to calculate coefficients
which give a more completely bias-free map. But this simplistic treatment is very
intuitive and easy to get a qualitative understanding of what is going on.
Now the other part of the OP's question:
> 丁玮 wrote:
>> Dear all,
>> Why the density map with coeffients |Fobs|-|Fcalc| show the density only
>> at half the height,
Again take the three cases above, but now we don't have the factor of 2.
If present in the real structure only, the Fo map will be ~half height
because not represented in the phases. The Fc map will be zero height,
so final result ~half height.
If present and modeled correctly, the maps will be the same so zero height.
If present in the model only, half height in Fo and full height in Fc,
so  negative half height in the difference.

Ed
>> and  with coeffients 2|Fobs|-|Fcalc| show the density at full height?
>> Thanks!
>>

```