Membrane Potential Calculator (Modified Nernst/GHK Equation)

A professional tool to calculate a membrane potential using the modified Nernst equation, more accurately known as the Goldman-Hodgkin-Katz (GHK) equation, which accounts for multiple ions.

Potassium (K⁺)

Sodium (Na⁺)

Chloride (Cl⁻)

Resting Membrane Potential (V_m)

— mV

---

E_K

— mV

E_Na

— mV

E_Cl

— mV

Intermediate values are the Nernst potentials for each individual ion.

Chart of Ion Equilibrium Potentials vs. Final Membrane Potential

What is the Resting Membrane Potential?

The resting membrane potential is the electrical potential difference across the plasma membrane of a cell when it is in a non-excited, or “resting,” state. For most neurons, this value is approximately -70 millivolts (mV), indicating that the inside of the cell is negatively charged relative to the outside. This potential is crucial for the function of excitable cells like neurons and muscle cells, as it provides the basis for generating electrical signals such as action potentials. The potential arises from two main factors: the differing concentrations of ions inside and outside the cell, and the selective permeability of the cell membrane to these ions. To properly calculate a membrane potential using the modified Nernst equation, one must consider the contributions of all major permeable ions.

The Goldman-Hodgkin-Katz (GHK) Equation Formula

While the standard Nernst equation calculates the equilibrium potential for a single ion, it is insufficient for real biological systems where membranes are permeable to multiple ions simultaneously. The Goldman-Hodgkin-Katz (GHK) equation, often referred to as the modified Nernst equation in this context, provides a more accurate calculation by considering the concentration gradients and relative permeabilities of the key ions involved: Potassium (K⁺), Sodium (Na⁺), and Chloride (Cl⁻).

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) )

This equation is essential for anyone needing to accurately calculate a membrane potential using the modified nernst equation. Learn more at our electrophysiology tools page.

Description of variables in the GHK equation. Units are based on standard physiological measurements.

Variable	Meaning	Unit	Typical Range
V_m	Membrane Potential	millivolts (mV)	-90 to -50 mV
R	Ideal Gas Constant	8.314 J/(K·mol)	Constant
T	Absolute Temperature	Kelvin (K)	~310 K (37°C)
F	Faraday Constant	96485 C/mol	Constant
P_ion	Relative membrane permeability for an ion	Unitless ratio	0.01 to 1.0
[Ion]_out	Extracellular ion concentration	millimoles/liter (mM)	e.g., [K⁺] ~4-5 mM
[Ion]_in	Intracellular ion concentration	millimoles/liter (mM)	e.g., [K⁺] ~140 mM

Practical Examples

Example 1: Typical Resting Neuron

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

• Inputs: T=37°C, P_K=1, P_Na=0.04, P_Cl=0.45, [K⁺]out=5mM, [K⁺]in=140mM, [Na⁺]out=145mM, [Na⁺]in=15mM, [Cl⁻]out=110mM, [Cl⁻]in=10mM.
• Result: Using these values, our Goldman-Hodgkin-Katz equation calculator yields a V_m of approximately -67.2 mV. This negative value confirms that the inside of the cell is more negative than the outside, primarily due to the high permeability and outward leak of K⁺ ions.

Example 2: Increased Sodium Permeability (Depolarization)

During the rising phase of an action potential, sodium channels open, drastically increasing sodium’s permeability.

• Inputs: Let’s increase P_Na to 20 while keeping other values constant (P_K=1, P_Cl=0.45).
• Result: The membrane potential shifts dramatically to approximately +33.5 mV. This demonstrates how a change in permeability for a single ion can completely alter the membrane potential, driving it towards that ion’s own equilibrium potential (E_Na).

How to Use This Calculator

This tool is designed to provide an intuitive way to explore the factors that determine resting membrane potential.

