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Why is there a negative membrane potential if a cell is at Gibbs-Donnan equibrilium?

Physics Asked by jte on April 2, 2021

So I was studying the Gibbs-Donnan equibrilium, which can exist in a cell when there are impermeable negatively charged proteins $mathrm{Pr^-}$ inside the cell and permeable cations $mathrm{K^+}$ and anions $mathrm{Cl^-}$ both inside and outside the cell.

Now I understand that these permeable ions move in and out of the cell and the driving force that causes this is determined by the concentration gradient of an ion and the electrical gradient across the membrane.

So, when the cell gets to the Gibbs-Donnan equibrilium, on each side of the membrane electrical neutrality is preserved:
$$
mathrm{[K^+]_{in} = [Cl^-]_{in} + [Pr^-]_{in}}
$$

$$
mathrm{[K^+]_{out} = [Cl^-]_{out}}
$$

Also, the products of permeable ion concentrations are the same on both sides:
$$
mathrm{[K^+]_{in} cdot [Cl^-]_{in} = [K^+]_{out} cdot [Cl^-]_{out}}
$$

It is stated that this will lead to a negatively charged membrane because of the impearable proteins inside the cell. Why is this the case? Shouldn’t there be a negative net charge inside the cell in relation to the outside of the cell for a negative membrane potential to occur?

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