Adaptive Biasing Force Method: Difference between revisions
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==Description== | ==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 <math>\frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi}</math>, which must be integrated to get the free energy: | |||
<center><math>\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 </math> ... (1)</center> | |||
The free energy gradient is calculated as a mean of instantaneous collective force <math>\bold F_{ICF}</math>: | |||
<center><math>\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*}} </math> ... (2)</center> | |||
with the instantaneous collective force calculated from the time evolution of the collective variable: | |||
<center><math>\bold F_{ICF} = \frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right)</math> ... (3)</center> | |||
where <math>\bold Z</math> is the matrix in the form: | |||
<center><math>[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}</math> ... (4)</center> | |||
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: | |||
<center><math>\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*} \pm \bold h/2} </math> ... (5)</center> | |||
with the standard error: | |||
<center><math>\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} </math> ... (6)</center> | |||
where the standard deviation is given by: | |||
<center><math>\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} }</math> ... (7)</center> | |||
where <math>\bold h</math> is the interval size also called a bin, <math>N_b</math> is the number of samples collected in a bin centered at <math>\boldsymbol \xi^{*}</math>, and <math>N_{corr}</math> 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 <math>\boldsymbol \xi</math>. 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 <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from interatomic interaction potential (V(x)). The application of the bias thus leads to the modified equations of motions: | |||
<center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} - \frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center> | |||
where <math>m_i</math> is mass of atom i, <math>\bold r_i</math> is atom position, and <math>t</math> is time. | |||
The free energy calculation is achieved by introducing a bias, which removes barriers or higher free energy regions alongside predefined collective variables . As a result, the system evolves alongside these collective variables by free diffusions. The bias is derived from the free energy (G(ξ)), which is projected in the form of biasing force (F_bias) to the Cartesian space and removed from force (F_pot) originated from interatomic interaction potential (V(x)). The application of the bias thus leads to the modified equations of motions: | |||
m_i (d^2 x_i)/(dt^2 )=F_(pot,i) (x_i )-F_(bias,i) (x_i )=-(∂V(x)/(∂x_i )-∂G(ξ)/∂ξ ∂ξ/(∂x_i )) (1) | |||
where m_i is mass of atom i, x_i is atom position, t is time. | |||
The Adaptive Biasing Force (ABF) method calculates the derivative of the free energy (PMF) using the following formula: | The Adaptive Biasing Force (ABF) method calculates the derivative of the free energy (PMF) using the following formula: | ||
<center><math>\frac{\partial A}{\partial \boldsymbol \xi} = - {\left \langle { \frac{d}{d t} \left( \bold {Z_{\xi}}^{-1} \frac{d \boldsymbol \xi}{d t} \right) } \right \rangle}_{\xi}</math> .... (1)</center> | <center><math>\frac{\partial A}{\partial \boldsymbol \xi} = - {\left \langle { \frac{d}{d t} \left( \bold {Z_{\xi}}^{-1} \frac{d \boldsymbol \xi}{d t} \right) } \right \rangle}_{\xi}</math> .... (1)</center> | ||
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In the ABF simulations, the estimated PMF is used to bias molecular dynamics simulations to improve sampling in the regions exhibiting large free energy barriers. | In the ABF simulations, the estimated PMF is used to bias molecular dynamics simulations to improve sampling in the regions exhibiting large free energy barriers. | ||
==References== | |||
<references /> | |||