Pump-Leak Membrane Potential: Goldman Equation, Na/K-ATPase, Donnan Swelling and Cell Volume Lab (2D)

Presets

Live readouts

Graphs

Membrane potential with E_K and E_Na

Intracellular Na+, K+ and Cl- concentrations

Relative cell volume (lysis threshold in red)

Current-voltage relation: leaks, pump and net

Numerical checks

running...

How to use

The cell shows K⁺ (purple), Na⁺ (yellow) and Cl⁻ (green) ions, leak channels, Na/K-ATPase pumps (orange), the charge layers and a micro-electrode reading the potential. The ladder (or bar on a phone) shows V relative to the three Nernst potentials. Dot densities follow the real concentrations; the transit rates follow the computed fluxes. Pick a preset or change any slider; the graphs show the time course, the ion concentrations, the volume and the current-voltage relation whose zero crossing is the resting potential. Try: (1) Normal cell: V is about −73 mV, between E_K and E_Na. (2) Ouabain: the cell swells and depolarises towards 0 mV over a few hours. (3) High external K⁺. (4) Cold: the pump slows and the cell gradually swells. (5) Raise the Na⁺ permeability: V moves towards E_Na. Space pauses, R restarts.

The science

This is an educational pump-leak model of an animal cell (in the spirit of Tosteson and Hoffman, and of Hodgkin and Katz). The numbers are round textbook values, not measurements of a particular cell.

Leaks. Na⁺, K⁺ and Cl⁻ cross the membrane through channels. For a constant field the outward flux of an ion of valence z is the Goldman-Hodgkin-Katz (GHK) flux

J = P x (c_i − c_o e^−x) / (1 − e^−x), x = zFV/RT

which vanishes at the Nernst potential E = (RT/zF) ln(c_o/c_i). RT/F is 26.7 mV at 37 °C.

Pump. The Na/K-ATPase exports 3 Na⁺ and imports 2 K⁺ per ATP, a net outward charge of one per cycle (electrogenic). Its rate depends on internal Na⁺ (Hill 3) and external K⁺ (Hill 2), scales with temperature with Q₁₀ = 3, and is blocked by ouabain. The slider maximum pump rate is the rate at the reference state (12 mM Na⁺, 5 mM K⁺, 37 °C).

Voltage. The membrane capacitance is so small that the potential settles within microseconds: V is the value at which the net charge flux is zero, P_NaJ_Na + P_KJ_K − P_ClJ_Cl + J_pump = 0 (found by bisection). With the pump at steady state, Na⁺ and K⁺ leaks balance the pump in the ratio 3:2, which gives the closed-form Mullins-Noda voltage equation

V = (RT/F) ln[ (r P_KK_o + P_NaNa_o + P_ClCl_i) / (r P_KK_i + P_NaNa_i + P_ClCl_o) ], r = 3/2

(r = 1 would be the plain GHK equation). Volume. Water crosses far faster than ions, so the cell is always in osmotic balance: its volume is the amount of solute divided by the bath osmolarity. The impermeant anions X⁻ (proteins, nucleotides) cannot leave, so without the pump Na⁺, K⁺ and Cl⁻ leak towards a Donnan distribution and the cell swells without limit (Donnan effect): the pump is a volume regulator as well as the source of the gradients. The page marks lysis above 1.6 times the starting volume.

Numerical checks run in the page. (1) Zero net charge flux at the solved V; (2) after 400 simulated hours V equals the Mullins-Noda formula within 0.05 mV (an independent closed form); (3) Na⁺ and K⁺ amounts have stopped changing; (4) charge neutrality Na + K − Cl − X is conserved; (5) internal osmolarity equals the bath osmolarity; (6) with only K⁺ permeable and no pump V equals the K⁺ Nernst potential; (7) after blocking the pump for 60 h the cell has swollen above 1.3.

Honest simplifications. Constant surface area and permeabilities (no voltage-gated channels, so no action potentials: see the Hodgkin-Huxley labs), one lumped impermeant anion, no Ca²⁺, no cotransporters (such as KCC2, NKCC1), HCO₃⁻ lumped into the fixed external anion, a simple pump rate law, and the mechanical strength of the membrane reduced to a single lysis threshold. Time is in hours and the permeabilities are per hour with the surface-to-volume ratio absorbed.