Adaptive Biasing Force Method

Navigation: Documentation / Methods / Adaptive Biasing Force Method

---

Contents

• ABF:Description
• ABF:Controls
• ABF:Collective variables
• ABF:Post-processing
• ABF:Multiple walkers approach
• ABF:Utilities
• ABF:Examples

Description

Theory

The Adaptive Biasing Force (ABF) method calculates the derivative of the free energy (PMF) using the following formula:

$\frac{\partial A}{\partial \mathit{\xi }}=-{⟨\frac{d}{dt}\left({{\mathbf{Z}}_{\mathit{\xi }}}^{-1}\frac{d\mathit{\xi }}{dt}\right)⟩}_{\xi }$ .... (1)

where ξ is the set of collective variables, t is time, and Z_ξ is the matrix defined as

${\left[{\mathbf{𝐙}}_{\xi }\right]}_{i,j}=\sum _{k=1}^N\frac{1}{m_k}\frac{\partial {\xi }_i}{\partial {\mathbf{𝐱}}_k}\frac{\partial {\xi }_j}{\partial {\mathbf{𝐱}}_k}$ .... (2)

where m_k is the mass of atom with cartesian coordinate x_k.

The averages over ξ are collected from unconstrained molecular dynamics. The free energy derivatives are practically calculated over discretized ranges of collective variables (CVs). Each CV_i is discretized into M_i bins leading into one-dimensional bins for one CV, two-dimensional bins for two CVs, etc. In each such bin, the vector of PMF is accumulated using the formula (1). For one CV this can be written as

$\frac{\partial A}{\partial \xi }{|}_{({\xi }_i+{\xi }_{i+1})/2}=-{⟨\frac{d}{dt}\left({Z_{\xi }}^{-1}\frac{d\xi }{dt}\right)⟩}_{({\xi }_i,{\xi }_{i+1}⟩}$ .... (3)

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.