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Theory#

The change in free energy of transferring a solute from one solvent to another \(\Delta G_{A->B}\), can be readily computed by constructing a thermodynamic cycle in which the solute in each solvent is alchemically transformed into a non-interacting state as shown below:

Namely,

\[ \Delta G_{A->B} = \Delta G_1 - \Delta G_3 \]

where here \(\Delta G_1\) and \(\Delta G_3\) are the change in free energy of alchemically transforming a solute (or perhaps multiple solutes in the case of charged molecules with counter ions) so that it no longer interacts with the surrounding corresponding 'solvent'.

Computing the solvation (/ hydration) free energy is a special case of this cycle when the first 'solvent' is vacuum.

This framework currently offers two routes to computing such a change in free energy as the solute is being alchemically transformed, a more commonly used 'equilibrium route' and a 'non-equilibrium' route.

Note

For more information about how the different inter- and intramolecular interactions are alchemically modified see the transformations page.

Equilibrium Calculations#

Within this framework we refer to free energy calculations that involve dividing the alchemical pathway into discrete windows at each value of the coupling parameter \(\lambda\), collecting equilibrium samples for each such discrete state, and using these samples to compute the free energy using an approach such as thermodynamic integration (TI) 1, BAR 2, and MBAR 3 as 'equilibrium' free energy calculations.

At present absolv does not offer functionality for computing the derivatives with respect to lambda required for TI, and only supports MBAR and technically BAR although this estimator is not recommended.

See the overview for more information on running equilibrium free energy calculations using absolv.

Non-equilibrium Calculations#

Within this framework we refer to free energy calculations that proceed by generating equilibrium configurations at both end states (with the solute fully interacting with the solute and with the solute-solvent interactions fully decoupled / annihilated), and then driving each configuration non-reverisbly along the alchemical pathway by scaling the coupling factor \(\lambda\) as a function of time 4.

From a practical perspective it is computationally more efficient and convenient to proceed along the alchemical pathway in a stepwise, rather than continuous fashion. More specifically, the protocol proceeds by making a perturbation to \(\lambda\), followed by several relaxation steps, and repeating these two sub-stebs until the alchemical transformation is complete.

in this way the required to transform the solute from the interacting to the non-interacting state and likewise from the non-interacting to the interacting state can be computed according to

\[W = \sum_{i=0}^{n-1} \left[u_{\lambda_{i+1}}\left(x_i\right) - u_{\lambda_{i}}\left(x_i\right)\right]\]

where here \(u_{\lambda_i}\left(x_i\right)\) is the reduced potential evaluated at configuration \(i\) and \(\lambda_i\).

The free energy is then estimated by solving

\[\sum^N_{i=1}\dfrac{1}{1+\exp(\beta(W^f_i-\Delta F))} = \sum^N_{i=1}\dfrac{1}{1+\exp(\beta(W^r_i+\Delta F))}\]

self consistently where \(W^f_i\) corresponds to work computed along the forward pathway going from the interacting to the non-interacting state and \(W^r_i\) to work computed along the reverse pathway going from the non-interacting to the interacting state. \(N\) refers to the total number of equilibrium snapshots that were generated at each end state.

  1. TP Straatsma and JA McCammon. Computational alchemy. Annual Review of Physical Chemistry, 43(1):407–435, 1992. ↩

  2. Charles H Bennett. Efficient estimation of free energy differences from monte carlo data. Journal of Computational Physics, 22(2):245–268, 1976. ↩

  3. Michael R Shirts and John D Chodera. Statistically optimal analysis of samples from multiple equilibrium states. The Journal of chemical physics, 129(12):124105, 2008. ↩

  4. Andrew J Ballard and Christopher Jarzynski. Replica exchange with nonequilibrium switches: enhancing equilibrium sampling by increasing replica overlap. The Journal of chemical physics, 136(19):194101, 2012. ↩