E9: Nernst and Donnan, and why cells need pumps

Week 14. Due Wed 14 Apr 23:59, reviewed Thu 15 Apr

About 2 hours. Theory: Donnan equilibrium and Membrane potentials and ion pumps. Use \(k_BT/e = 26.7\) mV (310 K).

1. How much trapped charge can a cell afford?

Outside a cell: 140 mM Na⁺, 10 mM K⁺, 150 mM Cl⁻. Inside, trapped macroions carry a charge density \(\rho_m\) (in mM of elementary charges, negative).

  1. Write a Python function that solves the Donnan equations for given outside concentrations and \(\rho_m\), and returns the inside concentrations, the membrane potential and the osmotic excess inside. Check that \(\rho_m/e = -125\) mM gives the result in the notes.
  2. Plot \(\Delta\mathcal V\) and the osmotic pressure \(\Delta P\) against \(-\rho_m/e\) from 0 to 200 mM.
  3. A lipid membrane ruptures at a tension of about 10 mN/m. With the Laplace pressure \(\Delta P = 2\gamma/R\), what pressure can a cell of radius 10 µm withstand? What is the largest trapped charge density it could have without pumping? Compare with the 125 mM of a real cell.
  4. Show that for small \(\rho_m\) the osmotic excess is \(\Delta c\approx \rho_m^2/(4c_s)\), where \(c_s = 150\) mM is the outside salt concentration. Why is it quadratic?

2. The pump–leak steady state

Use the squid axon data in the notes, and \(g_\text{K} = 25\,g_\text{Na}\).

  1. Derive the resting potential with a 3 Na⁺ : 2 K⁺ pump (notes) and calculate \(\Delta\mathcal V\).
  2. Some pumps exchange one ion for one ion and move no net charge. What would the resting potential be with a 1 Na⁺ : 1 K⁺ pump? Does the stoichiometry matter much for \(\Delta\mathcal V\)? What does it matter for?
  3. Calculate the free energy needed for one cycle of the 3:2 pump at \(\Delta\mathcal V = -60\) mV, in \(k_BT\), and compare it with the 19 \(k_BT\) from one ATP. What is the thermodynamic efficiency?
  4. Why must chloride, which is not pumped in this model, have its Nernst potential equal to the resting potential?