### Theory

In the Tanford-Kirkwood model (TK), the protein is treated by a spherical cavity with dielectric constant, $$\epsilon_{p}$$ and radius $$b$$, surrounded by an electrolyte solution modeled by the Debye-Hückel theory 1. Shire {\em et al} modified this model incorporating a solvent static accessibility correction for each ionizable residue (TKSA model)2. This modification was mostly introduced to takes into account the irregular protein-solvent interface. Thus, the model proposed is now referred to as the Tanford-Kirkwood model with a solvent accessibility (TKSA) 2-4. The interaction energy, between two charged residues is given by

\begin{equation} U_{ij}=e^{2}\left(\frac{A_{ij}-B_{ij}}{2b}-\frac{C_{ij}}{2a}\right)\left(1-SA_{ij}\right), \end{equation}

where $$e$$ is the elementary charge; $$b$$ is the radius of the protein; $$a$$ is the closest possible approach distance of an ion; $$A_{ij}$$, $$B_{ij}$$ and $$C_{ij}$$ are parameters obtained from the analytical solution of the Tanford and Kirkwood model, which are functions of the distance between ionizable residues, the dielectric constants, and the ionic strength 1,5,6; and $$SA_{ij}$$ is the average of the solvent accessible surface area of residues $$i$$ and $$j$$ 2,3,7.

Once the electrostatic energy between the titratable residues is determined, it is possible to calculate the free energy, $$\Delta G_{N}(\chi)$$, for the native state protein in a given state of protonation $$\chi$$:8,9

\begin{equation} \Delta G_{N}(\chi)=-RT(\ln 10)\sum_{i=1}^{n}(q_{i}+x_{i})\mathrm{p}K_\mathrm{a,ref,i}+\frac{1}{2}\sum_{i,j=1}^{n}U_{ij}(q_{i}+x_{i})(q_{j}+x_{j}), \end{equation}

where $$R$$ is the ideal gas constant; $$T$$ is the temperature; $$q_{i}$$ is the charge of the ionizable residue $$i$$ in its deprotonated state; $$x_{i}$$ is 0 or 1, according on the protonation state of the residue $$i$$; and $$\mathrm{p}K_\mathrm{a,ref,i}$$ is the reference $$pK_{a}$$. The probability that the protein is in its native state with a particular state of protonation, $$\rho_{N}(\chi)$$, is guven by

\begin{equation} \label{rho-equation} \rho_{N}(\chi)=\frac{1}{Z_{N}}\exp\left[-\frac{\Delta G_{N}(\chi)}{RT}-\nu(\chi)(\ln 10)pH\right], \end{equation}

where $$\nu(\chi)$$ is the number of ionizable residues that are protonated in the state of protonation $$\chi$$, and $$Z_{N}$$ is the partition function for the protein in its native state. In this server, it was assumed that the unfolded state does not contribute in the electrostatic interactions 10. The protonation states $$\chi$$ are calculated via Metropolis Monte Carlo. The mean total electrostatic energy, $$\langle W_{qq}\rangle$$ is computed by the average over these protonation states, taking into account Eq.\ref{rho-equation}:

\begin{equation} \label{charge-charge-equation} \langle W_{qq}\rangle=\sum_{\chi}\left[\frac{1}{2}\sum_{i,j=1}^n U_{ij}(q_{i}+x_{i})(q_{j}+x_{j})\right]\rho_{N}(\chi). \end{equation}

Ibarra-Molero and Makhatadze have shown that the contribution of electrostatic interaction energy to free energy, $$\Delta G_{qq}$$, can be described by the negative of the average total electrostatic energy of the native state, that is, $$\Delta G_{qq}\approx-\langle W_{qq}\rangle$$ 9.

## References

(1) Tanford, C.; Kirkwood, J. G. J. Am. Chem. Soc. 1957, 79, 5333–5339.

(2) Shire, S.; Hanania, G.; Gurd, F. N. Biochemistry 1974, 13, 2967–2974.

(3) Tanford, C.; Roxby, R. Biochemistry 1972, 11, 2192–2198.

(4) Orttung, W. H. Biochemistry 1970, 9, 2394–2402.

(5) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 351–361.

(6) Da Silva, F. L. B.; Jönsson, B.; Penfold, R. Protein Sci. 2001, 10, 1415–1425.

(7) Havranek, J. J.; Harbury, P. B. Proc. Natl. Acad. Sci. USA 1999, 96, 11145–11150.

(8) Bashford, D.; Karplus, M. J. Phys. Chem. 1991, 95, 9556–9561.

(9) Ibarra-Molero, B.; Loladze, V. V.; Makhatadze, G. I.; Sanchez-Ruiz, J. M. Biochemistry 1999, 38, 8138–8149.

(10) Strickler, S. S.; Gribenko, A. V.; Gribenko, A. V.; Keiffer, T. R.; Tomlinson, J.; Reihle, T.; Loladze, V. V.; Makhatadze, G. I. Biochemistry 2006, 45, 2761–2766.