Welcome to the SMPBICS Web Server

SMPBICS (Size Modified Poisson-Boltzmann Ion Channel Solver) is a finite element solver for the size modified Poisson-Boltzmann ion channel (SMPBIC) model. It has been recognized in the prestigious Journal of Computational Physics for the case of nonuniform ion sizes [1] and Journal of Computational Chemistry for the case of uniform ion sizes [2]. The SMPBICS web server is a platform for computing the electrostatic solvation energies of a voltage-dependent anion channel (VDAC) in a mixture of multiple ionic species [3].

Using the SMPBICS Web Server

To submit a job, a user can simply submit a VDAC Protein Data Bank Indentifier (PDB ID). The server then downloads the PDB file from the OPM Database and generates a PQR file by the built-in PDB2PQR package. The user can also upload their own PQR file or PDB file to let the server generate a PQR file.

To get started, please ensure that Javascript is enabled in your browser to access the web server's user-friendly interface and then click on the link below.

\( \def\Phit{\tilde{\Phi}} \def\es{\epsilon_s} \def\ep{\epsilon_p} \def\emm{\epsilon_m} \def\ez{\epsilon_0} \def\rr{{\mathbf r}} \def\s{{\mathbf s}} \def\nn{{\mathbf n}} \def\EE{{\mathbf e}} \def\PP{{\mathbf p}} \def\DD{{\mathbf d}} \def\rhot{\tilde{\rho}} \def\n{\mathbf{n}} \def\ei{\epsilon_\infty} \def\J{{\mathbf J}} \def\E{{\mathbf E}} \def\parn[1]{\left(#1\right)} \def\del[1]{\partial{#1}} \def\sdiv{{\nabla\cdot}} \def\epm{\epsilon_m} \def\R{\mathbb{R}} \def\zz{{\mathbf z}} \)

Calculate the electrostatic solvation energy of a VDAC

The SMPBIC Model

Let a simulation box domain, \(\Omega\), be partitioned into a protein reigon, \(D_p\), a membrane region, \(D_m\), and a solvent region, \(D_s\), as illustrated in Figure 1. By the notation and parameters of Tables 1 to 3, the nonuniform SMPBIC model is defined as a nonlinear system consisting of the \(n\) nonlinear algebraic equations,

\[ c_{i}(\rr) - {c}_{i}^b \left[1- \gamma \sum\limits_{j=1}^n v_j c_j(\rr) \right]^{\frac{ v_i}{v_0}} e^{-Z_{i} u(\rr)} =0, \quad \rr\in D_{s}, \quad i=1,2,\ldots,n, \tag{1} \]

the three Poisson equations,

\begin{align} - \epsilon_p\Delta u( \rr) =\alpha \sum\limits_{j=1}^{n_{p}}z_{j} \delta_{\rr_{j}}, & \quad \rr \in D_p, \\ - \epsilon_m\Delta u( \rr) =0, & \quad \rr \in D_{m},\tag{2} \\ \epsilon_s \Delta u( \rr) - \beta \sum\limits_{i=1}^n Z_i c_i(\rr) = 0, & \quad \rr \in D_s, \end{align}

the interface conditions,

\begin{eqnarray*} u(\s^-) = u(\s^+), \quad \ep \frac{\partial u(\s^-)}{\partial \nn_p(\s)} = \es \frac{\partial u(\s^+)}{\partial \nn_p(\s)}, & \quad \s\in\Gamma_p,\\ u(\s^-) = u(\s^+), \quad \emm \frac{\partial u(\s^-)}{\partial \nn_m(\s)} = \es \frac{\partial u(\s^+)}{\partial \nn_m(\s)} + \tau \sigma, & \quad \s\in\Gamma_m,\tag{3}\\ u(\s^-) = u(\s^+), \quad \ep \frac{\partial u(\s^-)}{\partial \nn_p(\s)} = \emm \frac{\partial u(\s^+)}{\partial \nn_p(\s)}, & \quad \s\in\Gamma_{pm}, \end{eqnarray*}

and the mixed boundary value conditions,

\[ u(\s) = g(\s), \quad \s \in \Gamma_D,\qquad \frac{\partial u( \s)}{\partial \nn_b(s)} = 0, \quad s\in \Gamma_N.\tag{4} \]

Here \(\alpha\), \(\beta\), \(\gamma\), \(\bar{v}\), and \(\tau\) are defined by

\[ \alpha = \frac{10^{10}e_{c}^{2}}{\ez k_{B}T}, \quad \beta = \frac{N_A e_{c}^{2}}{10^{17}\ez k_{B}T}, \quad \gamma = 10^{-27} N_A, \quad \bar{v} = \frac{1}{n}\sum_{i=1}^n v_i, \quad \tau = \frac{ 10^{-12} e_c}{\ez k_B T}. \]

