# Features of SMPBS

## Comparison of SMPBE with PBE

The SMPBE model is one simple modification of the classic Poisson-Boltzmann equation (PBE) model to reflect the size effects of ions and water molecules on the prediction of ionic concentrations. It is reduced to PBE by setting Λ = 0. For a salt (NaCl) solution, the concentrations *C _{Na}* and

*C*of sodium ions Na

_{Cl}^{+}and chloride ions Cl

^{-}are estimated by PBE and SMPBE (in moles per liter) as follows:

Using the above formulas, we predicted the ionic concentrations in tests on a central charged unit ball and a dipole with two opposite central charges in **[1]** and **[2]**, respectively, demonstrating that SMPBE can produce physically more reasonable ionic concentrations than PBE. Hence, SMPBE is a more reasonable model than PBE in such an application.

**Figure 1** shows that the concentrations *C ^{Na}* and

*C*of chloride and sodium ions produced by SMPBE reach a saturation value of 56 around the dipole with positive and negative central charges, respectively

^{Cl}**[2]**. As claimed by the law of electrostatic attraction, both

*C*and

^{Na}*C*were found to distribute symmetrically around these two balls.

^{Cl}
We also repeated these tests using PBE. Because it treats each ion as a point without any size, PBE produced unreasonably large values of *C ^{Na}* and

*C*(greater than 1000) around the two balls.

^{Cl}## References

- J. Li and D. Xie, An effective minimization protocol for solving a size-modified Poisson-Boltzmann
equation for biomolecule in ionic solvent,
*International Journal of Numerical Analysis and Modeling*, 12 (2015), pp. 286-301. - J. Ying and D. Xie, A hybrid solver of size modified Poisson-Boltzmann equation by domain decomposition, finite element, and finite difference. arXiv:1610.06173 [math.NA], 2016.