Membrane Transport: How Ion Channels Turn Concentration Gradients Into Voltage

A cell membrane is a selective wall, not a sealed box

A cell membrane is a lipid bilayer that is essentially impermeable to charged ions on its own — but it is studded with protein ion channels that open and close selectively, letting specific ions (Na⁺, K⁺, Ca²⁺, Cl⁻) cross at controlled rates. Ions do not sit still on either side either: they are actively pumped, most importantly by the sodium-potassium pump, which spends ATP to push 3 Na⁺ out for every 2 K⁺ it brings in, maintaining steep concentration gradients that the cell then uses as a stored form of electrical energy.

The Nernst equation: the voltage one ion alone would create

If a membrane were permeable to only a single ion species, that ion would diffuse down its concentration gradient until the resulting charge separation created an electric field strong enough to stop any further net flow — a genuine equilibrium where the electrical and chemical driving forces exactly cancel. The voltage at which that happens is the ion's equilibrium (Nernst) potential:

E_ion = (RT / zF) · ln([ion]_out / [ion]_in)

R = gas constant, T = temperature, z = ion charge, F = Faraday constant
at 37°C, for a monovalent ion:  E_ion ≈ 61.5 mV · log₁₀([out]/[in])

For a typical neuron, E_K ≈ −90 mV (potassium is concentrated inside, so it tends to leave, leaving the inside negative) and E_Na ≈ +60 mV (sodium is concentrated outside, so it tends to rush in). Neither of these is the actual resting membrane potential, because real membranes are never permeable to just one ion.

Goldman-Hodgkin-Katz: what happens with several ions at once

The real resting potential is a compromise set by the relative permeabilities of all the ions the membrane lets through at once, weighted by their concentration gradients — the Goldman-Hodgkin-Katz (GHK) equation:

V_m = (RT/F) · ln[ (P_K[K⁺]_out + P_Na[Na⁺]_out + P_Cl[Cl⁻]_in)
                    / (P_K[K⁺]_in  + P_Na[Na⁺]_in  + P_Cl[Cl⁻]_out) ]

P_ion = relative membrane permeability to that ion (note Cl⁻ terms are inverted — anion)

A resting neuron's membrane is far more permeable to K⁺ than to Na⁺, so V_m sits close to E_K (around −70 mV) but not exactly at it — the small leak of Na⁺ pulls it slightly positive of the pure-potassium equilibrium. Set P_Na to zero in the GHK formula and it collapses back exactly to the Nernst potential for potassium alone, showing the Nernst equation is simply the single-ion limit of the more general GHK result.

The action potential: when the permeabilities themselves flip

An action potential is not a change in concentration — it happens far too fast for that — it is a rapid, transient change in which permeability dominates. A depolarising stimulus opens voltage-gated Na⁺ channels; as Na⁺ permeability spikes, the GHK equation swings V_m sharply toward E_Na (the upstroke). Those channels then inactivate while voltage-gated K⁺ channels open, driving permeability back toward K⁺ dominance and pulling V_m back down toward E_K, typically overshooting slightly (the after-hyperpolarisation) before the resting balance of leak channels and the sodium-potassium pump restores the starting state.

rest:      P_K >> P_Na   →  V_m near E_K (~ −70 mV)
upstroke:  P_Na >> P_K   →  V_m swings toward E_Na (~ +60 mV, spike)
downstroke: Na channels inactivate, P_K rises again → V_m returns toward E_K

Why the sodium-potassium pump has to keep running

Every action potential lets a small number of Na⁺ and K⁺ ions leak across the membrane down their gradients, and left unchecked, thousands of spikes would eventually erase the concentration gradients the Nernst potentials depend on. The sodium-potassium pump continuously rebuilds those gradients at the cost of ATP — roughly half of a resting neuron's energy budget goes to this single pump — which is exactly why the excitability of a nerve cell is, at bottom, a story about actively maintained chemical disequilibrium, not a chemical reaction running to completion.

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