Ions in channels and bulk: mathematics and molecular biology

Abstract: Life and most of chemistry occurs in ionic solutions, but ionic solutions have only recently been recognized as the complex fluids that they are. The molecular view shows ions interacting with surrounding water and nearby ions. Everything is correlated in a complex way because ions and water have diameters comparable to their interaction length. The molecular scale shows only a small part of the correlation enforced by electrodynamics. Current defined as Maxwell did to include the ethereal current is exactly conserved, and therefore correlated, over all scales reaching to macroscopic boundary conditions some 1e9 larger than atoms crucial in batteries and nerve cells.

Jinn Liang Liu and I have built a molecular field theory PNPB Poisson Nernst Planck Bikerman that deals with water as molecules and describes local interactions with a steric potential that depends on the volume fraction of molecules and voids between them. The correlations of electrodynamics are described by a fourth-order differential operator that gives (as outputs) ion-ion and ion-water correlations; the dielectric response (permittivity) of ionic solutions; and the polarization of water molecules, all using a single correlation length parameter. The theory fits experimental data on activity and differential capacitance in ionic solutions of varying composition and content, including mixtures. Potassium channels, Gramicidin, L-type calcium channels, and the Na/Ca transporter are computed in three dimensions from structures in the Protein Data Bank.

Numerical analysis faces challenges:

1. Geometric singularities of molecular surfaces
2. strong electric fields (100 mV/nm) and resulting exponential nonlinearities, and the
3. enormous concentrations (> 10 M) often found where ions are important, for example, near electrodes in batteries, in ion channels, and in active sites of proteins.
4. Wide ranging concentrations of 1e-1 to 1e-8 M in and near almost every protein in biological cells make matters worse.

Challenges have been overcome using methods developed over many decades by the large community that works on the computational electronics of semiconductors.

mathematical physicsgeneral mathematics

Audience: researchers in the topic

( paper | slides )

Comments: meeting ended

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Global Seminar on Mathematical Modeling and Applications

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