CHARMM c30b1 pdetail.doc



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                Details about TSM Free Energy Calculations

* Menu:
 
* Introduction::           What will be covered.
* Theory and Methodology:: General discussion.
* Practice::               How to do it.




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                             Introduction
 

       For a good overview of free energy simulation methods, the follow-
ing references are suggested:  M. Mezei and D. L. Beveridge, in Annals of
the New York Academy of Sciences, chapter titled "Free Energy Simulations",
482 (1986) 1; T.  P. Straatsma, PhD dissertation, "Free Energy Evaluation
by  Molecular Dynamics Simulations", University of Groningen, Netherlands
(1987)  and  S.  H.  Fleischman  and C. L. Brooks III, "Thermodynamics of
Aqueous  Solvation:  Solution  Properties  of  Alchohols and Alkanes", J.
Chem. Phys., 87, (1987) p. 3029,  D.  J.  Tobias and  C.  L. Brooks  III,
J. Chem. Phys., 89, (1988) 5115-5127, and D.J. Tobias, "The Formation and 
Stability of  Protein Folding Initiation Structures",  Ph.D. dissertation
Carnegie Mellon University (1991).
 
         In the previous nodes we have generally referred to this area of
molecular  simulation  as a "perturbation" theory.  Actually, none of the
techniques  used  are  actually  perturbation methods.  The relationships
used  for computing the relative free energy differences are all exact in
the  statistical  mechanical  sense.  The use of the term perturbation in
this  context  arises  from  the  fact  that  in the pre-number crunching
supercomputer  days,  various  series  expansions were derived from these
equations and were in fact perturbation theories.  The name thermodynamic
integration  might  be used, however common practice has been to apply it
to  only  one  particular formulation (and furthermore not put that under
the  rubric of thermodynamic perturbation).  Finally, the use of the name
"free energy simulations" is another misonomer for two reasons: 1) we can
calculate  the  temperature  derivative  thermodynamic properties as well
(Delta  E  and  Delta  S) and the one thing we can't get is absolute free
energies  (as  van  Gunsterin  has  pointed  out , Mother  Nature doesn't
integrate all over phase space either). In fact, we generally are limited
to calculating relative changes in free energies, i.e. Delta Delta A's.
 
        In thermodynamic perturbation theory, a system with the potential 
energy function U0 is perturbed to one with the potential function U1, and 
the resulting free energy difference is calculated as

		A1 - A0 = -kT ln < exp[ -(1/kT)*(U1 - U0)] >

where k is Boltzmann's constant, T is temperature (degrees K), and A0 and 
A1 are the excess Helmholtz free energies of systems 0 and 1, respectively.  
Two methods of thermodynamic perturbation are implemented in CHARMM:

    1) Chemical perturbation, where the perturbation being considered is a 
       change in the system's potential function parameters and topology, 
       e.g., CH3OH is "mutated" to CH3CH3, and

    2) Internal coordinate perturbation, where the perturbation represents 
       a variation in configuration, and the potential function remains the 
       same for the perturbed and unperturbed systems.

Each of these is discussed separately below.



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                         THEORY AND METHODOLOGY

* Menu:

* Chemical::   Chemical Perturbation Theory and Methodology
* Internal::   Internal Coordinate Perturbation Theory and Methodology
* References:: Some References on Thermodynamic Perturbation




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                          Chemical Perturbation

        If  you  have  read  either  (Fleischman  and  Brooks,  1987)  or
(Straatsma,  1987)  or any of the McCammon or Kollman perturbation (oops!
that  word  again) papers, then you have seen the standard schpiel on why
getting  Delta  A's  (or  Delta G's) of solvation or drug/enzyme binding,
among  other  processes is so difficult and that if one is satisfied with
relative changes in free energies it is computationally more tractable to
"trans-mutate"  various  parts  of  a  system  in  a  way that is usually
physically     unreasonable     but    computationally    feasible    and
thermodynamically  equivalent to that obtained from the physical process.
Read some of the aforementioned references if this doesn't ring a bell.
 
        So  that's  what  we  are doing - calculating relative changes in
free  energies  (Delta  Delta  A) for solvation and small molecule/enzyme
binding, among other things.  In the rest of this node, we will discuss a
little  bit  of the theory (you're better off reading the papers) and lot
about  the  actual  how-to-do-it in our implementation.  Subsequent nodes
discuss  the  actual  implementation  and  some  issues  to consider when
attempting this type of calculation.
 
 
                              The Hamiltonian
 
        There are three basic techniques for calculating relative changes
in  free  energy  and  their  temperature  derivative  properties: 1) the
so-called "perturbation" approach  2) "Thermodynamic Integration" (TI) 3)
and  the  somewhat  dubious  "slow-growth" technique (which is actually a
step-child of the TI method).
 
        In all of the methods we use a hybrid hamiltonian,
 
                                             N           N
                H(lambda) = H  + (1 - lambda) H  + lambda H  .
                             o                 R           P
 
                where: H  = "Environment" part of the Hamiltonian
                        o
                       H  = "Reactant" part of the Hamiltonian
                        R
                       H  = "Product" part of the Hamiltonian
                        P
                       lambda = coupling parameter (extent of
                                                    transformation)
                       N = integer exponent
 
        The  various terms will be explained shortly.  First, a bit about
our  Weltanshauung  viz.  free energy simulations.  The system is divided
into  four  sets  of atoms: 1) The reactant atoms 2) the product atoms 3)
the  colo  atoms  and 4) the environment atoms.  The reactant and product
atoms  are  those  that are actually being changed. The colo atoms (short
for  co-located  charge)  are  those  in which only the charge changes in
going  from  reactant  to product.  The environment atoms are the rest of
the system (e.g. solvent; parts of a molecule common to both reactant and
product).   The  reactant  and product designations are arbitrary and are
used  as  a  convention  to denote the direction in which we are mutating
(i.e. start with reactant end up with product).
 
        A simple example, taken from (Fleischman and Brooks, 1987) is the
calculation  of  the  relative  change  in the solvation free energies of
methanol  and ethane.  This one has been done by virtually everybody that
has  written a free energy simulation code.  The system is represented by
the  water  molecules  (we used a box of 125 in our study) and the hybrid
methanol/ethane  system,  with  aliphatic  methyl  groups  represented as
extended atoms.
 
                        O1--H1
                       /          H
                      /          /
              H      C1--C2     O
               \                 \
                O                 H
               /
              H
 
Using  the  depiction  above, for the tranformation of methanol -> ethane
the  reactant  atoms  are the hybrid's O1 and H1; the product atom is the
hybrid's   C2  methyl  group.   The  hybrid  molecule's  C1 methyl  group
changes  charge  as  one  goes  from reactant to product.  This is a colo
atom,  in  going  from  methanol  ->  ethane  it starts with the methanol
methyl  group  charge  and ends up (at lambda = 1.) with the ethyl methyl
group  charge.   Otherwise,  C1  is  considered an environment atom.  The
atoms  of  the  water  molecules  constitute the actual environment atoms
in  this  system.   If  the  hybrid  molecule  was  larger  it  too could
contain environment atoms.
 
        All  potential energy terms involving the reactant atoms, as well
as  the  electrostatic  interactions  involving  colo  atoms  with  their
reactant charges, go into H .  The kinetic energies of the reactant atoms
                           R
also  are  included  in this term.  Similarly, the potential energy terms
involving  product  atoms  and  the  colo  product  charge  electrostatic
interactions along with the kinetic energies of the product atoms go into
H .  The rest of the energy terms are incorporated into H .
 P                                                       o
Note that for a potential energy term to be included in, say, H  only one
                                                               R
atom  in  the given interaction has to be a reactant atom (or in the case
of  a  electrostatic  interaction  a  colo  atom).  Similarly for product
terms.
 
        For  electrostatic terms involving colo atoms effectively what is
done  is  that  electrostatic  terms containing colo atoms are calculated
twice,  once  with  the  reactant charges and then again with the product
charges.  Terms  between  colo  reactant  charges  and reactant atoms are
avoided  and  similarly  for  product  atoms.   Actually, the programming
details  are a bit more complicated than that and if interested see *NOTE
implementation: (pimplem).   The  outcome is the  same as just described.
 
        It is assumed that when the hybrid molecule is constructed in the
residue  topology  file,  there  are  no internal coordinate energy terms
involving  reactant and product atoms.  As yet no checking is done in the
program.   Similarly,  it is assumed that non-bonded exclusions have been
specified between reactant and product atoms.
 
        In  our implementation, the Hamiltonian is constructed exactly as
specified in the equation above.  In many papers, that particular form of
the  Hamiltonian is given in the theoretical section (or more likely, the
form  with  N=1, i.e. linear) and in the actual implementation the lambda
dependence  of the Hamiltonian is quite a bit more complex.  This is done
in  those  implementations where the force constants and other parameters
in  the  energy  terms  are  factored  by  lambda rather than calculating
various energy terms and factoring them.
 
        In  a  statistical mechanical sense there is no particular reason
that  forces  one to factor the Hamiltonian consistently like we do.  The
thermodynamics  holds  regardless of path and the equipartion theorem for
obtaining  the  kinetic  energy  works  just  as  well  (though  in other
implementations  it  appears  that the factoring of the kinetic energy is
ignored anyway).   However,  we feel that there are certain advantages to
doing  it  this  way.  First, there is a certain conceptual simplicity in
factoring  the  Hamiltonian  consistently  for reactant and product terms
enmass.   Second,  it  makes obtaining the derivatives of the Hamiltonian
with  respect  to  lambda,  d E(lambda)/dlambda  programmatically simple.
These  derivatives  are  needed  in  the TI and, the related, slow growth
methods.   Actually,  the current algorithm for the slow growth method in
our  implementation  uses  finite  differences  for  the derivative as do
Kollman and van Gunsterin.  This  could easily be changed.  Factoring the
energy  terms  rather  than  functional parameters permits a more modular
design  and  makes  incorporating  changes  by others to energy functions
terms easier.
 
                      The Free Energy Equations
 
        As  we  said  there are two (maybe three, depending how you count
it)  different  ways  that  we   obtain the free energy changes.  For the
thermodynamic integration method (TI) the following expression is used:
 
                  _ 1
                 /
                 |
        /\ A =   |  < d H(lambda)/d lambda >       d lambda
        --       |                          lambda
                _/ 0
 
Expressions  for  energy  and  entropy  changes  can  be derived for this
equation  (Mezei and Beveridge, 1986) and have been incorporated into our
program.   They  suffer  from  very high uncertainties due to presence of
ensemble  averages  over  the  total  energy which are then multiplied by
ensemble  averages  over  d H/d lambda.   One  is  apparently  better off
getting average energies at the endpoints and subtracting.
 
