Goldman Equation Calculators

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The Goldman-Hodgkin-Katz (GHK) equation calculates the equilibrium membrane potential of a cell based on the concentrations of permeant ions on both sides of the membrane and the relative permeability of the membrane to each ion. Unlike the Nernst equation (which applies to a single ion), the Goldman equation accounts for multiple ions simultaneously. For most animal cells, the relevant ions are K⁺, Na⁺, and Cl⁻. The resting membrane potential (~−70 mV in neurons) results from the fact that the membrane is much more permeable to K⁺ than to Na⁺, so the resting potential is near the K⁺ equilibrium potential.

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Goldman Equation

V_m = (RT/F) × 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)]

R = 8.314 J/mol/K; T = temperature (K); F = 96,485 C/mol. Subscripts o = outside; i = inside. Note: Cl⁻ terms are reversed (outside in numerator, inside in denominator) because Cl⁻ is negatively charged.

Simplified form at 37°C: V_m = (25.7 mV) × ln [...]

Typical Mammalian Neuron Values

  • [K⁺]_i = 140 mM; [K⁺]_o = 5 mM
  • [Na⁺]_i = 15 mM; [Na⁺]_o = 145 mM
  • [Cl⁻]_i = 10 mM; [Cl⁻]_o = 120 mM
  • Permeability ratios at rest: P_K : P_Na : P_Cl = 1 : 0.04 : 0.45

Result: V_m ≈ −65 to −70 mV at rest (negative inside).

Nernst Equation (Single Ion)

E_ion = (RT/zF) × ln([ion]_o/[ion]_i). E_K = −90 mV (at 37°C, typical concentrations). E_Na = +60 mV. The Goldman equation takes a permeability-weighted average of single-ion Nernst potentials.

Action Potential

During action potential: P_Na increases dramatically (Nav opens) → V_m moves toward E_Na (+60 mV). Then Nav inactivates + Kv opens → V_m returns to E_K (−90 mV) → repolarization and brief hyperpolarization.

Glossary

Goldman Equation (GHK)
Calculates membrane potential from multiple ion concentrations and permeabilities: V_m = (RT/F) × 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)].
Nernst Equation
E = (RT/zF) × ln([ion]_o/[ion]_i); calculates equilibrium potential for a single ion; E_K ≈ −90 mV; E_Na ≈ +60 mV; Goldman equation is the multi-ion generalization.
Resting Membrane Potential
The electrical potential across a resting cell membrane (~−65 to −70 mV in neurons); determined primarily by high K⁺ permeability through leak channels; maintained by Na⁺/K⁺-ATPase gradients.

Frequently Asked Questions

The Goldman-Hodgkin-Katz (GHK) equation calculates the steady-state membrane potential when multiple ions can permeate the membrane: V_m = (RT/F) × 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)]. R = 8.314 J/mol/K; T = 310 K (37°C); F = 96,485 C/mol. It gives a permeability-weighted combination of individual ion Nernst potentials. At rest: K⁺ dominates (high permeability) → V_m near E_K ≈ −90 mV, but offset toward more positive values by Na⁺ and Cl⁻ contributions → typical resting V_m ≈ −65 to −70 mV.

Nernst equation: E = (RT/zF) × ln([ion]_o/[ion]_i). Calculates the equilibrium potential for a SINGLE ion — the membrane voltage at which there is no net flow of that ion (electrochemical equilibrium). E_K ≈ −90 mV; E_Na ≈ +60 mV; E_Cl ≈ −65 mV (typical mammalian neuron). Goldman equation: accounts for MULTIPLE ions simultaneously, weighted by their relative membrane permeabilities. When permeability to one ion dominates, V_m approaches that ion's Nernst potential. At rest (P_K >> P_Na): V_m ≈ E_K but slightly more positive. During action potential (Nav opens, P_Na >> P_K): V_m moves toward E_Na.

The resting membrane potential (~−65 to −70 mV) reflects the balance of ion gradients and permeabilities: K⁺ dominates the resting membrane permeability (K⁺ leak channels are open; Nav and Cav channels are mostly closed). K⁺ diffuses out of the cell (down its concentration gradient) carrying positive charge → inside becomes negative. Equilibrium at −90 mV (E_K) — but Na⁺ permeability (even at rest) pulls V_m slightly toward E_Na (+60 mV). Net result: V_m ≈ −70 mV. Na⁺/K⁺ ATPase pumps maintain the concentration gradients (3 Na⁺ out / 2 K⁺ in per cycle) — it is electrogenic (net outward current) contributing slightly to the negative resting potential.

During an action potential: Depolarization phase: Voltage-gated Na⁺ channels (Nav) open → P_Na increases dramatically (P_Na:P_K changes from 0.04:1 to ~20:1) → Goldman equation predicts V_m moves toward E_Na (+60 mV) → depolarization to +30 to +40 mV (overshoot). Repolarization phase: Nav rapidly inactivates → P_Na returns to resting level. Voltage-gated K⁺ channels (Kv) open (delayed) → P_K increases further → V_m moves toward E_K (−90 mV) → repolarization. Afterhyperpolarization: Kv channels close slowly → brief hyperpolarization below resting potential. The Goldman equation at each phase predicts V_m from the changing permeability ratios — beautifully explaining the entire action potential waveform.