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The Goldman–Hodgkin–Katz equation estimates membrane potential from ion concentrations and relative permeabilities.
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Formula
Vm in V · R = 8.314 J/(mol·K) · T in K · F = 96,485 C/mol · ion concentrations in one common unit · Pi dimensionless relative permeabilities
| Symbol | Meaning | Unit |
|---|---|---|
| Inside potential relative to outside in the steady GHK model | V | |
| Universal gas constant | J/(mol·K) | |
| Absolute temperature | K | |
| Faraday constant | C/mol | |
| PK, PNa, PCl | Relative membrane permeabilities for potassium, sodium and chloride | dimensionless |
| [Ion]_i, [Ion]_o | Inside and outside ion concentrations; chloride positions are reversed in the quotient | mol/L or mol/m3 |
V_m = \frac{RT}{F} \ln \frac{P_K[K^+]_o+P_{Na}[Na^+]_o+P_{Cl}[Cl^-]_i}{P_K[K^+]_i+P_{Na}[Na^+]_i+P_{Cl}[Cl^-]_o}The GHK voltage equation extends the single-ion Nernst case to multiple permeant ions under the constant-field approximation. For monovalent K+, Na+ and Cl−, the chloride concentrations reverse position in the quotient. Relative permeability weights each ion’s contribution. The model assumes steady concentration profiles and a constant electric-field distribution across the membrane; it is not a complete description of channel dynamics or active pumps. If only one ion is permeant, the result approaches that ion’s Nernst potential.
Describes the steady membrane potential for several permeant ions at constant temperature and relative permeabilities; concentrations must be inserted on the correct inside/outside sides.
The potential settles so that the permeability-weighted ionic currents balance in the steady state.
Simplified Nernst limit for one ion
Follows when practically only this ion is permeant; sign and concentration ratio depend on z.
Which single-ion limit results if only K+ is permeant?
The GHK equation reduces to the Nernst potential for K+.
At 37 °C and [K+] outside/inside = 5/140, it is about 61.5 mV·log10(5/140) ≈ −89 mV.
Using Cl− in the same concentration ratio as K+.
In the usual GHK expression, chloride reverses inside and outside because it has a negative charge.
Formula cluster
GHK extends Nernst to multiple permeant ions and their relative permeabilities.
At 37 °C, let PK=1, PNa=0.04, PCl=0.45; K outside/inside 5/140, Na 145/15 and Cl 110/10 mmol/L. The quotient is (5+5.8+4.5)/(140+0.6+49.5)=0.0805, so Vm=(RT/F)ln(0.0805)≈−67 mV. Check: the inside is negative because potassium permeability dominates and outside K is much lower.
Neuronal resting potential, effects of altered ion gradients and how relative membrane permeabilities shift voltage
Here is how to work through a typical Goldman–Hodgkin–Katz Equation (Vₘ = (RT/F)·ln(ΣPᵢ·ionₒ / ΣPᵢ·ionᵢ)) task step by step:
Task
Which single-ion limit results if only K+ is permeant?
Solution path
The GHK equation reduces to the Nernst potential for K+.
At 37 °C and [K+] outside/inside = 5/140, it is about 61.5 mV·log10(5/140) ≈ −89 mV.
It estimates the steady membrane potential when several ions can cross the membrane. Unlike the Nernst equation, it does not treat only one ion, but combines potassium, sodium and chloride with their relative permeabilities. The inside and outside concentrations refer to the two sides of the same membrane, not arbitrary sampling locations. The result is a voltage, usually in millivolts. It is a model value under stated conditions, not a direct measurement of an arbitrary action potential. For biological applications, record cell type, temperature, concentration sides and relative permeabilities. The calculated voltage can be meaningfully compared with a resting potential only under comparable conditions.
Potassium and sodium carry positive charge, while chloride is negative. In deriving the GHK expression, this sign means chloride has its intracellular concentration in the numerator and extracellular concentration in the denominator. Treating every ion identically can produce a substantially wrong potential. Before substituting, check both the charge and the membrane side. The familiar form also applies to monovalent ions; multivalent ions require a suitable formulation. For biological applications, record cell type, temperature, concentration sides and relative permeabilities. The calculated voltage can be meaningfully compared with a resting potential only under comparable conditions.
Permeabilities weight how readily each ion crosses the membrane in the state being considered. They are neither concentrations nor simply the number of open channels. If PK is much larger than PNa, potassium influences the potential more strongly even when extracellular sodium concentration is high. Values are often normalized to one permeability, for example PK=1; the others are then relative ratios. Actual values depend on cell type, channel state and measurement conditions. For biological applications, record cell type, temperature, concentration sides and relative permeabilities. The calculated voltage can be meaningfully compared with a resting potential only under comparable conditions.
The Nernst equation gives the equilibrium potential for a single ion. In GHK, several ions contribute according to their permeability. If the membrane is practically permeable to only one ion, the GHK prediction reduces to that ion’s Nernst limit. For K+ at 37 °C and an outside-to-inside ratio of 5/140, this is about 61.5 mV·log10(5/140), or roughly −89 mV. Both models assume electrochemical equilibrium or a steady state. For biological applications, record cell type, temperature, concentration sides and relative permeabilities. The calculated voltage can be meaningfully compared with a resting potential only under comparable conditions.
No. The static GHK equation gives a steady potential for specified concentrations and permeabilities. During an action potential, channel gating and therefore permeabilities change rapidly over time; membrane capacitance and ionic currents also matter. Dynamic models such as Hodgkin-Huxley equations, or direct measurements, are needed for that. GHK remains useful for understanding the resting potential qualitatively and predicting how a change in permeability shifts it. Its conclusions remain tied to its assumptions. For biological applications, record cell type, temperature, concentration sides and relative permeabilities. The calculated voltage can be meaningfully compared with a resting potential only under comparable conditions.
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Curated exam set for "Goldman–Hodgkin–Katz Equation":
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Which formula describes Goldman–Hodgkin–Katz Equation?
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How do you rearrange Goldman–Hodgkin–Katz Equation for Simplified Nernst limit for one ion?
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Which common mistake happens with Goldman–Hodgkin–Katz Equation?
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