Here, our NEQ approach provided estimates of both D40 and C106 that were in closer agreement with the experimental values while exhibiting comparable performance on the other three residues. Considering the overall performance we found that the introduction of coupling dramatically improves agreement with experiment reducing our ), which varies as a function of pH between its asymptotic values (dotted). The zero point of this curve (single dot) is used to resolve the corresponding pK a value. The middle plots (dashed) depict the pH-dependent pK a value (Equations 9). Each residue has two limiting pK a values (dotted) which correspond to the cases when the other coupled residue is protonated or deprotonated. As the pH changes, the probability that the other residue is deprotonated shifts, resulting in a pH-dependent pK a . The lower plots depict several probabilities. Solid lines correspond to the protonation probabilities of the individual sites; these are a composite probability of the doubly protonated (i.e., ⟨A H B H ⟩) and singly protonated (i.e., ⟨A H B -⟩) microstates (see Equation ).