Resting Membrane Potential Calculator

Goldman-Hodgkin-Katz (GHK) Calculator

This tool calculates the resting membrane potential (V_m) of a cell by using the GHK equation, which considers the concentrations and relative permeabilities of the key ions: Potassium (K⁺), Sodium (Na⁺), and Chloride (Cl⁻).

Comparison of Ion Equilibrium Potentials (Nernst) vs. Calculated Resting Potential (GHK)

What is Calculating Resting Membrane Potential Using Permeabilities?

The resting membrane potential is the electrical voltage difference across the plasma membrane of a cell when it is not stimulated or excited. For nerve and muscle cells, this potential is crucial for their ability to send signals. Calculating the resting membrane potential using permeabilities involves the Goldman-Hodgkin-Katz (GHK) equation. This powerful formula considers the two main factors that establish the potential: the concentration gradients of ions and the selective permeability of the membrane to those ions. While simpler models like the Nernst equation calculate the equilibrium potential for a single ion, the GHK equation provides a more accurate picture by accounting for the simultaneous influence of multiple ions—primarily potassium (K⁺), sodium (Na⁺), and chloride (Cl⁻)—and their different abilities to cross the membrane.

The Goldman-Hodgkin-Katz (GHK) Formula and Explanation

The GHK equation calculates the membrane potential (V_m) by weighting the contribution of each ion based on its permeability (P) and its concentration gradient across the membrane ([Ion]_out vs [Ion]_in). The formula is:

V_m = (RT/F) * ln( (P_K[K⁺]_out + P_Na[Na⁺]_out + P_Cl[Cl⁻]_in) / (P_K[K⁺]_in + P_Na[Na⁺]_in + P_Cl[Cl⁻]_out) )

Note that for the chloride ion (Cl⁻), a negative ion, the intracellular and extracellular concentrations are inverted in the equation to correctly account for its charge.

GHK Equation Variables

Variable	Meaning	Unit	Typical Range
Vm	Membrane Potential	millivolts (mV)	-40 to -90 mV
R	Ideal Gas Constant	Joules / (Kelvin·mol)	8.314 J·K⁻¹·mol⁻¹
T	Absolute Temperature	Kelvin (K)	~310 K (37 °C)
F	Faraday’s Constant	Coulombs / mol	96,485 C·mol⁻¹
Pion	Relative Permeability	Unitless	PK: 1, PNa: 0.01-0.05, PCl: 0.4-0.5
[Ion]	Ion Concentration	millimolar (mM)	5-150 mM

For more details on how these factors interact, you might want to read about the {related_keywords}.

Practical Examples

Example 1: Typical Resting Neuron

Let’s calculate the resting potential for a standard neuron with typical physiological values at 37 °C.

• Inputs: T=37°C, P_K=1, P_Na=0.04, P_Cl=0.45, [K⁺]_out=5, [K⁺]_in=140, [Na⁺]_out=145, [Na⁺]_in=15, [Cl⁻]_out=110, [Cl⁻]_in=10.
• Calculation: Using the GHK equation, the combination of high K⁺ permeability and its gradient strongly pushes the potential negative, while the small Na⁺ permeability slightly counteracts this.
• Result: The resulting V_m is approximately -65.3 mV.

Example 2: Increased Sodium Permeability

Imagine a scenario where some voltage-gated sodium channels begin to open, increasing sodium’s permeability. This is the beginning of an excitatory signal.

• Inputs: Same as above, but let’s increase P_Na from 0.04 to 0.2.
• Calculation: The increased influence of the Na⁺ gradient (which has a positive equilibrium potential of about +61 mV) pulls the total membrane potential towards a more positive value.
• Result: The resulting V_m becomes approximately -50.6 mV. This depolarization is a key step towards firing an action potential. Understanding this shift is key to grasping the {related_keywords}.

How to Use This Resting Membrane Potential Calculator

1. Set Temperature: Enter the cell’s temperature and select the correct unit (°C or K). The calculation automatically converts to Kelvin, which the GHK equation requires.
2. Input Relative Permeabilities: Enter the relative permeability for each ion (P_K, P_Na, P_Cl). These are unitless ratios, typically normalized to Potassium (P_K = 1).
3. Enter Ion Concentrations: Fill in the intracellular and extracellular concentrations for K⁺, Na⁺, and Cl⁻. Ensure the units are in millimolar (mM).
4. Interpret the Results: The primary result is the Resting Membrane Potential (V_m) in millivolts (mV). This is the overall potential predicted by the GHK equation.
5. Review Intermediate Values: The calculator also shows the Nernst equilibrium potential for each individual ion (E_K, E_Na, E_Cl). This helps you see the “goal” potential for each ion, and how the final V_m is a weighted average based on permeability.
6. Analyze the Chart: The bar chart provides a visual comparison, showing how the final Vm is a compromise between the different Nernst potentials, pulled closest to the ion with the highest permeability (usually K⁺ at rest). This helps in understanding topics like {related_keywords}.

