Adaptive Biasing Force Method: Difference between revisions
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==Description== | ==Description== | ||
===Free Energy from Unconstrained MD Simulations=== | ===Free Energy from Unconstrained MD Simulations=== | ||
The Adaptive Biasing Force (ABF) | 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> | <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> | ||
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<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> | <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 [[Adaptive Biasing Force Method# | 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 [[Adaptive Biasing Force Method#Instantaneous Collective Forces|finite-difference approach]]. | ||
===Sampling Space Discretization=== | ===Sampling Space Discretization=== | ||
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===Adaptive Bias=== | ===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 by free | 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 by free diffusion alongside these collective variables. 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 inter-atomic interaction potential <math>V(\bold R)</math>. 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> | <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> | ||
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Since the biasing force is not known prior to the simulation, it is calculated during the simulation and adaptively applied. To accelerate sampling, the biasing force is applied even if an inadequate number of samples is collected in a bin. In this case, the biasing force is scaled in the early stages to avoid artifacts from applications of overestimated biasing forces. The biasing force can also be smoothed to decrease noise in collected data. For further details, see feimode in [[ABF:Controls]]. | Since the biasing force is not known prior to the simulation, it is calculated during the simulation and adaptively applied. To accelerate sampling, the biasing force is applied even if an inadequate number of samples is collected in a bin. In this case, the biasing force is scaled in the early stages to avoid artifacts from applications of overestimated biasing forces. The biasing force can also be smoothed to decrease noise in collected data. For further details, see feimode in [[ABF:Controls]]. | ||
=== | ===Instantaneous Collective Forces=== | ||
PMFLib implements two algorithms to calculate Equation 3: simplified and original ABF algorithms. The original ABF algorithm is theoretically more accurate than the simplified one but it works only for MD without constraints. Therefore, the simplified algorithm is the default one as it is numerically more robust at all MD setups. | PMFLib implements two algorithms to calculate Equation 3: simplified and original ABF algorithms. The original ABF algorithm is theoretically more accurate than the simplified one but it works only for MD without constraints. Therefore, the simplified algorithm is the default one as it is numerically more robust at all MD setups. | ||
====Simplified ABF Algorithm==== | ====Simplified ABF Algorithm==== | ||
The simplified ABF algorithm (fmode=1) uses the result of the product rule for derivatives applied on Equation 3 leading into the two distinct contributions into the | The simplified ABF algorithm (fmode=1) uses the result of the product rule for derivatives applied on Equation 3 leading into the two distinct contributions into the instantaneous collective forces: | ||
<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = + \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t) \right) \cdot \bold a(t) + \frac{d}{dt} \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t)) \right) \cdot \bold v(t) = \bold F_{ICF,pot}(t) + \bold F_{ICF,kin}(t)</math> ... (9)</center> | <center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = + \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t) \right) \cdot \bold a(t) + \frac{d}{dt} \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t)) \right) \cdot \bold v(t) = \bold F_{ICF,pot}(t) + \bold F_{ICF,kin}(t)</math> ... (9)</center> | ||
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<center><math>\bold F_{ICF,kin}(t-dt/2) = \frac {\bold Z^{-1}(t) \nabla \boldsymbol \xi (t) - \bold Z^{-1}(t-dt) \nabla \boldsymbol \xi (t-dt)}{dt} \bold v(t-dt/2) </math> ... (11)</center> | <center><math>\bold F_{ICF,kin}(t-dt/2) = \frac {\bold Z^{-1}(t) \nabla \boldsymbol \xi (t) - \bold Z^{-1}(t-dt) \nabla \boldsymbol \xi (t-dt)}{dt} \bold v(t-dt/2) </math> ... (11)</center> | ||
Finally, to get <math>\bold F_{ICF,kin}</math> at the same time as <math>\bold F_{ICF,kin}</math>, two values are averaged: | |||
<center><math>\bold F_{ICF,kin}(t) = \frac {\bold F_{ICF,kin}(t+dt/2) + \bold F_{ICF,kin}(t-dt/2) }{2}</math> ... (12)</center> | <center><math>\bold F_{ICF,kin}(t) = \frac {\bold F_{ICF,kin}(t+dt/2) + \bold F_{ICF,kin}(t-dt/2) }{2}</math> ... (12)</center> | ||
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The algorithm uses a history of values collected in two consecutive time steps. But, the first result is available from the fourth time step to be compatible with the original algorithm. | The algorithm uses a history of values collected in two consecutive time steps. But, the first result is available from the fourth time step to be compatible with the original algorithm. | ||
====Original ABF Algorithm==== | ====Original ABF Algorithm==== | ||