Bennett Acceptance Ratio

Free Energy Fundamentals
Theory Definitions Exponential Averaging / Zwanzig Thermodynamic Integration Bennett Acceptance Ratio WHAM MBAR
Methods of Free Energy Simulations Thermodynamic Cycle Intermediate States (Constructing) Soft Core Potentials Running Simulations Extracting Simulation Information Analysis / Bootstrap Sampling
Improving Sampling and Accelerating Simulations Umbrella Sampling Hamiltonian Replica Exchange [math]\displaystyle{ \lambda }[/math]-dynamics / Expanded Ensemble

Free Energy How-to's
Example Calculations Absolute Solvation Free Energy Relative Binding Affinity Absolute Binding Affinity
Tutorials Ethanol Solvation - GROMACS Expanded Ensemble - GROMACS Absolute Binding Free Energy - GROMACS

The Bennett Acceptance Ratio (BAR) is one of the earliest free energy methods which draws on data from multiple states to estimate the free energy difference; This method has significantly improved results over EXP. Both EXP and TI require the ensemble average from a single state to estimate free energies. Although TI needs the derivatives at state [math]\displaystyle{ k }[/math], it does not require the configurations from any neighboring state; BAR however, requires configuration information from two states to estimate free energy differences.

BAR works under the principal that at the same configuration, [math]\displaystyle{ \vec{q} }[/math], at two separate states, [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math], there is a pathway connecting the two potentials and a difference of [math]\displaystyle{ \Delta U_{ij}(\vec{q}) }[/math]. Because the states are in the same configuration, there is a exact relationship between the distributions of potential energy differences [math]\displaystyle{ \Delta U_{ij}(\vec{q}) }[/math] of states sampled from [math]\displaystyle{ i }[/math] and [math]\displaystyle{ \Delta U_{ji}(\vec{q}) }[/math] the distribution of potential energy differences sampled from the state [math]\displaystyle{ j }[/math].. Since its an exact function of distributions, statistics can be applied to find the optimal way to use the information between two states, improving the free energy estimate. This is where Bennett started his derivation and, since he was the first to derive this, the method was named after him.

Derivation

This derivation starts from a slightly modified version of the core free energy equation. Taking the properties of expectation values, we can write the free energy difference as

[math]\displaystyle{ \Delta A_{ij} = -k_{B} T \ln \frac{Q_j}{Q_i} = k_{B}T \ln \frac{\left\langle\alpha(\vec{q}) \exp[-\beta \Delta U_{ij}(\vec{q})]\right\rangle_j}{\left\langle\alpha(\vec{q}) \exp[-\beta \Delta U_{ji}(\vec{q})]\right\rangle_i} }[/math]

which is true for any [math]\displaystyle{ \alpha(\vec{q})\gt 0 }[/math] for all [math]\displaystyle{ \vec{q} }[/math]. This was Bennett's start point and the he used variational calculus to select the [math]\displaystyle{ \alpha(\vec{q}) }[/math] which minimized the variance of the free energy. The end result is an implicit function of the free energy, and the number of samples at each state, [math]\displaystyle{ n_i }[/math] and [math]\displaystyle{ n_j }[/math], and is

[math]\displaystyle{ \sum_{i=1}^{n_i} \frac{1}{1 + \exp(\ln(n_i/n_j) + \beta \Delta U_{ij} - \beta \Delta A))} - \sum_{i=1}^{n_j} \frac{1}{1 + \exp(\ln(n_j/n_i) - \beta \Delta U_{ji} + \beta \Delta A))} = 0 }[/math]

which must be solved numerically. This is the full BAR equation and its full derivation can be found in Bennett's paper.

Comparison with Other Methods

BAR has been shown to be superior to EXP in every way (save maybe simplicity). BAR is not only better from a practical and theoretical aspect, and it is shown to converge to EXP in assuming all samples come from one state. Even so, they less phase overlap is needed to run BAR effectively.

Comparing TI and BAR is not such a simple thing since each approach requires very different information. Based on experience, BAR will do better than TI on average, although more details are needed to make a better comparison. One such detail is that BAR can give in fewer intermediate states the same statistical precision as TI. However, if the integrand is very smooth for TI, it will perform at the same level as BAR (examples of this are changes, not removing, non- and bonded parameters). One advantage that could be attributed to BAR over TI is that you do not need to calculate [math]\displaystyle{ \frac{dU}{d\lambda} }[/math] and so you do not have to modify your code to do so.

A variant of BAR which takes in data from more than two states has been developed, it is coincidentally called the Multistate Bennett Acceptance Ratio or MBAR for short.