        The   method   we   have   used  the  most  is the  thermodynamic
perturbation  technique.   For  this,  the free energy change is given as
follows:
 
      /\ A (lambda ->  lambda') = - kT ln < exp - (V - V ) >
      --                                            R   P   lambda
 
Notice that all of the averages are at lambda.  To get the total delta A
                  ___
                  \
        /\ A  =   /       /\ A (lambda -> lambda')
        --        ---     --          i         i
                   i
 
the  pieces  are  added  up.   The user must insure that the whole lambda
range is covered.  For example, in the methanol -> ethane calculation, we
ran dynamics at 3 points: lambda = .125, .5 and .875.  To cover the range
we calculated delta A's as follows:
 
          lambda'      lambda       lambda'
 
           0.000       0.125        0.250
           0.250       0.500        0.750
           0.750       0.875        1.000
 
I.e.,  for  each  lambda  in  which  dynamics were run two delta A's were
calculated, one lower and one higher than the corresponding lambda.  This
has  been  termed  "double-wide" sampling.  Note that the pieces all join
up.
 
        In  our  implementation we have the capability of calculating the
temperature  derivative  related  thermodynamic  properties,  delta E and
delta  S.   This  is  effected by the use of equations derived by Brooks.
 
See Fleischman and Brooks, 1987 for the corrected set of equations.  They
use a finite difference approximation to the derivatives that avoids then
necessity  of  taking the differences of large averages that would result
from using the explicit temperture derivatives.
 
        With   both  of  the  aformentioned  methods  the  technique  for
accomplishing the simulations is called the "window" procedure.  In these
methods  simulations  are  run  at a discrete number of lambda points (we
generally  use  3  - 6 and long trajectories; other workers use up to 100
lambda points and very short trajectories).  In the case of thermodynamic
perturbation  the  total  free  energy  change  is  pieced  together from
perturbations  done  with  each  "window".   In the case of thermodynamic
integration  the  integration  is  done  by  a quadrature method.  In our
implementation,  we fit the ensemble average as a function of lambda to a
cubic  spline  polynomial and then integrate the polynomial analytically.
No  extrapolation to endpoints is done.  So if you start at lambda = .125
and  end  at  lambda  =  .875  (like  we  do)  you  can use thermodynamic
perturbation  to get the end points (.125 -> 0 and .875 -> 1.) and TI for
the middle.
 
        An   alternative   sampling   method  is  termed  "slow  growth".
It  is  more  or  less  an approximation to the thermodynamic integration
method.   In  this  case  instead  of lambda being a constant for a given
trajectory  (as  in  the  window  method),  instead  the parameter varies
monotonically with each time step.
 
 
                 n steps
                 ----
                 \
        /\ A =   /    H(lambda + delta lambda) - H(lambda )
        --       ----         i                          i
                  i
and
        lambda  = lambda    + delta lambda
              i         i-1
 
Where H is the Hamiltonian and delta lambda = 1/nstep.




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                      Internal Coordinate Perturbation


    According to the thermodynamic perturbation (TP) theory, the Helmholtz 
free energy difference, A1 - A0, between system 0, in which the 
conformational coordinate of interest (e.g. an internal coordinate) is equal 
to x, and another system, in which the coordinate has been "perturbed" by 
the amount dx, is given by the equation

        A1 - A0 = A(x + dx) - A(x) 

                = kT ln < exp [-kT (U(x + dx) - U(x)) ] >           (1)
                                                         x

where U is the potential energy k is Boltzmann's constant and T is 
temperature (degrees K).  The <...>x notation denotes a canonical ensemble 
average over the "reference" ensemble in which the coordinate is equal to x.  
Although the potential energy may depend on many degrees of freedom, for the 
sake of simplicity we have only explicitly indicated its dependence on x.  
If we assume that the ergodic hypothesis holds, we can equate the ensemble 
average appearing in equation (1) to the time average computed from an MD 
simulation, e.g.

<exp [ -(1/kT) (U(x + dx)) ]> = 

             (1/N) * Sum   { exp [ -(1/kT)*(Ui(x+dx) - Ui(x)) ] }   (2)
                       1->N

where Ui is the value of the potential energy at the ith timestep and N is 
the number of timesteps in the simulation.  Since the average is over the 
reference ensemble, we must constrain the system so the value of the 
coordinate of interest is x at each step of the simulation.  In other words, 
we must impose the holonomic constraint

                      sigma = x(t) - x(0) = 0                       (3)

during the integration of the equations of motion.  The conformational 
coordinate may correspond to a set of internal coordinates.  In that case, 
equation (3) implies a set of holonomic constraints.  In addition to 
enforcing the conformational constraint, we need to carry out the 
perturbation (x -> x + dx), calculate the potential energy difference, U(x + 
dx) - U(x), and restore the constraint at each step of the simulation.

	With the above considerations, the following pseudo-computer code 
illustrates schematically the implementation of the TP method into an MD 
simulation:

	set up dynamics; specify constraint and perturbation
	do i = 1,N
		compute potential energy and forces
		take unconstrained dynamics step
		satisfy constraints
		perform perturbation
		compute potential energy
		restore constraints
	end do
	compute averages and thermodynamics

A detailed description of an algorithm for satisfying internal coordinate 
constraints is given in (Tobias, 1991).  We concentrate here on the tasks of 
specifying and performing the perturbation, and computing the difference in 
the potential energy of the perturbed and reference systems.

	The specification of the perturbation consists of identifying the 
degree(s) of freedom to be perturbed and the atoms whose positions change as 
a result of the perturbation.  Our implementation allows for perturbations 
of distances, angles, and torsions between groups of atoms.  For example, we 
may use a distance perturbation to study the breakup of a salt-bridge (ion 
pair) formed by the sidechains of lysine and glutamic acid, where x might be 
the distance between the N atom in the lysine sidechain and the carboxyl C 
atom in the glutamic acid sidechain, and the perturbation would consist of 
moving the entire glutamic acid residue.  Alternatively, we could use an 
angle perturbation to study the angular dependence of the strength of a 
hydrogen bond between two amides, where x is the O...H-N angle, and the 
perturbation moves the entire hydrogen bond donor molecule.  Or, we could 
use a torsional perturbation to study the trans-gauche isomerization in 
butane, where x is the dihedral angle for methyl group rotation about the 
central C-C bond, and the perturbation moves a terminal methyl group and the 
hydrogen atoms on the adjacent methylene group.  In addition to simple 
perturbations of a single internal coordinate, we can define more 
complicated perturbations involving more than one internal coordinate in 
order to study correlated conformational transitions.  For example, we could 
combine perturbations of the phi and psi dihedral angles to study backbone 
conformational equilibria in peptides (see *Notes implementation: 
(pimplem).).

	The procedure for carrying out internal coordinate perturbations 
during molecular dynamics simulations may be summarized as follows: after 
choosing an internal coordinate to perturb, and deciding which atoms will be 
moved by the perturbation, we compute a Cartesian displacement vector which 
changes the internal coordinate by a specified amount, and add the displace-
ment vector to the positions of the atoms to be moved.  Thus, in our imple-
mentation, the perturbation can be described as a rigid body movement of the 
perturbed atoms relative to the unperturbed atoms.

	Once we have moved all of the atoms involved in the perturbation, we 
need to compute the potential energy difference, delta U = U1 - U0 = U(x + 
dx) - U(x).  To do this, we could compute U(x), carry out the perturbation 
and compute U(x + dx), and simply take the difference.  However, this direct 
route is computationally inefficient, because interaction energies between 
atoms which are not moved by the perturbation are unnecessarily recomputed.  
To minimize the computational effort required to compute delta U, we only 
consider the interactions which change as a result of the perturbation.  For 
this purpose, we partition the system into two parts: the atoms which are 
moved by the perturbation (denoted by "s" for "solute"), and those which are 
not (denoted by "b" for "bath").  With this partitioning, we can write the 
potential energy as a sum of three contributions:

    U(x) = Uss(x) + Usb(x) + Ubb(x),                                (4)

where Uss, Usb, and Ubb are the solute-solute, solute-bath, and bath-bath 
interaction energies, respectively.  Clearly, Ubb(x + dx) - Ubb(x) = 0 since 
the positions of the bath atoms are not changed by the perturbation.  Thus,

   U1 - U0 = Uss(x + dx) - Uss(x) + Usb(x+dx) - Usb(x).             (5)

	Since, in typical applications, the number of solute atoms is much 
smaller than the number of bath atoms, equation (5) represents a large 
reduction in computational effort over the direct route.  Before we proceed, 
we point out that when we need U(x + dx) in addition to delta U (e.g. for 
computation of conformational entropies using finite difference temperature 
derivatives of the TP free energy (see *Note description: (perturb).), we 
can use the following expression:

     U(x + dx) = Uss(x + dx) + Usb(x + dx) + Ubb(x)

               = Uss(x + dx) + Usb(x + dx) + U(x) - Uss(x) - Usb(x),   (6)

since Ubb(x + dx) = Ubb(x).  We assume that U(x) is computed when the forces 
required for the propagation of the dynamics are computed.  Thus, we still 
only need to compute the changes in the solute-solute and solute-bath 
interaction energies which result from the perturbation.

	In general, when we have a choice, we partition the system so that the 
solute consists of the smallest possible number of atoms.  There are two 
good reasons for this.  First, smaller solute partitions require less effort 
to compute the interaction energies.  Second, with smaller solute 
partitions, there is less of a chance that the solute atoms will "run into" 
bath atoms as a result of the perturbation.  When solute and bath atoms run 
into one another, there is a large, positive van der Waals contribution to 
deltaU.  This is undesireable because large delta U values lead to poorer 
convergence of the average in equation (2).  The partitioning of the system 
is especially important when the perturbations are carried out in "crowded" 
environments, such as in solution or in the interior of a protein.

	In some cases it is useful to divide the solute partition into two 
sections, and accomplish the desired perturbation by moving each section by 
half the perturbation.  For example, to perturb the dihedral angle in butane 
by dx, we could include both methyl carbons and all of the hydrogens in the 
solute partition, with the C1 methyl group and C2 methylene hydrogens in one 
section, and the C4 methyl group and C3 hydrogens in the other section, and 
move each section by dx/2.  This "double move" strategy is useful when the 
perturbation is carried out on a small molecule in a crowded environment 
where the movement of n atoms by dx/2 is more favorable than the movement of 
approximately n/2 atoms by dx.  The option to perform perturbations in this 
fashion is available in our implementation.

	In principle, we could get the free energy difference between any two 
conformations, x0 and x1, in a single simulation using the TP theory 
expression:

       delta A = A1 - A0 = A(x1) - A(x0) 

               = - kT ln < exp [ -(1/kT)*(U(x1) - U(x0)) ] >   .     (7)
                                                            x0

However, in practice, for typical simulation lengths, the average in 
equation (7) exhibits acceptable convergence only when deltaA <= 2kT 
(Beveridge & DiCapua, 1989).  Thus, if the free energy difference between 
the conformations x0 and x1 or the free energy barrier separating them, is 
more than about 2kT, then a single simulation is not sufficient to determine 
accurately the free energy difference.  This problem is circumvented by 
breaking up the range of the coordinate, x1 - x0 into n segments or 
"windows", y(i),

dy = (x1 - x0)/(n + 1);  y(i) = x0 + (i - 1) dy; i = 1,...,n,          (8)

and running a series of n simulations where the free energy differences,

   delta A(i) = A(y(i+1)) - A(y(i) 

              = -kT ln < exp [ -(1/kT)*(U(y(i+1)) - U(y(i))) ] >       (9)
                                                                y(i)

are computed.  Then the free energy difference between x1 and x0 is obtained 
by summing the results from the n windows, e.g.

