MathBench > Cellular Processes

Nernst Potential

 

The Goldman Equation

Our story so far: The sodium-potassium pump is building up a high concentration of potassium inside your cells, and the cell membranes contain potassium channels which are continually opening and closing, allowing up to a million potassium ions back out every second. BUT, we know that as potassium leaves, the charge also builds up on the outside of the cell, which ... stops them from leaving. Eventually there is a balance, or steady state, and what we want to know is:

WHERE DOES THAT STEADY STATE HAPPEN?

Another way of saying this is:

WHAT IS THE VOLTAGE DIFFERENCE THAT BALANCES DIFFUSION?

Before I show you the actual equation, let's think about what SHOULD affect the equilibrium voltage difference, or equilibrium potential:

So to summarise, we need the concentrations of all of the relevant ions, both inside and outside the cell, somehow including their permeabilities, and involving a ratio (or fraction) of inside to outside. (This reminds me of a cartoon where a student says he's sure he can write his English paper by morning, since the dictionary contains all the words he needs, and he just needs to put them in the right order...)

And now for the right order... (don't worry, I'll show you how to derive a simpler version of this equation in a few screens, for now let's just take it as a gift)

goldman equation

so let's see, we have

Looking good so far. There are also a few features we didn't anticipate, such as the constant (k) and the log. We'll talk about them on the next page...