Energy variational approach to study charge inversion (layering) near charged walls

Hyon, Yun Kyong; Fonseca, James; Eisenberg, Robert S.; Li, Chun

doi:10.3934/dcdsb.2012.17.2725

Cited by 54 publications

(58 citation statements)

References 41 publications

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“…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%

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.

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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Section: Introductionmentioning

confidence: 99%

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.

121

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“…Then, the ionic flux resulting from the averaged force is

{\overset{J}{true}}_{i} MathClass-rel= MathClass-bin- \frac{D_{i}}{k_{B} T} c_{i} MathClass-rel\nabla μ_{i}

for i = 1, ⋯ , N , where

μ_{i} MathClass-rel= \frac{δ E_{normalPNP}^{normaltotal}}{δ c_{i}}

is the chemical potential for the i th ion species. D i is the diffusion constant of i th ion species . Remark The second term in the total energy is equal to

MathClass-op\int ϵ \frac{MathClass-rel| MathClass-rel\nabla φ {MathClass-rel|}^{2}}{2} normald \overset{x}{true}

using the appropriate boundary condition for the electrostatic potential.…”

Section: Dynamics Of Ion Transport: Pnp and Mpnp Systemsmentioning

confidence: 99%

“…Combining with , , we have the PNP equation as follows:

\begin{matrix} leftalign \\ rightalign-odd \frac{\partial c_{i}}{∂t} & align-even = \nabla \cdot (D_{i} (\nabla c_{i} + \frac{z_{i} q}{k_{B} T} c_{i} \nabla φ)), & rightalign-label (2.2) \end{matrix}

\begin{matrix} leftalign \\ rightalign-odd \nabla \cdot MathClass-open(ϵ \nabla φ MathClass-close) & align-even = - (ρ_{0} + \sum_{i = 1}^{N} z_{i} q c_{i}) . & rightalign-label (2.3) \end{matrix}

The detailed derivation of the PNP system can be found in . The PNP system , satisfies the following dissipative energy law:

\begin{matrix} aligned \\ right & left \frac{d}{d t} \int \{k_{B} T \sum_{i = 1}^{N} c_{i} \log c_{i} + \frac{1}{2} (ρ_{0} + \sum_{i = 1}^{N} z_{i} q c_{i}) φ\} d \overset{MathClass-op\to}{x} & right \\ right & left = - \int \{\} \end{matrix}

…”

Section: Dynamics Of Ion Transport: Pnp and Mpnp Systemsmentioning

confidence: 99%

An energetic variational approach to ion channel dynamics

Hyon

Eisenberg

2013

Math. Meth. Appl. Sci.

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We introduce a mathematical model to study the transport of ions through ion channels. The system is derived in the framework of the energetic variational approach, taking into account the coupling between electrostatics, diffusion, and protein (ion channel) structure. The geometric constraints of the ion channel are introduced through a potential energy controlling the local maximum volume inside the ion channel. A diffusive interface (labeling) description is also employed to describe the geometric configuration of the channels. The surrounding bath and channel are smoothly connected with the antechamber region by this label function. A corresponding modified Poisson–Nernst–Planck channel system for ion channels is derived using the variational derivatives of the total energy functional. The functional consists of the entropic free energy for diffusion of the ions, the electrostatic potential energy, the repulsive potential energy for the excluded volume effect of the ion particles, and the potential energy for the geometric constraints of the ion channel. For the biological application of such a system, we consider channel recordings of voltage clamp to measure the current flowing through the ion channel. The results of one‐dimensional numerical simulations are presented to demonstrate some signature effects of the channel, such as the current output produced by single‐step and double‐step voltage inputs. Copyright © 2013 John Wiley & Sons, Ltd.

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“…The inhomogeneous simulation domain is divided into a number of homogeneous regions, each with constant permittivity. This approach is particularly convenient for simulations of ions in electrolytes, because the number of ions is orders of magnitude smaller than the number of solvent molecules, enabling faster computation [3,6,[10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

“…For example, simulations of ions near and within nanometer molecular structures such as ion channels have been performed with a high accuracy with this approach [10,61,67,68].…”

Section: Introductionmentioning

confidence: 99%

Comparison of three-dimensional Poisson solution methods for particle-based simulation and inhomogeneous dielectrics

et al. 2012

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Particle-based simulation represents a powerful approach to modeling physical systems in electronics, molecular biology, and chemical physics. Accounting for the interactions occurring among charged particles requires an accurate and efficient solution of Poisson's equation. For a system of discrete charges with inhomogeneous dielectrics, i.e., a system with discontinuities in the permittivity, the boundary element method (BEM) is frequently adopted. It provides the solution of Poisson's equation, accounting for polarization effects due to the discontinuity in the permittivity by computing the induced charges at the dielectric boundaries. In this framework, the total electrostatic potential is then found by superimposing the elemental contributions from both source and induced charges. In this paper, we present a comparison between two BEMs to solve a boundary-integral formulation of Poisson's equation, with emphasis on the BEMs' suitability for particle-based simulations in terms of solution accuracy and computation speed. The two approaches are the collocation and qualocation methods. Collocation is implemented following the induced-charge computation method of D. Boda et al. [J. Chem. Phys. 125, 034901 (2006)]. The qualocation method is described by J. Tausch et al. [IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 20, 1398 (2001)]. These approaches are studied using both flat and curved surface elements to discretize the dielectric boundary, using two challenging test cases: a dielectric sphere embedded in a different dielectric medium and a toy model of an ion channel. Earlier comparisons of the two BEM approaches did not address curved surface elements or semiatomistic models of ion channels. Our results support the earlier findings that for flat-element calculations, qualocation is always significantly more accurate than collocation. On the other hand, when the dielectric boundary is discretized with curved surface elements, the two methods are essentially equivalent; i.e., they have comparable accuracies for the same number of elements. We find that ions in water--charges embedded in a high-dielectric medium--are harder to compute accurately than charges in a low-dielectric medium.

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Energy variational approach to study charge inversion (layering) near charged walls

Cited by 54 publications

References 41 publications

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

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

An energetic variational approach to ion channel dynamics

Comparison of three-dimensional Poisson solution methods for particle-based simulation and inhomogeneous dielectrics

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