# What's a cells membrane potential without any leak channels?

Consider the following ion concentrations on either side of a cell membrane (in = inside cell, out = outside cell):

$$[\text{Na}^+]_{\text{in}} = 10mM$$,
$$[\text{Na}^+]_{\text{out}} = 142mM$$,
$$[\text{K}^+]_{\text{in}} = 148mM$$,
$$[\text{K}^+]_{\text{out}} = 5mM$$,
$$[\text{Cl}^-]_{\text{in}} = 4mM$$,
$$[\text{Cl}^-]_{\text{out}} = 103mM$$.

Let us further say that there are no leak channels or voltage-gated channels in the cell membrane, and also that ion pumps stop working once they achieve these concentration gradients (i.e. the state is at equilibrium with respect to the ion pump.

There is a greater net positive charge on the inside compared to the outside. This should give me a positive membrane potential when I think about the cell membrane as a capacitor.

I know, Nernst and Goldman equations say otherwise. (For my example it would be zero membrane potential due to zero permeability).

Can someone explain why there is or is not a positive membrane potential, with appropriate sources?

There are already multiple questions with a similar background but none of the answers are satisfactory.

• can you say anything about why the other answers are not satisfactory, and link to those questions? Nov 17, 2020 at 19:48

Goldman equation is not applicable in this case. If you have a non-permeable membrane, all the permeabilities of the ionic species ($$P_x$$) are zero, which leads to an indeterminate form of the argument of the logarithm in the Goldman equation. The logarithm does not equate to $$0$$. Consequently, Goldman equation cannot be used to find the membrane potential ($$E_m$$) in this case.