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How to cite ABF
The references for the ABF
method are Darve and
Pohorille, J. Chem.
Phys. 2001 and Darve, RodriguezGomez and
Pohorille, J. Chem.
Phys. 2008.
The reference for the new NAMD formulation
and
implementation is Hénin, Fiorin, Chipot,
and Klein, J. Chem.
Theory Comput. 2010.
References
Calculating
free energies using average force
Eric Darve and Andrew Pohorille
A
new, general formula that connects the derivatives of the free energy
along the selected, generalized coordinates of the system with the
instantaneous force acting on these coordinates is derived. The
instantaneous force is defined as the force acting on the coordinate of
interest so that when it is subtracted from the equations of motion the
acceleration along this coordinate is zero. The formula applies to
simulations in which the selected coordinates are either unconstrained
or constrained to fixed values. It is shown that in the latter case the
formula reduces to the expression previously derived by den Otter and
Briels [Mol. Phys. 98, 773 (2000)]. If simulations are carried out
without constraining the coordinates of interest, the formula leads to
a new method for calculating the free energy changes along these
coordinates. This method is tested in two examples — rotation
around the C–C bond of 1,2dichloroethane immersed in water
and transfer of fluoromethane across the waterhexane interface. The
calculated free energies are compared with those obtained by two
commonly used methods. One of them relies on determining the
probability density function of finding the system at different values
of the selected coordinate and the other requires calculating the
average force at discrete locations along this coordinate in a series
of constrained simulations. The free energies calculated by these three
methods are in excellent agreement. The relative advantages of each
method are discussed.
J. Chem. Phys. 2001, 115, 91699183.




Calculating
free energies using a scaledforce molecular dynamics algorithm
Eric
Darve, Michael A. Wilson and Andrew Pohorille
We
propose and test a family of methods to
calculate the free energy along a generalized coordinate, ξ,
based
on computing the force acting on this coordinate. First, we derive a
formula that connects the free energy in unconstrained simulations with
the force of constraint that can be readily calculated numerically.
Then, we consider two methods, which improve the efficiency of the free
energy calculation by yielding uniform or nearly uniform sampling
of ξ. Both rely on modifying the force acting on ξ.
In one method,
this force is replaced by a force with zero mean and ξ is
advanced
quasistatically. In the second method, the force is augmented
adaptively by a biasing force. We provide formulas for calculating the
free energy of the unmodified system from the forces acting in these
modified, nonHamiltonian systems. Using conformational transitions in
1,2dichloroethane as a test case, we show that both methods perform
very well.
Mol.
Sim. 2001, 28, 113144.






Overcoming
free energy barriers using unconstrained molecular dynamics simulations
Jérôme
Hénin and Christophe Chipot
Association
of
unconstrained molecular dynamics (MD) and the formalisms
of thermodynamic integration and average force [Darve and Pohorille, J.
Chem. Phys. 115, 9169 (2001)] have been employed to determine
potentials of mean force. When implemented in a general MD code, the
additional computational effort, compared to other standard,
unconstrained simulations, is marginal. The force acting along a chosen
reaction coordinate ξ is estimated from the individual forces exerted
on the chemical system and accumulated as the simulation progresses.
The estimated free energy derivative computed for small intervals of ξ
is canceled by an adaptive bias to overcome the barriers of the free
energy landscape. Evolution of the system along the reaction coordinate
is, thus, limited by its sole selfdiffusion properties. The
illustrative examples of the reversible unfolding of decaLalanine,
the association of acetate and guanidinium ions in water, the
dimerization of methane in water, and its transfer across the water
liquidvapor interface are examined to probe the efficiency of the
method.
J. Chem. Phys. 2004, 121,
29042914






Exploring
the freeenergy landscape of a short peptide using an average force
Christophe Chipot and
Jérôme Hénin
The
reversible
folding of decaalanine is chosen as a test case for
characterizing a method that uses an adaptive biasing force (ABF) to
escape from the minima and overcome the barriers of the freeenergy
landscape. This approach relies on the continuous estimation of a
biasing force that yields a Hamiltonian in which no average force is
exerted along the ordering parameter ξ. Optimizing the parameters that
control how the ABF is applied, the method is shown to be extremely
effective when a nonequivocal ordering parameter can be defined to
explore the folding pathway of the peptide. Starting from a
βturn
motif and restraining ξ to a region of the conformational space that
extends from the αhelical state to an ensemble of extended
structures, the ABF scheme is successful in folding the peptide chain
into a compact alpha helix. Sampling of this conformation is, however,
marginal when the range of ξ values embraces arrangements of greater
compactness, hence demonstrating the inherent limitations of
freeenergy methods when ambiguous ordering parameters are utilized.
J. Chem. Phys. 2005, 123, 244906




