Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

Liu, Weishi; Tu, Xuemin; Zhang, Mingji

doi:10.1007/s10884-012-9278-x

Cited by 52 publications

(37 citation statements)

References 54 publications

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“…This is the final result for the shape of the Coulomb staircase of occupancy as a function of Q f . Figure 2(b) shows qualitative agreement of P Q ( ) f shapes with (13), including the concentration-dependent shift between curves with different P b . We will make a more detailed comparison below.…”

Section: Shapes Of the Coulomb Blockade Oscillationsmentioning

confidence: 57%

“…Ions in solution are surrounded by hydration shells with associated dehydration potential barriers that are also crucial for selectivity in many cases [5][6][7][8][9]. Selectivity frequently involves a 'knock on' mechanism or, more generally, the correlated motion of several ions [10][11][12][13]. The protein residues forming the 'locus' of the selectivity filter are amino acids, of which aspartate (D) and glutamate (E) have negatively charged side chains (Q e 1 f = − where e is the proton charge), lysine (K) has a positively charged side chain (Q e 1 f =+ ), and alanine (A) has a neutral side chain.…”

Section: Introductionmentioning

confidence: 99%

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Coulomb blockade model of permeation and selectivity in biological ion channels

2015

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Biological ion channels are protein nanotubes embedded in, and passing through, the bilipid membranes of cells. Physiologically, they are of crucial importance in that they allow ions to pass into and out of cells, fast and efficiently, though in a highly selective way. Here we show that the conduction and selectivity of calcium/sodium ion channels can be described in terms of ionic Coulomb blockade in a simplified electrostatic and Brownian dynamics model of the channel. The Coulomb blockade phenomenon arises from the discreteness of electrical charge, the strong electrostatic interaction, and an electrostatic exclusion principle. The model predicts a periodic pattern of Ca 2+ conduction versus the fixed charge Q f at the selectivity filter (conduction bands) with a period equal to the ionic charge. It thus provides provisional explanations of some observed and modelled conduction and valence selectivity phenomena, including the anomalous mole fraction effect and the calcium conduction bands. Ionic Coulomb blockade and resonant conduction are similar to electronic Coulomb blockade and resonant tunnelling in quantum dots. The same considerations may also be applicable to other kinds of channel, as well as to charged artificial nanopores.

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Section: Shapes Of the Coulomb Blockade Oscillationsmentioning

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Coulomb blockade model of permeation and selectivity in biological ion channels

2015

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“…Without permanent charges, the flux of one ion species is independent of the other based on the classical PNP models for dilute mixtures (as is well known); for PNP with hard-sphere potentials, the flux of one ion species does depend on the other in the first order of characteristic ionic radius (see, e.g., [27,34,38]) due to ion-to-ion interactions. In this case, for both classical PNP and PNP with hard-sphere potentials studied in the abovementioned papers, only the average quantity H(1) of the channel geometry affects the fluxes.…”

Section: 3mentioning

confidence: 92%

“…A strategy for analyzing this connecting problem of classical PNP models was developed in [14] (in [36] for a general setting), which has been successfully extended to handle PNP with hard-sphere ions in [27,34,38]. The classical PNP system is first reduced to two subsystems: the limiting fast and the limiting slow system.…”

Section: Geometric Singular Perturbation Theory For (21)-(22)mentioning

confidence: 99%

“…Numerical simulations of PNP with approximated models of μ ex k (x) have been conducted for ion channel problems in comparison with experimental data and have shown great successes for properties such as ion permeation and ion selectivity (e.g., [18,19,20,21]). Other important phenomena involving μ ex k (x) such as steric effects, layering, charge inversion, and critical potentials have also been studied [3,16,23,24,25,27,30,31,33,34,38,56].…”

Section: Introductionmentioning

confidence: 99%

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Effects of (Small) Permanent Charge and Channel Geometry on Ionic Flows via Classical Poisson--Nernst--Planck Models

Ji¹,

Liu²,

Zhang³

2015

SIAM J. Appl. Math.

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121

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Abstract. In this work, we examine effects of permanent charges on ionic flows through ion channels via a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP) model. The geometry of the three-dimensional channel is presented in this model to a certain extent, which is crucial for the study in this paper. Two ion species, one positively charged and one negatively charged, are considered with a simple profile of permanent charges: zeros at the two end regions and a constant Q 0 over the middle region. The classical PNP model can be viewed as a boundary value problem (BVP) of a singularly perturbed system. The singular orbit of the BVP depends on Q 0 in a regular way. Assuming |Q 0 | is small, a regular perturbation analysis is carried out for the singular orbit. Our analysis indicates that effects of permanent charges depend on a rich interplay between boundary conditions and the channel geometry. Furthermore, interesting common features are revealed: for Q 0 = 0, only an average quantity of the channel geometry plays a role; however, for Q 0 = 0, details of the channel geometry matter; in particular, to optimize effects of a permanent charge, the channel should have a short and narrow neck within which the permanent charge is confined. The latter is consistent with structures of typical ion channels.Key words. ionic flow, permanent charge, channel geometry AMS subject classifications. 34A26, 34B16, 34D15, 37D10, 92C35 DOI. 10.1137/140992527 Introduction.In this work, we analyze effects of permanent charges on ionic flows through ion channels, based on a quasi-one-dimensional classical PoissonNernst-Planck (PNP) model. The geometry of the three-dimensional channel is presented in this model to a certain extent, which is crucial for the study in this paper. We start with a brief discussion of the biological background of ion channel problems, a quasi-one-dimensional PNP model, and the main concern of our work in this paper.

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Effects of ion sizes on Poisson–Nernst–Planck systems with multiple ions

Lin

2022

Math Methods in App Sciences

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Poisson-Nernst-Planck models for ionic flow through a membrane channel including ionic interactions modeled by the density functional theory. The model includes an arbitrary number of positively changed ions with the same valences and one negatively charged ion and ignores the permanent charge. The model can be viewed as a singularly perturbed differential system; therefore, our analysis is mainly based on the geometric singular perturbation theory.The existence and the uniqueness result for small ion sizes are established, and treating the sizes as small parameters, we also derive an approximation of the individual flux, the I-V (current-voltage) relation and the individual flux difference. Critical potentials are identified, and their roles in characterizing ionic flow properties are studied.

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Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

Cited by 52 publications

References 54 publications

Coulomb blockade model of permeation and selectivity in biological ion channels

Coulomb blockade model of permeation and selectivity in biological ion channels

Effects of (Small) Permanent Charge and Channel Geometry on Ionic Flows via Classical Poisson--Nernst--Planck Models

Effects of ion sizes on Poisson–Nernst–Planck systems with multiple ions

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