Subject: Histogram method, redux

From: [email-masked]
Date: Thu, 4 Feb 1999 15:41:42 -0500 (EST)

Thanks for the replies to my earlier query about the histogram method.
I have attempted to read some of the references suggested (given
below, for those interested, along with other references requested),
but I must confess that I can barely understand them.  Before spending
any more time trying to make sense of them, it would be wise to
determine whether these methods would actually serve my purposes.  So
here I give a schematic description of the problem I'm dealing with,
in the hope that someone may be able to tell me whether the "multiple
histogram method" would be of any use.

Consider a model of protein folding dynamics, with the following
features:

1. The folding is perfectly two-state, the two states being designated
U (unfolded) and F (folded).  Moreover, the two states are well
separated in terms of energy; i.e., if we were able to produce an
energy histogram at the folding transition temperature (Tf), the
distribution would be distinctly bimodal, with a largely unpopulated
region separating the peaks corresponding to F and U.

2. Let t_F(T) and t_U(T) be the mean passage times for the folding
(U->F) and the unfolding (F->U) transitions, respectively, as
functions of the temperature T.

3. We distinguish three temperature regimes I < II < III.  In regime
I, for all practical purposes, t_U(T) is infinite, while t_F(T) is
"reasonably small".  Conversely, in regime III, t_F(T) is practically
infinite, while t_U(T) is small.

4. In regime II, a narrow neighborhood of the folding transition
temperature Tf, both t_F(T) and t_U(T) are roughly equal, and both are
extremely large (of the order of 2-3 CPU-weeks on our workstations).

(Of course, no sharp boundaries separate these three temperature
regimes; t_F(T) and t_U(T) grow rapidly, as one approaches regime II
>from regime I and III, respectively.)

5. Because of the slowing down described in 4, it is extremely
difficult to determine Tf.

The single histogram method (which I described in my original post)
would require a single run near Tf, long enough to sample both the F
and U states in a way that actually reflects their relative
probabilities.  For the reason mentioned in (4) and (5) this is
impractical, which seems to rule out this method.

But it was suggested by a few responders to my earlier post that the
*multiple* histogram method may be useful in this case.  The only way
I can see to apply this method is by producing two histograms, one at
temperature T1 in regime I to sample F, and one at temperature T3 in
regime III to sample U.  Then, if we "absorb" the temperature into the
definition of the Hamiltonian (as Ferrenberg and Swendsen do, see
reference below), then this could be construed a case of having two
"Hamiltonians", H/(k*T1) and H/(k*T2), (where H is the actual
Hamiltonian, in the standard nomeclature).  Then I would use the
method of Bennett (see reference below) to obtain a free energy
difference between these two states (as a function of T).  Is this
correct?

(The method of Ferrenberg and Swendsen is a generalization of
Bennett's method to more than two histograms, if I've understood them
correctly).

These multiple histogram methods require a certain amount of overlap
between the histograms, which leads, for the system described above,
to a somewhat paradoxical situation: the greater the overlap between
the two states (meaning that neither P(U;T)/P(F;T) nor P(F;T)/P(U;T)
is too small), the closer the simulation temperatures have to be to
Tf, and therefore the longer it would take to adequately sample the
phase space (as remarked in (4) above).  This suggests that the
multiple histogram method would not be very useful either.  Am I
right?

Thanks for your patience.  Below are the references mentioned above.
Regards,

KJ

Papers that report using the histogram method:

@ARTICLE{klimov-thirumalai98,
		AUTHOR	= {D. K. Klimov and M. R. Betancourt and
		D. Thirumalai},
		TITLE	= {Virtual atom representation of hydrogen
		bonds in minimal off-lattice models of alpha helices:
		effect on stability, cooperativity and kinetics},
		JOURNAL	= {Folding and Design},
		VOLUME	= 3,
		NUMBER	= 6,
		PAGES	= {481--496},
		YEAR	= 1998
		}

@ARTICLE{guo-brooks97,
		AUTHOR  = {Zhuyan Guo and Charles L. {Brooks~III}},
		TITLE   = {Thermodynamics of Protein Folding:  A Statistical
		Mechanical Study of a Small All-$\beta$ Protein},
		JOURNAL	= {Biopolymers},
		VOLUME	= 42,
		PAGES	= {745--757},
		YEAR	= 1997
		}

@ARTICLE{berriz-shakhnovich97,
		AUTHOR	= {Gabriel F. Berriz and Alexander M. Gutin and
		Eugene I. Shakhnovich},
		TITLE	= {Cooperativity and Stability in a {L}angevin Model
		of Proteinlike Folding},
		JOURNAL	= {Journal of Chemical Physics},
		VOLUME	= 106,
		NUMBER	= 22,
		PAGES	= {9276--9285},
		YEAR	= 1997
		}

@ARTICLE{socci-onuchic95,
		AUTHOR	= {Nicholas D. Socci and Jos\'{e} N. Onuchic},
		TITLE	= {Kinetics and thermodynamic analysis of
		proteinlike heteropolymers: Monte Carlo histogram
		technique},
		JOURNAL	= {Journal of Chemical Physics},
		VOLUME	= 103,
		NUMBER	= 11,
		PAGES	= {4732--4744},
		YEAR	= 1995
		}

@ARTICLE{sali-karplus94,
		AUTHOR  = {Andrej \u{S}ali and Eugene Shakhnovich and
		Martin Karplus},
		TITLE   = {How Does a Protein Fold?},
		JOURNAL	= {Nature},
		VOLUME	= 369,
		PAGES	= {248--251},
		YEAR    = 1994
		}

Methodological papers:

@ARTICLE{bennett76,
		AUTHOR  = {Charles H. Bennett},
		TITLE   = {Efficient Estimation of Free Energy Differences
		from {M}onte {C}arlo Data},
		JOURNAL = {Journal of Computational Physics},
		VOLUME  = 22,
		PAGES   = {245--268},
		YEAR    = 1976
		}

@ARTICLE{ferrenberg-swendsen88,
		AUTHOR  = {Alan M. Ferrenberg and Robert H. Swendsen},
		TITLE   = {New {M}onte {C}arlo Technique for Studying
		Phase Transitions},
		JOURNAL = {Physical Review Letters},
		VOLUME  = 61,
		NUMBER  = 23,
		PAGES   = {2635--2638},
		YEAR    = 1988
		}

@ARTICLE{ferrenberg-swendsen89,
		AUTHOR  = {Alan M. Ferrenberg and Robert H. Swendsen},
		TITLE   = {Optimized {M}onte {C}arlo Data Analysis},
		JOURNAL = {Physical Review Letters},
		VOLUME  = 63,
		NUMBER  = 12,
		PAGES   = {1195--1198},
		YEAR    = 1989
		}

@ARTICLE{kumar-rosenberg92
		AUTHOR	= {Shankar Kumar and Djamal Bouzida and
		Robert H. Swendsen and Peter A. Kollman and
		John M. Rosenberg},
		TITLE	= {The Weighted Histogram Analysis Method for
		Free-Energy Calculations on Biomolecules.
		{I}. {T}he Method},
		JOURNAL	= {Journal of Computational Chemistry},
		VOLUME	= 13,
		NUMBER	= 8,
		PAGES	= {1011--1021},
		YEAR	= 1992
		}