Resolving coupled pH titrations using alchemical free energy calculations

Abstract

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 $\text{p}{\text{K}}_{\text{a}}$ . Here, we derive a formalism based on double free energy differences ( $\Delta \Delta \text{G}$ ) for quantifying the individual site $\text{p}{\text{K}}_{\text{a}}$ values of coupled residues. As $\Delta \Delta \text{G}$ values can be obtained by means of alchemical free energy calculations, the presented approach allows for a convenient estimation of coupled residue $\text{p}{\text{K}}_{\text{a}}$ s in practice. We demonstrate that our approach and a previously proposed microscopic $\text{p}{\text{K}}_{\text{a}}$ formalism, can be combined with alchemical free energy calculations to resolve pH-dependent protein $\text{p}{\text{K}}_{\text{a}}$ 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 naïvely computed $\text{p}{\text{K}}_{\text{a}}$ 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 $\text{p}{\text{K}}_{\text{a}}$ values of both uncoupled and coupled residues.

Keywords: computational alchemy; free energy calculations; molecular dynamics; pKa calculations; residue coupling.