                        x1
      delta A = Integral   (p(deltaA(y))/py) dy
                        x0
                   
              = Sum    delta A(i),                                     (10)
                  1->n

where p(z) denotes the partial derivative of z.  Aside from yielding more 
accurate free energy differences, the window method is attractive because it 
allows us to map out the free energy surface as a function of the 
conformational coordinate.

	By far the most time consuming task in a molecular dynamics simulation 
is the evaluation of the forces necessary to propagate the equations of 
motion.  The additional work required for computing the interaction energies 
needed for the TP free energy differences is relatively small.  Thus, it is 
advantageous to get more than one free energy difference from a single 
simulation.  This is the motivation for using the so-called "double-wide" 
sampling method (Beveridge & DiCapua, 1989), where the free energy 
differences A(y + dy) - A(y) and A(y - dy) - A(y) are obtained in one 
simulation.  Furthermore, we can divide dx into m subintervals, dy(m) = 
dy/m, and compute 2m free energy differences,

           +/-
delta A(i,k)  =  A(y(i) +/- dy(m)) - A(y(i))

              = -kT ln < exp[ -(1/kT)*(U(y(i) +/- kdy(m)) - U(y(i))) ] >
                                                                       y(i)

                                                   k = 1,...,m;        (11)

over the range y(i) - dy <= y <= y(i) + dy from a single simulation with x = 
y(i).  Then we sum the free energy differences from the various subintervals 
(in analogy with equation (10)) to get a free energy surface for each 
window.  This "double-wide, multiple-point" window method allows a higher 
resolution mapping of the free energy surface with little additional 
computational effort.

	Let us now comment on how dy is chosen.  As we have already said, dy 
should be chosen so that the free energy change in a given window is not 
more than a couple of kT.  In addition, the shape of the free energy surface 
in a given window can be used to determine a good choice for dy.  A 
reasonable choice for a given system can be made by considering results from 
short simulations with a modest dy and several subintervals at a couple of 
values of y in the range of interest.  In general, for perturbations in 
crowded environments (e.g. in solution or the interior of a protein), 
excessively large values of dy always result in positive free energy 
differences.  This is because the perturbation results in repulsive van der 
Waals interactions of the atoms in the solute partition with those in the 
bath partition.  The value of dy where the free energy difference begins to 
sharply increase can then be regarded as the upper bound on acceptable dy 
values.  Of course, it is possible that the underlying free energy surface 
really does rise sharply beyond the second subinterval in both directions.  
That is why we suggest running another test at a different value of y.  In 
addition to running short test calculations, it is also useful to consult 
previous work to get a preliminary estimate for an acceptable size of a 
perturbation in a similar system (for several examples, see (Tobias, 1991)).

	In our implementation, the information needed to calculate 
conformational thermodynamics (free energies, internal energies, entropies, 
average interaction energies), and their associated statistical 
uncertainties, is written to a datafile during a simulation.  The data file 
is subsequently "post-processed" to yield the quantities of interest.  The 
alternative approach is to calculate the average properties of interest as 
the simulation progresses, and simply write out the final results at the end 
of the simulation.  The latter approach has the advantage that large, 
cumbersome data files do not need to be saved on a mass-storage device (e.g. 
disk or tape).  However, we prefer the post-processing approach because of 
the flexibility it gives us in the analysis of the data.  For example, we 
can: examine the time evolution, and hence the "convergence", of the average 
properties; carry out the averaging on an arbitrary amount of the data; 
compare various protocols for computing the statistical uncertainties or 
finite-difference temperature derivatives, etc.

	We use the method of block averages (a.k.a. batch averages) to compute 
the average properties and their uncertainties (Wood, 1968).  In this 
method, the total number of samples, N, is divided into m "batches" of n 
samples (mn = N), and the average of the property of interest, <O>i, is 
computed for each batch i:

     <O>i = (1/n) Sum      O(k,i),                                  (12)
                    k=1->n

where O(k,i) is the kth observation of O in the ith batch.  The average of 
the N samples, <O>, is simply the average of the batch averages:

     <O> = (1/m) Sum      <O> ;                                     (13)
                   i=1->m    i


and the "uncertainty", std<O>, is estimated from the standard deviation in 
the batch averages:

    std<O> = ( Sum    [ (<O> - <O>)**2 / m(m - 1)] )**1/2           (14)
               i=1->n       i


We use equation (14) to compute the uncertainty in the average of the 
exponential in equation (1).  Then we obtain the uncertainty in the free 
energy (and other thermodynamic functions) by error propagation (Young, 
1962), e.g.

	std(delta A)**2 = (p(delta A)/p(z))**2 (std(z))**2

                    = (kT*std(z)/z)**2                               (15)

where z is the average of the exponential in equation (1).

	In order for the uncertainty given by equation (14) to be a good 
estimate of the "true" uncertainty (e.g. in a large number of random 
samples), the block size must be chosen so that the block averages are 
uncorrelated (randomly distributed), and the number of blocks is not too 
small for the evaluation of a meaningful standard deviation.  The block size 
n is typically chosen arbitrarily and possible correlations in the data are 
ignored.  More refined uncertainties can be obtained by considering the 
actual correlation of the data determined explicitly from the 
autocorrelation function (Straatsma, et al., 1986).  However, we presently 
have no facility for carrying out the correlation function analysis.




File: PDETAIL ]-[ Node: References
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References

	Beveridge, D. L. & DiCapua, F. M. (1989), in "Computer Simulations of 
Biomolecular Systems", eds. van Gunsteren, W. F. & Weiner, P. K. (Escom, 
Leiden).

	Straatsma, T. P. (1987). "Free Energy Evaluation by Molecular Dynamics 
Simulations" (Ph.D. dissertation, Department of Physical Chemistry, 
University of Groningen).

	Tobias, D. J. (1991). "The Formation and Stability of Protein Folding 
Initiation Structures" (Ph.D. Dissertation, Department of Chemistry,  
Carnegie Mellon University).

	Wood, W. W. (1968), in "Physics of Simple Liquids", eds. Rowlinson, J. 
S. & Rushbrooke, G. S. (North-Holland, Amsterdam).

	Young, H. D. (1962). "Statistical Treatment of Experimental Data" 
(McGraw-Hill, New York).




File: PDETAIL ]-[ Node: Practice
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                                  Practice
 

        In  this  node  we  tell  you how to actually set up and run free
energy simulations.
 
        The  calculation  is  done  in  three steps.  The first two steps
occur  in  the  same  input  file  -  perturbation set up and running the
dynamics.   The  last step, the post-processing, is generally done with a
separate  input file since the output of several trajectories are usually
used.
 
        To set up the free energy simulation dynamics input file you start
with the usual set up for a dynamics run: psf, coordinates, image input or
stochastic  boundary  condition input etc..  In addition you have to issue
free energy  simulation (FES) set up commands.  Currently the set up input
is    initiated    by   the   TSM   command   (*Note  syntax:  (perturb).)
(*Note description: (perturb).).   For chemical perturbations,  these com-
mands define the reactant, product and colo lists; the type of simulation:
slow growth or window procedure  (both the thermodynamic  perturbation and 
the  thermodynamic integration methods can be done with the  window proce-
dure).   For internal coordinate perturbations,  the setup commands define
the  internal coordinate(s) to be perturbed, the set of atoms moved by the
perturbation, and how and where the thermodynamic results will be written.
 

* Menu:

* CPrac::  Chemical Perturbation Practice
* IPrac::  Internalal Coordinate Perturbation Practice




File: PDETAIL ]-[ Node: CPrac
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                      CHEMICAL PERTURBATION - PRACTICE


        As  currently  configured , most of the minimization routines will
work   using   the  hybrid  V(lambda)  potential.   We  generally  do  any
minimization  prior to dynamics with the hybrid molecule unperturbed since
we  are really concerned with removing bad contacts.  It is not guaranteed
in  the  future  that  the V(lambda) will be available to the minimization
routines.
 
        After  the  FES set up has been entered flags have been set in the
program  and  data  structures  created  and  dynamics  can be run with no
changes in the commands used in any other dynamics run.  One will normally
run  some  thermalization  runs  with  the  data  being  discarded.  For a
thermalization  run  the  SAVE  command in the FES set up is generally not
used.   For  production  runs  for  TI  or Thermodynamic Perturbation (TP)
the  SAVE option must be issued in the FES set up input.  This will result
in  the  output of V(R) and V(P), lambda among other things in a formatted
file.  All this will be discussed below with examples.
 

* Menu:

* SetUp::     Setting Up the FES Simulation and Running Dynamics
* PostD::      Post-processing the Data
* Optional::  Using Some Optional FES Set Up Commands



File: PDETAIL ]-[ Node: SetUp
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          Setting Up the FES Simulation and Running Dynamics
 
Below  is  a  fragment of the input file for setting up the thermalization
of  the  ethanol  ->  propane  hybrid.   Windowing will be used and we can
decide  at  the  end  whether  to  post-process the output using TI or TP.
Using the representation below the system is partioned as follows:
 
                             O1--H1
                            /
                           /
                     C1--C2
                           \
                            \
                             C3
 
The  reactant  atoms are O1 and H1; the only product atom in this example
is  C3  and  there  is  one  COLO  atom,  C2.   The methyl group C1 is an
"environment"  atom.   It  is present in both reactant and product and in
our model its charge does not change in going from reactant to product.
 
* Ethanol -> Propane
*
 
! Read topology file
READ RTF CARD
* TOPOLOGY FILE ethanol -> propane
*
   20    1                ! Version number
MASS     1 H      1.00800 ! hydrogen which can h-bond to neutral atom
MASS    13 CH2E  14.02700 !   -    "    -           two
MASS    14 CH3E  15.03500 !   -    "    -           three
MASS    53 OH1   15.99940 ! hydroxy oxygen
 
! This is put in to force the necessity of using a GENERATE Noangles
! in the input file.  The standard topology files use this statement.
AUTOGENERATE ANGLEs
 
RESI ETP 0.000
GROU
C1 CH3E 0.    ! environment atom
C2 CH2E 0.265 ! COLO atom the charge is the reactant charge
O1 OH1 -0.7   ! reactant atom
H1 H    0.435 C3 ! reactant atom note the non-bonded exclusion with
GROU
C3 CH3E 0.    ! product atom
 
BOND C1 C2       !environment term
BOND C2 O1 O1 H1 !reactant terms
BOND C2 C3       !product term
 
! the angles MUST be specified
! note the absence of O1 C2 C3 between reactant and product atoms
ANGLe  C1 C2 C3  !product term
ANGLe  C1 C2 O1  C2 O1 H1  !reactant terms
 
! this will be a V(R) term.
DIHED C1 C2 O1 H1
 
! don't really need it but what the heck.
DONO H1 O1
ACCE O1
 
IC C1 C2 O1 H1  1.54 111. 180. 109.5 0.96
IC C2 O1 H1 BLNK 0. 0. 0. 0. 0.
IC C3 C2 C1 BLNK 0. 0. 0. 0. 0.
 
