Three great ways to calculate Born radii beyond the Coulomb Field Approximation

By
Peter Fedichev, Quantum CTO
– November 9, 2009Posted in: Generalized Born, polar liquid, solvation energy

There is no need to explain how important are electrostatics interactions in Biology. Unfortunately electrostatics forces are long range and the interactions led by the interaction of molecular charges with each other and the charges of the water molecules are hard to calculate fast and accurately. On the other hand, continuous electrostatics is fast and simple. Moreover, electrostatic energy of a system of charges in a solute can be approximately calculated within the sol called Generalized Born approximation (GB). It’s not accurate, it’s wrong but it’s fast! That’s why we decided to walk an extra mile to see how far we can take GB seriously.

Generalized Born calculation for a system of charges q_i placed at the positions r_i proceeds as follows. The solvation energy is given by the approximate (e.g. variational)Kirkwood-like anzatz:


E_{Solv} = -\frac{1}{2}\sum_{i,j}\frac{q_i q_j}{\sqrt{(r_i-r_j)^2+R(r_i)R(r_j)}},

where R(r) is a carefully chosen variational function. The classic approximation (often called the Coulomb Field Approximation, or CFA) is given by

\frac{1}{R(r)} =\frac{1}{4\pi}\int_{W}\frac{1}{s_{i}^{4}}d^{3}r^{\prime}=\frac{1}{4\pi}\int_{\Gamma_{W}}\frac{\left(\mathbf{n^{\prime}}\mathbf{s}_{i}\right)}{s_{i}^{4}}df^{\prime},

where the integration is assumed either over the macromolecule excluded volume or over the water-macromolecule interface \Gamma_W. This approximation is known to fail miserably even for a single charge within a sphere. The surface integral implementations of Born radii calculations are more stable than the volume integration and we’ll stick to the surface integral formalism.

Is there an approximate R(r) giving the exact solution at least in a simple geometry? It turns out the answer is “yes” and in fact there are three different ways to get R(r) right. One is similar to CFA and reads:

\frac{1}{R(r)^{3}}=\frac{1}{4\pi}\int_{\Gamma_{W}}\frac{\left(\mathbf{n^{\prime}}\mathbf{s}_{i}\right)}{|r-r^\prime|^{6}}df^{\prime}.

Two more approximations are given by the generic formula:

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

with \alpha=1,2. It’s easy to check by a direct calculation that all the tree suggested expressions for the Born radii give the same EXACT result R(r)=(a^2-r^2)/a for any system of charges placed at positions r within a sphere of a radius a.

In practice it means that there are at least 3 different GB approaches capable to recover Kirkwood’s result for the solvation energy of a spherical protein with charged groups inside. Which of the approximation works better for an arbitrary (not spherical geometry)? Remains to be seen in our future posts.

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About Peter Fedichev, Quantum CTO

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