# Why is it possible to calculate the equilibrium potential of an ion using the Nernst equation from empirical measurements in the cell at rest?

In trying to understand the Nernst and GHK equations, I've hit upon a snag somewhere deep in my understanding of the subject matter.

Scenario 1: when calculating the membrane potential of a living cell at rest we use the Goldman equation. This requires us to measure both the concentrations of the ions and the membrane's permeability to said ions, at rest (at least for the concentrations).

Scenario 2: given two cells separated by a membrane (as in Figure 3 here), with a different concentration of KCl in each cell, and having the membrane permeable only to K ions, we can compute the equilibrium membrane potential using the concentrations of the K ions - after the system has reached equilibrium.

So far so good. Here's my question:

Given a living cell at rest with all its ion concentrations, the Nernst equation is used to compute the equilibrium potential of a single ion, using the concentrations at rest. Why is this allowed? As I understand it, the concentrations as measured are what the cell "decides" them to be - it can change the permeabilities of the channels, or create more pumps, etc. - and then we'll measure different ion concentrations at rest. And these concentrations will change if a cell decides to open or close some channels (which is what happens to create the action potential). But the concentrations that are used as input for Nernst's equations should be measured at the actual, not just the steady state, equilibrium (right?).

So, why can we use the steady state measurements to derive the equilibrium potentials?

Excellent question! The answer is actually a really cool and crucially important detail about membrane potentials and the movement of ions. Basically, the electric force is very strong. You don't actually need to move many ions for the membrane potential to fluctuate dramatically. The number of ions that "move" to establish equilibrium is tiny compared to the actual concentrations. That's why the Nernst and GHK equations are so powerful: what really matters is the relative permeability of different ions and the gross concentration differences, not the changes in their concentration that occur when, for example, voltage-gated channels open during an action potential.

You can calculate just how many ions have to move, which depends on the size of the cell. Here is a nice example from the reference I have linked below:

For a typical cell, 1 microcoulomb of charge (6 × 10^12 monovalent ions) per square centimeter of membrane, transferred from one side of the membrane to the other, changes the membrane potential by roughly 1 V. This means, for example, that in a spherical cell of diameter 10 μm, the number of K+ ions that have to flow out to alter the membrane potential by 100 mV is only about 1/100,000 of the total number of K+ ions in the cytosol.

Therefore to answer your question: why is it allowed to use the resting concentrations? Because the concentrations of the ions don't change by more than 1/100,000 fold with a 100mV swing in potential, and there isn't much impact on the GHK/Nernst equation from a change this small. For example, as the voltage goes from ~-70mV when a fairly typical neuron is at rest (and the membrane is most permeable to K+), up to maybe +30mV at the peak of an action potential (and the membrane is most permeable to Na+), only a "few" sodium ions actually cross the membrane, so maybe the internal sodium concentration changes from 10mM to 10.0001mM. The thing that changes most during an action potential is not the actual concentrations, only the permeability to different ions.

Maybe an additional misconception you have is about the cell "deciding" what the concentrations are. Indeed, if the concentrations of particular ions changed dramatically, it would be necessary to recompute equilibrium potentials. The key concept is that those concentrations don't change much during normal functioning of the cell, and they don't have to change much to observe big changes in membrane potential. An excitable cell uses those swings in membrane potential as the signal, rather than the concentrations of the ions. One exception is calcium, see the note at the end.

Reference:

Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walter, P. (2002). Ion channels and the electrical properties of membranes.

(as a side note, movement of ions does matter over a longer time scale, which is why it is necessary for the Na+/K+ pump to constantly operate to keep the membrane potential steady, and some ions like Ca2+ are at such low concentrations normally that the influx of calcium can be substantial relative to the resting concentration, which is partly why calcium is so powerful as a second messenger and why calcium chelating components of the cell are important)

• Thank you for your prompt response, but unfortunately I don't understand how this relates to the question. Can you elaborate more, please? Mar 7, 2017 at 18:45
• @VladimirGritsenko I made an edit to try to connect it more explicitly to your original question. Does this make sense? If not, perhaps I am misunderstanding your question, can you clarify? Mar 7, 2017 at 19:26
• I think I understand now - since the factor that most affects membrane potential are permeabilities, not concentrations, (and thus concentrations change very little during an AP), we can treat the steady state as a very close approximation to equilibrium? Mar 8, 2017 at 8:22
• @VladimirGritsenko Mostly right - steady state and equilibrium are the same thing, but you are probably thinking about "rest" or a cell in the absence of external input. There is a new steady state/equilibrium voltage any time something happens to change the permeabilities. Ions will flow until that new equilibrium is reached, but it doesn't take the movement of very many ions and it isn't enough to appreciably change the concentrations. Mar 8, 2017 at 19:10