What makes a correct binding energy calculation?

A correct calculation of a ligand binding affinity requires proper account of a few large contributions at the same time. The final value of the binding free energy is the sum of multiple sign-alternating entries: electrostatic (ES), solvation (ESolv), exchange-van der Waals (vdW) and entropic (TdS) term.

Common wisdom claims that ES and ESolv nearly compensate each other (up to a few h-bond energies), the remaining (normally large) vdW contribution is nearly canceled by TdS term. In fact, there is another important anti-correlation: strongly burried ligands with large vdW contributions are normally both entropycally and electrostatically constrained: The deeper a ligand gets into a protein (low dielectric constant medium) from water (high dielectric costant medium), the more the dessolvation energy (ES+ESolv) is, the larger the entropy losses normally are.


To clarify this issue, a simple calculation with 32 pdb structures with knows binding affinities was performed (1AU0, 1AU3, 1AU4, 1BR5, 1BR6, 1C83, 1C84, 1C85, 1C86, 1C87, 1EFY, 1EJN, 1JIJ, 1JIK, 1MS6, 1N3W, 1NAX, 1NLI, 1PWY, 1Q6K, 1RRI, 1RRW, 1RRY, 1RS2, 1TOW, 1TSM, 1URG, 1V2M, 2CBR, 2CBS, 3CBS, 4ERK). The binding affinities were taken from the PDBBind database. For every structure both the complex, the ligand and the protein was initially optimized in QUANTUM force field, then a thermodynamic integration was used to establish the free binding energies. The correlation between the calculated and measured (reported in the literature) values is shown on the left Figure 1. The free binding energies are reported in kJ/mol.


First we check the sanity of our solvation energy calculations. The following graph represents the anti-correlation between the electrostatic energy ES and the solvation contribution ESolv (both polar and non-polar, no additional dielectric constant for the protein was used since the protein is fully flexible). The two values nearly cancel each other, as it should be the case.
The only physically meaningful combination is the dessolvation punishment ESolv+ES, which is plotted against the vdW contribution (see Figure 3).


Remarkably, the dessolvation energy anti-correlates with the vdW term, which is a good measure of the contact surface. In fact, many scoring functions are pair sums of relatively short range potentials and hence correlate well with the contact surface. Non-additive desolvation energy does not correlate with the contact terms and (together with entropy losses) provides a limitation on binding energy for large and deeply burried ligands. TdS depends on the details of interactions in a smooth logarithmic way, which means that the dessolvation punishment is a leading effect.

The whole Molecular Mechanics (MM) Hamiltonian was used to generate MM trajectories for thermodynamic integration for each of the complexes. The results of the calculations are represented on the Figure 1. Root mean squared error of the calculations 6.1 kJ/mol. The dessolvation energy is a big contribution and removes (or better to say improves over) the correlation between the binding free energy and contact surface.
Conclusion 1: Binding free energy of a ligand is not a contact surface (or anything close to that approximable with a sum of pairwise short-range potential)
Conclusion 2: Desolvation energy is a major limiting factor restricting the binding energy of large ligands.