adaptive biasing force
Overcoming free energy barriers in molecular simulations using an average force

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Theoretical foundations

Strictly speaking, a potential of mean force (PMF) is the reversible work supplied to the system to bring two solvated particles, or ensembles of particles, from an infinite separation to some contact distance:

w(r) = −1/β
ln g(r)                                                                                                                                    (1)

Here, g(r) is the pair correlation function of the two particles, or ensembles thereof. The vocabulary “PMF” has, however, been extended to a wide range of order parameters, or reaction coordinate surrogates, that go far beyond simple interatomic or intermolecular distances. In this perspective, generalization of equation (1) is not straightforward. This explains why it may be desirable to turn to a definition suitable for any type of parameter, ξ:

A) = -1/β ln P(ξ)  + A                                                                                                                          (2)

A(ξ) is the free energy of the state defined by a particular value of ξ, which corresponds to an iso-hypersurface in phase space. A0 is a constant and P(ξ)  is the probability density to find the chemical system of interest at ξ. The connection between the derivative of the free energy with respect to the order parameter, dA(ξ)/dξ, and the forces exerted along the latter may be written as:

dA(ξ)/dξ = 〈∂V(x) / ∂ξ〉ξ - 1/β 〈∂ ln |J| / ∂ξ〉ξ = -Fξξ                                                                               (3)

where |J| is the determinant of the Jacobian for the transformation from generalized to Cartesian coordinates. The first term of the ensemble average corresponds to physical forces exerted on the system, derived from the potential energy function, V(x). The second contribution is a purely geometric correction that accounts for difference in phase space availability as the order parameter varies. It is worth noting, that, contrary to its instantaneous component, Fξ, only the average force, 〈Fξξ is physically meaningful.

In the framework of the average biasing force (ABF) approach, Fξ is accumulated in small windows or bins of finite size, δξ, thereby providing an estimate of the derivative dA(ξ)/dξ defined in equation (3). The force applied along the order parameter to overcome free energy barriers is defined as:

FABF = -
Fξξ  ξ ξ                                                                                                                                     (4)

As sampling of the phase space proceeds, the estimate of dA(ξ)/dξ is progressively refined. The biasing force, FABF, introduced in the equations of motion guarantees that in the bin centered about ξ, the force acting along the order parameter averages to zero over time. Evolution of the system along ξ is, therefore, governed mainly by its self- diffusion properties.

To illustrate how ABF works, let us consider the naive, prototypical example of the reversible unfolding of a short hydrophobic peptide, deca-alanine, in vacuo. The surrogate reaction coordinate chosen to investigate this process is the distance separating the N- and the C-termini of the peptide. Around 14 Å, the peptide chain is folded in an α-helical conformation.  As the order parameter is increased, the intramolecular hydrogen bonds that form the scaffold of the α-helix are progressively disrupted as the peptide unfolds. At this stage, it should be clearly understood that an ABF simulation is distinct from a non-equilibrium pulling experiment, whereby the peptide chain is driven to a random coil. ABF ensures that folded and unfolded conformations remain in a thermodynamic equilibrium, interconversion being fully reversible.

Starting from the α-helical conformation, samples of Fξ are accrued in small, 0.1-Å wide bin. When a reasonable distribution of Fξ is obtained in the visited bins and, hence, an acceptable estimator of Fξξ at a given value of ξ, FABFis applied to advance sampling in regions of the order parameter that have not been visited hitherto on account of the free energy barriers. Note that as the free energy increases, the distribut ion of Fξ is offset on the negative side; Fξξ is negative. Symmetrically, when the free energy decreases, Fξξ becomes positive. As the simulation progresses, the different bins of the order parameter are continuously filled until uniform sampling is achieved. The complete, optimally converged PMF is recovered within 5 ns.