[gmx-users] number of lambda values

[David Mobley]

Soren,

>  So if I understand correctly, there is no way, apart from from the indirect
>  method you describe below, that can say how big a part of the sampled
>  configurations that overlap?

Well, it is hard. I think there is a paper on "Phase space overlap
measures" which I can track down and send you a reference for if you
can't find it. (Let me know).

The other way is error analysis. For example, FEP (with both
exponential averaging and BAR) uses the potential energy difference of
a single simulation snapshot evaluated in two different potentials. It
is those snapshots which have small potential energy differences that
contribute the most to the relevant  averages. If you can get an idea
of how many (or how few) snapshots have these small potential energy
differences, you have an idea of how good your overlap is. For
example, you can use a block bootstrap type method (break your
timeseries up into blocks the length of the correlation time;
construct a new timeseries by randomly selecting blocks from the
original timeseries; do the analysis again) to see if there are some
particular snapshots which are dominating the averages you compute (if
there are, the result will vary wildly from one bootstrap iteration to
the next). I don't know how to do this sort of thing for thermodynamic
integration though.

Sorry I don't have a cure-all answer. Again,  if you want to use TI,
my recommendation is to start out with a lot of lambda values, and
then cut back based on the shape of dV/dlambda. Where it is smooth you
can get by with fewer lambda values; where it isn't smooth, you need
more.

>  What is it that makes the phase space change a great deal for a "hard"
>  transformation? (And why do you consider it particularly "hard"?)
>  In other words, why would turning of LJ interactions be more difficult than
>  other transformations? And, forgive my ignorance which other kinds of
>  transformations could such other transformations be?

OK, for an example of a "hard" transformation, think about
disappearing an uncharged sphere ("LJ sphere") in water. At lambda=0,
you have a fully interacting sphere that can't overlap with water; at
lambda=1, you have a non-interacting sphere that will often overlap
with water. In between, you have to switch between these two extremes.
This is "hard" because the two extremes are very unlike one another.
That is, normal water has a *very* small probability of sampling
states where there is a cavity the size of an LJ sphere of any
reasonable size at the point we happen to be inserting our LJ sphere.
That is, if you think about the possible water configurations at the
non-interacting extreme, only a very small fraction of these possible
configurations have room for an LJ sphere.
But free energy calculations require you to transform between these
two extremes in a way that has good overlap everywhere (that is,
simulations run at one lambda value occupy very similar configurations
to those at adjoining lambda values). Since the endpoints are so
different, this means you will need a lot of lambda values.

Modifying electrostatics interactions is an example of a
transformation that is much "easier", typically. Consider a pair of
charged spheres joined by a bond in water (a dipole). Turning off its
charges won't lead to overlap with water -- it will just change the
average water structure around the dipole somewhat, since when it has
a net dipole moment, water tends to respond to offset this a bit.  In
the fully charged state, water will reasonably often occupy
configurations that are somewhat similar to those in the uncharged
state. There's no hard, steric barrier (like in the LJ sphere case)
preventing it from happening, just a small (depending on the dipole
moment) energetic penalty which thermal energy fights against.

There are lots of other transformations people do. For example, in
absolute binding free energy calculations, I and some others restrain
ligands in the binding site prior to turning off electrostatic and LJ
interactions. This is a fairly easy transformation in terms of phase
space overlap, in my opinion -- but it's a hard one in terms of
correlation times (that is, the relevant motions that need to be
sampled can be pretty slow).

If you want to be any more specific about what you are trying to do, I
and others on the list may be able to be more helpful about providing
suggestions that are specific to your particular problem, or our sense
of how "hard" a transformation it is likely to be, and so on.
Otherwise, a general suggestion is to just look at the literature
where people have done something similar to what you are trying to do,
and get a sense of (a) whether what they did worked, and (b) if so,
how hard it was.

David