Hi Rob,
>> This also means that the position of the minima of the target function
>> are not changed by the DEN (gamma=1) restraints.
> I would have thought that changing the value and gradient of the
> target function had the potential to alter the minima?
Indeed, the target function is changed during the search, but
once a stable minimum is found, the DEN potential needs to
be zero by definition and the coordinates have to sit in a minimum
of the original target function.
First we note that if the DEN potential minimum is at the same position
as the atomic coordinates, the potential and the first derivative are zero.
Assume the atoms are at a stable minimum of the combined energy function
(original target function + DEN potential with gamma=1) AND the DEN potential
minimum is different from the atomic positions. Then the DEN potential
minimum would move towards the atomic positions, which would change the
combined energy function and its derivative. The atoms would not be in a
stable minimum anymore, which contradicts the assumption and proofs that
the DEN potential is always zero if the atoms are in a stable minimium of the
combined energy function.
> Here are my thoughts: since the DEN update formula is recursive, the
> equilibrium distance can also be written in > terms of the Dij alone (still
> assuming gamma=1):
> dij(t+1) = Dij(0)*(1-kappa)^(t+1) + kappa*sum_n=0^t{Dij(t+1-n)*(1-kappa)^n}
> This means that the equilibrium distance is indeed dependent on the initial
> distance Dij(0) for all times t. …
I hope I do not get you wrong, but with this argument
aren't you just saying that the path/trajectory (of both the atomic coordinates
and the DEN potential) depends on the starting point?
Every simulation/minimization depends on the starting point.
In a steepest descent minimization the step size determines
how long it takes to move away from the starting point, just like
the parameter kappa determines how long it takes for the DEN potential
and the atomic coordinates to move away from the starting
model. I do not see the difference? Am I missing something here?
The important point is that the decision on how to move the DEN
minimum from one iteration (at time t) to the next (at time t+1)
depends only on where the atoms are at t+1 and where the DEN minimum was
at time t. If we assume that there is a second starting point which results
in a minimization path that happens to cross exactly the path from the first
starting point (same atomic coordinates and same position of DEN minimum)
at some time t'. Then the new position of the DEN minimum at time t'+1 would
be
exactly at the same position that you get from the first path at time t+1.
This
shows that the DEN update does not depend on the starting point.
Cheers,
Gunnar
PS: Just for the record, here we only discuss DEN refinement for gamma=1.
On Aug 31, 2012, at 11:30 AM, Robert Nicholls wrote:
> Hi Gunnar,
>
> I generally agree with your comments. However, I'd like to clarify a couple
> of points:
>
>> For gamma=1 the DEN potential can follow anywhere, the entire conformational
>> space is accessible and dij(t+1) depends only on Dij(t) and dij(t).
> ...
>> But, again, the starting (or reference)
>> model is completely forgotten and never used after the first iteration.
>
>
> Certainly, the entire conformational space is accessible. However, I'm not so
> sure about the starting model being completely forgotten and never used after
> the first iteration. Here are my thoughts: since the DEN update formula is
> recursive, the equilibrium distance can also be written in terms of the Dij
> alone (still assuming gamma=1):
> dij(t+1) = Dij(0)*(1-kappa)^(t+1) + kappa*sum_n=0^t{Dij(t+1-n)*(1-kappa)^n}
> This means that the equilibrium distance is indeed dependent on the initial
> distance Dij(0) for all times t. For values of kappa in (0,1), this
> dependency will diminish with time t, but will always exist. In fact, the
> equilibrium distance dij(t) is dependent on the whole history of the distance
> throughout the procedure, i.e. Dij(n) for n=0…t. Of course, the degree of
> influence of the historical information is controlled by kappa. Values of
> kappa~=0 would mean that the initial distance has very high weight
> (equilibrium distance dij(t) = Dij(0) in the limit kappa=0), and kappa~=1
> would mean that the most recent distances have very high weight (equilibrium
> distance dij(t) = Dij(t) in the limit kappa=1, as you have already stated).
> Intermediate values of kappa will give various non-zero weights to the
> historical values of Dij.
>
>> This also means that the position of the minima of the target function
>> are not changed by the DEN (gamma=1) restraints.
>
>
> I would have thought that changing the value and gradient of the target
> function had the potential to alter the minima?
>
>> It is therefore usually useful to run a final minimization without
>> restraints to test whether the refinement reached a stable minimum of the
>> target function.
>
> I agree. In the context of REFMAC5, my current favourite strategy at low
> resolution is to first use external restraints in order to aid the structure
> to adopt a more sensible conformation, but then subsequently release the
> external restraints and replace them with jelly-body restraints towards the
> final refinement stages.
>
>> From the user perspective, I think the main difference is that DEN is
>> designed
>> to be used in simulated annealing MD refinement, whereas jelly-body is
>> designed
>> to be used in minimization (and cannot be used for MD refinement as there
>> are
>> no second derivatives).
>
> I agree. Since the second derivative is utilised in ML refinement, it is
> possible to design a regulariser that has the desirable properties X=0 and
> X'=0 (e.g. jelly-body refinement) in the absence of any externally-derived
> prior information. Since this is not possible in simulated annealing MD
> refinement, the analogous solution will undoubtedly have to alter X and/or
> X'. Either way, all of these 'tricks' are just designed to aid robustness and
> combat overfitting! Certainly, both approaches can give positive results when
> refining at low resolution.
>
> Cheers
> Rob
>
--
_____________________________________________________________
Gunnar F. Schröder
Computational Structural Biology Group
Institute of Complex Systems (ICS-6)
Forschungszentrum Jülich
tel +49-2461-61-3259
http://www.schroderlab.org
