Adaptive Biasing Force Method

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Description

Free Energy from Unconstrained MD Simulations

The Adaptive Biasing Force (ABF) Method calculates the free energy as a function of selected collective variables from unconstrained molecular dynamics (MD) simulations. The method does not provide the free energy directly, but instead, it provides the free energy gradient $\frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi}$ , which must be integrated to get the free energy:

\Delta G = G(\boldsymbol \xi_2) - G(\boldsymbol \xi_1) = \int_{\boldsymbol \xi_1}^{\boldsymbol \xi_2} \frac{\partial G(\boldsymbol \xi)} {\partial \boldsymbol \xi} \boldsymbol d \xi

... (1)

The free energy gradient is calculated as a mean of instantaneous collective force $\bold F_{ICF}$ :

\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*}}

... (2)

with the instantaneous collective force calculated from the time evolution of the collective variable:

\bold F_{ICF} = \frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right)

... (3)

where $\bold Z$ is the matrix in the form:

[Z]_{ij}=\sum_{k=1}^{N_{atoms}} \frac{1}{m_k} \frac{\partial \xi_i}{\partial \bold x_k} \frac{\partial \xi_j}{\partial \bold x_k}

... (4)

The analytical calculation of instantaneous collective force by Equation 3 requires the second derivatives of collective variables with respect to Cartesian coordinates. Since this can be prohibitive for complex collective variables such as the simple base-pair parameters, Equation 3 is evaluated numerically by a finite-difference approach, as suggested by Darve et al.

Sampling Space Discretization

Due to numerical reasons, mean forces are collected on a regular grid. The averaging of instantaneous collective force is then done in small intervals centered at discrete CV values:

\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2}

... (5)

with the standard error:

\sigma_e \left( \frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} \right) = \sqrt{ \frac {N_{corr}} {N_b} } \sigma(\bold F_{ICF})_ {\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2}

... (6)

where the standard deviation is given by:

\sigma(\bold F_{ICF})_ {\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2} = \sqrt { \frac{1}{N_b} \sum_{i=1}^{N_b} \left( F_{ICF}(\boldsymbol \xi_i) - \overline {F_{ICF}} \right)^2 \vert _{\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2}  }

... (7)

where $\bold h$ is the interval size also called a bin, $N_b$ is the number of samples collected in a bin centered at $\boldsymbol \xi^{*}$ , and $N_{corr}$ is a statistical inefficiency due to correlation in time series.

Adaptive Bias

ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables $\boldsymbol \xi$ . As a result, the system evolves alongside these collective variables by free diffusions. The bias is derived from the free energy, which is projected in the form of biasing force $\bold F_{bias}$ to the Cartesian space and removed from force $\bold F_{pot}$ originated from interatomic interaction potential $V(\bold R)$ . The application of the bias thus leads to the modified equations of motions: