where βα i = ln 10(pH -pK (i) a ) and pK a are the AA dissociation constants. For the CR part of the free energy we have chosen the form corresponding to the Langmuir-Davies dissociation isotherm (related to the Henderson-Hasselbalch isotherm) that is typically used to model the CR process in proteins . Other forms, such as the Frumkin-Fowler-Guggenheim isotherm, would correspond to more detailed models of the CR process . The thermodynamic equilibrium solutions for η k and the corresponding charges q k are then obtained from an implicit solution of