Assumptions for Nernst Equation

Question

While I am familiar with some of the conditions for the Nernst Equation, for example:

1) "The membrane is only permeable to one ion even if there are several other ions in the system"

there is an assumption/condition that I have yet to see mentioned (and I am pretty sure that it is an essential one that must be included). That assumption is as follows:

The two solutions separated by the selectively permeable membrane must initially have the same total charge...i.e. there is no initial potential difference between the 2 sides of the membrane

I believe this must be true because of the following 3 cases:

for all 3 cases, only potassium is permeable, and there is a "Left Side" and a "Right Side" on either side of the selectively permeable membrane.

A) Left Side: 25 mM K+, 125 mM Na+ |&| Right Side: 125 mM K+, 25 mM Na+

B) Left Side: 25 mM K+, 10000000000 mM Na+ |&| Right Side: 125 mM K+, 25 mM Na+

C) Left Side: 25 mM K+, 0.00000000001 mM Na+ |&| Right Side: 125 mM K+, 25 mM Na+

Without using my proposed assumption, in all 3 cases, the Nernst equation for potassium would be the same:

(RT)/(ZF) * ln (25/125)

However, I fail to see how case B & C could have the same Nernst potential for potassium as case A when the electrical forces on the left side of the membrane are completely different! Let's use case B as an example. Clearly, the potassium concentration differences exist between the two sides (25 mM vs 125 mM), BUT there is an enormous electrical repulsive force imposed by the sodium (1000000000 mM Na+) so there is no way that the same amount of potassium will pass over to the left side as it would in Case A, right?

Could someone please clarify?

Answer 1

Assumption (1) is not really even true. The Nernst equation is always valid to find the reversal potential for a single ion: that is its purpose. To determine the equilibrium potential for a number of ions, you use the Goldman/GHK equation. It happens to be that the Goldman equation reduces to the Nernst in case of single-ion permeability, which of course never happens for real but is a reasonable approximation in some cases.

As far as your additional assumption, no, that assumption is not required. The Nernst equation doesn't tell you which directions ions flow, it tells you what the reversal potential is for an ion. If you add a voltage from some other source (like your example of adding a bunch of mysterious unaccounted for positive charges outside the membrane) that doesn't change the reversal potential, it changes the voltage. If you put a large positive charge on the left side somewhat like your situation (B), potassium would flow up its concentration gradient towards the Nernst potential for potassium.

Note also that your example is utterly ridiculous: if you want to know what happens when you have 10000000 M Na+ ions together on one side of a membrane you need to consult a physicist, not a biologist. I don't know where you put all the other negative charges, but you're probably about to have a) A very expensive electric bill, and b) a very big hole in your lab.

In real biology, the number of charges on each side of the membrane is similar enough that you can call it equal (see for example 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 any case where someone describes "adding sodium" to one side of the membrane, they don't mean just sodium, they mean sodium as a salt with some negative ion, it's just that the identity of that negative ion doesn't matter if it isn't membrane permeable (relative to the other ions involved).