How to use Born surface charges to calculate solvation energy?

By
Peter Fedichev, Quantum CTO
– November 17, 2009Posted in: Generalized Born, solvation energy, water

We have recently established a relation between the Generalized Born methods and finite elements surface electrostatics. Indeed, if the Generalized Born solvation energy is interpreted as a (though approximate) definition of the reaction field potential, then the boundary condition defines the surface chages in terms of the Born radii of the charges:


\sigma_{S}(\mathbf{r}^{\prime})=-\frac{1}{4\pi}\sum_{j}q_{j}\frac{R_{j}}{\left|\mathbf{r}^{\prime}-\mathbf{r}_{j}\right|^{3}}. (1)

Once the surface charges are known, we can try to interpret them as approximate water polarization charges and express the solvation energy in terms of the reaction field potential of the surface charges:

E_{Solv} = \frac{1}{2}\sum_i\int_{\Gamma_W} d^2f^\prime\frac{\sigma_S(r^\prime)q_i}{|r_i-r^\prime|}. (2)

The expression comes from classic electrostatics and does not depend on the definition of the Born radii. There are many ways to get the radii, three of them give exact result for a (system of) charge(s) within a spherical cavity. One of them,

\frac{1}{R_{i}}=\frac{1}{4\pi}\int_{\Gamma_{w}}\frac{d^{2}f^{\prime}}{|\mathbf{r}_{i}-\mathbf{r}^{\prime}|^{3}}, (3)

is special and guarantees the overall neutrality of the system: \int d^2f \sigma_S = -\sum_i q_i. To see the idea works we took a realistic protein (H1N1 neuromidase, pdb accession code 2ht7) and calculated Born radii of each of the atoms by charging it with a single charge q=1 and calculating the solvation energy either exactly (using a finite element method) and approximately, using Eq.(1) – (2). The Born radii are obtained from the definition R_i = -q^2/2E_{Solv} and correlated against each other on the graph below:



The correlation shows that although the two quantities are not exactly similar, the surface charges Generalized Born radii are nearly identical to exact ones when the radii are small (the atoms near the protein surface). The correlation fails for deeply buried atoms, as it often happens to Born models. What remains to be seen though if the correlation holds, at least approximately, for a multiply charged molecules (see our future posts).

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About Peter Fedichev, Quantum CTO

Peter Fedichev, Ph.D., Chief Scientific Officer, co-founder