Water polarization charges in Generalized Born models.

Solvation energy is one of the major contribution to binding energy of a ligand to a biologically active protein and is the contribution mostly responsible for the protein selectivity. Although nothing can, probably, better approximate the interaction with water molecules than a sufficiently long molecular dynamics run in an all-atom force field, out of necessity we decided to try our best in ascribing as much physical meaning to Generalized Born models as it can be possibly made.

To do that (see our recent “Fast Surface Based Electrostatics for biomolecules modeling” preprint) we start from the classic GB approximate expression for the solvation energy and re-interpret it as a direct definition of the reaction field potential:

$\varphi_{1}(\mathbf{r})=-\sum_{j}\frac{q_{j}}{\sqrt{\left(\mathbf{r}-\mathbf{r}_{j}\right)^{2}+R\left(\mathbf{r}\right)R_{j}}}.$

Here $q_i$ are the charges of the atoms, $r_i$ are the charges positions and $R(r_i)$ are the Born radii. This is enough to establish the water polarization charge density $\sigma_S$ at the molecule boundary at a point $r^\prime$ , no matter which specific approach for the Born radii calculation is chosen:

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

Therefore, the polarization charge density can be a sanity check both for the direct reaction field potential interpretation and for an estimate of the Born radii calculation accuracy. The obvious requirement is the overall neutrality of the combined molecule and water system: $\int d^2f \sigma_S = -\sum_i q_i$ . Below we provide the calculations of the total water polarization charges vs. the macromolecule charge using our FSBE algorithm for roughly 430 different macromolecules from a pdb databank:

Well, it does not look good at all. The results on the graph above show not a correlation, but rather a distribution. The total water polarization charge anti-correlates with the macromolecule charge (as it should), though not ideally. This is a common deficiency of all Generalized Born models, but one. In fact it is possible to construct an approximation for the Born radius and enforce the neutrality at once. We will discuss it in our future posts.