Source-linked AI summary
Computing the Field in Proteins and Channels
Bob Eisenberg
TL;DR
The paper reviews how PNP theory describes ion-channel electrodiffusion by coupling charge-generated electrical potential with ion migration and diffusion. It synthesizes the governing equations, their nonlinear numerical solution, and the resulting prediction of channel current and I–V curves for specified bath conditions.
Problem
Ion-channel current requires a framework that jointly represents electrical potential generated by charges and ion transport through concentration and potential gradients.
Method
The review explains PNP equations and solves their coupled Poisson and integrated Nernst–Planck system iteratively, beginning with a constant-field potential estimate.
Results
The framework predicts ion concentrations, fluxes, and channel current, with repeated applied-potential calculations producing I–V curves for specified bath concentrations.
Takeaways & Limitations
PNP treats channel occupancy and conduction as consequences of the self-consistent electrical potential and concentration profiles rather than as independently fixed quantities.
Abstract
from arXiv · showhide
This is an early but comprehensive review of the PNP Poisson Nernst Planck theory of ion channels. Extensive reference is made to the earlier literature. The starting place for this theory of open channels is a theory of electrodiffusion rather like that used previously to describe membranes. The theory uses Poisson's equation to describe how charge on ions and the channel protein creates electrical potential; it uses the Nernst-Planck equations to describe migration and diffusion of ions in gradients of concentration and electrical potential. Combined, these are also the "drift-diffusion equations" of solid state physics, which are widely, if not universally used to describe the flow of current and the behavior of semiconductors.
APPENDIX: PNP THEORY
The PNP framework combines Poisson’s equation for electrical potential with Nernst–Planck equations for ion concentrations and fluxes, solving their nonlinear coupling to predict channel current. Numerical iteration produces current–voltage curves for specified bath concentrations and applied potentials.
- Historical context: The review places PNP in a broader electrodiffusion literature, relating it to Poisson–Boltzmann theory and earlier analytical and one-dimensional treatments of channel electrostatics.Earlier work addressed finite-length cylinders, perturbation expansions, and conditions under which constant-field and constant-concentration-gradient approximations apply.
- Theory and equations: PNP combines Poisson’s equation, which determines electrical potential from charge, with Nernst–Planck equations governing ion migration, diffusion, concentration, and flux.The equations are coupled because concentrations determine potential through Poisson’s equation, while concentrations depend exponentially on potential.
- Theory and equations: The potential difference inside the pore is distinct from the applied trans-membrane potential because the baths are assumed to remain at equilibrium during current flow.Permanent end charges also create Donnan or built-in bath potentials.
- Theory and equations: Assuming a potential profile fixes both the channel’s concentration distribution and its conducting state, because concentration and flux depend exponentially on that profile.The flux expressions include unidirectional tracer fluxes and can also be represented using conditional diffusion probabilities under concentration boundary conditions.
- Numerical solution: The coupled nonlinear equations are solved iteratively by alternating concentration updates from integrated Nernst–Planck equations with potential updates from discretized Poisson’s equation.The procedure starts from a constant-field potential estimate and typically converges in ten iterations in less than one second on the reported workstation.
- Numerical solution: Repeating the converged calculation across applied potentials generates an I–V curve for each specified pair of bath concentration sets.The reported calculation evaluates 100 applied-potential values and takes about 100 seconds per curve.