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1. An efficient finite element iterative method for solving a nonuniform size modified Poisson-Boltzmann ion channel model

   https://doi.org/10.1016/j.jcp.2022.111556  

   Xie, Dexuan (December 2022, Journal of Computational Physics)

   In this paper, a nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is presented as a nonlinear system of an electrostatic potential and multiple ionic concentrations. It mixes nonlinear algebraic equations with a Poisson boundary value problem involving Dirichlet-Neumann mixed boundary value conditions and a membrane surface charge density to reflect the effects of ion sizes and membrane charges on electrostatics and ionic concentrations. To overcome the difficulties of strong singularities and exponential nonlinearities, it is split into three submodels with a solution of Model 1 collecting all the singular points and Models 2 and 3 much easier to solve numerically than the original nuSMPBIC model. A damped two-block iterative method is then presented to solve Model 3, along with a novel modified Newton iterative scheme for solving each related nonlinear algebraic system. To this end, an effective nuSMPBIC finite element solver is derived and then implemented as a program package that works for an ion channel protein with a three-dimensional molecular structure and a mixture of multiple ionic species. Numerical results for a voltage-dependent anion channel (VDAC) in a mixture of four ionic species demonstrate a fast convergence rate of the damped two-block iterative method, the high performance of the software package, and the importance of considering nonuniform ion sizes. Moreover, the nuSMPBIC model is validated by the anion selectivity property of VDAC. 

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2. An efficient finite element solver for a nonuniform size-modified Poisson-Nernst-Planck ion channel model

   https://doi.org/10.1016/j.jcp.2024.113393  

   Xie, Dexuan (December 2024, Journal of Computational Physics)

   This paper presents an efficient finite element iterative method for solving a nonuniform size- modified Poisson-Nernst-Planck ion channel (SMPNPIC) model, along with an SMPNPIC program package that works for an ion channel protein with a three-dimensional crystallographic structure and an ionic solvent with multiple ionic species. In particular, the SMPNPIC model is constructed and then reformulated by novel mathematical techniques so that each iteration of the method only involves linear boundary value problems and nonlinear algebraic systems, circumventing the numerical difficulties caused by the strong nonlinearities, strong asymmetries, and strong differential equation coupling of the SMPNPIC model. To further improve the method’s efficiency, an efficient modified Newton iterative method is adapted to the numerical solution of each related nonlinear algebraic system. In addition, a uniform SMPNPIC model is introduced and solved, numerically, as a special case of the SMPNPIC model. Numerical results for a voltage-dependent anion channel (VDAC) and two mixture solutions of four ionic species, including ATP4− ions, demonstrate the method’s convergence, the package’s high performance, and the importance of considering nonuniform ion size effects. They also partially validate the SMPNPIC model by the anion selectivity property of VDAC. 

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3. Nonlocal nonuniform size modified Poisson–Boltzmann model and efficient finite element solver for protein electrostatics and ion distributions

   https://doi.org/10.1016/j.cnsns.2026.109823  

   Xie, Dexuan (July 2026, Communications in Nonlinear Science and Numerical Simulation)

   This paper presents a novel nonlocal, nonuniform size modified Poisson–Boltzmann (NNSMPB) model that simultaneously accounts for nonuniform ion size effects and nonlocal dielectric correlations of water molecules to predict protein electrostatics and ion distributions. An efficient finite element solver is developed and implemented as a modular software package for proteins with three-dimensional molecular structures and ionic solutions containing multiple species with distinct ion sizes. The solver can also be applied to existing PB-type models, including nonlocal uniform SMPB, nonlocal modified PB, local uniform/nonuniform SMPB, and classical PB models as special cases. Six physiologically relevant ionic solutions comprising up to six species (sodium, potassium, calcium, chloride, magnesium, and bicarbonate) are constructed as benchmarks for numerical tests. Numerical results demonstrate the convergence and efficiency of the solver, confirm the high performance of the package, and validate the significant influence of nonuniform ion size effects on protein electrostatics and ion distribution profiles. 

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4. An efficient finite element solver for a nonlocal size modified Poisson–Boltzmann model of proteins in multi-species ionic solutions

   https://doi.org/10.1016/j.cam.2025.117026  

   Xie, Dexuan; Jemison, Liam; Jiang, Yi (March 2026, Journal of Computational and Applied Mathematics)

   This paper introduces a nonlocal size-modified Poisson–Boltzmann (NSMPB) model and an effi- cient finite element iterative solver for proteins with three-dimensional molecular structures in ionic solutions containing multiple ion species. This model is the first Poisson–Boltzmann variant to simultaneously account for both nonlocal dielectric correlations and ionic size effects. The solver is constructed from a novel solution decomposition to overcome the numerical difficulties caused by the strong singularity, nonlinearity, and nonlocality of the model. It features an efficient modified Newton iterative method, an effective damping parameter selection scheme, and two good initial iteration strategies. To facilitate and streamline protein simulations, the solver is implemented as a software package that integrates a mesh generation tool, a protein data bank file retrieval program, and the PDB2PQR package. Numerical experiments involving an ionic solution with four species, three proteins with up to 11,439 atoms, and irregular interface-fitted tetrahedral meshes with up to 1,188,840 vertices demonstrate the fast convergence of the modified Newton iterative method, the efficiency of the solver, and the high performance of the package. This package provides a valuable tool for protein simulations. 

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5. A Poisson-Nernst-Planck single ion channel model and its effective finite element solver

   https://doi.org/10.1016/j.jcp.2023.112043  

   Xie, Dexuan; Chao, Zhen (May 2023, Journal of Computational Physics)

   A single ion channel is a membrane protein with an ion selectivity filter that allows only a single species of ions (such as potassium ions) to pass through in the “open” state. Its selectivity filter also naturally separates a solvent domain into an intracellular domain and an extracellular domain. Such biological and geometrical characteristics of a single ion channel are novelly adopted in the construction of a new kind of dielectric continuum ion channel model, called the Poisson-Nernst-Planck single ion channel (PNPSIC) model, in this paper. An effective PNPSIC finite element solver is then developed and implemented as a software package workable for a single ion channel with a three-dimensional X-ray crystallographic molecular structure and a mixture of multiple ionic species. Numerical results for a potassium channel confirm the convergence and efficiency of the PNPSIC finite element solver and demonstrate the high performance of the software package. Moreover, the PNPSIC model is applied to the calculation of electric current and validated by biophysical experimental data. 

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