Subject: SUMMARY-Global minimum philosophical stuff

From: "Victor M. Rosas Garcia" <[email-masked]>
Date: Mon, 4 Dec 1995 11:45:31 -0500

Hi there,

	Well, it took me some time to assimilate all the answers I got to my
original posting (see below for the original question).  I received a good
number of answers that I summarize and integrate below.  Thanx to:

Georg Schreckenbach
Mike Gilson
Gregory V. Nikiforovich
Rene Fournier
Jan H. Jensen
Robert W. Zoellner
Nathalie Meurice
Matt (/G=Matthew/S=Harbowy/OU=LIPTONUS-EC02/O=TMUS.TJL/@LANGATE.gb.sprint.com
and
Eugene Leitl

for all their input.

My original posting was as follows:

********************************************************************************
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
********************************************************************************

	I think I can divide the topics addressed in the answers in three
categories:

a) problems in making sure that we found a global minimum
b) approaches to make sure that we found a global minimum
c) applications

a) problems in making sure that we found a global minimum
	In general, people agree that it is very difficult to assess if we have
a global minimum from the mathematical point of view.  Examples show that
certain functions can be very pathological and it can be very difficult to find
a minimum.  Fortunately, in chemistry we are restricted to functions of a
particular kind, and those are usually well behaved.  Even in that case, it is
not so much the form of the functions that makes the problem difficult, but the
huge number of minima in molecules like proteins is what kills us.  Basically
it is a war of attrition of the proteins against our algorithms, and the
proteins have always won, so far :).  A point was made that if we knew the
shape for the whole energy hypersurface it *would* be possible to say which one
is the global minimum.  I guess that answers my question that *in principle*
(logically), it is possible to say that a global minimum has been found.  Now
I'm thinking about the same question as applied to statistical methods for the
search, precluding any knowledge of the energy hypersurface.

b) approaches to make sure that we found a global minimum
	There were as many different approaches as answers, I'd say.  The
methods mentioned were:
-Run several statistical searches with different starting points, counting
repetitions of the lowest energy conformer until some criterion for repetition
is satisfied.  The problems here are how to quantify the level of confidence
for a given conformer to be the global minimum, and how different conformers
have to be to be considered "different".
- Use of genetic algorithms.
- Use of Explicit Statespace Trasversal Modeling which is:

 >An attempt to model the system by representing it by a
 >vector of numbers to be parallelly (there's a trick to it) changed
 >by a transition table. The local Hamiltonian is computed each time
 >anew from elementary contributions.
 This approach would use Cellular Automata-like encoding.

- Also Genetic programming calibration (this is very related to genetic
           algorithms)

An interesing point was that energy minimization is just a way to find a 3D
arrangement.  A comment to this is that the set of 3D arrangements is a
superset of the set of conformers and it includes isomers of the molecule of
interest.  Usually one tries to avoid isomerizations during the search.

c) Applications

	This got to be interesing, in the particular case of proteins, some
people questioned the validity of the assumption that proteins are at the
global minimum.  After checking a couple of references (mostly Anfinsen's
seminal paper in Science and Ken Dill's review in Biochemistry) I sort of
confirmed that there is little evidence for that.  So the search for the global
minimum of a protein *could* be a waste of time.  Also, an important point was
raised, in that it is more important to consider the applicability of the
geometries to experimental problems, than to be concerned about the ranking in
energy.

	I think that's it.  I hope summary was clear and that I did not
misunderstood the answers I got.  See ya'

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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