# Why don't ions have one equilibrium potential? (Nernst equation)

I know that equilibrium potentials are dependent on the ratio of ion concentrations inside and outside of the cell and temperature. I also know that the equilibrium potential is reached when there is no net movement of ions across the membrane, i.e. the chemical and electrical gradients are equal. Why don't the ions move until they reach the same ratio of inside and outside concentration every time? For example, why doesn't ion "x" always diffuse until the ratio of inside to outside concentration is 15 to 1? Thanks in advance!

Good question. The answer (as I understand it) is that if there were only a single type of permeable ion, then there would be only one equilibrium ratio, and it would depend on the transmembrane potential.

However, in real life, there (a) is usually more than one membrane permeable ion, and more importantly (b) the transmembrane potential is not a fully independent variable but is affected by the transmembrane permeation of all those ions.

Imagine sovling a single instance of the Nernst equation when you know the electrical potential $\Delta \psi$:

• $\frac{\Delta G}{F}=\text{ion motive force}=0=-m \Delta \psi+ RT\ln{\frac{[X_{out}]}{[X_{in}]}}$

If you know $\Delta \psi$ (and $R$ and $T$) and you are at equilibrium (so $\Delta G$ is 0), then the ion concentration ratio will always go to the same value, as you suggested in your question.

But now suppose there are many equilibrating ions, say sodium, potassium, protons, and chloride. And suppose you don't know $\Delta \psi$. Now, there is a Nernst equation for each species, as well as Gauss's law that $\nabla \centerdot \bf{E} = \frac{\rho}{\epsilon}$. And probably also total material balances like $V_{out}[X_{out}] + V_{in}[X_{in}] = N_{tot}$ for each ion. To fully solve the system to determine equilibrium, all equations have to be solved at once! I hope you can see that (a) doing this is complicated and (b) if parameters like the volume of the cell $V_{in}$ or even more likely the total amount $N_{tot}$ of ions in the system changes for one ion, it could completely change the ratio $\frac{[X_{out}]}{[X_{in}]}$ for other ions. That's why in general, the ratio is not the same every time.

• Even when there is only one permeable ion, my professors have led me to believe that the equilibrium potential for that one ion is not a constant. Also, the Goldman equation is used when there are multiple ions that are permeable. – Penguiness Feb 11 '15 at 5:29
• The Goldman equation is a more general case that can be used when ion fluxes are not zero, i.e. the membrane is not at equilibrium. For equilibrium calculations, the Nernst equation applies. – Curt F. Feb 11 '15 at 5:38
• Looking at the wiki page for the Goldman equation, I now see that the membrane reaches its reversal potential (net current from all ions is 0) but doesn't necessarily mean that the ions themselves are at equilibrium. Can you see any way for there to be multiple equilibrium potentials for a specific ion when only that ion is permeable? – Penguiness Feb 11 '15 at 17:33