PATCH FIRST NONE LAST NONE
!
END
 
! Read parameter file
READ PARAM CARD
* parameter file for ETP hybrid.
*
 
BOND
CH2E CH3E  225.0  1.54
CH2E OH1   400.0  1.42
OH1  H     450.0  0.96
 
THETA
CH3E CH2E CH3E  45.0 112.5
CH3E CH2E OH1   45.0 111.0
CH2E OH1  H     35.0 109.5
 
PHI
CH3E CH2E OH1 H  0.5   3   0.0
 
NONBONDED  NBXMOD 5  ATOM CDIEL SHIFT VATOM VDISTANCE VSWIT -
     CUTNB 8.0  CTOFNB 7.5  CTONNB 6.5  EPS 1.0  E14FAC 0.4  WMIN 1.5
 
!                  Emin       Rmin
!                  (kcal/mol) (A)
H        0.0440    -0.0498    0.8000
CH2E     1.77      -0.1142    2.235  1.77 -0.1 1.9
CH3E     2.17      -0.1811    2.165  1.77 -0.1 1.9
OH1      0.8400    -0.1591    1.6000
 
HBOND AEXP 4 REXP 6 HAEX 0 AAEX 0   NOACCEPTORS  HBNOEXCLUSIONS  ALL  -
   CUTHB 0.5 CTOFHB 5.0 CTONHB 4.0  CUTHA 90.0  CTOFHA 90.0  CTONHA 90.0
!
H*    N%      -0.00      2.0 ! WER potential adjustment
H*    O*      -0.00      2.0
 
END
 
! read the sequence of one residue
read sequence card
* ETP
*
1
ETP
 
! Generate the  hybrid molecule.  Note that we use the NOANGLE command
! because of the AUTOGENERATE ANGLES command in the RTF file.
GENERATE ETP SETUP NOANGLE
 
! determine the geometry and coordinates
IC SEED  1 C1 1 C2 1 O1
IC PARAM
IC PURGE
IC BUILD
 
! The Hybrid molecule is built. Now set up the FES stuff.
TSM
! Assign reactant list:
REAC sele etp 1 O1 .or. etp 1 H1 end
! Assign product list:
PROD sele etp 1 C2 end
! Set lambda - we will use TI or TP.
! The lambda dependence of the Hamiltonian will be linear.
! This is the default and the POWEr 1 command is actually unecessary.
LAMBda .125 POWEr 1
! The common methyl group is a colo atom.  Since the charge in the
! rtf was for the reactant the RCHArge command is actually unecessary.
COLO ETM 1 C2  PCHArge 0. RCHArge 0.265
!
! This is a thermalization run - so no save statement.
! Just terminate the FES setup with an END statement.
END
 
! Set up dynamics.
! Since we are interested in the thermodynamic properties and not
! the dynamics, we can use Langevin heat bath dynamics to maintain
! temperature equilibration. Lambda is .125.
 
title
* etp: Ethanol To Propane
* FES run
*
!a simple expedient
shake bond angle
!  Set-up Langevin dynamics for temperature control
scalar fbeta set 50.0 sele .not. hydrogen end
!
! open restart file for output
open unit 3 write form name etp0.res
!
dynamics langevin timestep 0.001 nstep 10 nprint 2 iprfrq 2 -
     firstt 298.0 finalt 298.0 twindl -5.0 twindh 5.0 -
     ichecw 1 teminc 60 ihtfrq 20 ieqfrq 200 -
     iasors 0 iasvel 1 iscvel 0 -
     iunwri 3 nsavc 0 nsavv 0 iunvel 0 -
     iunread -1                           - !{* Nonbond options *}
     inbfrq 10 imgfrq 10 ilbfrq 0 tbath 300.0 rbuffer 0.0 -
     eps 1.0 cutnb 8.0 cutim 8.0 ctofnb 7.75
stop
*END of INPUT*
 
        This  file  has  everything  you  need  to  run the example.  The
topology  and parameter input are included.  The FES set up was initiated
with the TSM command.  The reactant and product lists were specified with
REAC  and  PROD  commands  that  use  the  standard CHARMM atom selection
syntax.   Had  their  been either no reactant atoms or product atoms then
the  command  would  have been REAC NONE or PROD none as the case may be.
Note that specifying both would have resulted in an error condition being
flagged.   Since  we  are  using the window method we specified LAMBda as
being  0.125.   We also explicitly specified the lambda dependence of the
Hamiltonian  as  being  (1-lambda)**1 for the reactant part and lambda**1
for  the  product  part.  Since not entering the POWEr parameter causes a
default of 1 for the exponent in was unecessary to actually enter it.
 
        There  is one COLO atom in the system.  The product charge of the
C2  methylene  extended atom was 0.  In the RTF the charge was .265 which
is  the  reactant  (ethanol)  charge.   Since that's what we want for the
reactant  charge  there  was  actually  no  need  to  enter  the  RCHArge
parameter.  Again, we put it there for illustrative purposes, the default
is to assume that for any COLO atom the charge in the RTF is the reactant
charge  unless  the  RCHArge  parameter  is included in the COLO command.
Note that charges can also be changed with the SCALAR command.
 
        We  could  have  chosen a value of the POWEr parameter other than
one  (i.e.  non-linear  lambda scaling).  This is potentially useful when
using   the TI method for the free energy change.  Non-linear scaling has
one  major  advantage.   At  lambda = 0 the components of the derivatives
dH(lambda)/dlambda  due  to  the  product  part  of  the  Hamiltonian are
identically  zero  and similarly, at lambda = 1 the components due to the
reactant   part   are  zero.   This  solves  the  "lambda  goes  to  zero
catastrophe" problem.  This is the problem that as lambda approaches zero
or  one  the positions of the atoms affected (mostly product or reactant,
respectively, and sometimes environment atoms bonded to them) feel forces
that  approach  a  constant or zero value (zero potential energy) and can
thus have positions anywhere in phase space.
 
        Since  the  approximations  to the ensemble averages are obtained
from  finite  length trajectories, determining values of those quantities
becomes  a computationally intractable proposition.  The TI integral over
dlambda  will  tend  to diverge when linear scaling is used.  In both the
TI  and TP methods actually calculating the dynamics trajectory generally
will be problematical, with large movements of the atoms resulting in bad
van  der  Waals  contacts  (the  r**12  repulsion eventually is felt) and
fraying  of  bonds  with  lambda approaching zero or one.  Another way of
viewing  the situation is that at lambda = 0 or 1 the product or reactant
atoms, respectively, do not exist yet.  Doing the perturbation to lambda'
(or equivalently viewing the derivative, dH/dlambda, as a perturbation to
lambda  +  dlambda)  requires having the coordinates of atoms that do not
exist  yet  or  any  longer.  Non-linear scaling and the TI method can be
used  to  avoid  this  difficulty  for  the reasons given in the previous
paragraph.   Another  way  is to scale the TI integral by a function that
reduces  the  weight  of  the  integrand  as  lambda  -> 0 or 1.  This is
discussed in Mezei and Beveridge.
 
        For  lambda  =  0, if use of the TI method with non-linear lambda
scaling  was planned we would issue a command, prior to the FES setup, to
delete the product atoms from the hybrid molecule rtf:
 
   DELEte ATOMs SELEct etp 1 C2 END
 
This  is  a standard CHARMM PSF modification command and would be issued
after  the  segment  generation.  Alternatively, we could have just used
an RTF for ethanol.
 
        The  FES  setup command sequence would be modified slightly from
the previous example:
 
   TSM
   REAC sele etp 1 O1 .or. etp 1 H1 end
   ! no product atom at lambda = 0
   PROD NONE
   ! non-linear lambda scaling
   LAMBda .125 POWEr 2
   END
 
Note  that  since there are no product atoms at lambda = 0, the PROD NONE
command  is  issued.   Also  there  is no need for the COLO command.  For
lambda at 1 we can use an equivalent procedure (left as an assignment for
the reader).
 
        In  most of our work to date, we have used linear scaling and the
TP method.  To get around the catastrophe problem, we do not run dynamics
at  lambda  =  0  or  1.  Instead we run them at values of lambda a small
distance  away  from  0  or  1  and "perturb" down to the endpoints.  One
potential  problem may occur with this procedure.  In cases, such as that
of  the  transformation  hydrophobic  ->  hydrophobic  solute  in aqueous
solution,  where water structure rearrangements around the solute are the
major  contributing  factor  to  the  free energy change, not sampling at
lambda = 0 or 1 may mean that the significant part of phase space for the
rearrangement  is  not  adequately  sampled. If in going from reactant ->
product  (or vice versa) a significant volume becomes newly accessible to
the solvent, the presence of the r**-12 repulsive forces from the "almost
but  not  completely  disappeared"  atoms  may  conceivably  prevent  the
necessary  configurations  of  the  water molecules from appearing in the
finite length trajectory.  This problem has not been investigated yet.
 
        Non-linear  scaling  may  be preferred for sampling efficiency, a
debatable  point  that  has  been  discussed  by a number of researchers.
Problems  can  result  since  the monotonicity of the integrand in the TI
intregral  is  no  longer  assured.   In  the  case of the TP method, the
non-linear scaling forces the use of very small "perturbations" lambda ->
lambda'.  The  non-linear  exponent  makes the delta V(lambda -> lambda')
very  large.   For  example,  if  the  exponent  is 6 and lambda = .5 and
lambda' = .25, a not unreasonable "window", the potential energy term for
the  product  gets  multiplied  by  .5**6  = 0.16 for lambda and .25**6 =
0.00024 for lambda'.  So one has terms of exp(-beta(.15V  - 0.00024V ) in
                                                        P           P
the   ensemble   average  for  the  TP  method,  causing  extremely  slow
convergence.  For the temperature derivative related properties, one runs
into problems with floating point overflows.  This is probably the reason
why Kollman uses his convoluted "charge decoupling" technique whereby the
van  der Waals interaction contribution to the perturbation is calculated
with  slow  growth  and the charge interaction contribution is calculated
with windows.
 
        For the production runs we merely add a SAVE statement to the FES
set  up  commands, any place before the END statement.  In it the FORTRAN
unit number for the output file that will contain the FES information and
an output frequency (we generally use 1).
 
    SAVE UNIT 10 FREQ 1
 
The  FES  output  file must be opened for formatted write access prior to
invoking the dynamics command.  Use the unit number specified in the SAVE
statement.
 
     OPEN UNIT 10 WRITE FORM NAME ETP1.PRT
 
Production  dynamics  are  run  in  the usual way.  To run the production
dynamics  the  the command used for the previous (thermalization) example
is slightly modified.
 
! open restart file for output
open unit 4 write form name etp1.res
! open restart file for input
open unit 3 write form name etp0.res
!
dynamics langevin rest timestep 0.001 nstep 10 nprint 2 iprfrq 2 -
     firstt 298.0 finalt 298.0 twindl -5.0 twindh 5.0 -
     ichecw 1 teminc 60 ihtfrq 20 ieqfrq 200 -
     iasors 0 iasvel 1 iscvel 0 -
     iunwri 4 nsavc 0 nsavv 0 iunvel 0 -
     iunread 3                            - !{* Nonbond options *}
     inbfrq 10 imgfrq 10 ilbfrq 0 tbath 300.0 rbuffer 0.0 -
     eps 1.0 cutnb 8.0 cutim 8.0 ctofnb 7.75
 
We  opened  the  restart file from the thermalization for input on unit 3
and opened the a file for restart file output on unit 4.  In the dynamics
command  we  modified  iunread  and  iunwrite  to deal with this. We also
specified  that  this  is a dynamics restart.  All this has nothing to do
specifically with FES simulations.
 
