SMPBICS

Calculate the electrostatic solvation energy of a VDAC

The SMPBIC Model

Let a simulation box domain, $Ω$ , 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,

$\begin{matrix} (1) & c_{i} (r) - c_{i}^{b} {[1 - γ \sum_{j = 1}^{n} v_{j} c_{j} (r)]}^{\frac{v_{i}}{v_{0}}} e^{- Z_{i} u (r)} = 0, r \in D_{s}, i = 1, 2, \dots, n, \end{matrix}$

the three Poisson equations,

$\begin{aligned} - ϵ_{p} Δ u (r) = α \sum_{j = 1}^{n_{p}} z_{j} δ_{r_{j}}, & r \in D_{p}, \\ (2) & - ϵ_{m} Δ u (r) = 0, & r \in D_{m}, \\ ϵ_{s} Δ u (r) - β \sum_{i = 1}^{n} Z_{i} c_{i} (r) = 0, & r \in D_{s}, \end{aligned}$

the interface conditions,

$\begin{array}{rc} u (s^{-}) = u (s^{+}), ϵ_{p} \frac{\partial u (s^{-})}{\partial n_{p} (s)} = ϵ_{s} \frac{\partial u (s^{+})}{\partial n_{p} (s)}, & s \in Γ_{p}, \\ (3) & u (s^{-}) = u (s^{+}), ϵ_{m} \frac{\partial u (s^{-})}{\partial n_{m} (s)} = ϵ_{s} \frac{\partial u (s^{+})}{\partial n_{m} (s)} + τ σ, & s \in Γ_{m}, \\ u (s^{-}) = u (s^{+}), ϵ_{p} \frac{\partial u (s^{-})}{\partial n_{p} (s)} = ϵ_{m} \frac{\partial u (s^{+})}{\partial n_{p} (s)}, & s \in Γ_{p m}, \end{array}$

and the mixed boundary value conditions,

$\begin{matrix} (4) & u (s) = g (s), s \in Γ_{D}, \frac{\partial u (s)}{\partial n_{b} (s)} = 0, s \in Γ_{N} . \end{matrix}$

Here $α$ , $β$ , $γ$ , $\bar{v}$ , and $τ$ are defined by

$α = \frac{10^{10} e_{c}^{2}}{ϵ_{0} k_{B} T}, β = \frac{N_{A} e_{c}^{2}}{10^{17} ϵ_{0} k_{B} T}, γ = 10^{- 27} N_{A}, \bar{v} = \frac{1}{n} \sum_{i = 1}^{n} v_{i}, τ = \frac{10^{- 12} e_{c}}{ϵ_{0} k_{B} T} .$

With the values of physical parameters given in Table 2 and the default value of T given in Table 3, we can estimate $α$ , $β$ , $γ$ and $τ$ as follows

$α = 7042.9399, β = 4.2414, γ = 6.0221 \times 10^{- 4}, τ = 4.392 .$

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:

$c_{i} (v) = \frac{c_{i}^{b} e^{- Z_{i} u (v)}}{1 + γ \frac{{\bar{v}}^{b}}{v_{0}} \sum_{j = 1}^{n} c_{j}^{b} e^{- Z_{j} u (v)}}, i = 1, 2, \dots, n .$

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

${\begin{array}{cl} - ϵ_{p} Δ u (r) = α \sum_{j = 1}^{n_{p}} z_{j} δ_{r_{j}}, & r \in D_{p}, \\ Δ u (r) = 0, & r \in D_{m}, \\ ϵ_{s} Δ u (r) + β \frac{\sum_{i = 1}^{n} Z_{i} c_{i}^{b} e^{- Z_{i} u (r)}}{1 + γ \frac{{\bar{v}}^{2}}{v_{0}} \sum_{i = 1}^{n} c_{i}^{b} e^{- Z_{i} u (r)}} = 0, & r \in D_{s}, \\ u (s^{-}) = u (s^{+}), ϵ_{p} \frac{\partial u (s^{+})}{\partial n_{p} (s)} = ϵ_{s} \frac{\partial u (s^{-})}{\partial n_{p} (s)}, & s \in Γ_{p}, \\ u (s^{-}) = u (s^{+}), ϵ_{m} \frac{\partial u (s^{+})}{\partial n_{m} (s)} = ϵ_{s} \frac{\partial u (s^{-})}{\partial n_{m} (s)} + τ σ, & s \in Γ_{m}, \\ u (s^{-}) = u (s^{+}), ϵ_{p} \frac{\partial u (s^{-})}{\partial n_{p} (s)} = ϵ_{m} \frac{\partial u (t, s^{+})}{\partial n_{p} (s)}, & s \in Γ_{p m}, \\ u (s) = g (s), & s \in Γ_{D}, \\ \frac{\partial u (t, s)}{\partial n_{b} (s)} = 0, & s \in Γ_{N}, \end{array}$

Protein region illustration — **Figure 1.** An illustration of the simulation box $Ω$ partition $Ω = D_{p} \cup D_{m} \cup D_{s}$ in a diagram for the cross section $(x = 0)$ of $Ω$ .

**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 $\frac{e_{c}}{k_{B} T} Φ,$ , with $Φ$ being an electrostatic potential in volts.
$D_{p}$	A protein region with permittivity constant $ϵ_{p}$ and containing an ion channel molecular structure with $n_{p}$ atoms.
$D_{m}$	A membrande region with permittivity constant $ϵ_{m}$ and a membrane surface charge density $σ$ .
$D_{s}$	A solvent region with permittivity constant $ϵ_{s}$ containing a solution of $n$ ionic species.
$Γ_{p}$	Interface between $D_{p}$ and $D_{s}$
$Γ_{m}$	Interface between $D_{p}$ and $D_{m}$
$Γ_{p m}$	Interface between $D_{m}$ and $D_{s}$
$Γ_{D}$	Top & Bottom surfaces of $Ω$
$Γ_{N}$	Side surfaces of $Ω$
$Ω$	A simulation box domain satisfying $Ω = D_{p} \cup D_{m} \cup D_{s} \cup Γ_{m} \cup Γ_{p} \cup Γ_{p m}$ , constructed by $Ω = {(x, y, z) ∣ L_{x 1} < x < L_{x 2}, L_{y 1} < y < L_{y 2}, L_{z 1} < z < L_{z 2}}$ .
$\partial Ω$	Boundary of $Ω$ : $Γ_{D} \cup Γ_{N}$
$g$	Boundary value function
$r_{j}$	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: $v_{i} = min {v_{0} ∣ i = 1, 2, . . ., n}$
$δ r_{j}$	Dirac-delta distribution at atomic position $r_{j}$
$n_{p}$	Unit outward normal vector of $D_{p}$
$n_{m}$	Unit outward normal vector of $D_{m}$
$n_{s}$	Unit outward normal vector of $D_{s}$
$n_{b}$	Unit outward normal vector of $Ω$

**Table 2.** Physical constants of the SMBBIC model.
Parameter	Value	Unit (abbr.)	Description
$ϵ_{0}$	$8.854187817 \times 10^{- 12}$	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
$ϵ_{p}$	$2.0$	Unitless	Protein region dielectric permittivity constant
$ϵ_{m}$	$2. .0$	Unitless	Membrane region dielectric permittivity constant
$ϵ_{s}$	$80.0$	Unitless	Solvent region dielectric permittivity constant
$σ$	$0.2$	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

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.
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.
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.
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.
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/
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.

SMPBICS

Welcome to the SMPBICS Web Server

Using the SMPBICS Web Server

The SMPBIC Model

Credits

Update History

References