## The Nernst Equation

The past two weeks I have learned a lot about how we look at neurons and, more specifically, the neuronal membrane as electrical circuits. It turns out that the largest factor influencing the membrane potential is the conductance of the ion channels embedded in the membrane. Perhaps a future post will be dedicated to understanding the contribution of ion channels to membrane potential. In order to understand the resting state of a membrane, however, we do not need to worry about the effects of the channels. Last week we learned the derivation for an important equation used to predict membrane potentials for different ions. This is called the Nernst Equation. The information and equations given here are taken from notes of a lecture given by Dr. John White to our cellular biophysics class.

We begin with Fick's First Law, which describes random thermal motion

where phi is the flux of the concentration due to pure diffusion, c is the concentration, and D is the diffusion coefficient.

In one dimensional steady-state, this equation turns into

The diffusion coefficient relies on the lipid solubility of the species being examined. Now, let's add in the flux due to external force

We can substitute

Now we add the two fluxes together

In the case of ions in a neuron, the force is applied via an electric field.

From here, we convert the force per charge of voltage to force per mole by multiplying by zF, where z is the charge per molecule and F is the Faraday constant (about 9.65x10^4 C/mol). Doing this gives us

Therefore, our sum of fluxes is now

Granted, this equation only takes into account an electric field of one-dimension and a concentration gradient of one-dimension. To make this more useful, we will assume the membrane is of width d and that the flux across the membrane is zero.

The diffusion coefficient is related to molar molecular mobility by the relationship

which leads us to

This is equivalent to

Almost done! If we then integrate both sides of the equation over the interval [0,d],

Assuming both c(x) and psi(x) are continuous at the boundaries, this leaves us with

or the Nernst Equation!

[...] potentials for the sodium, potassium, and leak currents, respectively, and are found using the Nernst Equation. and are the maximal conductances of their respective ions, and is the leak conductance, which [...]