        The  above  procedure is repeated for more lambda points.  In our
study  we  used  lambda  =  .5  and  .875.   At  each  value  of lambda a
thermalization run is done followed by production runs.  One advantage of
this implementation as compared to some others is that one can always run
additional  trajectories  at  any  given value of lambda and add tack the
output  onto  that  of  the  previous  runs.   Similarly,  we  can insert
trajectories  at other values of lambda and recalculate the thermodynamic
properties.
 


File: PDETAIL ]-[ Node: PostD
Up: CPrac -=- Previous: SetUp -=- Next: Optional\n


                         Post-Processing the Data
 
        Assume  that  we  now  have  three  FES  output  files: etp1.prt,
etp2.prt  and  etp3.prt for lambda = .125, .5 and .875, respectively.  To
make  things  interesting  let  us  assume  that  we  went  back  and ran
additional trajectories at each value of lambda and these files are named
etp4.prt,  etp5.prt  and  etp6.prt  for  lambda  =  .125,  .5  and  .875,
respectively.   We  used  the  window  sampling method with linear lambda
scaling.   The  next  step is to post-process the output so as to compute
the free energy changes.  The following input file will do the job:
 
* Post-processing Example ETP: ethanol -> propane vacuum.
* TP method linear lambda scaling.
*
! open FES data files for input.
open unit 10 form read etp1.prt
open unit 11 form read etp2.prt
open unit 12 form read etp3.prt
open unit 13 form read etp4.prt
open unit 14 form read etp5.prt
open unit 15 form read etp6.prt
!
! now the post-processing input
!
TSM  POST PSTAck 6 PLOT
! lambda = .125 -> lambda' = 0.
PROC FIRST 10 NUNIT 2 LAMB 0.0 TEMP 298.0 DELT 10. BINS 100 CTEM
! lambda = .125 -> lambda' = 0.25
PROC FIRST 10 NUNIT 2 LAMB 0.25 TEMP 298.0 DELT 10. BINS 100 CTEM
! lambda = .5 -> lambda' = 0.25
PROC FIRST 12 NUNIT 2 LAMB 0.0 TEMP 298.0 DELT 10. BINS 100 CTEM
! lambda = .5 -> lambda' = 0.75
PROC FIRST 12 NUNIT 2 LAMB 0.0 TEMP 298.0 DELT 10. BINS 100 CTEM
! lambda = .875 -> lambda' = 0.75
PROC FIRST 14 NUNIT 2 LAMB 0.0 TEMP 298.0 DELT 10. BINS 100 CTEM
! lambda = .875 -> lambda' = 1.0
PROC FIRST 14 NUNIT 2 LAMB 0.0 TEMP 298.0 DELT 10. BINS 100 CTEM
!
! the END command tells the post-processor to tally everything up.
END
STOP
 
        This  is  an complete input file for post-processing using the TP
method.  First,  all of the data files that were needed were opened prior
to  invoking the post-processor.  The command to initiate post processing
was:
 
TSM  POST <parameters>
 
For a full explanation of the parameters see *Note PDESC: (perturb). Note
that   by   not   specifying   the   sub-command  TI  (for  thermodynamic
integration),  the  TP  (thermodynamic  perturbation) method is selected.
The  PSTAck  command allocates space for various temporary arrays. PSTAck
should be equal to the number of PROCess commands.  In the case of the TP
method  they  are  used  for  holding  plotting  values  until  the final
processing is done.  The PLOT parameter indicates that plotting output is
to  be  generated.   This is written into the CHARMM output file (unit 6)
and can be extracted with an editor.  The format will serve as data input
to the PLT2 program.  For the TP method delta A, delta E and delta S as a
function  of  lambda  are  output  with  lambda  points marked by X's and
lambda' points marked by O's.
 
        The  PROCess commands cover the complete range from 0 to 1.  This
example  shows  how we can tack on trajectories.  The post-processor will
read  each  file  until it encounters and end of file (EOF) condition and
count the steps itself.  The nstep parameter at the beginning of the file
is  ignored.  Often this value is not accurate since it is something that
is written out before the dynamics starts; not an actual count of records
written.   The  FIRSt parameter specifies the initial FORTRAN unit number
for  the  first  file  for  a  given PROCess command; NUNIt specifies the
number  of  files  to  be read.  It is assumed that they were opened with
contiguous  unit  numbers.   The  NUNIt parameter includes the FIRSt unit
number  in  the count.  If there is only one file, NUNIt can be left out.
The  parameter  LAMBDa indicates lambda prime; TEMP is the temperature to
be   used  in  the  calculations  (all  those  1/kT's).   DELTat  is  the
temperature  increment  for the finite difference temperature derivatives.
We  normally  use  10  degrees.   The  parameter  BINSize  is used in the
estimation  of  the  statistical  uncertainty  (error)  of  the a various
thermodynamic properties.
 
        So  here  is  a  good  point  at  which  to discuss how the error
estimation is calculated (at least I think it's a good point).  The major
problem  is that the data is very highly correlated.  To get around this,
in  many implementation including ours, the data is divided into bins and
deviations  of  the  bin  averages  from the total trajectory average are
calculated to get the variance.  The idea is that the use of bin averages
will  remove  the correlated dependence of the variance.  We always use a
bin  size  of  100.  The estimated error will depend on the choice of bin
size.   One  would hope that it would converge as a function of bin size.
Our trajectory lengths are generally not long enough to ensure this.
 
        A  method  for  determining  the variance of a property that uses
correlation  functions  has  been  developed by T. P. Straatsma, H. J. C.
Berendsen  and A. J. Stam ( T. P. Straatsma, H. J. C. Berendsen and A. J.
Stam,  Molecular  Physics,  57  p89  (1986)  also in T. P. Straatsma' PhD
dissertation  referenced  above).  This method looks promising as long as
one  visually monitors the correlation function behavior and extrapolates
at  the  longer  lag  times.   The  problem  is  that  the  error  in the
approximation  of  the  correlation  function is introduced.  The authors
used  an  exponential extrapolation to overcome this.  This algorithm has
been implemented by a member of the research group in another context and
is being tested.
 
        Once  the  variance  of the ensemble averages are calculated, the
uncertainty in the thermodynamic properties are determined by propagation
of errors.  For the TI method the necessary derivatives are determined by
numerical  differentiation.   Total  uncertainties  are  determined  from
summing the variances for each window (or lambda point in the TI method).
 
        One  final  remark  on about the PROCess command line.  The CTEMp
parameter sets a flag to calculate the average temperature at each lambda
point.   This  can  be  done since the FES data file contains the kinetic
energy  at  each  step.   The  calculated  temperature  is  not  used  in
calculating  any of the properties, however.  To use it one would have to
reprocess  the  data  with  the  average  temperature determined from the
previous iteration entered manually in the TEMP command.
 
        Note  that the user does not specify the lambda scaling.  This is
determined by reading  a  header  lineas in the first file in the series.
Furthermore,  the  user  is responsible for ensuring that this value does
not  change  among  files  in  the  series  as  well  as in the different
processing commands (unless it is intentional for some reason).
 
        As   mentioned   above   the   final  command,  END,  causes  the
post-processor  to  tally up the averages and calculated the final values
of  the  thermodynamic  properties  and to write out the plotting output.
 
        We  could  have  chosen to use the TI method.  As an illustrative
example,  let us assume that we used non-linear lambda scaling, say POWEr
2  and  ran  simulations  at  lambda = 0.0, 0.2, 0.4, 0.8 and 1.0 and the
output  files  (one  per  lambda) are in etp1.prt, etp2.prt, etp3.prt and
etp4.prt  and etp5.prt.  The input file would a title and open statements
for  each  file  as  before.   The  post-processing  commands would be as
follows:
 
* Post-processing Example ETP: ethanol -> propane vacuum.
* TI method with non-linear lambda scaling N = 2.
*
! open FES data files for input.
open unit 10 form read etp1.prt
open unit 11 form read etp2.prt
open unit 12 form read etp3.prt
open unit 13 form read etp4.prt
open unit 14 form read etp5.prt
 
TSM  POST PSTAck 5 TI PLOT
! lambda = 0.0
PROC FIRST 10 NUNIT 2 ZERO TEMP 298.0 BINS 100 CTEM
! lambda = 0.2
PROC FIRST 11 NUNIT 2 TEMP 298.0  BINS 100 CTEM
! lambda = 0.4
PROC FIRST 12 NUNIT 2 TEMP 298.0 BINS 100 CTEM
! lambda = 0.8
PROC FIRST 13 NUNIT 2 TEMP 298.0 BINS 100 CTEM
! lambda = 1.0
PROC FIRST 14 NUNIT 2 ONE TEMP 298.0 DELT 10. BINS 100 CTEM
! the END command tells the post-processor to tally everything up.
! sorts the lists of ensemble averages and does the final integration.
END
 
Note  that  there  is  no  LAMBdaprime  parameter as there was for the TP
post-processing.   Also  since  the temperature derivative properties are
calculated  in  a  different  fashion, there  is  no DELTatemp parameter,
either.   Also, for  the points at lambda = 0 and 1 the sub-commands ZERO
and  ONE are introduced.  These tell the program that the lambda value in
the  data  file  is  exactly  zero  or  one.   This is necessary to avoid
floating point errors at the endpoints.
 
        When  the  END  command  is  issued, the post-processor sorts the
ensemble  averages  as  a  function  of  lambda, fits the points to cubic
spline   polynomials   and   integrates   the  resulting  function.   The
propagation  of  error  is determined by numerical differentiation of the
integrals with respect to each ensemble average data point.
 
        Instead  of delta A, delta E and delta S as a function of lambda,
the integrands for the TI integrals are written out for plotting, namely,
<dH/dlamba> and d<H(lambda)>/dlambda.
 
        If we wanted to use the same data as in the first post-processing
example, i.e., linear lambda scaling and no dynamics run at lambda = 0 or
1;  post-processing  would have had to be done in two steps if use of the
TI  method  was  desired.   First,  the  thermodynamic properties for the
interval  between the lowest and highest values of lambda (.125 and .875)
would  be calculated using the TI method.  The endpoint pieces would have
to  be  computed in a separate invocation of the post-processor using the
TP  method  (i.e.  .125 -> 0 and .875 -> 1).  One would then manually add
up  the  pieces.  It is not clear what the point would be to doing such a
calculation.
 



File: PDETAIL ]-[ Node: Optional
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              Using Some of the Optional FES Set Up Commands
 
        We  will  now  discuss  some of the set up options that make life
interesting here in Perturbationland.  There  are  several  commands that
are not  used  much,  PIGGyback,  GLUE  and  NOKE  that are  described in
*Note description: (perturb).   They were  put  in  for  some specialized
purposes and may be useful for weird things.
 