With the values of physical parameters given in Table 2 and the default value of T given in Table 3, we can estimate \(\alpha\), \(\beta\), \(\gamma\) and \(\tau\) as follows

\[ \alpha = 7042.9399, \quad \beta = 4.2414, \quad \gamma = 6.0221 \times 10^{-4}, \quad \tau = 4.392. \]

In a uniform ion size case, we set \(v_i = \bar{v}\) for all \(i\) to get an expression of each \(c_i\) from Eq. \((1)\) as follows:

\[ c_{i}(v)=\frac{c_{i}^{b}e^{-Z_{i}u(v)}}{1 + \gamma\frac{\bar{v}^b}{v_0} \sum\limits_{j=1}^n c_{j}^{b} e^{-Z_{j}u(v)}}, \quad i= 1,2, \ldots, n. \]

Applying the above expressions to Eq. \((2)\), we get the uniform SMPBIC model as the following nonlinear interface boundary value problem,

\begin{equation} \label{SMPBIC model} \left\{\begin{array}{cl} - \epsilon_p\Delta u(\rr) =\alpha \sum\limits_{j=1}^{n_{p}}z_{j} \delta_{\rr_{j}}, & \rr \in D_{p}, \\ \Delta u(\rr) =0, & \rr \in D_{m}, \\ \epsilon_s \Delta u(\rr) + \beta \frac{ \sum\limits_{i=1}^{n}Z_{i} c_{i}^{b}e^{-Z_{i}u(\rr)}}{1+ \gamma \frac{\bar{v}^2}{v_0} \sum\limits_{i=1}^n c_{i}^{b}e^{-Z_{i}u(\rr)}} =0, & \rr \in D_{s}, \\ u(\s^-) = u(\s^+), \quad \ep \frac{\partial u(\s^+)}{\partial \nn_p(\s)} = \es \frac{\partial u(\s^-)}{\partial \nn_p(\s)}, & \s\in\Gamma_{p},\\ u(\s^-) = u(\s^+), \quad \emm \frac{\partial u(\s^+)}{\partial \nn_m(\s)} = \es \frac{\partial u(\s^-)}{\partial \nn_m(\s)} + \tau \sigma, & \s\in\Gamma_{m},\\ u(\s^-) = u(\s^+), \quad \ep \frac{\partial u(\s^-)}{\partial \nn_p(\s)} = \emm \frac{\partial u(t,\s^+)}{\partial \nn_p(\s)}, & \s\in\Gamma_{pm}, \\ u(\s) ={g}(\s), &\s\in \Gamma_{D}, \\ \frac{\partial u(t, \s)}{\partial\nn_b(\s)} = 0, & \s \in \Gamma_{N}, \end{array} \right. \end{equation}

Protein region illustration

Figure 1. An illustration of the simulation box \(\Omega\) partition \(\Omega = D_p \cup D_m \cup D_s\) in a diagram for the cross section \((x=0)\) of \(\Omega\).

Table 1. The notation used in the SMPBIC model.
Parameter Description
\(c_i\) An ionic concentration function of species \(i\) in moles per liter (mol/L)
\(u\) An electrostatic potential function defined by \( \frac{e_c}{k_B T} \Phi, \), with \(\Phi\) being an electrostatic potential in volts.
\(D_p\) A protein region with permittivity constant \(\ep\) and containing an ion channel molecular structure with \(n_p\) atoms.
\(D_m\) A membrande region with permittivity constant \(\emm\) and a membrane surface charge density \(\sigma\).
\(D_s\) A solvent region with permittivity constant \(\es\) containing a solution of \(n\) ionic species.
\(\Gamma_p\) Interface between \(D_p\) and \(D_s\)
\(\Gamma_m\) Interface between \(D_p\) and \(D_m\)
\(\Gamma_{pm}\) Interface between \(D_m\) and \(D_s\)
\(\Gamma_D\) Top & Bottom surfaces of \(\Omega\)
\(\Gamma_N\) Side surfaces of \(\Omega\)
\(\Omega\) A simulation box domain satisfying \(\Omega = D_p \cup D_m \cup D_s \cup \Gamma_m \cup \Gamma_p \cup \Gamma_{pm}\), constructed by \(\Omega = \{(x, y, z) \mid L_{x1} < x < L_{x2}, L_{y1} < y < L_{y2}, L_{z1} < z < L_{z2}\}\).
\(\partial\Omega\) Boundary of \(\Omega\): \(\Gamma_D \cup \Gamma_N\)
\(g\) Boundary value function
\(\rr_j\) Position of atom \(j\) in angstroms (Å)
\(z_j\) Charge number of atom \(j\)
\(c_{i}^{b}\) Bulk concentration of ionic species \(i\)
\(Z_i\) Charge number of ionic species \(i\)
\(v_i\) Ion volume of ionic species \(i\)
\(v_0\) A size scaling parameter: \(v_i = \min \{v_0 \mid i = 1, 2, ..., n\}\)
\(\delta\rr_j\) Dirac-delta distribution at atomic position \(\rr_j\)
\(\nn_p\) Unit outward normal vector of \(D_p\)
\(\nn_m\) Unit outward normal vector of \(D_m\)
\(\nn_s\) Unit outward normal vector of \(D_s\)
\(\nn_b\) Unit outward normal vector of \(\Omega\)

