Resolving coupled pH titrations using alchemical free energy calculations

In a protein, nearby titratable sites can be coupled: the (de)protonation of one may affect the other. The degree of this interaction depends on several factors and can influence the measured pKa. Here, we derive a formalism based on double free energy differences (ΔΔG) for quantifying the individual site pKa values of coupled residues. As ΔΔG values can be obtained by means of alchemical free energy calculations, the presented approach allows for a convenient estimation of coupled residue pKas in practice. We demonstrate that our approach and a previously proposed microscopic pKa formalism, can be combined with non-equilibrium (NEQ) alchemical free energy calculations to resolve pH-dependent protein pKa values. Toy models and both, regular and constant-pH molecular dynamics simulations, alongside experimental data, are used to validate this approach. Our results highlight the insights gleaned when coupling and microstate probabilities are analyzed and suggest extensions to more complex enzymatic contexts. Furthermore, we find that naïvely computed pKa values that ignore coupling, can be significantly improved when coupling is accounted for, in some cases reducing the error by half. In short, our results suggest that free energy methods can resolve the pKa values of both uncoupled and coupled residues.