A conservative finite difference scheme for Poisson–Nernst–Planck equations

Flavell, Allen; Machen, Michael; Eisenberg, Robert S.; Kabre, Julienne; Li, Chun; Li, Xiaofan

doi:10.1007/s10825-013-0506-3

Cited by 81 publications

(63 citation statements)

References 30 publications

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“…In addition to the BL analysis, many (conservative) numerical schemes have been developed for PNP systems, such as finite element method [17], finite difference scheme [18] and finite volume method [19,20], in one or high-dimensional spaces [21,22]. Due to the presence of BL, computation of the PNP system needs to accurately capture the behaviour of solution in BL.…”

Section: Introductionmentioning

confidence: 99%

Electroneutral models for dynamic Poisson-Nernst-Planck systems

2018

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The Poisson-Nernst-Planck (PNP) system is a standard model for describing ion transport. In many applications, e.g., ions in biological tissues, the presence of thin boundary layers poses both modelling and computational challenges. In this paper, we derive simplified electro-neutral (EN) models where the thin boundary layers are replaced by effective boundary conditions. There are two major advantages of EN models. First of all, it is much cheaper to solve them numerically. Secondly, EN models are easier to deal with compared with the original PNP, therefore it is also easier to derive macroscopic models for cellular structures using EN models.Even though the approach is applicable to higher dimensional cases, this paper mainly focuses on the one-dimensional system, including the general multi-ion case. Using systematic asymptotic analysis, we derive a variety of effective boundary conditions directly applicable to EN system for the bulk region. This EN system can be solved directly and efficiently without computing the solution in the boundary layer. The derivation is based on matched asymptotics, and the key idea is to bring back higher order contributions into effective boundary conditions. For Dirichlet boundary conditions, the higher order terms can be neglected and classical results (continuity of electrochemical potential) are recovered. For flux boundary conditions, however, neglecting higher contribution leads to physically incorrect solutions since they account for accumulation of ions in boundary layer. The validity of our EN model is verified by several examples and numerical computation. In particular, our EN model is much more efficient than the original PNP model when applied to the computation of membrane potential. Implemented with the Hodgkin-Huxley model, the computational time for solving the EN model is significantly reduced without sacrificing the accuracy of the solution due to the fact that it allows for relatively large mesh and time step sizes.

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Section: Introductionmentioning

confidence: 99%

Electroneutral models for dynamic Poisson-Nernst-Planck systems

2018

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“…Biosensor models are typically based on lumped element circuits fitted to experiments which provide limited insight in the device physics. On a more physical ground, the Poisson-Boltzmann (PB) [6] equations for steady-state (DC) conditions and the Poisson-Nerst-Planck (PNP) equations for the sinusoidal small-signal (AC) regime [7]. Oxide-electrolyte interfaces are treated according to the site-binding model [6], [8] and corrections are sometimes included to account for steric effects stemming from the finite size of anions and cations that adhere to the interface.…”

Section: Introductionmentioning

confidence: 99%

Derivation and Numerical Verification of a Compact Analytical Model for the AC Admittance Response of Nanoelectrodes, Suitable for the Analysis and Optimization of Impedance Biosensors

Pittino

Scarbolo

Widdershoven

et al. 2015

IEEE Trans. Nanotechnology

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This paper presents a compact analytical model for the AC response of nanoelectrode-based impedimetric biosensors to dielectric nanoparticles suspended in the electrolyte. The model highlights the functional dependence of the impedance change on the nanoparticle and the system geometrical and physical parameters. The model is carefully verified by means of 2-D simulations carried out with an ad hoc numerical solver of the Poisson–Nernst–Planck (Poisson-Drift-Diffusion) equations. The results can be useful to determine optimum detection conditions for impedimetric nanobiosensors, and to interpret experimental results

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“…Classical Poisson-Nernst-Planck (PNP) system has been widely applied to model ionic transport in biological setting as well as other areas [10,28,18]. Various analysis and computation [37,38,8] regarding this system have been attempted in the literature.…”

Section: Introductionmentioning

confidence: 99%

Selectivity of the KcsA potassium channel: Analysis and computation

et al. 2019

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Potassium (K + ) channels regulate the flux of K + ions through cell membranes and plays significant roles in many physiological functions. This work studies the KcsA potassium channel, including the selectivity and current-voltage (IV) relations. A modified Poisson-Nernst-Planck system is employed, which include the size effect by Bikerman model and solvation energy by Born model. The selectivity of KcsA for various ions (K + , Na + , Cl − , Ca 2+ and Ba 2+ ) is studied analytically, and the profiles of concentrations and electric potential are provided. The selectivity is mainly influenced by permanent negative charges in filter of channel and the ion sizes. K + is always selected compared with Na + (or Cl − ), as smaller ion size of Na + causes larger solvation energy. There is a transition for selectivity among K + and divalent ions (Ca 2+ and Ba 2+ ), when negative charge in filter exceeds a critical value determined by ion size. This explains why divalent ions can block the KcsA channel. The profiles and IV relations are studied by analytical, numerical and hybrid methods, and are cross-validated. The results show the selectivity of the channel and also the saturation of IV curve. A simple strategy is given to compute IV relations analytically, as first approximation. The numerical method deals with general structure or parameters, but the limitations and difficulties of pure numerical 1 arXiv:1902.04634v1 [physics.bio-ph] 12 Feb 2019 simulation are also pointed out. The hybrid method provides IV relations most effectively for comparison. The reason for saturation of IV relation is illustrated, and the IV curve shows agreement with the profile and scale of experimental results.

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A conservative finite difference scheme for Poisson–Nernst–Planck equations

Cited by 81 publications

References 30 publications

Electroneutral models for dynamic Poisson-Nernst-Planck systems

Electroneutral models for dynamic Poisson-Nernst-Planck systems

Derivation and Numerical Verification of a Compact Analytical Model for the AC Admittance Response of Nanoelectrodes, Suitable for the Analysis and Optimization of Impedance Biosensors

Selectivity of the KcsA potassium channel: Analysis and computation

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