Table 2. Physical constants of the SMBBIC model.
Parameter Value Unit (abbr.) Description
\(\ez\) \(8.854187817 \times 10^{-12}\) Farad/meter (F/m) Permittivity of vacuum
\(e_c\) \(1.602176565 \times 10^{-19}\) Coulomb (C) Elementary charge
\(k_B\) \(1.380648813 \times 10^{-23}\) Joule/Kelvin (J/K) Boltzmann constant
\(N_A\) \(6.0221409 \times 10^{23}\) Mole-1 (mol-1) Avogadro constant

Table 3. Default values and descriptions of model parameters.
Parameter Default Value Unit (abbr.) Description
\(\ep\) \(2.0\) Unitless Protein region dielectric permittivity constant
\(\emm\) \(2..0\) Unitless Membrane region dielectric permittivity constant
\(\es\) \(80.0\) Unitless Solvent region dielectric permittivity constant
\(\sigma\) \(0.2\) Microcoulomb/cm2 (μC/cm2) Surface charge density
\(T\) \(298.15\) Kelvin (K) Absolute temperature

Credits

The SMPBICS web server is developed by Professor Dexuan Xie and his students Max A. Dreher, Andrew Ly and Matthew Stahl, under the support of National Science Foundation (award DMS-2153376). Max and Andrew was also supported by SURF (Support for Undergraduate Research Fellows) awards. This project is based on the SMPBS web design and programming implemented by Jeremy Streich and Yang Xie [3]. The web server is hosted by the University of Wisconsin-Milwaukee L&S IT Office.

Update History

  • 07/27/23 - Start of development
  • 08/07/23 - Homepage re-done.
  • 10/05/23 - Add PDB2PQR as a built-in package to generate PQR files.
  • 10/15/23 - Download PDB file from the OPM database.
  • 11/01/23 - Add Kitware's Paraview Glance [5] , [6].
  • 04/15/24 - Adopt the mesh generation package ICMPv3 [3] , [4]
  • 06/15/24 - Pages finalized.
  • 07/15/24 - Initial release.

References

  1. D. Xie. An Efficient Finite Element Iterative Method for Solving a Nonuniform Size Modified Poisson-Boltzmann Ion Channel Model, Vol. 470, DOI 19.1016/j.jcp.2023.112043, Journal of Computational Physics, pages 111556: 1-15, 2023.
  2. D. Xie, S.H. Audi, and R.K. Dash. A Size Modified Poisson-Boltzmann Ion Channel Model in a Solvent of Multiple Ionic Species: Application to VDAC, Journal of Computational Chemistry, Vol. 41 (3), pages 218-231, 2020.
  3. Liam Jemison, Matthew Stahl, Ranjan K. Dash, and Dexuan Xie. VDAC Solvation Free Energy Calculation by a Nonuniform Size Modified Poisson-Boltzmann Ion Channel Model. arXiv:2407.01569v1 May 2024.
  4. Z. Chao, S. Gui, B. Lu, and D. Xie (corresponding author): Efficient Generation of Membrane and Solvent Tetrahedral Meshes for Finite Element Ion Channel Calculation, International Journal of Numerical Analysis & Modeling, Vol. 19 (6), pages 885-904, 2022.
  5. Ahrens, James, Geveci, Berk, Law, Charles, ParaView: An End-User Tool for Large Data Visualization, Visualization Handbook, Elsevier, 2005, ISBN-13: 9780123875822. Website: https://www.paraview.org/
  6. Glance (https://kitware.github.io/glance/app/): A general purpose standalone web application and a framework for building custom viewers on the web which can involve remote services.