Goldman-Hodgkin-Katz Equation Calculator

Find the resting membrane potential Vm from the relative permeabilities and concentrations of K+, Na+, and Cl- using the Goldman-Hodgkin-Katz equation.

⚡ Goldman-Hodgkin-Katz Equation Calculator
K+ permeability (PK)
Na+ permeability (PNa)
Cl- permeability (PCl)
[K+]out / [K+]in
mM out
mM in
[Na+]out / [Na+]in
mM out
mM in
[Cl-]out / [Cl-]in
mM out
mM in
Temperature
°C
Membrane potential (Vm)
—
Absolute temperature (T)
—
Step-by-step working

⚡ What is the Goldman-Hodgkin-Katz Equation Calculator?

The Goldman-Hodgkin-Katz (GHK) equation calculator finds a cell's resting membrane potential Vm from the relative permeabilities and inside/outside concentrations of several ions at once, most commonly potassium (K+), sodium (Na+), and chloride (Cl-). Unlike the single-ion Nernst equation, the GHK equation is a permeability-weighted combination of every permeant ion's contribution, which is what actually sets the resting potential of a real cell membrane.

Neuroscience and physiology courses use the GHK equation to explain why a resting neuron's membrane potential sits close to, but not exactly at, the potassium Nernst potential. Electrophysiologists use it to model how changing ion channel permeabilities, whether through voltage-gated channel opening during an action potential or through pharmacological block, shifts the membrane potential. Cardiac and renal physiology also rely on GHK-style reasoning to explain resting potentials in non-neuronal excitable and transport tissues.

A common misconception is that the resting membrane potential is simply the average of the individual ions' Nernst potentials. It is not, each ion's contribution is weighted by its relative permeability, so an ion with a small concentration gradient but high permeability can dominate the result just as much as an ion with a large gradient but low permeability. Another point of confusion is the sign convention for chloride: because Cl- is an anion, its inside concentration goes in the numerator and its outside concentration goes in the denominator, the opposite placement from the cations K+ and Na+.

This calculator removes the arithmetic complexity of combining six concentrations and three permeabilities inside a single logarithm, letting students verify homework answers and researchers quickly explore how a permeability change shifts Vm.

📐 Formula

Vm  =  (RT / F) × ln[ (PK[K+]out + PNa[Na+]out + PCl[Cl-]in) / (PK[K+]in + PNa[Na+]in + PCl[Cl-]out) ]
R = gas constant = 8.314 J/(mol·K)
T = absolute temperature in Kelvin (°C + 273.15); this calculator defaults to 37°C (310.15 K), human body temperature
F = Faraday's constant = 96,485 C/mol
PK, PNa, PCl = relative membrane permeability to K+, Na+, and Cl- (unitless ratios)
Note the sign convention: cations (K+, Na+) contribute their outside concentration to the numerator and inside concentration to the denominator; the anion (Cl-) does the reverse.
The result of (RT/F) × ln(...) is in volts; this calculator multiplies by 1000 to display Vm in millivolts (mV).
Example: Standard mammalian concentrations with PK:PNa:PCl = 1:0.04:0.45 at 37°C give Vm ≈ −72.51 mV.

📖 How to Use This Calculator

Steps

1
Enter the ion permeabilities. Type the relative permeabilities PK, PNa, and PCl (unitless ratios; defaults use the classic Hodgkin-Katz ratio 1 : 0.04 : 0.45).
2
Enter the ion concentrations. Type the outside and inside concentration for K+, Na+, and Cl-, in millimolar (mM).
3
Enter the temperature and read the result. Enter the temperature in degrees Celsius (defaults to 37 degrees C body temperature) and read the membrane potential Vm in millivolts.

💡 Example Calculations

Example 1 — Resting Mammalian Neuron

Standard resting concentrations and Hodgkin-Katz ratio: PK=1, PNa=0.04, PCl=0.45, 37°C

1
Numerator = 1×5 + 0.04×145 + 0.45×4 = 5 + 5.8 + 1.8 = 12.6
2
Denominator = 1×140 + 0.04×12 + 0.45×110 = 140 + 0.48 + 49.5 = 189.98
3
Vm = (8.314 × 310.15 / 96485) × ln(12.6 / 189.98) = −72.51 mV, close to the textbook −70 mV neuron resting potential
Vm = −72.51 mV
Try this example →

Example 2 — Action Potential Upstroke (Na+ Channels Open)