Assessing
the
efficiency of free energy calculation
methods
David
RodriguezGomez,
Eric Darve and Andrew Pohorille
The
efficiencies of two recently developed methods for calculating free
energy changes along a generalized coordinate in a system are discussed
in the context of other, related approaches. One method is based on
Jarzynski's identity [Phys. Rev. Lett. 78, 2690 (1997)]. The second
method relies on thermodynamic integration of the average force and is
called the adaptive biasing force method [Darve and Pohorille, J. Chem.
Phys. 115, 9169 (2001)]. Both methods are designed such that the system
evolves along the chosen coordinate(s) without experiencing free energy
barriers and they require calculating the instantaneous, unconstrained
force acting on this coordinate using the formula derived by Darve and
Pohorille. Efficiencies are analyzed by comparing analytical estimates
of statistical errors and by considering two numerical
examples—internal rotation of hydrated 1,2dichloroethane and
transfer of fluoromethane across a waterhexane interface. The
efficiencies of both methods are approximately equal in the first but
not in the second case. During transfer of fluoromethane the system is
easily driven away from equilibrium and, therefore, the performance of
the method based on Jarzynski's identity is poor.
J.
Chem. Phys. 2004, 120,
35633578




Computation of
free energy profiles with parallel adaptive dynamics
Tony
Lelièvre, Mathias Rousset, and Gabriel Stoltz
We propose a
formulation of an adaptive computation of free energy differences, in
the adaptive biasing force or nonequilibrium metadynamics spirit, using
conditional distributions of samples of configurations which evolve in
time. This allows us to present a truly unifying framework for these
methods, and to prove convergence results for certain classes of
algorithms. From a numerical viewpoint, a parallel implementation of
these methods is very natural, the replicas interacting through the
reconstructed free energy. We demonstrate how to improve this parallel
implementation by resorting to some selection mechanism on the
replicas. This is illustrated by computations on a model system of
conformational changes.
J. Chem. Phys. 2007, 126, 134111




Adaptive biasing
force method for scalar and vector free energy calculations
Eric
Darve, David RodriguezGomez and Andrew Pohorille
In free energy
calculations based on thermodynamic integration, it is necessary to
compute the derivatives of the free energy as a function of one (scalar
case) or several (vector case) order parameters. We derive in a compact
way a general formulation for evaluating these derivatives as the
average of a mean force acting on the order parameters, which involves
first derivatives with respect to both Cartesian coordinates and time.
This is in contrast with the previously derived formulas, which require
first and second derivatives of the order parameter with respect to
Cartesian coordinates. As illustrated in a concrete example, the main
advantage of this new formulation is the simplicity of its use,
especially for complicated order parameters. It is also straightforward
to implement in a molecular dynamics code, as can be seen from the
pseudocode given at the end. We further discuss how the approach based
on time derivatives can be combined with the adaptive biasing force
method, an enhanced sampling technique that rapidly yields uniform
sampling of the order parameters, and by doing so greatly improves the
efficiency of free energy calculations. Using the backbone dihedral
angles Phi and Psi in NacetylalanylN'methylamide as a numerical
example, we present a technique to reconstruct the free energy from its
derivatives, a calculation that presents some difficulties in the
vector case because of the statistical errors affecting the derivatives.
J. Chem. Phys. 2008, 128(14), 144120






Exploring
multidimensional free energy landscapes using timedependent biases on
collective variables
Jérôme Hénin,
Giacomo Fiorin, Christophe
Chipot and Michael L. Klein
A new
implementation of the adaptive biasing force (ABF) method is described.
This implementation supports a wide range of collective variables and
can be applied to the computation of multidimensional energy profiles.
It is provided to the community as part of a code that implements
several analogous methods, including metadynamics. ABF and metadynamics
have not previously been tested side by side on identical systems.
Here, numerical tests are carried out on processes including
conformational changes in model peptides and translocation of a halide
ion across a lipid membrane through a peptide nanotube. On the basis of
these examples, we discuss similarities and differences between the ABF
and metadynamics schemes. Both approaches provide enhanced sampling and
free energy profiles in quantitative agreement with each other in
different applications. The method of choice depends on the dimension
of the reaction coordinate space, the height of the barriers, and the
relaxation times of degrees of freedom in the orthogonal space, which
are not explicitly described by the chosen collective variables.
J. Chem. Theory Comput. 2010,
6(1),
3547


