Here, ∆∆G 0 corresponds to ∆∆G env : the free energy of deprotonating residue A while in the presence of protonated B. Similarly, ∆∆G 3 corresponds to the deprotonation of A in the presence of deprotonated B. For all branches in the cycle, the remaining protonatable sites in the protein are fixed to their model states at pH 7.4. Notice that ∆∆G 1 and ∆∆G 2 will shift the populations of "reactants" and "products" with respect to ∆∆G 0 and ∆∆G 3 . Following a derivation provided in Appendix A, we resolve a pH-dependent ∆∆G(pH):