Same concentrations, but PNa rises to 20 as voltage-gated Na+ channels open

1
Numerator = 1×5 + 20×145 + 0.45×4 = 5 + 2900 + 1.8 = 2906.8
2
Denominator = 1×140 + 20×12 + 0.45×110 = 140 + 240 + 49.5 = 429.5
3
Vm = (8.314 × 310.15 / 96485) × ln(2906.8 / 429.5) = +51.10 mV, a dramatic swing toward the positive Na+ Nernst potential, this is the action potential upstroke
Vm = +51.10 mV
Try this example →

Example 3 — Resting Potential at Room Temperature (25°C)

Same resting concentrations and permeabilities as Example 1, but at 25°C instead of 37°C

1
T = 25°C + 273.15 = 298.15 K, numerator and denominator are unchanged from Example 1: 12.6 and 189.98
2
Vm = (8.314 × 298.15 / 96485) × ln(12.6 / 189.98)
3
Vm = −69.71 mV, slightly less negative than the 37°C result, since the RT/F prefactor scales directly with absolute temperature
Vm = −69.71 mV
Try this example →

❓ Frequently Asked Questions

What is the Goldman-Hodgkin-Katz equation?+
The Goldman-Hodgkin-Katz (GHK) equation calculates the resting membrane potential Vm of a cell from the relative permeabilities and the inside/outside concentrations of multiple ions (typically K+, Na+, and Cl-) simultaneously, unlike the Nernst equation which handles only one ion at a time.
What is the GHK equation formula?+
Vm = (RT/F) x ln[(PK[K+]out + PNa[Na+]out + PCl[Cl-]in) / (PK[K+]in + PNa[Na+]in + PCl[Cl-]out)], where R is the gas constant, T is absolute temperature, F is Faraday's constant, and PK, PNa, PCl are relative permeabilities.
Why is chloride handled differently from potassium and sodium in the GHK equation?+
Chloride is an anion, so its inside concentration appears in the numerator and its outside concentration appears in the denominator, the reverse of potassium and sodium, which are cations. This sign convention keeps the equation consistent with the underlying Nernst-Planck flux derivation for ions of opposite charge.
What is a typical resting membrane potential from the GHK equation?+
Using typical mammalian neuron concentrations ([K+]out=5mM, [K+]in=140mM, [Na+]out=145mM, [Na+]in=12mM, [Cl-]out=110mM, [Cl-]in=4mM) with the classic Hodgkin-Katz permeability ratio PK:PNa:PCl of 1:0.04:0.45 at 37 degrees C, the GHK equation gives Vm of about -72.5 mV, close to the textbook -70 mV resting potential of a real neuron.
How does the GHK equation explain the action potential upstroke?+
At rest PK is much larger than PNa, so Vm sits close to the K+ Nernst potential. When voltage-gated Na+ channels open during an action potential, PNa increases by more than a hundredfold, and the GHK equation shows Vm swinging sharply toward the positive Na+ Nernst potential, which is the depolarization phase.
How is the GHK equation different from the Nernst equation?+
The Nernst equation gives the equilibrium potential for one ion species in isolation. The GHK equation is a permeability-weighted combination of the Nernst potentials for several ions at once, and it is this weighted combination, not any single ion's Nernst potential, that determines a real cell's actual resting membrane potential.
What units do the permeability values PK, PNa, PCl use?+
They are relative, unitless ratios to each other, not absolute permeability coefficients in cm/s. Only the ratios PK:PNa:PCl matter for the calculated Vm, so scaling all three permeabilities by the same factor leaves the result unchanged.
What happens to Vm if permeability to only one ion is nonzero?+
If, for example, PNa and PCl are both 0 and only PK is nonzero, the GHK equation reduces exactly to the single-ion Nernst equation for potassium, since every term involving Na+ and Cl- drops out of both the numerator and denominator.
Why does this calculator default to 37 degrees C?+
37 degrees C (310.15 K) is human body temperature and the physiologically relevant value for mammalian neuron membrane potentials. Using a different temperature, such as 25 degrees C room temperature, changes the RT/F prefactor and therefore the final millivolt value.
Can the GHK equation be extended beyond K+, Na+, and Cl-?+
Yes, the general GHK equation sums a term for every permeant ion species, with cations contributing to the numerator via their outside concentration and anions contributing via their inside concentration. K+, Na+, and Cl- are used here because they dominate the resting and action potentials of most animal cells.
Tags: Goldman Hodgkin Katz Equation Calculator Ghk Equation Calculator Resting Membrane Potential Calculator Goldman Equation Membrane Potential Calculator Formula Free Online

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What is the Goldman-Hodgkin-Katz equation?

