Update Theory

[fjclark]

Proposed Change

Update Theory section to align with Gilson et al. 1997. Theory to be framed in the most general way possible while remaining accessible to beginners.

Copying @egallicc's comment from Overleaf:

I plan to summarize Gilson et al's derivation of the statistical mechanics expression for the binding constant starting from the chemical potentials of the species. I don't think it will be too heavy. But it's the only derivation I know that rigorously introduces the standard state. I can do the same for hydration free energies and other unimolecular equilibria, such as solvent partitioning. Let me know.

Toni and I agree that it would be helpful to have a very general theory section which can be specilaised e.g. in the "What simulation protocol should I use section". It was suggested during the steering meeting to expand discussion on applications other than just binding, so including hydration free energies etc seems sensible. However, maybe Darrin/ others can comment.

Proposed Authors and Reviewers

Please confirm (before the end of January) that you are happy to author/ review and state when you're available. Authors, please say roughly when you plan to contribute the changes (no later than the end of May).

Proposed author: @egallicc (already confirmed via Overleaf)
Proposed reviewer: Darrin York (unable to find GitHub handle)

[egallicc]

Limitations of the current theory section (in my opinion):

• solvation and solvent partitioning equilibria are not covered
• the explanation of the origin of the standard state and binding site volume term and its treatment using restraining potentials is not always correct.

For example, in Eq.4, V should be the binding site volume Vsite, not the volume of the simulation box. The latter would make the definition of the binding free energy dependent on the simulation settings. The C0 Vsite term in Eq. 4 does not correct for the difference in standard vs. simulated concentration as stated below Eq. 4. It is the ideal term, the standard binding free energy, when receptor and ligand do not interact. In Eq. 18 the volume is denoted by V_L and is defined as restraint dependent, suggesting that V_L = Vbox could be a viable choice.

The issues above can be addressed, I think, in a unified way starting with well-established statistical mechanics expressions of the chemical potentials of the species involved following reference 60 (Gilson et al., 1997) for binding, and Widom, Ben-Naim, and others for solvation and solvent partitioning.

The theory of Gilson et al., in particular, clarifies the role of restraints in binding calculations. Those that are required by the theory and the auxiliary restraints introduced and then removed to aid convergence. This should help frame issue #146.

I intend to write a draft of the theory section deriving statistical mechanics expressions for the standard free energy of solvation, partitioning, and binding, which can serve as a starting point to derive all the computational protocols discussed in the paper. We can then discuss whether it is helpful and worthy of inclusion.