Thermodynamics of Protonation Equilibria

Therefore, the pK of group AH in water can be obtained from the computed value of AG , when the remaining free energy terms shown in the thermodynamic cycle are known. The AGg values can be obtained from ab initio methods or experiments, and the solvation free energy values can be calculated by Monte Carlo simulation methods combined with a reaction field approach, integral equation techniques, or continuum dielectric methods. However, because the proton solvation free energy, AG , (H+), is a quantity about which there exists considerable experimental uncertainty and for which there are no reliable calculations, frequently only the difference in the pfC s of two groups, AH and BH can be obtained  

assuming that the concentrations are proportional to the probabilities of being in a given state, and with the use of a Boltzmann distribution. 

AG can be considered to be the free energy level of the protonated state relative to the deprotonated one. Our method for calculating apparent pK s is based on these energy levels. 

For a protein with M ionizable groups, the free energy can be expressed as a sum of terms shown by Eq. [9]. If we choose the neutral state of all ionizable residues as the reference state, the free energy of the ionization state (Xj,. . . , x ) of the whole molecule is  

The intrinsic pK thus represents the of a group in the protein when all other titratable amino acids are in their neutral state. Sometimes, it is useful to have all sites deprotonated as the reference state for the system in this case the free energy of the molecule with the protonation state (ar j,. . . , of the whole molecule is 

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