The Goldman-Hodgkin-Katz (GHK) equation calculates the resting membrane potential Vm of a cell from the relative permeabilities and the inside/outside concentrations of multiple ions (typically K+, Na+, and Cl-) simultaneously, unlike the Nernst equation which handles only one ion at a time.

What is the GHK equation formula?

Vm = (RT/F) x ln[(PK[K+]out + PNa[Na+]out + PCl[Cl-]in) / (PK[K+]in + PNa[Na+]in + PCl[Cl-]out)], where R is the gas constant, T is absolute temperature, F is Faraday's constant, and PK, PNa, PCl are relative permeabilities.

Why is chloride handled differently from potassium and sodium in the GHK equation?

Chloride is an anion, so its inside concentration appears in the numerator and its outside concentration appears in the denominator, the reverse of potassium and sodium, which are cations. This sign convention keeps the equation consistent with the underlying Nernst-Planck flux derivation for ions of opposite charge.

What is a typical resting membrane potential from the GHK equation?

Using typical mammalian neuron concentrations ([K+]out=5mM, [K+]in=140mM, [Na+]out=145mM, [Na+]in=12mM, [Cl-]out=110mM, [Cl-]in=4mM) with the classic Hodgkin-Katz permeability ratio PK:PNa:PCl of 1:0.04:0.45 at 37 degrees C, the GHK equation gives Vm of about -72.5 mV, close to the textbook -70 mV resting potential of a real neuron.

How does the GHK equation explain the action potential upstroke?

At rest PK is much larger than PNa, so Vm sits close to the K+ Nernst potential. When voltage-gated Na+ channels open during an action potential, PNa increases by more than a hundredfold, and the GHK equation shows Vm swinging sharply toward the positive Na+ Nernst potential, which is the depolarization phase.

How is the GHK equation different from the Nernst equation?

The Nernst equation gives the equilibrium potential for one ion species in isolation. The GHK equation is a permeability-weighted combination of the Nernst potentials for several ions at once, and it is this weighted combination, not any single ion's Nernst potential, that determines a real cell's actual resting membrane potential.

What units do the permeability values PK, PNa, PCl use?

They are relative, unitless ratios to each other, not absolute permeability coefficients in cm/s. Only the ratios PK:PNa:PCl matter for the calculated Vm, so scaling all three permeabilities by the same factor leaves the result unchanged.

What happens to Vm if permeability to only one ion is nonzero?

If, for example, PNa and PCl are both 0 and only PK is nonzero, the GHK equation reduces exactly to the single-ion Nernst equation for potassium, since every term involving Na+ and Cl- drops out of both the numerator and denominator.

Why does this calculator default to 37 degrees C?

37 degrees C (310.15 K) is human body temperature and the physiologically relevant value for mammalian neuron membrane potentials. Using a different temperature, such as 25 degrees C room temperature, changes the RT/F prefactor and therefore the final millivolt value.

Can the GHK equation be extended beyond K+, Na+, and Cl-?

Yes, the general GHK equation sums a term for every permeant ion species, with cations contributing to the numerator via their outside concentration and anions contributing via their inside concentration. K+, Na+, and Cl- are used here because they dominate the resting and action potentials of most animal cells.

📌 Quick Tips

💡The GHK equation is the multi-ion generalization of the single-ion Nernst equation, it weights each ion's contribution to the membrane potential by that ion's relative permeability, not just its concentration gradient.
💡A resting neuron is far more permeable to K+ than to Na+ (typical ratio PK:PNa:PCl of about 1:0.04:0.45), which is why the resting potential sits close to the K+ Nernst potential rather than halfway between K+ and Na+.
💡During the action potential upstroke, voltage-gated Na+ channels open and PNa rises by more than a hundredfold, temporarily making PNa the dominant term and pulling Vm toward the Na+ Nernst potential near +60 mV.
💡Chloride enters the equation differently from the cations, its intracellular concentration appears in the numerator and its extracellular concentration in the denominator, the opposite arrangement from K+ and Na+, because Cl- is an anion.
💡Permeability values PK, PNa, and PCl are relative (unitless) ratios, not absolute diffusion constants, only their ratios to each other matter for the calculation, so any consistent scaling gives the same Vm.