Although ∆G protein (pH) does report a non-linear ), 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 (Equation ). 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. The 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 ).