Membrane Transport 🧬

Ion channels, pumps, and the Goldman-Hodgkin-Katz membrane potential

Membrane Potential Vm
−70.1 mV
ENa Nernst (Na⁺)
+67 mV
EK Nernst (K⁺)
−90 mV
ECl Nernst (Cl⁻)
−65 mV
Net Na⁺ Driving Force
−137 mV
Net K⁺ Driving Force
+20 mV
Physics & equations

Goldman-Hodgkin-Katz (GHK) equation gives the membrane potential when multiple ion species are permeable:

Vm = (RT/F) · ln[(PNa[Na⁺]out + PK[K⁺]out + PCl[Cl⁻]in) / (PNa[Na⁺]in + PK[K⁺]in + PCl[Cl⁻]out)]

Nernst equation for a single ion X of valence z: EX = (RT/zF) · ln([X]out/[X]in)

The Na⁺/K⁺-ATPase pump actively moves 3 Na⁺ out and 2 K⁺ in per ATP cycle, maintaining the resting concentration gradients: [Na⁺] 145 mM outside / 12 mM inside; [K⁺] 4 mM / 140 mM; [Cl⁻] 120 mM / 4 mM (typical mammalian neuron).

About this simulation

This simulation computes a real neuron's resting membrane potential live using the Goldman-Hodgkin-Katz (GHK) equation, driven by Na⁺, K⁺ and Cl⁻ ions crossing a membrane with adjustable permeabilities. Animated ions cross through open channels while the Na⁺/K⁺-ATPase pump actively works against their natural gradients, and every readout — membrane potential, per-ion Nernst potentials, and net driving forces — updates in real time as you change the underlying physiology.

🔬 What it shows

V_m = (RT/F)·ln[(P_Na[Na⁺]_out + P_K[K⁺]_out + P_Cl[Cl⁻]_in) / (P_Na[Na⁺]_in + P_K[K⁺]_in + P_Cl[Cl⁻]_out)] — the GHK equation weights each ion's contribution to membrane voltage by its permeability, not just its concentration gradient. The Na⁺/K⁺-ATPase pump counteracts diffusion by actively moving 3 Na⁺ out and 2 K⁺ in per ATP cycle, maintaining the resting gradients ([Na⁺] 145 mM outside / 12 mM inside; [K⁺] 4 mM / 140 mM) that diffusion alone would otherwise erase.

🎮 How to use

Drag P_Na, P_K and P_Cl to change each ion's membrane permeability and watch V_m shift accordingly, Temperature to change the thermal energy term in the GHK equation, and Pump rate to speed up or slow down the Na⁺/K⁺-ATPase. Live readouts show V_m, each ion's individual Nernst equilibrium potential (E_Na, E_K, E_Cl), and the net driving force pushing Na⁺ and K⁺ across the membrane.

💡 Did you know?

A real resting neuron's membrane is roughly 25 times more permeable to K⁺ than to Na⁺, which is precisely why the resting potential (around −70 mV) sits so much closer to E_K (≈−90 mV) than to E_Na (≈+67 mV) — the GHK equation is a weighted average, and K⁺'s far larger permeability dominates the weighting.

Frequently asked questions

Why does the GHK equation weight permeability, not just concentration?

Concentration gradients alone (as in the simpler Nernst equation) only tell you where an ion "wants" to go, but a membrane impermeable to that ion won't let it move regardless of the gradient. The GHK equation multiplies each ion's concentrations by its permeability so that ions the membrane actually lets through dominate the resulting membrane potential.

Why is E_Na positive but E_K negative?

Each ion's Nernst potential is the voltage at which its electrical driving force exactly balances its concentration gradient. Na⁺ is far more concentrated outside the cell, so a positive interior voltage is needed to stop it flowing in; K⁺ is far more concentrated inside, so a negative interior voltage is needed to stop it flowing out.

What does "net driving force" actually mean?

Net driving force is the difference between the actual membrane potential V_m and that ion's own Nernst equilibrium potential. A large driving force on Na⁺ (V_m far from E_Na) means Na⁺ is strongly pulled to flow in whenever channels open; a small driving force on K⁺ (V_m close to E_K) means K⁺ is nearly at equilibrium already.

Why is the Na⁺/K⁺-ATPase pump necessary if diffusion already sets a resting potential?

Every open channel lets Na⁺ leak in and K⁺ leak out along their gradients, which would slowly dissipate the concentration differences that create the resting potential in the first place. The pump actively moves 3 Na⁺ out and 2 K⁺ in per ATP cycle, continuously restoring the gradients that diffusion is always trying to erase.

What happens to V_m if you increase P_Na relative to P_K?

Since the GHK equation weights each ion's contribution by its permeability, increasing P_Na relative to P_K shifts V_m away from the resting potential (near E_K) and toward E_Na — this is exactly what happens during an action potential, when voltage-gated Na⁺ channels open and briefly make the membrane far more Na⁺-permeable.