Equilibrium Potentials II Answer 1

Here are the answers for Figure 7:

• For bulk electroneutrality to apply, the extracellular concentration of chloride ions, which are negatively charged, has to be equal to the concentration of the positively charged sodium and potassium ions, and thus $$\left[\text{Cl}^-\right]_\text{out} = \left[\text{Na}^+\right]_\text{out} + \left[\text{K}^+\right]_\text{out} = 120\ \text{mM} + 5\ \text{mM} = 125\ \text{mM}.$$
• The total number of particles on the outside is thus 125 + 120 + 5 = 250 milliosmolar.
• The equation describing the total number of particles on the inside of the cell (the left hand side) is then $$250\ \text{mOsm} = \left[\text{K}^+\right]_\text{in} + \left[\text{Cl}^-\right]_\text{in} + [P].$$
• The requirement for bulk electroneutrality implies that the intracellular concentration of potassium ions has to be equal to the concentration of chloride ions, or that $$\left[\text{K}^+\right]_\text{in} = \left[\text{Cl}^-\right]_\text{in}.$$
• The fact that the Nernst potentials for chloride and potassium must be equal to one another at equilibrium leads to Equation 7: $$\left[\text{Cl}^-\right]_\text{in}\ \left[\text{K}^+\right]_\text{in} = \left[\text{K}^+\right]_\text{out}\ \left[\text{Cl}^-\right]_\text{out}.$$
• Since we have worked out the concentrations of the potassium and chloride ions on the outside, we can substitute in those values to obtain $$\left[\text{Cl}^-\right]_\text{in}\ \left[\text{K}^+\right]_\text{in} = 5\ \text{mM} \times 125\ \text{mM} = 625\ {\text{mM}^2}.$$
• But since we have just established, based on bulk electroneutrality, that the concentrations of chloride and potassium ions inside the cell are identical, we can further simplify this to $$\left[\text{Cl}^-\right]_\text{in}^2 = 625\ {\text{mM}^2}.$$
• This immediately implies that $$\left[\text{Cl}^-\right]_\text{in} = 25\ \text{mM}.$$
• From bulk electroneutrality, this also implies that $$\left[\text{K}^+\right]_\text{in} = 25\ \text{mM}.$$
• Osmotic balance then implies that $$\begin{align} 250\ \text{mOsm} &= \left[\text{K}^+\right]_\text{in} + \left[\text{Cl}^-\right]_\text{in} + [P] \\ &= 25\ \text{mOsm} + 25\ \text{mOsm} + [P], \end{align}$$ so that $[P] = 200\ \text{mM}$.
• This solves the problem: the membrane is in osmotic balance, has bulk electroneutrality on both sides, and is at an equilibrium.
• What is the value of the equilibrium potential across the membrane? We could calculate either the chloride ion or the potassium ion equilibrium; both should give the same value if we have done our calculations correctly. To use Equation 6, we will calculate the potassium ion equilibrium: $$E_{\text{K}^+} = 58\ \text{mV} \times \log_{10} \frac{5\ \text{mM}}{25\ \text{mM}} = -40.5\ \text{mV}.$$
• To check our work, set up the equivalent Nernst equation for chloride ions; you will immediately see whether we've gotten the right answer or not.