# 6.021/Notes/2006-10-04

### From OpenWetWare

## Contents |

## Ion Transport

- Major constituents of cells
- important functions
- charge is substrate for neural communication
- every charged particle (in principle) affects every other ion
- more complicated than other mechanisms

## Mechanisms

2 distinct mechanisms of diffusion and drift

### Diffusion

Given by Fick's law

### Drift

- Effect of electrical forces on montion of charged particles.
- Electric Field (vector field)
*E*(*x*,*t*) - force on particle
*f*_{p}=*Q**E*(*x*,*t*) =*z*_{n}*e**E*(*x*,*t*) where*z*_{n}is valence and C. - Motions of small particles in water are viscosity dominated (Stokes 1855)

Forces | Size scale | Time scale |

inertial (F=ma) | radius^{3}
| acceleration |

viscosity | radius
| velocity |

where *u*_{p} is mechanical mobility in units of velocity/force, *u*_{n} is the molar mechanical mobility and *f* becomes the force on a mole of particle.

For charged particles:
*v* = *u*_{n}*z*_{n}*e**N*_{A}*E*(*x*,*t*) = *u*_{n}*z*_{n}*F**E*(*x*,*t*) (*F* = *e**N*_{A} which is Faradya's number) = charge/mole about 96500 C/mol.

*D*_{n} = *u*_{n}*R**T*: Einstein's relation

Flux due to drift:

where (electric field depends on the potential gradient)

The flux of ions is the current density given by *J*_{n} = *z*_{n}*F*φ_{n} This is in units of current/area and is easier to measure than flux.

### Combined transport

Combining diffusion and drift to get **Nernst-Planck Equation**:

Note that this is really just a combination of Fick's and Ohm's Laws.

**Continuity:** (needed to solve equations just like in other transport mechanisms)

or equivalently

Unlike diffusion, also need one more equation for ψ but this electric potential depends on all particles.

From Gauss' law: where ε is the permitivity and ρ is the charge density.

This leads to **Poisson's Equation**

## Electroneutrality

- In a solution with some charge, after some time, all charges go to the edges away from each other.
- τ
_{r}is the relaxation time and is on the order of nanoseconds for physiological salines - Similarly, in space, a region around the charge is formed that negates the charge. This is known as the Debye layer and has a thickness of around a nanometer.
- Thus for times much greater than the relaxation time and distances much greater than the Debye distance, we can assume electroneutrality of the solution.