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
On 30 Aug 2012, at 19:43, Gunnar Schroeder wrote:
> Hi Rob,
>
> thank you, your comments helped a lot.
>
> From the Refmac5 paper I did not get the fact that d is set to d_current
> after each step. In that case you are right, jelly-body corresponds rather to
> DEN with gamma=1 than to gamma=0.
>
> And of course, a very important difference is, as you said, the fact that
> jelly-body is applied only to the second derivative.
>
> However, I would like to clarify this one point you made:
> 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).
> The update formula is (again, for gamma=1):
> dij(t+1) = (1-kappa)*dij(t) + kappa * Dij(t+1)
>
> Dij(t) : distance between atom i and j and time t.
> dij_ref : distance between atom i and j in the reference structure.
> dij(t) : equilibrium distance of restraint between atom i and j at time t.
>
> The parameter kappa just defines how quickly dij(t) changes,
> i.e. kappa=1 sets dij(t+1)= Dij(t+1) at each time step.
>
> The parameter kappa is usually set to 0.1, which means the restraints
> slowly follow the atomic coordinates. But, again, the starting (or
> reference)
> model is completely forgotten and never used after the first iteration.
> This also means that the position of the minima of the target function
> are not changed by the DEN (gamma=1) restraints. It could just take longer
> to get there as the restraints need to be dragged along.
>
> For gamma<1, the situation is different, there are additional forces toward
> the reference (could be the starting) model, in which case dij(t+1)
> additionally
> depends on dij_ref. This also changes the position of the minima of the
> target
> function. 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.
>
> 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).
>
> Cheers,
> Gunnar