Key Factors That Affect Resting Membrane Potential

• Potassium (K⁺) Gradient: This is the most significant factor in most resting cells. Because resting membrane permeability to K⁺ is high, its strong outward concentration gradient is the primary driver of the negative potential.
• Potassium (K⁺) Permeability: The large number of open K⁺ “leak” channels at rest makes the membrane highly permeable to it, giving K⁺ the dominant influence on the resting potential.
• Sodium (Na⁺) Gradient & Permeability: Although the Na⁺ concentration gradient is very strong, its permeability at rest is very low. This small inward “leak” of positive charge is what keeps the resting potential slightly more positive than the K⁺ equilibrium potential (-70 mV vs -90 mV, for example).
• The Na⁺/K⁺ Pump: This active transport pump is vital. It uses ATP to pump 3 Na⁺ ions out for every 2 K⁺ ions it pumps in. This action maintains the concentration gradients that the GHK equation depends on. Without it, the gradients would eventually dissipate.
• Chloride (Cl⁻) Gradient: In many neurons, the Cl⁻ equilibrium potential is very close to the resting membrane potential. As a result, its net effect on the resting state is often minimal, but it plays a key role in cell inhibition.
• Impermeant Anions: Large, negatively charged proteins and organic phosphates are trapped inside the cell. They contribute to the overall negative charge of the intracellular fluid but are not part of the GHK equation as they cannot permeate the membrane. A good resource for this is the guide on {related_keywords}.

Frequently Asked Questions (FAQ)

1. Why is the resting membrane potential negative?

It’s primarily negative because of the efflux (outward movement) of positively charged potassium (K⁺) ions down their steep concentration gradient, leaving behind a net negative charge inside the cell. The membrane is much more permeable to K⁺ at rest than to other ions.

2. What is the difference between the GHK equation and the Nernst equation?

The Nernst equation calculates the equilibrium potential for a *single* ion, assuming the membrane is only permeable to that ion. The GHK equation is more comprehensive; it calculates the overall membrane potential by considering the contributions of *multiple* ions simultaneously, weighted by their respective permeabilities.

3. What do the permeability units (P_K, P_Na, P_Cl) mean?

They are relative and unitless. They represent how easily one ion can cross the membrane compared to another. By convention, permeability is often expressed relative to potassium, so P_K is set to 1. A P_Na of 0.04 means sodium is only 4% as permeable as potassium.

4. Why are the Chloride concentrations ([Cl⁻]_in and [Cl⁻]_out) flipped in the GHK equation?

Because Chloride (Cl⁻) is a negative ion (anion), its effect on the voltage is opposite to that of positive ions (cations) like K⁺ and Na⁺. Flipping its concentrations in the formula is a mathematical way to account for its negative charge without changing the structure of the equation.

5. What happens if I set a permeability to 0?

If you set an ion’s permeability to 0, it means the membrane is completely impermeable to that ion. That ion will then have no influence on the membrane potential, and its terms will effectively be removed from the GHK calculation.

6. How does temperature affect the resting potential?

Temperature is part of the (RT/F) term in the equation. Higher temperatures increase the kinetic energy of ions, leading to a slightly larger potential difference for the same concentration gradients. However, changes in concentration or permeability usually have a much more significant impact.

7. Does this calculator work for an action potential?

Yes, you can model an action potential by drastically changing the permeabilities. For example, at the peak of an action potential, P_Na becomes much larger than P_K (e.g., P_K😛_Na = 1:20). You can simulate this with the calculator to see the V_m shoot up towards the sodium equilibrium potential (+60mV). This is a core part of understanding the {related_keywords}.

8. Where do the default concentration values come from?

The default values are typical physiological concentrations found in mammalian neurons. [K⁺] is high inside the cell while [Na⁺] and [Cl⁻] are high outside. These gradients are actively maintained by ion pumps like the Na⁺/K⁺-ATPase.

Related Tools and Internal Resources

Explore these related concepts to deepen your understanding of neurophysiology and cell excitability.

• {related_keywords}: Calculate the equilibrium potential for a single ion.
• {related_keywords}: Learn about the rapid changes in membrane potential that allow nerve cells to communicate.
• {related_keywords}: Understand the forces that drive ions across the membrane.