Subject: Global minimum (philosophical) problems...

From: "Victor M. Rosas Garcia" <[email-masked]>
Date: Tue, 7 Nov 1995 09:41:44 -0500

Hi everybody,
	I'm just getting into the free energy calculation business, so I've
been reading the most basic references (by van Gunsteren, Jorgensen, Mitchell
and people like them).  The first thing I noticed is that it is extremely
important to make sure that one has a global minimum, together with all the
conformers within certain energy; and it is very obvious that it is not easy to
make sure that a conformer is "the" global minimum.  I wonder if it is
theoretically IMPOSSIBLE to prove that a given conformation is the global
minimum.  In my mind, is the same problem found when we want to say that we
have found *all* the conformers within a certain energy range.  Both (I think)
boil down to proving a negative proposition (something that we *logically*
cannot do), e.g.

For the global minimum:
"There are no other conformers with a lower energy than this one."
For a distribution:
"There are no other conformers that should be included in the population."

Of course, it is always possible to *disprove* or *falsify* a negative
proposition by showing a counterexample, but being unable to find a
counterexample doesn't prove the truth of a negative proposition.

	OTOH, both of the propositions above can be cast in an affirmative form
if we restrict them to a particular case, e.g.:
For the global minimum, we could easily prove or disprove:
"This particular conformer is higher in energy than the one we call 'global
minimum'."
By the same token, for a distribution:
"This particular conformer has an energy higher than our cutoff limit of
interest, (so we don't have to include it in the distribution)."

	So, I guess the point I'm trying to make is that we will *never ever*
be sure of having "the" global minimum or a complete population because it is
logically impossible to prove it.  By running several conformational searches
with different starting conditions we are probably doing the best we can.

I'd like to know your opinions on this matter.

PS Sorry if this stuff is too obvious, it isn't to me.

Victor

-- 
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Victor M. Rosas Garcia                   * "How can we contrive to be 
[email-masked]                *  at once astonished at the  
Virginia Tech doesn't necessarily share  *  world and yet at home in it?"
the opinions you just read.	         *  G. K. Chesterton
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From: Gregory Nikiforovich <[email-masked]>
Date: Tue, 7 Nov 1995 15:08:03 -0600 (CST)

> interest, (so we don't have to include it in the distribution)."

	Since I like philosophical problems, let me comment. As far as we
 are interested in conformers (like in conformational search problem), energy
 is just a target function, and energy minimization is a way to find a
 conformer, i.e., a particular 3D arrangement of atoms/functional groups. If
 you look at the problem from this side, you would notice that several very
 similar 3D arrangements may possess different energies. On the other hand,
 some very different 3D arrangements can possess almost identical energies
 (e.g., enantiomers). So the true task of conformational search is not to find
 all conformers within a given cutoff; it is to find all different 3D 
 arrangements within a given energy cutoff. This task is much more realistic,
 since the possibility of missing a really different 3D arrangement during any
 search is less than to miss a minimum close to "the global" one in energy.
 However, it is not philosophy any more...

 	Best wishes.

Gregory V. Nikiforovich                                   Phone (314) 935-4677
Research Professor                                        Fax   (314) 935-4979
Center for Molecular Design                            E-mail address:
Washington University                                  [email-masked]
Lopata Hall, Box 1099
St.Louis, MO 63130