Ions in solutions and protein channels

A problem presented at the UK MMSG Oxford 2005.

Presented by:
Prof Robert Eisenberg (Department of Molecular Biophysics and Physiology, Rush University Medical Center)
J Chapman, R Eisenberg, J Norbury, C Please, G Richardson

Problem Description

Ion channels are proteins of enormous biological and clinical importance because they control such a wide range of cellular function. Channels are also much simpler than many other proteins: they perform an important part of their function without changing structure (on the biological time scale > 0.1 msec, say) using only electrodiffusion. No conformation changes or chemistry (specifically, no change in the shape of electron orbitals) are involved. For example, channels select between hard charged spheres (ions) that are chemically very similar. The challenge is to understand this selectivity using the usual language of physics without invoking ill-defined properties of the channel protein.

The key to the solution, in my view, is the treatment of the electrical and chemical properties of concentrated solutions of salt. The permanent (i.e., fixed) charge on the channel protein ensures a very large local concentration of mobile ions, say 20 Molar (pure water is 55 Molar). Those ions form a crowded plasma of variable number density and our challenge is to determine by theory, computation, and simulation the free energy per mole and conductivity of that crowded plasma.

Preliminary work using physical theories of concentrated salt solutions shows that quite simple treatments do surprisingly well. The challenge is to replace those physical theories with sound mathematics and to extend the mathematics to other related problems concerning plasmas of ions in proteins and solution.

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Study Group Report

Flow of charge within an ion channel is considered. Continuum models which describe the distribution of mobile ionic charge are presented with dimensional analysis used to derive simplified solutions. The question of accounting for the transition from regions where continuum models are adequate to regions where individual molecules must be considered is also discussed.

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Follow-Up Activities

The following publications have been written as a result of this problem:

Singular perturbation analysis of the steady-state Poisson–Nernst–Planck system: Applications to ion channels
A Singer, D Gillespie, J Norbury & RS Eisenberg (2008)
European Journal of Applied Mathematics 19 (5), 541–560.
A Poisson–Nernst–Planck Model for Biological Ion Channels—An Asymptotic Analysis in a Three-Dimensional Narrow Funnel
A Singer & J Norbury (2009)
SIAM Journal on Applied Mathematics 70 (3), 949–968.

The following follow-up meetings have occured to continue work on aspects of this problem:

2006 Ion Channels Followup Meeting
Thursday 30th to Friday 31st March 2006, University of Oxford