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Subsections


Alchemical Free Energy Perturbation Calculations

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 Vand\oeuvre-lès-Nancy cedex, France

© 2001-2006, CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

Introduction

Theoretical background

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, $a$ and $b$, is expressed by:


\begin{displaymath}
\Delta A_{a \rightarrow b} = -\frac{1}{\beta} \ \ln
\left\l...
...l H}_a({\bf x}, {\bf p}_x)
\right]
\right\}
\right\rangle_a
\end{displaymath} (8)

Here, $\beta^{-1} \equiv k_B T$, where $k_B$ is the Boltzmann constant, $T$ is the temperature. ${\cal H}_a({\bf x}, {\bf p}_x)$ and ${\cal H}_b({\bf x}, {\bf p}_x)$ are the Hamiltonians characteristic of states $a$ and $b$, respectively. $\left\langle \cdots \right\rangle_a$ denotes an ensemble average over configurations representative of the initial, reference state, $a$.

Figure 5: Convergence of an FEP calculation. If the ensembles representative of states $a$ and $b$ are too disparate, equation (8) will not converge (a). If, in sharp contrast, the configurations of state $b$ form a subset of the ensemble of configurations characteristic of state $a$, the simulation is expected to converge (b). The difficulties reflected in case (a) may be alleviated by the introduction of mutually overlapping intermediate states that connect $a$ to $b$ (c). It should be mentioned that in practice, the kinetic contribution, ${\cal T}({\bf p}_x)$, is assumed to be identical for state $a$ and state $b$.


\includegraphics[width=4cm]{figures/overlap1} (a) \includegraphics[width=4cm]{figures/overlap2} (b) \includegraphics[width=4cm]{figures/overlap3} (c)

Convergence of equation (8) implies that low-energy configurations of the target state, $b$, are also configurations of the reference state, $a$, 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 $a$ to $b$. This pathway is characterized by a general extent parameter, often referred to as ``coupling parameter'' [4,21,17,18], $\lambda $, that makes the Hamiltonian and, hence, the free energy, a continuous function of this parameter between $a$ and $b$:


\begin{displaymath}
\Delta A_{a \rightarrow b} = -\frac{1}{\beta} \ \sum_{i = 1}...
... x}, {\bf p}_x; \lambda_i)
\right]
\right\}
\right\rangle_i
\end{displaymath} (9)

Here, $N$ stands for the number of intermediate stages, or ``windows'' between the initial and the final states -- see Figure 5.

The dual-topology paradigm

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, $a$, of the system, and (iii) the atoms that correspond to the target state, $b$, at the end of the alchemical transformation. The atoms representative of state $a$ should never interact with those of state $b$ 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, $\lambda $, that connects smoothly state $a$ to state $b$, is calculated as a linear combination of the corresponding Hamiltonians:


\begin{displaymath}
{\cal H}({\bf x}, {\bf p}_x; \lambda)
= {\cal H}_0({\bf x},...
...f x}, {\bf p}_x)
+ (1-\lambda) {\cal H}_a({\bf x}, {\bf p}_x)
\end{displaymath} (10)

where ${\cal H}_a({\bf x}, {\bf p}_x)$ describes the interaction of the group of atoms representative of the reference state, $a$, with the rest of the system. ${\cal H}_b({\bf x}, {\bf p}_x)$ characterizes the interaction of the target topology, $b$, with the rest of the system. ${\cal H}_0({\bf x}, {\bf p}_x)$ 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$_\alpha$ in glycine co-exist throughout the simulation (see Figure 6), yet without actually seeing each other.

Figure 6: Dual topology description for an alchemical simulation. Case example of the mutation of alanine into serine. The lighter color denotes the non-interacting, alternate state.
\includegraphics[width=12.5cm]{figures/dual_top}

The energy and forces are defined as a function of $\lambda $, in such a fashion that the interaction of the methyl group of alanine with the rest of the protein is effective at the beginning of the simulation, i.e.$\lambda $ = 0, while the glycine C$_\alpha$ hydrogen atom does not interact with the rest of the protein, and vice versa at the end of the simulation, i.e.$\lambda $ = 1. For intermediate values of $\lambda $, 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 $\lambda $. 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].

Implementation of free energy perturbation in NAMD

The procedure implemented in NAMD is particularly adapted for performing free energy calculations that split the $\lambda $ 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.

Examples of input files for running FEP alchemical calculations

The first example illustrates the use of TCL scripting for running an alchemical transformation with the FEP feature of NAMD. In this calculation, $\lambda $ is changed continuously from 0 to 1 by increments of $\delta \lambda $ = 0.1.

fep             on
fepfile         ion.fep
fepCol          X
fepOutfile      ion.fepout
fepOutFreq      5
fepEquilSteps   5000

set step        0.0
set dstep       0.1

while {$step <= 1.0} {
  lambda $step
  set step [expr $step + $dstep]
  lambda2 $step
  run  10000
}
Turn FEP functionality on.
File containing the information about growing/shrinking atoms described in column X.
Output file containing the free energy.
Frequency at which fepOutFreq is updated.
Number of equilibration steps per $\lambda $-state.
Starting value of $\lambda $.
Increment of $\lambda $, i.e. $\delta \lambda $.
TCL script to increment $\lambda $:
(1) set lambda value;
(2) increment $\lambda $;
(3) set lambda2 value;
(4) run 10,000 MD steps.

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, $\lambda $-states may be declared explicitly, avoiding the use of TCL scripting:

lambda          0.0
lambda2         0.1
run             10000
(1) set lambda value;
(2) set lambda2 value;
(3) run 10,000 MD steps.

This option is generally preferred to set up windows of diminishing widths as $\lambda \rightarrow$ 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).

Description of FEP simulation output

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 $\Delta A$. 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:


\begin{displaymath}
{\cal H}({\bf x}, {\bf p}_x; \lambda_i) - {\cal H}({\bf x}, ...
...a_{i+2}) - {\cal H}({\bf x}, {\bf p}_x; \lambda_{i+1})
\right]
\end{displaymath} (11)

The free energy difference between states $\lambda_i$ and $\lambda_{i+1}$ may then be expressed as:


$\displaystyle \exp(-\beta \Delta A_{i \rightarrow i+1})$ $\textstyle =$ $\displaystyle \frac{\displaystyle
\left\langle \exp\left\{-\frac{\beta}{2}
\lef...
...cal H}({\bf x}, {\bf p}_x; \lambda_{i+1})
\right]
\right\}
\right\rangle_{i+1}}$ (12)
  $\textstyle =$ $\displaystyle \frac{\displaystyle
\left\langle \exp\left\{-\frac{\beta}{2}
\lef...
...cal H}({\bf x}, {\bf p}_x; \lambda_{i+1})
\right]
\right\}
\right\rangle_{i+1}}$  


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Next: Locally Enhanced Sampling Up: Additional Simulation Parameters Previous: Adaptive Biasing Force Calculations   Contents   Index
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