Welcome to the SMPBS Web Server

SMPBS (Size Modified Poisson-Boltzmann Solvers) is a library of size modified Poisson-Boltzmann equation (SMPBE) solvers that incorporates finite element, finite difference, solution decomposition, domain decomposition, and multigrid techniques. It has been published in the Journal of Computational Chemistry [7]. A sister site, the SDPBS web server, uses the classic Poisson-Boltzmann model.

Applications of the SMPBS web server

Currently, the SMPBS web server can be used to predict the electrostatic solvation energy of a biomolecule and the binding energy of a biomolecule complex in a symmetric 1:1 ionic solvent. Click the links below to get started (note: Javascript must be enabled in your browser).

The SMPBE Model

In the SMPBS web server, the electrostatics of a biomolecule in a symmetric 1:1 ionic solvent is predicted through solving a size modified Poisson-Boltzmann equation (SMPBE) as follows:

SMPBE equations

where u is an electrostatic potential function in units kBT/ec, Λ is a parameter for characterizing the size effects of ions and water molecules on electrostatics, ∂Ω denotes the boundary of a sufficiently large bounded domain Ω for calculation, Dp, Ds, and Γ denote a solute region hosting the biomolecule, a solvent region, and an interface between the solute and solvent regions, respectively, which satisfy that Ω = DpDs ∪ Γ as illustrated in Figure 1. The other parameters are given in Tables 1,2, and 3.

Protein region illustration

Figure 1. An illustration of a protein region Dp surrounded by solvent region Ds.
Γ denotes the interface and the two species of ions are colored in red and blue.

Table 1. Physical constants of SMPBE.
Parameter Value Unit (abbr.) Description
ε0 8.854187817 × 10-12 Farad/meter (F/m) Permittivity of vacuum
ec 1.602176565 × 10-19 Coulomb (C) Elementary charge
kB 1.380648813 × 10-23 Joule/Kelvin (J/K) Boltzmann constant
NA 6.0221409 × 1023 Mole-1 (mol-1) Avogadro constant

Table 2. SMPBE model parameters.
Parameter Default Value Unit (abbr.) Description
Λ 3.11 Angstrom (Å) Uniform ionic size parameter
εp 2.0 Unitless Biomolecular region dielectric constant
εs 80.0 Unitless Solvent region dielectric constant
T 298.15 Kelvin (K) Absolute temperature
Is 0.1 Mole/Liter (mol/l) Ionic strength

Table 3. Other SMPBE notation.
Parameter Description
Dp Protein region
Ds Solvent region
Γ Interface between Dp and Ds
Ω Computational domain satisfying Ω = DpDs ∪ Γ
∂Ω Boundary of Ω
rj Position of atom j (in angstroms)
zj Charge number of atom j
n(s) Unit outward normal vector of Dp
g Boundary value function
δrj Dirac-delta distribution at atomic position rj

The SMPBE solver and program package

The SMPBE model can be solved numerically in the SMPBS web server by a finite element solver [1] or a finite element and finite difference hybrid solver [4]. Both solvers were developed in Prof. Dexuan Xie's research group by using advanced techniques of solution decomposition, domain decomposition, and multigrid. Their computer programs were written in C++, C, Fortran, and Python based on the state-of-the-art finite element library DOLFIN from the FEniCS Project [2]. Their finite element meshes are generated by a molecular surface-fitted tetrahedral mesh generator called GAMer-II, which is an extension of a molecular surface and volumetric mesh generation program package reported in [5]. GAMer-II can generate tetrahedral meshes for a rectangular or spherical computational domain and three molecular interfaces - the Gaussian surface, the solvent-excluded surface (SES), and the solvent-accessible surface (SAS) [5, 3]. It can adaptively generate the seven overlapped boxes of the computational domain and a mesh of the central box that mixes an uniform Cartesian mesh with an unstructured finite element mesh for the finite element and finite difference hybrid solver [4].

Basic usage of the SMPBS web server

First, select either solvation energy or binding energy calculation. Before submitting a job, prepare PQR file(s) of a biomolecule. This can be done by first downloading a PDB file of the biomolecule from the RCSB Protein Data Bank (PDB), and then converting it to a PQR file by using the web server PDB2PQR. Next, adjust calculation parameters as desired, and submit the job. A typical work flow for a job submission in calculating solvation energy is illustrated in Figure 2.

User workflow diagram

Figure 2. Diagram of of SMPBS Solvation Energy user workflow.


The SMPBS web server is developed by Professor Dexuan Xie's research group in the Department of Mathematical Sciences and Yang Xie of the Department of Computer Science at University of Wisconsin-Milwaukee (UWM). It is hosted and maintained by the Information Technology Office in the College of Letters and Science at UWM. Development was partially supported by the National Science Foundation, USA, through grant DMS-1226259. Please contact Dexuan Xie via email (dxie@uwm.edu) with any questions regarding the server.


If this web server is useful in your work, please use the following citation:

  • Yang Xie, J. Ying, and D. Xie. SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation, Journal of Computational Chemistry, 38 (8) (2017) 541-552.
  • Additional Acknowledgements

    • Jeremy Streich - initial implementation of SDPBS web server and socket programming
    • UWM L&S IT Office - internal web development framework and libraries
    • Professor David Koes - guidance on integrating 3Dmol.js [6]
    • Drs. Yi Jiang and Jinyong Ying - help with SMPBE solver libraries

    Update History

    • 10/25/16 - Initial release.
    • 12/05/17 - Add new reference 7.


    1. 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) 286-301. [PDF]
    2. A. Logg, K.-A. Mardal, and G. N. Wells, eds., Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, Springer Verlag, 2012.
    3. D. Xu and Y. Zhang. Generating triangulated macromolecular surfaces by Euclidean distance transform. PloS ONE, 4 (12) (2009) e8140s.
    4. J. Ying and D. Xie, A hybrid solver of size modified Poisson-Boltzmann equation by domain decomposition, finite element, and finite difference, Applied Mathematical Modelling, (2017) available on line: https://doi.org/10.1016/j.apm.2017.09.026
    5. Z. Yu, M.J. Holst, Y. Cheng, and J.A. McCammon. Feature-preserving adaptive mesh generation for molecular shape modeling and simulation. Journal of Molecular Graphics and Modelling, 26 (8) (2008) 1370-1380.
    6. Nicholas Rego and David Koes. 3Dmol.js: molecular visualization with WebGL. Bioinformatics 31 (8) (2015) 1322-1324 doi:10.1093/bioinformatics/btu829.
    7. Yang Xie, J. Ying, and D. Xie. SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation, Journal of Computational Chemistry, 38 (8) (2017) 541-552. [PDF]