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$$ \mathrm{p}{K}_{\mathrm{a}} $$. Here, we derive a formalism based on double free energy differences (Delta Delta G$$ \Delta \Delta \mathrm{G} $$) for quantifying the individual site pKa$$ \mathrm{p}{K}_{\mathrm{a}} $$ values of coupled residues. As Delta Delta G$$ \Delta \Delta \mathrm{G} $$ values can be obtained by means of alchemical free energy calculations, the presented approach allows for a convenient estimation of coupled residue pKa$$ \mathrm{p}{K}_{\mathrm{a}} $$s in practice. We demonstrate that our approach and a previously proposed microscopic pKa$$ \mathrm{p}{K}_{\mathrm{a}} $$ formalism, can be combined with alchemical free energy calculations to resolve pH-dependent protein pKa$$ \mathrm{p}{K}_{\mathrm{a}} $$ 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 naively computed pKa$$ \mathrm{p}{K}_{\mathrm{a}} $$ values that ignore coupling, can be significantly improved when coupling is accounted for, in some cases reducing the error by half. In short, alchemical free energy methods can resolve the pKa$$ \mathrm{p}{K}_{\mathrm{a}} $$ values of both uncoupled and coupled residues. A pair of coupled residues can exist in four possible pH-dependent protonation states. We describe a formalism for resolving the pKa values of such residues via double free energy difference computed using alchemical-based molecular dynamics simulation. Such free energy differences can also be used to compute pH-dependent protonation curves and may provide insights into the importance of residues at enzymatic active sites in the presence or absence of ligands. image