Healthcare & Clinical
Goldman-Hodgkin-Katz (GHK) Membrane Voltage Equation (K+, Na+, Cl-)
Calculate mammalian neuron resting membrane potential and action-potential overshoot at 37 C (R*T/F = 26.7267 mV) from relative K+, Na+, and Cl- permeabilities and extra/intracellular ion concentrations.
Model resting membrane voltage and voltage-gated Na+ depolarization spikes!
Whereas the single-ion Nernst equation predicts equilibrium potential when only one ion can cross the lipid bilayer, real excitable cell membranes are simultaneously permeable to Potassium (K+, z = +1), Sodium (Na+, z = +1), and Chloride (Cl-, z = -1)! Because Cl- carries a negative valence, its concentration ratio flips upside-down inside the logarithm—placing intracellular [Cl-]_i in the numerator alongside extracellular [K+]_o and [Na+]_o!
At human physiological temperature T = 310.15 K (37 C), the thermal voltage factor R * T / F equals 8.31446 * 310.15 / 96485.33 * 1000 = 26.7267 mV. In a resting neuron (P_K : P_Na : P_Cl = 1 : 0.04 : 0.45), high K+ permeability anchors V_m at -67.33 mV; when voltage-gated Na+ channels snap open (P_Na jumping 500x to 20), V_m surges to +51.13 mV toward the Na+ Nernst potential!
| Membrane State (37 C, R*T/F = 26.7267 mV) | Relative P_K : P_Na : P_Cl | Numerator / Denominator | GHK Potential V_m (mV) |
|---|---|---|---|
| Resting Neuron ([K]o=5, [K]i=140, [Na]o=145, [Na]i=12) | 1.00 : 0.04 : 0.45 | 15.30 / 189.98 = 0.08053 | -67.33 mV |
| Action Potential Peak (Na+ Channels Open) | 1.00 : 20.00 : 0.45 | 2909.50 / 429.50 = 6.7742 | +51.13 mV |
| Hyperkalemia Emergency ([K+]_o = 9 mM) | 1.00 : 0.04 : 0.45 | 19.30 / 189.98 = 0.10159 | -61.12 mV (Depolarized!) |
Goldman-Hodgkin-Katz Membrane Potential V_m (mV at 37 C)
Evaluates 26.7267 * LN((P_K*[K]o + P_Na*[Na]o + P_Cl*[Cl]i) / (P_K*[K]i + P_Na*[Na]i + P_Cl*[Cl]o)), noting the inverted [Cl]i / [Cl]o placement due to z = -1.
=IF(OR((B2 * B5 + B3 * B7 + B4 * B10) <= 0, (B2 * B6 + B3 * B8 + B4 * B9) <= 0), "Invalid Ion Permeability or Concentration", ROUND(26.7267 * LN((B2 * B5 + B3 * B7 + B4 * B10) / (B2 * B6 + B3 * B8 + B4 * B9)), 2))Compare Against Pure Potassium Nernst Potential E_K (mV)
Shows how leak Na+ permeability pulls resting V_m (-67.33 mV) slightly positive of pure K+ equilibrium E_K = 26.7267 * LN([K]o / [K]i) = -89.06 mV.
=ROUND(26.7267 * LN(B5 / B6), 2)Try the example
Use the original values or change them to test your understanding.