Goldman Equation and Resting Membrane Potential for the MCAT

The Nernst equation gives the equilibrium potential of one ion. The Goldman equation gives the membrane potential when several ions can cross, weighting each ion by its permeability.

The idea

V_m = 61.5 log [ (P_K[K⁺]_out + P_Na[Na⁺]_out + P_Cl[Cl^-]_in) / (P_K[K⁺]_in + P_Na[Na⁺]_in + P_Cl[Cl^-]_out) ] mV at 37 C.

Anions have inside and outside swapped. You do not need to memorize it. You need the logic: the membrane potential is pulled toward the equilibrium potential of the most permeable ion.

Typical equilibrium potentials

At rest the neuron is far more permeable to K⁺ (leak channels) than Na⁺, so V_m sits near -70 mV, close to E_K but not equal to it.

Why it is not exactly E_K

A small Na⁺ leak pulls V_m up from -90 mV. That is the whole difference.

Role of Na⁺/K⁺ ATPase

It pumps 3 Na⁺ out and 2 K⁺ in per ATP. It contributes a few mV of negative charge, but its main job is to maintain the gradients that the leak channels use. Block it and the gradients slowly run down, not instantly.

What shifts V_m

• Open voltage-gated Na⁺ channels: P_Na rises, V_m moves toward +60 (depolarization).
• Open K⁺ channels more: toward -90 (hyperpolarization).
• Raise extracellular K⁺: E_K becomes less negative, so the cell depolarizes.

Common traps

• Saying V_m equals E_K. It is close, not equal.
• Using Nernst for a multi-ion membrane. Nernst needs a membrane permeable to one ion.
• Blaming the pump alone for the resting potential. Gradients plus K⁺ permeability set it.

One-minute recap

• Goldman: permeability-weighted version of Nernst.
• Resting potential near -70 mV because K⁺ permeability dominates.
• Raising extracellular K⁺ depolarizes the cell.