* Menu:

* SLOWG::      Slow Growth Method
* DONT::       The DONT Command
* UmSam::      Umbrella Sampling in TSM



File: PDETAIL ]-[ Node: SLOWG
Up: Optional -=- Previous: Optional -=- Next: DONT\n


                             SLOW GROWTH
 
        An  alternative  method  for  free energy simulations is the slow
growth    technique.     For    a    discussion   of   the   method   see
*Note  Theory and Methodology::.  The  syntax  of the command is given in
*Note  Description: (Perturb).   There  are  some  disadvantages to using
this  technique.   Since  lambda  is changed with every step, there is no
way  to  tack  on  additional trajectories.  In general the paths are not
reversible  (lambda  0  ->  1  vs.  1  -> 0) and the general, not totally
satisfactory,  procedure  has  been  to  average  the two directions.  In
our  implementation  the  the free energy change is calculated during the
dynamics  run  and  so a temperature must be assumed a priori.  Actually,
since  the  SAVE  command  still  works  with this method, an addition to
the  post-processor  could  be  made  to  allow  the  reprocessing of the
data.   Currently  the  only  property  that  is  calculated  is the free
energy  change.   Also currently, the derivative dH/dlambda is determined
by  finite  difference H(lambda')-H(lambda) ( a "perturbation" to lambda'
at each step.  As mentioned in *Note Theory and Methodology::, analytical
differentiation   could   be  easily  implemented.   The  most  egregious
shorcoming is that error analysis is not easily accomplished.
 
        There  are  some  advantages to the method.  The actual input and
management of the problem is generally easier than with the window method
in  that the free energy is determined in one trajectory.  However, since
one  generally  has  to  run  the  transformation  in both directions the
simplification  is  not  all that great.  One has only two thermalization
runs  to  do  (at  each  end point).  It is arguable that there are cases
where  it  is  better  to  use slow growth and that the steady continuous
change  in lambda is a more stable way to achieve the transformation than
the  discrete  jumps of the window method.  When using non-linear scaling
the  window  technique  is  problematical  if  the TP method is used (see
*Note Theory and Methodology::)  and  it is not clear yet whether one can
get  away with using a few lambda points and numberical quadrature in the
TI window method.
 
        I  will  finish  this discussion of the slow growth method with a
few pointers.  One, the final free energy change is written at the end of
the  dynamics  run,  buried  in  other  output.   Two,  if there are both
reactant  and products, set up the thermalization with the LAMBDA command
not  the  SLOWgrowth  and use a lambda value a small bit away from 0 (for
the 0 -> 1 transformation) or 1 (for the 1 -> 0 case).  Then run the slow
growth  production  dynamics.   Three, there are two way in which you can
switch directions.  The easiest way is to issue the SLOWgrowth command as
follows:
 
   SLOW LFROM 1. LTO 0. TEMP 298.
 
However,  this  may  offend  your  conseptual sensibilities since you are
transforming  FROM  the  product  TO the reactant in that case.  The more
difficult alternative is to switch the reactant and product designations.
However,  if  you  do  that  and  have  COLO atoms remember to switch the
PCHArge  and  RCHArge  values.  Note that in this case use of the RCHArge
parameter  is mandatory since the default is to assume that the charge in
the  RTF  (or set by a SCALAR command) is the REACTANT charge.  Do it the
first way.  Fourth, and last, note that the command allows you to specify
non-linear lambda scaling with the POWEr parameter.
 



File: PDETAIL ]-[ Node: DONT
Up: Optional -=- Previous: SLOWG -=- Next: UmSam\n


                            The DONT command
 
        Note that in the ethanol -> propane input file, we used the SHAKE
BOND  ANGLE  command.   This effectively removes those degrees of freedom
from  consideration  in  the perturbation.  We assume that they would not
have significant effect on the solvation thermodynamics.  We chose to use
the  SHAKE  constraints  so  that the calculation would be similar to the
Monte  Carlo  simulations  run  by  Jorgensen (for methanol -> ethane, an
other  part of our study).  It most situations it is not desirable to use
SHAKE  on  bonds other than to hydrogen.  However, if all of the internal
energy  terms  are  scaled  by  lambda,  at  small  lambda  values  large
fluctuations in the energy may result as bond stretching (especially) and
angle  bending terms are weakened for interactions involving reactant and
product  atoms.   Therefore  we  have implemented an alternative means of
dealing   with   this   situation:    the   DONT   command   (see   *Note
Description(PERTURB).)   For  reactant  and product separate commands are
issued  that  specify  which  internal energy terms are to be ignored for
purposes of the lambda scaling:
 
        DONT REAC BOND THETA
 
        DONT PROD BOND THETA
 
The  full  syntax  and explanation of the parameters are described in the
node  mentioned  above.   One  can  specify bond stretching (BOND); angle
bending  (ANGLE or THETA); torsion (DIHEdral or PHI) or improper dihedral
(IMPHI  or  IMPRoper)  terms.   We generally specify only BOND and THETA.
Torsional  motions do play significant role in the free energy changes we
are interested in.
 
        When  a  DONT  command  is  issued  it  means that during the FES
dynamics  run  the  specified  terms  are  calculated and included in the
environment part of the hamiltonian without lambda scaling.  Thus we have
some overcounting of terms.  I.e., carbon atoms can end up with more than
a valence of 4.  However, we have found that this does not greatly change
the overall free energy changes (for a complete cycle).
 
        Note  that  reissuing  a given DONT command (REAC or PROD) clears
all of the flags first.  This is a command that we use alot.
 



File: PDETAIL ]-[ Node: UmSam
Up: Optional -=- Previous: DONT -=- Next: Optional\n


                            Umbrella Sampling
 
        When  we  wish  to  sample torsional minima separated by barriers
that  make  transitions  infrequent (> 1 kT), we can resort to "umbrella"
or  "importance"  sampling.   We  adopt  a  modified potential with lower
barriers  and  then the correct the approximation to the ensemble average
with the "actual" potential energy at the end.
 
        <A> = <A/w> /<1/w>
                   w      w
where,
        w = exp(-beta (V        - V         ))
                        actual     surrogate
 
<A>  on  the  lhs  of  the  equation is the corrected ensemble average of
property  A.   The ensemble averages on the rhs of the equation are those
that  result  from  having  the  surrogate  potential  energy term in the
Hamiltonian.
 
        In   the   current  implementation  the  umbrella  correction  is
available  for  the  ensemble averages used to calculate delta A, delta E
and  delta  S in the free energy simulations.  It is presently limited to
modifying the three-fold torsional term.
 
        It  is  assumed  that  the  surrogate potential is the one in the
parameter  file.   Hence, if you have dihedral angles of the same type as
the  one  that is to be subjected to umbrella sampling and you donot want
them  treated  the  same way the atoms must be given different type names
and the parameter file modified (see *Note description: (perturb).).
 
        Given  below  is  an example command.  Assume that the problem is
the transformation of n-butane into propane in aqueous solution or vacuum
and the hybrid molecule has a segment name of btp (guess what that stands
for).  The umbrella command might look like this:
 
        UMBR btp 1 C1 btp 1 C2 btp 1 C3 btp 1 C4 VACT 1.6
 
The  four  atoms  of  the  butane  dihedral  are specified and the 3-fold
term  for  the  "actual"  potential  is  given.   The "surrogate" term is
present  in  the  parameter  file and might be something like .5.  If the
molecule  had  more  than one dihedral angle around the same central axis
all of them would be specified in separate UMBRella commands.
 
        Invoking  the  UMBRella  command causes the FES output to have an
additional  field,  the  w term in equation above.  The header line after
the  title  in  the  data  file  has  a  field  that  indicates that this
additional field is present.
 
        In  the  post-processing,  the  flag parameter UMBR must be added
to each PROCess command to indicate that the umbrella correction is to be
applied.



File: PDETAIL ]-[ Node: IPrac
Up: Practice -=- Previous: CPrac -=- Next: Practice\n


                INTERNAL COORDINATE PERTURBATION - PRACTICE


	Below we show a CHARMM22 input file which computes thermodynamic 
surfaces as functions of a reaction coordinate connecting ideal right- and 
left-handed alpha helical conformations (aR and aL, respectively) of the 
alanine dipeptide in vacuum.  The conformational transition involves 
changes in both the backbone phi and psi dihedral angles.  The ideal aR and 
aL conformations have phi,psi = -60,-60 and 60,60 degrees, respectively.  
We defined the reaction coordinate to lie on the line phi = psi.  Thus, the 
transition involves 120 degree changes in both the dihedral angles.  We 
have divided the overall transition into ten windows, with 12 degree 
perturbations (6 degrees in both the positive and negative directions) of 
phi and psi in each window.  Furthermore, each window is divided into four 
subintervals, yielding thermodynamic data at 3 degree intervals.

	All ten simulations are performed with one input file using the 
CHARMM looping capability.  The first simulation is performed at phi,psi = 
-54,-54 degrees so that the aR endpoint is reached by perturbation in the 
negative direction.  After each structure is built, it is minimized to 
relieve any bad contacts that may have resulted from the building, while 
keeping the dihedral angles in the vicinity of the desired values with 
harmonic constraints.  Then the dihedral angles are set back to the desired 
values using IC EDIT and the structure is rebuilt using IC BUILD.  Next, 
the i.c. constraints are specified and the peptide is equilibrated.  
Following equilibration, the perturbation is specified and the production 
dynamics is run.  Note that the BY values for the phi and psi perturbations 
have opposite signs.  This is because the atoms to be moved by the 
perturbations (e.g. the atoms in the INTE selection) occur at one end of 
the two central atoms in the phi dihedral angle and the other end in the 
psi dihedral angle.  Hence, the direction of rotation which corresponds to 
an increase in phi corresponds to a decrease in psi (see the discussion of 
the MOVE command at *Note implementation: (pimplem).).  Therefore, since we 
want phi and psi to increase and decrease together, we use BY values with 
opposite signs.  After each perturbation simulation, the constraints and 
perturbations are cancelled using TSM CLEAR in preparation for the next 
simulation.  Finally, after all ten simulations have been carried out, the 
data is post-processed.  In this example, we have requested construction of 
the thermodynamic surfaces (free energy, internal energy, and entropy) at 
300 K using a temperature increment of 10 K in the finite-difference 
derivatives.  After the END command terminates the post-processing, the 
thermodynamic surfaces are printed out, first as functions of phi (since it 
was specified in the first MOVE command), and then as functions of psi.

