On Sat, 23 Oct 2010 10:05:15 -0700
Pavel Afonine <[log in to unmask]> wrote:

> Hi Tim,
> 
>  ...but I hope this answers the question:
> Babinet's vs. the flat model?  Use them together!  ;)
> 
> 
> thanks a lot for your reply.
> 
> Could you please explain the *****physical***** meaning of using both
> models together?

I can try!  Typically, we model the bulk solvent using a real space
mask that is set to 1 in the bulk solvent region and 0 in the protein.
This gets Fourier transformed, symmetrized and added in to the
scattering factors from the molecule (Equation 1 in the paper, page 6
in your presentation):

Ftot = Fc + ks*Fs*exp(-Bs*s^2/4)

which works great and is how things are usually coded in most
macromolecular software, no problems or arguments there.  However,
we can come from the opposite - but equivalent! - direction of
Babinet's principle, which tells us the bulk solvent can also be
modeled by inverting everything: set the bulk solvent region to 0 and
the protein region to 1 in the real space mask, apply a Fourier
transform to that and then invert the phase:

Ftot = Fc - ks*Fm*exp(-Bs*s^2/4)

(I'm using Fm to distinguish it from Fs, due to the inversion of 0's
and 1's in the real space mask)  This is equation 2 in the paper.

So we're still using the flat model to compute Fm, and we're using
Babinet's principle to add it in to the structure factors - although
its better described as adding the inverse (thus the minus sign in the
second equation) of the complement (Fm rather than Fs). These two
equations are exactly equivalent, without any loss of generality. So, I
would argue the flat model and Babinet's are very much congruous.  Also
take a look at the description/discussion in the paper regarding Figure
2 (which helped me think about things at first).

The big difference is that Babinet's is usually applied as:

Ftot = Fc - ks*Fc*exp(-Bs*s^2/4)

which, I would argue, isn't quite right - the bulk solvent doesn't
scatter like protein, but it does get the shape right.  Which I think
is why Fokine and Urzhumtsev point out that at high resolution this
form would start to show disagreement with the data.  I haven't looked
at this explicitly though, so we still haven't answered that question!
We didn't want to spend much time on it in the paper, our main goal was
to try out the differentiable models we describe.  The Babinet trick
was a convenient way to make coding easier.

Anyway, I hope this helps explain it a bit more, and again: sorry for
the long-windedness.

Regards,
Tim

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        Tim Fenn
        [log in to unmask]
        Stanford University, School of Medicine
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        Phone:  (650) 736-1714
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