A finite element iterative solver for a PNP ion channel model with Neumann boundary condition and membrane surface charge

Xie, Dexuan; Chao, Zhen

doi:10.1016/j.jcp.2020.109915

Cited by 6 publications

(33 citation statements)

References 33 publications

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“…In this work, we define σ as a piecewise function as follows:

σ ((), s) = ({, \begin{array}{l} σ_{t}, & bolds \in Γ_{mt}, \\ σ_{b}, & bolds \in Γ_{mb}, \end{array})

where σ b and σ t denote the surface charge density functions defined on the bottom surface Γ mb and top surface Γ mt of membrane, respectively. This membrane surface charge density function is an improvement of the one used in Reference 20. It enables us to deal with more membrane charge cases.…”

Section: Methodsmentioning

confidence: 99%

“…We developed effective iterative schemes for solving the linear finite element Equation () and the nonlinear finite element system (), respectively, and implemented them in Python and Fortran as a software package based on our previous work 20 . We omit the description of these iterative schemes here since these schemes are very similar to those given in Reference 20.…”

Section: Methodsmentioning

confidence: 99%

“…In Step 1, we use the Slotboom variable transformation

c_{i} = e^{- Z_{i} u} {\overset{true¯}{c}}_{i}

to transform J i as

J_{i} = e^{- Z_{i} u} \frac{\partial {\overset{true¯}{c}}_{i} ((), s)}{\partial z}, i = 1, 2, …, n,

where

{\overset{true¯}{c}}_{i}

denotes the i ‐th Slotboom variable, which has been calculated and saved during a search for a PNPic finite element solution 20,24 . Because of the transformation (), we now only need to calculate the partial derivative

\frac{\partial {\overset{true¯}{c}}_{i} ((), s)}{\partial z}

.…”

Section: Methodsmentioning

confidence: 99%

“…To this end, a PNPic model is defined as a boundary value problem, where Dirichlet boundary value conditions are commonly used due to their simplicity in implementation and flexibility in application. Hence, in this work, we develop an improved PNPic model using Dirichlet boundary value conditions and its finite element solver based on our recent work, 20 in which we constructed a PNPic model using Neumann boundary value conditions on the four side surfaces of Ω to avoid the difficulties of selecting boundary value functions. With our new PNPic solver, in this paper, we will carry out numerical experiments to explore the PNPic solution differences caused by Dirichlet and Neumann boundary value conditions.…”

Section: Introductionmentioning

confidence: 99%

“…From biochemistry and physiology it has been known that charges from membrane can have significant effects on electrostatic and ionic concentration. To reflect such effects, we recently added a membrane surface charge density function, σ , to the both sides of a membrane in a PNPic model 20 and a size modified PBE ion channel model 21 . But, due to the interactions of a charged membrane with cations and anions surrounding the membrane, charges on each side surface of a membrane may have different values 22 .…”

Section: Introductionmentioning

confidence: 99%

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An improved Poisson‐Nernst‐Planck ion channel model and numerical studies on effects of boundary conditions, membrane charges, and bulk concentrations

Chao

Xie

2021

J Comput Chem

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In this paper, an improved Poisson‐Nernst‐Planck ion channel (PNPic) model is presented, along with its effective finite element solver and software package for an ion channel protein in a solution of multiple ionic species. Numerical studies are then done on the effects of boundary value conditions, membrane charges, and bulk concentrations on electrostatics and ionic concentrations for an ion channel protein, a gramicidin A (gA), and five different ionic solvents with up to four species. Numerical results indicate that the cation selectivity property of gA occurs within a central portion of ion channel pore, insensitively to any change of boundary value condition, membrane charge, or bulk concentration. Moreover, a numerical scheme for computing the electric currents induced by ion transports across membrane via an ion channel pore is presented and implemented as a part of the PNPic finite element package. It is then applied to the calculation of current–voltage curves, well validating the PNPic model and finite element package by electric current experimental data.

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“…In this work, we define σ as a piecewise function as follows:

σ ((), s) = ({, \begin{array}{l} σ_{t}, & bolds \in Γ_{mt}, \\ σ_{b}, & bolds \in Γ_{mb}, \end{array})

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…In Step 1, we use the Slotboom variable transformation

c_{i} = e^{- Z_{i} u} {\overset{true¯}{c}}_{i}

to transform J i as

J_{i} = e^{- Z_{i} u} \frac{\partial {\overset{true¯}{c}}_{i} ((), s)}{\partial z}, i = 1, 2, …, n,

where

{\overset{true¯}{c}}_{i}

\frac{\partial {\overset{true¯}{c}}_{i} ((), s)}{\partial z}

.…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An improved Poisson‐Nernst‐Planck ion channel model and numerical studies on effects of boundary conditions, membrane charges, and bulk concentrations

Chao

Xie

2021

J Comput Chem

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Integral equation method for the 1D steady-state Poisson-Nernst-Planck equations

Chao,

Geng,

Krasny

2023

J Comput Electron

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An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green’s 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by a combination of midpoint and trapezoid rules, and the resulting algebraic equations are solved by Gummel iteration. Numerical tests for electroneutral and non-electroneutral systems demonstrate the method’s 2nd order accuracy and ability to resolve sharp boundary layers. The method is applied to a 1D model of the K$$^+$$ + ion channel with a fixed charge density that ensures cation selectivity. In these tests, the proposed integral equation method yields potential and concentration profiles in good agreement with published results.

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A finite element iterative solver for a PNP ion channel model with Neumann boundary condition and membrane surface charge

Cited by 6 publications

References 33 publications

An improved Poisson‐Nernst‐Planck ion channel model and numerical studies on effects of boundary conditions, membrane charges, and bulk concentrations

An improved Poisson‐Nernst‐Planck ion channel model and numerical studies on effects of boundary conditions, membrane charges, and bulk concentrations

Integral equation method for the 1D steady-state Poisson-Nernst-Planck equations

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

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