* Input file for i.c. constraint and perturbation example.
* In this example, we compute the relative thermodynamics
* of the right- and left-handed alpha helical conformations
* of the alanine dipeptide in vacuum.  The phi,psi values
* of the right- and left-handed conformations are -60,-60
* and 60,60 degrees, respectively.  We compute the thermodynamic
* surfaces using 10 windows with perturbations dphi,dpsi = +/- 3,6
* degrees.
*

! Read topology and parameter files
open unit 1 read form name toph19.inp
read rtf card unit 1
close unit 1

open unit 1 read form name param19.inp
read param card unit 1
close unit 1

! Read the sequence and generate the psf
read sequence card
* alanine dipeptide (blocked alanine residue)
*
3
amn ala cbx

generate pept setup

! Set window counter to be used in loop
set 1 1

! Set phi and psi dihedral angles for first window (we'll perturb
! back to right-handed alpha)
set 2 -54.0 ! phi
set 3 -54.0 ! psi

! Do all of the windows in a loop
label LOOP

ic param

! Set phi and psi to the desired values and build
! the dipeptide
ic edit
 dihe 1 c 2 n 2 ca 2 c @2 ! phi
 dihe 2 n 2 ca 2 c 3 n @3 ! psi
end
ic seed 1 cl 1 c 2 n
ic build

! Minimize with harmonic dihedral constraints to relieve
! any bad contacts that might result from building
cons dihe pept 1 c pept 2 n pept 2 ca pept 2 c min @2 force 100.0
cons dihe pept 2 n pept 2 ca pept 2 c pept 3 n min @3 force 100.0
update atom cdie shif vdis vswi cutnb 10.0
minimization sd nstep 50 tolgrd 0.01 inbfrq 50
cons cldh

! Reset phi and psi to the desired values before setting the
! i.c. constraints
ic fill
ic print
ic edit
 dihe 1 c 2 n 2 ca 2 c @2 ! phi
end
coor init sele ((atom pept 2 h) .or. (segi pept .and. resi 1)) end
ic build
ic edit
 dihe 2 n 2 ca 2 c 3 n @3 ! psi
end
coor init sele ((atom pept 2 o) .or. (segi pept .and. resi 3)) end
ic build

! Set the i.c. constraints; maximum number of iterations = 100;
! tolerances = 1.0e-5 degrees
tsm
 maxi 100
 fix dihe pept 1 c pept 2 n pept 2 ca pept 2 c toli 1.0e-5 ! phi
 fix dihe pept 2 n pept 2 ca pept 2 c pept 3 n toli 1.0e-5 ! psi
end

! Run 10 steps equilibration dynamics
dynamics verlet strt nstep 10 timestep 0.0005 firstt 300.0 -
 inbfrq 10 nprint 1 iprfrq 10 iasors 1 ichecw 1 iasvel 1 -
 atom cdie shif vdis vswi cutnb 10.0

! Open perturbation data file
open unit 10 write form name example_@1.icp

! Set up perturbations; save data on unit 10 every step; use
! 2 subintervals in each window (nwin 2), e.g., change dihedrals by
! +/- 3,6 degrees; note "by" values have opposite signs so the moves
! both go in the same direction
tsm
 savi icun 10 icfr 1 nwin 2
 move dihe pept 1 c pept 2 n pept 2 ca pept 2 c by  6.0 -
  inte sele ((atom pept 2 h) .or. (atom pept 2 n) .or.  -
            (segi pept .and. resi 1)) end ! phi
 move dihe pept 2 n pept 2 ca pept 2 c pept 3 n by -6.0 -
  inte sele ((atom pept 2 o) .or. (atom pept 2 c) .or.  -
            (segi pept .and. resi 3)) end ! psi
end

! Run 20 steps perturbation dynamics
dynamics verlet strt nstep 20 timestep 0.0005 firstt 300.0 -
 inbfrq 20 nprint 1 iprfrq 20 iasors 1 ichecw 1 iasvel 1 -
 atom cdie shif vdis vswi cutnb 10.0

! Close data file and clear constraints and perturbations
close unit 10
tsm clear

! Get ready for next window
incr 1 by 1
incr 2 by 12.0
incr 3 by 12.0
coor init sele all end
if 1 le 10 goto LOOP

! At this point, all ten simulations have been carried out

! Open the data files for post-processing
open unit 10 read form name example_1.icp
open unit 11 read form name example_2.icp
open unit 12 read form name example_3.icp
open unit 13 read form name example_4.icp
open unit 14 read form name example_5.icp
open unit 15 read form name example_6.icp
open unit 16 read form name example_7.icp
open unit 17 read form name example_8.icp
open unit 18 read form name example_9.icp
open unit 19 read form name example_10.icp

! Post-process the data; use binsize of 5 datasets
! in error calculation; calculate avg. temp.; use
! T = 300 K and deltaT = 10 K in thermodynamics;
! construct thermodynamic surfaces
tsm post ic maxp 2 maxw 2 surf maxs 100
 proc first 10 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 11 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 12 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 13 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 14 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 15 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 16 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 17 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 18 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
 proc first 19 nunits 1 bins 5 ctem temp 300.0 delt 10.0 -
  begin 1 stop 20
end

stop


Lambda-Dynamics approach to free energy calculations:


	In this node, we will focus on aspects of the free energy 
calculations which are not covered in TSM discussion.  However, it is
recommended to go through details of TSM approach first since the new
dynamics is based on same underlying principles as TSM does and we
will use those basic concepts and terminologies without further 
explanation.

 
(i) Potential of Mean Force

	In addition to TI and thermodynamic perturbation method, an 
alternative approach to the free energy calculations is to develop 
the free energy A along a "reaction coordinate", zeta (to avoid
confusion, we will use "zeta" instead of "xi" used in the reference [1]). 
Zeta is typically a configurational coordinate. Thus, the reactant 
configuration and the product configuration are represented by
zeta = 0 and zeta = 1 respectively. For a continuous coordinate zeta, 
the Helmholtz free energy difference, or the reversible work required to 
carry the system from the reactant configuration to the product 
configuration, is often referred to as the "potential of mean force" (PMF), 
W(zeta). The exact connection between W(zeta) and the probability density 
of the system rho(zeta) is expressed as 

              W(zeta) = -KT ln[rho(zeta)]

That leads to the free energy difference  DeltaA(zeta) = W(zeta), 
when rho(zeta) is normalized at a particular value of zeta, e.g., 
rho(zeta=0) = 1.

	It is well known that conventional importance MD or MC sampling 
techniques will be prone to produce an inadequate estimation of W(zeta) 
in situations where the potential of mean force differs by more than 
a few kcal/mol over the range of  0 <= zeta <= 1.0 .  To expand the range 
of DeltaA(zeta), the umbrella sampling method is frequently employed. Here 
the basic idea is to introduce a biasing or umbrella potential U*(zeta), 
thereby replacing the original potential energy  V(x,y,z) by a modified 
potential V(x,y,z) + U*(zeta).  The ultimate goal is to use the auxiliary 
potential U*(zeta) to "flatten" out the energy barriers along the coordinate
zeta.  In so doing, a more uniformly distributed density function rho(zeta) 
can be generated with a fixed amount of sampling because transitions 
between the reactant and product configuration are now more facile.

	From statistical mechanics, the explicit connection between the 
biased probability density function rho*(zeta) and the true density 
rho(zeta) of the system can be easily made. It is simply stated as  
                             
                          rho*(zeta)exp{U*(zeta)/kT}
              rho(zeta) = --------------------------
                          <exp{U*(zeta)/kT}>*

where the notation " * " emphasizes that the ensemble average is being 
taken over the modified potential function.

	Owing to the complexity of many problems of interest, it is 
apparent that a single biasing potential is usually not sufficient enough 
to cover the whole range of the reaction coordinate zeta, and 
simultaneously produce good sampling.  Thus, a set of restraining 
potentials, U*(zeta), is used to shift the local minima in the desired 
direction.  In this "window" approach, the potential of mean force,     
W(zeta), in each window takes the form of

         W (zeta) = -kT ln[rho*(zeta)] - U*(zeta) + C
          i                   i           i          i

         C = kTln <exp[U*(zeta)]>*
          i             i

The direct evaluation of the constants C's , being of exponential form, 
is susceptible to large numerical fluctuations. These fluctuations in turn 
make the accurate calculation of these constants very difficult. In order 
to achieve an uniformly good potential of mean force, W(zeta), the different 
constants, C's , from successive simulation windows have to be perfectly 
matched so as to make W(zeta) agree in the overlapping regions.  The 
implementation of many conventional matching schemes such as the "splicing" 
method is very difficult, if not impossible, when generating two or higher 
dimensional potential of mean force surfaces.

(ii) Lambda Dynamics Methodology I

Thus far, two auxiliary quantities have been introduced to facilitate 
free energy calculations: a coupling parameter lambda (extent of 
transformation) in TSM; a "reaction coordinate" zeta in the PMF.  On 
the one hand, the coupling parameter approach takes advantage of the 
fact that the free energy difference, DeltaA, is a function of the 
state only. Thus, DeltaA can be computed along any reversible path 
connecting the initial and final states.  This added flexibility 
certainly hold  great potential for designing new approaches to free 
energy calculations.  On the other hand, the power of specific biasing 
potentials used in the umbrella sampling method can be fully utilized 
to combat the difficulties encountered in conventional free energy 
calculations.  To make full use of the coupling parameter lambda, two 
crucial questions, among others, pertinent to the lambda will be focused 
on: (1) What is the best way to deal with the lambda variable in 
developing a new method? In other words, rather than being separated from 
the configurational coordinate set {x,y,z}, is it more efficient to treat 
the lambda variable on the same footing as an ordinary space coordinate? 
(2) If so, what  governs the dynamics of the lambda variable?  Based on 
the assumption that the lambda variable can be treated as an ordinary 
coordinate, a natural connection to the umbrella sampling method may be made 
immediately. Identifying the lambda variable as the "reaction coordinate" 
zeta, the problem of calculating chemical free energy differences is 
isomorphic with that of computing potentials of mean force in the lambda 
variable.  This transformation brings up a point we have discussed earlier,
namely, the matching problem of the constant C's associated with the 
umbrella sampling has to be attended to. Therefore, another important 
question: (3) What is the optimal way of processing the simulation data?  
This question is of great importance for the generation of a multidimensional 
free energy surface.
 
	As noted before, any two equilibrium states can be generally 
connected by reversible transition pathways.  A typical approach is to
partition the system Hamiltonian as a linear function of a coupling 
parameter lambda

     H(lambda) = (1 - lambda) H  + lambda H  + H 
                               R           P    Env

Instead of using a single variable, consider a general case where the 
pathway itself is characterized by a set of coupling variables, 
{lambda(i)}, i = 1,...,n.  This represents the situation where 
"alchemical" mapping of one molecule into another, for the purpose 
of computing a free energy difference, changes different components of 
the interaction terms with different scaling factors. We can control the 
transition between the two molecules by partitioning a Hamiltonian of 
the general form

H   ({lambda(i); i=1,n}) = H ({lambda(i); i=1,n}) + H ({lambda(i); i=1,n})
 R*n                        R                        P
                          
                         + H
                            Env

where H    stands for the Hamiltonian of the atoms not being transformed
       Env        
while H    ({lambda(i); i=1,n}) is the reactant (product) Hamiltonian.  
       R(P)
H    ({lambda(i); i=1,n}) is a legitimate mapping provided that
 R*n

H    ({lambda(i) = 0; i=1,n}) and H    ({lambda(i) = 1; i=1,n})
 R*n                               R*n

correspond to the Hamiltonians for the reactant and product states 
respectively.  Because the two states, R  and P, are separated by many 
intermediate states with {0} < {lambda(i)} < {1}, and there exist generally 
many conceivable potential pathways (and barriers) on the potential 
energy surface, a natural question arises: what is the optimal transition 
pathway? In the language of the perturbation calculations, how does one 
select values of {lambda(i)}, i = 1,...,n, so that efficient ensemble 
averaging can be performed? 

