# Theory¶

Electrostatic Charge Density Starting from version 3.0 EON, incorporates two types of electrostatic description: charge density and potential. The charge density is modeled using partial charges and smearing them with a Gaussian function with a prefactor that is proportional to the partial atomic charge. This method has advantage of fast speed compared to more expensive potential calculations; therefore, it is set as a default behavior of EON.

For two charge distributions A and B:

$ChargeTanimoto_{A,B} = \frac{O(A, B)}{O(A, A) + O(B, B) - O(A, B)}$

Charge Tanimoto similarly to potential Tanimoto below behaves for equal and opposite charge distributions similarly

$\begin{split}ChargeTanimoto_{A,A} &= 1.0, \\ ChargeTanimoto_{A,-A} &= -\frac{1}{3}\end{split}$

For hit list ranking, we report a score (Shape + Charge Tanimoto Combo) that is the sum of the Shape Tanimoto (ST) and the Charge Tanimoto.

Electrostatic Potential

EON uses a field-based measure of Tanimoto to compare the electrostatic potential of two small molecules. This electrostatic potential is calculated internally using Zap, OpenEye’s Poisson-Boltzmann (PB) electrostatics toolkit.

The basic equation for a field Tanimoto is:

$PotentialTanimoto_{A,B} = \frac{\int A(\vec{r})*B(\vec{r})}{\int A(\vec{r})*A(\vec{r}) + \int B(\vec{r})*B(\vec{r}) - \int A(\vec{r})*B(\vec{r})}$

The two boundary cases for Electrostatic Tanimoto occur when B = A:

$\begin{split}PotentialTanimoto_{A,A} & = \frac{\int A(\vec{r})*A(\vec{r})}{\int A(\vec{r})*A(\vec{r}) + \int A(\vec{r})*A(\vec{r}) - \int A(\vec{r})*A(\vec{r})} \\ & = 1\end{split}$

and the opposite case, when B = -A:

$\begin{split}PotentialTanimoto_{A,-A} & = \frac{\int A(\vec{r})*-A(\vec{r})}{\int A(\vec{r})*A(\vec{r}) + \int -A(\vec{r})*-A(\vec{r}) - \int A(\vec{r})*-A(\vec{r})} \\ & = \frac{- \int A(\vec{r})*A(\vec{r})}{\int A(\vec{r})*A(\vec{r}) + \int A(\vec{r})*A(\vec{r}) + \int A(\vec{r})*A(\vec{r})} \\ & = -\frac{1}{3}\end{split}$

In EON in the potential mode, we report electrostatic potential similarity based on the outer dielectric used in the PB calculation. We use an outer dielectric of 80. The rationale for using a PB electrostatic field is that the external potential is dampened by orientation of the aqueous solvent. It is a common observation that proteins essentially act to reproduce the aqueous desolvation of well-bound ligands. As a result a PB electrostatic field is more likely to correctly capture the essential elements of binding than that from the Coulombic field.

For hit list ranking, we report a score (Shape + Potential Tanimoto Combo) that is the sum of the Shape Tanimoto (ST) and the PB Electrostatic Potential Tanimoto.