This feature has been contributed to NAMD by the following authors:
Surjit B. Dixit, Jérôme Hénin and Christophe Chipot
Equipe de dynamique des assemblages membranaires,
UMR CNRS/UHP 7565,
Université Henri Poincaré,
BP 239,
54506 Vanduvre-lès-Nancy cedex, France
© 2001-2006, CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
A method to perform alchemical free energy perturbation (FEP) [32,4,31,29,19,14,21,9,10] is available in NAMD. Within the
FEP framework, the free energy difference between two alternate
states, and
, is expressed by:
Here,
, where
is the Boltzmann constant,
is the temperature.
and
are the Hamiltonians characteristic of states
and
, respectively.
denotes an ensemble average over configurations
representative of the initial, reference state,
.
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Convergence of equation (8) implies that low-energy
configurations of the target state, , are also configurations
of the reference state,
, thus resulting in an appropriate overlap
of the corresponding ensembles -- see Figure 5.
In practice, transformation between the two thermodynamic states
is replaced by a series of transformations between non-physical,
intermediate states along a well-delineated
pathway that connects
to
.
This pathway is characterized by a general extent parameter, often
referred to as ``coupling parameter'' [4,21,17,18],
, that makes the Hamiltonian and, hence, the free energy,
a continuous function of this parameter between
and
:
Here, stands for the number of intermediate stages, or ``windows''
between the initial and the final states -- see Figure 5.
In a typical FEP setup involving the transformation of one chemical
species into an alternate one in the course of the simulation, the atoms
in the molecular topology can be classified into three groups, (i) a
group of atoms that do not change during the simulation -- e.g.the
environment, (ii) the atoms describing the reference state, , of the
system, and (iii) the atoms that correspond to the target state,
, at the end of the alchemical transformation. The atoms
representative of state
should never interact with those of state
throughout the MD simulation. Such a setup,
in which atoms of both the initial and the final states of the
system are present in the molecular topology file -- i.e.the psf file -- is characteristic of the so-called ``dual topology''
paradigm [13,23,2]. The hybrid Hamiltonian of
the system, which is a function of the general extent parameter,
,
that connects smoothly state
to state
, is calculated as a linear
combination of the corresponding Hamiltonians:
where
describes the interaction of the group of
atoms representative of the reference state,
, with the rest of the system.
characterizes the interaction of the target topology,
, with the rest of the system.
is the Hamiltonian describing those atoms that do not
undergo any transformation during the MD simulation.
For instance, in the point mutation of an
alanine side chain into that of glycine, by means of an FEP
calculation, the topology of
both the methyl group of alanine and the hydrogen borne by the
C in glycine co-exist throughout the simulation (see
Figure 6), yet without actually seeing each
other.
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The energy and forces are defined as a function of , in
such a fashion that the interaction of the methyl group of alanine
with the rest of the protein is effective at the beginning of the
simulation, i.e.
= 0, while the glycine C
hydrogen
atom does not interact with the rest of the protein, and vice versa
at the end of the simulation, i.e.
= 1. For intermediate
values of
, both the alanine and the glycine side chains
participate in non-bonded interactions with the rest of the
protein, scaled on the basis of the current value of
. It
should be clearly understood that these side chains never
interact with each other. Construction of an appropriate
list of excluded atoms, common to the two alternate topologies,
is, therefore, necessary.
It is also worth noting that the free energy calculation does not alter intramolecular potentials, e.g.bond stretch, valence angle deformation and torsions, in the course of the simulation. In calculations targeted at the estimation of free energy differences between two states characterized by distinct environments -- e.g.a ligand, bound to a protein in the first simulation, and solvated in water, in the second -- as is the case for most free energy calculations that make use of a thermodynamic cycle, perturbation of intramolecular terms may, by and large, be safely avoided [5].
The procedure implemented in NAMD is particularly
adapted for performing free
energy calculations that split the
reaction path into a number of non-physical,
intermediate states, or ``windows''. Separate simulations
can be started for each window.
Alternatively, the TCL scripting ability of
NAMD can be employed advantageously
to perform the complete simulation in a single run.
An example making use of such script is supplied at the end
of this user guide.
The following keywords can be used to control the alchemical free energy calculations.
The first example illustrates the use of TCL scripting for running
an alchemical transformation with the FEP feature of NAMD. In this
calculation, is changed continuously from 0 to 1
by increments of
= 0.1.
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The user should be reminded that by setting run 10000, 10,000 MD steps will be performed, which includes the preliminary fepEquilSteps equilibration steps. This means that here, the ensemble average of equation (9) will be computed over 5,000 MD steps.
Alternatively, -states may be declared
explicitly, avoiding the use of TCL scripting:
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This option is generally preferred to set up windows of diminishing
widths as
0 or 1 -- a way to circumvent
end-point singularities caused by appearing atoms that may
clash with their surroundings. It may be used in conjunction
with a soft-core potential (see relevant section).
The fepOutFile contains electrostatic and van der Waals energy data calculated for lambda and lambda, written every fepOutFreq steps. The column dE is the energy difference of the single configuration, dE_avg and dG are the instantaneous ensemble average of the energy and the calculated free energy at the time step specified in column 2, respectively. The temperature is specified in the penultimate column. Upon completion of fepEquilSteps steps, the calculation of dE_avg and dG is restarted. The accumulated net free energy change is written at each lambda value and at the end of the simulation. The cumulative average energy dE_avg value may be summed using the trapezoidal rule to obtain an approximate thermodynamic integration (TI) estimate for the free energy change during the run.
Whereas the FEP module of NAMD supplies free energy differences
determined from equation (8), the wealth of information
available in fepOutFile may be utilized profitably to
explore different routes towards the estimation of . As
commented on previously, TI may constitute one such route. The
simple overlap sampling (SOS) represents an interesting alternative,
that combines advantageously forward and reverse
transformations to improve convergence and accuracy of the
calculation [20]. The linear scaling of the Hamiltonian
highlighted in equation (10) obviates the need for
explicit simulation of the reverse transformation, because:
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(11) |
The free energy difference between states and
may then be expressed as: