Subject: Re: CCL:G:Histogram method, redux (fwd)

From: David Stewart <[email-masked]>
Date: Fri, 5 Feb 1999 12:47:09 -0500

I am forwarding this to the list on behalf of my colleague, Alan
  Ferrenberg.

From: Alan M. Ferrenberg <[email-masked]>
Subject: Re: CCL:G:Histogram method, redux (fwd)

-------------------------------------------------------------------------
> Date: Thu, 4 Feb 1999 15:41:42 -0500 (EST)
> From: [email-masked]
> To: [email-masked]
> Subject: CCL:G:Histogram method, redux
> Sender: Computational Chemistry List <[email-masked]>
> 
> 
> 
> 
> 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
> 

As much as I'd like to push the merits of multiple histograms for this
problem, there are actually some much more efficient approaches that you
might want to consider.  The "multicanonical" simulation method was
designed to work well for magnetic systems with probabilitiy
distributions that have the same features as your distributions of folding
states.  In the multicanonical approach, instead of simulating at
different temperatures and combining the results later (like the F-S
multihistogram method), you bias your sampling with a distribution
function that preferentially samples in the "dead area" between the two
peaks.  This allows the simulation to cross from one state to the other
without a huge free-energy barrier.  The result is that the mean passage
times are dramatically reduced.  With some experimentation, you can get a
reasonably flat distribution of states, and then the passage from one
state to the other is like a random walk.  To get back to the canonical
ensemble, you reweight this "flat" distribution using the inverse of the
biasing function.

Here are some references to early papers describing the multicanonical
method:

  B. Berg and T. Neuhaus, Phys. Lett. B267, 249 (1991).
  B. Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992).
  B. Berg, U. Hansmann and T. Neuhaus, Z. Phys. B90, 229 (1993).
  W. Janke, B. Berg and M. Katoot, Nucl. Phys. B382, 649 (1992).

There have been more recent papers, but these should be enough to give you
a feel for the method and decide if it will meets your needs.

Happy reweighting!

amf

***************************************************************************
Alan M. Ferrenberg ([email-masked])
Academic Professional, University Computing and Networking Services
Adjunct Asst. Professor, Department of Physics
Computational Physics Consultant

"Our great and glorious masterpiece is to live appropriately"

       -- Michel de Montaigne (1533-1592)
***************************************************************************

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From: [email-masked] (Max Vasquez)
Date: Fri, 5 Feb 1999 10:17:18 -0800

>a feel for the method and decide if it will meets your needs.

Hansmann has applied the multicanonical method to study folding of
atomistic models of relatively short peptides; see
J. Phys. Chem. B 102, 653-656 (1998) for a recent reference, and J.
Comp. Chem. 14, 1333-1338 (1993) for one of the first (if not *the*
first) applications of multicanonical methods to protein/peptide
folding.  The work of Hao and Scheraga with lattice models of proteins
is also worth checking (note that they call the procedure "entropy
sampling", but this is equivalent to multicanonical); see for example,
J. Phys. Chem. 98, 4940-4948 (1994).

***********************************
Max Vasquez
Protein Design Labs, Inc.
(510) 574-1477
(510) 574 1500 (FAX)