The Simulation of Ionic Charge Transport in Biological Ion Channels: An Introduction to Numerical Methods

Saraniti, Marco; Aboud, Shela; Eisenberg, Robert S.

doi:10.1002/0471780367.ch4

Cited by 32 publications

(31 citation statements)

References 191 publications

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“…Whatever the reason, it is a fact that nearly all biology occurs in ultrafiltrates of blood called plasmas, in which ions move much as they move in gaseous plasmas, or as quasi-particles move in semiconductors [21,22,23,24]. 16,38,49,4,17,66,36,37,40,2,20,31,55,56,1,13,70,65,29,30,43,7]. Computational and experimental experience with a variety of PNP-like systems shows that the existing mathematical analysis is unsatisfactory.…”

mentioning

confidence: 99%

Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges

Eisenberg¹,

Liu²

2007

SIAM J. Math. Anal.

130

218

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Abstract. Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-NernstPlanck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of piecewise constant permanent charge, assuming the Debye number is large, because the electric field is so strong compared to diffusion. Reservoirs are represented by the outer regions with permanent charge zero. If the reciprocal of the Debye number is viewed as a singular parameter, the PNP system can be treated as a singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow systems in geometric singular perturbation theory). A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. Application of the exchange lemma from geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the solution of the singular boundary value problem near each singular orbit. A set of simultaneous equations appears in the construction of singular orbits. Multiple solutions of such equations in this or similar problems might explain a variety of multiple valued phenomena seen in biological channels, for example, some forms of gating, and might be involved in other more complex behaviors, for example, some kinds of active transport. Key words. singular perturbation, boundary layers, internal layers AMS subject classifications. 34A26, 34B16, 34D15, 37D10, 92C35DOI. 10.1137/060657480 1. Introduction. Electrodiffusion, the diffusion and migration of electric charge, plays a central role in a wide range of our technology and science [53,11,54,14,15,67,41]: semiconductor technology controls the migration and diffusion of quasi-particles of charge in transistors and integrated circuits [75,62,71], chemical sciences deal with charged molecules in water [11,19,8,26,9,10], all of biology occurs in plasmas of ions and charged organic molecules in water [3,16,33,72]. It is no coincidence that the physics of electrodiffusion is of such general importance: systems of moving charge have a richness of behavior that can be sometimes easily controlled by boundary conditions [67,71], and the goal of technology (and much of physical science) is to control systems to allow useful behavior.Control is important to the medical and biological sciences as well. Medicine seeks to control disease and help life. Evolution controls life by selecting those organisms that successfully reproduce. Organisms control their internal environment and external behavior to make reproduction possible, often using electrodiffusion for the mechanism of control [72,33]. Whatever the reason, it is a fact that nearly all biology occurs in ultrafiltrates of blood called plasmas, in which ions move much as they move in gaseous plasmas, or ...

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mentioning

confidence: 99%

Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges

Eisenberg¹,

Liu²

2007

SIAM J. Math. Anal.

130

218

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“…(1) to estimate current through a channel, instead of counting particles that cross boundaries [14,[37][38][39][40][41].…”

Section: Discussionmentioning

confidence: 99%

Shockley-Ramo theorem measures conformation changes of ion channels and proteins

Eisenberg

Nonner

2007

J Comput Electron

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“…The topic of ion transfer free energy across the liquid/liquid interface is closely related to the problem of ion channels in biological membranes, 312,313 but this is outside the scope of this chapter.…”

Section: 298mentioning

confidence: 99%

Reactivity and Dynamics at Liquid Interfaces

Benjamin¹

2015

Reviews in Computational Chemistry

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The Simulation of Ionic Charge Transport in Biological Ion Channels: An Introduction to Numerical Methods

Cited by 32 publications

References 191 publications

Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges

Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges

Shockley-Ramo theorem measures conformation changes of ion channels and proteins

Reactivity and Dynamics at Liquid Interfaces

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