	We propose a new mapping scheme for performing free energy 
calculations.  We permit the control variables {lambda(i)} for the chemical 
transformation to evolve as dynamic variables, under the influence of both 
the hybrid Hamiltonian above, and added biasing potentials. The dynamic 
treatment of the {lambda(i)} variables is directed at the first two 
questions raised earlier.  Since the biasing potentials are at our disposal, 
the new method will allow us more control over the sampling space, thereby 
enhancing our control over the convergence of ensemble sampling.

	We formulate the lambda-dynamics by utilizing an extended 
Hamiltonian.  In this formalism, the {lambda(i)} are considered as a set 
of fictitious "particle" coordinates, 

        H        ({lambda(i)}) = H   ({lambda(i)})  
         Extended                 R*n

                                 n
                                ____        __          __
                            +   \     M(i)  | dlambda(i) |**2
                                /    ------ |---------   |
                                ----   2.0  | dt         |
                                i = 1       |__        __|

                            +   U*({lambda(i)})

where the lambda-dependent potential term, U*({lambda(i)}), will serve as 
an umbrella or a biasing potential to limit the range of {lambda(i)}. 
Moreover, a kinetic energy term is associated with a set 
of fictitious masses {M(i)}.  The coupling between spatial coordinates 
and energy parameters is through the lambda dependence of H   ({lambda(i)}). 
    	                                                   R*n 

	The introduction of the umbrella potential term U*({lambda(i)})       
in H         ({lambda(i)}) provides a great deal of freedom in improving 
    Extended

efficiency in sampling.  By using a well designed U*({lambda(i)}), we can 
achieve the following objectives: (1) limit the range of {lambda(i)} 
sampled in independent simulations or windows to some particular set of
 values; (2) supply the boundary conditions for the overall range of 
{0} <= {lambda(i)} <= {1}; and more importantly, (3) increase transition 
rates among potential wells separated by high energy barriers. By utilizing 
the second set of adjustable parameters {M(i)}, we should be able to gain 
more control over the evolution of the {lambda(i)} variables.
 	
	Generally speaking, the new method will generate a multidimensional 
probability density distribution. This entails a multidimensional potential 
of mean force surface in the {lambda(i)} variable space. This leads us to 
another important aspect of our approach, namely to apply a more powerful 
analysis technique to the resulting distributions.  This is directed 
towards the third question raised above.  Optimal histogram analysis 
method, developed by Ferrenberg and Swendsen for studying phase transitions, 
and further extended to biological simulations by Kumar and the Swendsen 
group, and by Boczko and Brooks, has proven to be very successful in 
generating free energy surfaces for physical, chemical and biological 
processes.  In addition to producing the best possible estimation of free 
energies and optimizing links (constant C's) between simulations, the 
weighted histogram analysis method (WHAM) is a multidimensional method and 
has a built-in estimate of sampling errors.  These characteristics serve 
well the needs of our new lambda-dynamics.


	Our lambda dynamics approach also offers better control over a 
couple of technical problems in FEP.  The first is related to the 
{lambda(i)} --> {0} or {lambda(i)} --> {1} limiting situation, often 
referred to as the "close encounters" problem in the literature. In MC 
simulations, this generally does not cause any real problem since random 
numbers are used to sample the configurational variable and stability 
is not an issue. However, in the case of MD simulations, large numerical 
errors, often leading to instability of the numerical solution to 
the dynamics equations, take place when either {lambda(i)} --> {0} or 
{lambda(i)} --> {1} occur.  Conventional perturbation calculations often 
avoid the problem by running simulations at a small distance away from {0} or 
{1}.  Nonetheless, this approach is not appropriate for the formalism of 
lambda-dynamics because the lambda variable itself evolves dynamically 
during the course of the simulation. The umbrella potential of our 
formulation offers a convenient solution to this problem.  Simple biasing 
potentials are generally sufficient to prevent the boundary crossing 
{lambda(i)} < {0} or {lambda(i)} > {1}.  To further restrain the range of 
lambda variables, we have implemented a different partition scheme of the 
system Hamiltonian in CHARMM, i.e. a linear lambda variable is replaced by
a quadratic form lambda**2, as will be seen shortly.  This is an extremely 
helpful technique at the {lambda(i)} --> {0} end.   The second issue is 
that the extra degrees of freedom associated with the fictitious masses 
provide additional control over the dynamics of the lambda variables.  


(A) Application for relative free energy calculations:
 
         To avoid redundancy, we will only outline the partitioning schemes
of the system Hamiltonian for each application.  Please refer to BLOCK.DOC
and DYNAMC.DOC for detailed CHARMM commands when performing lambda dynamics
simulation runs.

	The lambda-dynamics formalism is applicable to typical free energy
calculation problems, e.g, a mutation of one molecule into another.  This 
is a prototype for the majority of free energy calculations carried out to 
study relative solubility, relative binding affinity or protein stability. 
We will outline our implementation using the same example as TSM does.

        To calculate the relative solvation free energies of methanol and 
ethane, we write the extended system Hamiltonian as   
 

     H        ({lambda(i)}) = [lambda(2)**2] H  + [lambda(3)**2] H  + H 
      Extended                                R                   P    Env

                                  3
                                ____        __          __
                            +   \     M(i)  | dlambda(i) |**2
                                /    ------ |---------   |
                                ----   2.0  | dt         |
                                i = 2       |__        __|

                            +   U*({lambda(i)})

subject to the condition

        3
       ____  
       \
       /     [lambda(i)**2] = 1.0            
       ----
       i = 2

    
This system has three distinguished blocks: a reactant (OH group of the 
methanol), a product (CH3 of the ethane) and environment (unchanged CH3 
group and solvent).  By default, block one is assigned to the environment,
which leads to lambda(1) = 1.0 in our implementation.  Please refer to 
*note block:(block.doc) for detailed CHARMM commands.

(iii) Lambda Dynamics Methodology II

By using a different partitioning scheme in H        ({lambda(i)}) 
                                    	     Extended
                      N
                     ____  
                    \
V(x,{lambda(i)}) =  /     [lambda(i)**2] (V (x)-F ) +  V
                    ----                   i     i      Env
                    i = 2

one can perform free energy calculations for multiple ligands simultaneously. 
Here one lambda(i) is assigned to each potential, V (x), which represents a
 				                   i
distinct solute molecule (or ligand), F  is the biasing potential for ligand i.
				       i
If the {lambda(i)} variables are subject to a holonomic condition

       N 
       ____  
       \
       /     [lambda(i)**2] = 1.0            
       ----
       i = 2

the probability distributions of the system at given lambda(i) values are a 
manifestation of the free energy differences between different solute 
molecules. For example, 


       P[lambda(i) = 1, {lambda(m =/= i} = 0]           D_Delta A
       ______________________________________ = Exp{-  --------}
                                                          kT
       P[lambda(j) = 1, {lambda(m =/= j} = 0]

where D_Delta A is the total free energy difference between molecules i and j.
Since the free energy difference between molecules for half of the
thermodynamic cycle (F , e.g. ligands in the unbound state) is incorporated 
                      i
into the Hamiltonian of the system for the other half
of the cycle (e.g. protein-liand bound state), D_Delta is the relative binding 
affinity of the ligands to the protein receptor.

With this formalism of Hamiltonian l-dynamics can evaluate relative binding
free energy of multiple ligands simultaneously, providing either qualitative
ranking within short simulation time or quatitative results with desired 
precision. In the former case (or the screening calculation), 
the ligands with favorable binding affinity
are identified and relative ordering of those ligands can be obtained from
the probability distribution of the ligands in the [lambda(i)**2]=1 state. 
The running averages of each [lambda(i)**2] can also be used to provide the
ranking. In the latter case, the free energy difference between molecules  
for half of the thermodynamic cycle will be evaluated. Here F acts as a
						             i		 
potential. It corresponds to the estimated free energy of the ligand from 
previous calculations. Therefore the estimated free energy can be improved
iteratively. The WHAM method can be used to combine data from different
simulations to get optimal estimate of the free energy.

There exists an analogy between our screening calculations and competitive
binding experiments carried out in the laboratory.  In fact, a competitive
binding experiment usually consists of different ligands and a single receptor
in solution. By determining the relative concentrations of constituent ligands
in solution, the relative binding affinity of ligands can be inferred.
Thus, a "best" ligand can be selected accordingly. Similarly, by determining
the running averages or the relative populations of each lambda(i), we can
distinguish favorable ligand molecules from unfavorable ones.
 
In hybrid MC/MD method, Metropolis Monte Carlo method is used to evolve the 
lambda-space and molecular dynamics method is used for evolving the atomic
coordinates for generating the canonical ensemble of the ligands instead of
using molecular dynamics method with extended Hamiltonian in the
lambda-dynamics method. Both methods gave the same configurational partition
functions. The relative free energy can be calculated from the probability
distrubitions in MC/MD method. The MC/MD method was originally presented by
Bennett and Tidor, independently. The straightforward extension of this
method for multiple ligands, which was named CMC/MD, was achieved by
Pitera & Kollman.
One advantage of MC/MD is that sampling of {lambda} can be determined by the
user, while the proper choice of the biased potential is required in the
lambda-dynamics method. 
For example, it is possible for MC/MD to sample only chemically important end
points. However, the acceptance ratio may be too small when the ligands have
the large and different parts. 
The disadvantage of MC/MD is that the sudden change in chemical space evolved
by MC step produces a discontinuous effect in the atomic momentum space so
that the atoms of the newly selected ligand do not have velocities appropriate
for the environmental conditions.
The re-assignment of the velocities to the ligands is not carried out in this
version of CHARMM.

	The holonomic constraint used above, which is necessary to ensure 
the physical nature of the end points, is treated using the Lagrange 
multiplier method.  Moreover, a renormalization of the {lambda(i)} variables 
at each MD step is performed to guarantee that the condition is satisfied. 
The reason is that the equation is always solved approximately in any 
simulation algorithm, and the renormalization prevents small errors  from 
propagating.
 

REFERENCES

X. Kong and C. L. Brooks III, J. Chem. Phys., 105, 2414 (1996).

C. L. Brooks III, M. Karplus, and B. M. Pettitt, In Advances in Chemical 
Physics,  Vol. LXXI, edited by I. Prigogine and S. A. Rice (Wiley, New York, 
1988).

Z. Liu and B. J. Berne, J. Chem. Phys., 99, 6071 (1993).

A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett., 63, 1195 (1989).

A. M. Ferrenberg, Ph.D Thesis, Carnegie Mellon University, 1989.

S. Kumar, D. Bouzida, R. H. Swendsen, P. A. Kollman, and J. Rosenberg, J. 
Comput. Chem., 13, 1011 (1992).

E. M. Boczko and C. L. Brooks III, J. Phys. Chem., 97, 4509 (1993).

B. Tidor, J. Phys. Chem., 97, 1069 (1993).

B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, 
and M. Karplus, J. Comput. Chem., 4, 187 (1983).

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