1. Set Temperature: Begin by entering the system’s temperature in Celsius. For mammalian cells, 37°C is standard.
2. Enter Ion Permeabilities: Input the relative permeabilities for K⁺, Na⁺, and Cl⁻. These are unitless ratios, typically normalized to K⁺ (P_K = 1).
3. Enter Ion Concentrations: Provide the intracellular and extracellular concentrations for each ion in millimoles (mM). The default values represent a typical mammalian neuron.
4. Interpret the Results: The primary result is the overall membrane potential (V_m). The intermediate values show the Nernst potential for each ion, representing the potential the membrane would have if it were only permeable to that single ion. A similar tool can be found at our Nernst potential calculator page.
5. Analyze the Chart: The chart visually compares the individual equilibrium potentials (E_ion) to the final calculated V_m, helping to illustrate which ion has the dominant influence.

Key Factors That Affect Resting Membrane Potential

• Potassium (K⁺) Permeability: At rest, the membrane is most permeable to K⁺. This is the single most important factor in establishing the negative resting potential.
• Sodium-Potassium Pump: This pump actively transports 3 Na⁺ ions out of the cell for every 2 K⁺ ions it brings in. This maintains the steep concentration gradients necessary for the potential to exist.
• Ion Concentration Gradients: The difference in ion concentrations between the inside and outside of the cell is the driving force. Without these gradients, there would be no potential.
• Sodium (Na⁺) Permeability: While low at rest, the slight inward leak of Na⁺ makes the resting potential slightly more positive than the equilibrium potential for K⁺ alone.
• Chloride (Cl⁻) Permeability: The movement of chloride also contributes to the final resting potential, typically helping to stabilize it.
• Temperature: Temperature influences the kinetic energy of ions and is a key component of the RT/F term in the GHK equation. Higher temperatures increase the rate of ion movement. For a deep dive, see our article on the electrochemical gradient.

Frequently Asked Questions (FAQ)

1. Why is the resting potential negative?

It is primarily due to the cell membrane being most permeable to potassium (K⁺) ions at rest. K⁺ ions, which are highly concentrated inside the cell, leak out down their concentration gradient, leaving behind unbalanced negative charges (mainly proteins) inside the cell.

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

The Nernst equation calculates the equilibrium potential for a *single* ion. The GHK (modified Nernst) equation calculates the overall membrane potential when the membrane is permeable to *multiple* ions, providing a more realistic value for a living cell.

3. Why is the permeability for Cl⁻ inverted in the GHK formula?

Chloride (Cl⁻) is an anion (negatively charged). Its contribution to the potential is opposite to that of cations (like K⁺ and Na⁺). To account for this, its intracellular and extracellular concentrations are swapped in the numerator and denominator of the equation.

4. What happens if I set the permeability of an ion to zero?

Setting an ion’s permeability to zero effectively removes it from the GHK calculation, meaning it has no influence on the final membrane potential. You can test this with the resting membrane potential simulator.

5. How does the Na⁺/K⁺ pump contribute to the potential?

Its primary role is indirect: it uses ATP to maintain the high [K⁺] inside and high [Na⁺] outside. This concentration gradient is the battery that the diffusion potential runs on. Its direct electrogenic effect (pumping 3 positive charges out for every 2 in) contributes only a few millivolts to the total potential.

6. What does it mean to “calculate a membrane potential using the modified nernst equation”?

This phrase refers to using the Goldman-Hodgkin-Katz (GHK) equation. It’s considered a “modified” version of the Nernst principle because it extends the concept from one ion to multiple ions, which is necessary for a biological context.

7. What are typical permeability ratios?

In a typical resting neuron, the relative permeabilities are approximately P_K : P_Na : P_Cl = 1 : 0.04 : 0.45. This shows the dominant role of potassium.

8. Can this calculator model an action potential?

Partially. You can manually change the permeability values to simulate different phases. For example, dramatically increasing P_Na mimics depolarization, and then increasing P_K mimics repolarization. However, a true neuron action potential involves dynamic, voltage-dependent changes in permeability, which this static calculator does not automate.

Related Tools and Internal Resources

Explore more concepts in cellular biophysics with our collection of specialized calculators:

• Nernst Potential Calculator: Calculate the equilibrium potential for a single ion.
• Osmolarity Calculator: Determine the osmolarity of solutions.
• Reversal Potential Calculator: A tool similar to our GHK calculator for exploring ion flow.
• Electrophysiology Tools Hub: A central collection of our neuroscience-related calculators.
• Cellular Biology Calculators: More tools for exploring the fundamental processes of life.
• Goldman-Hodgkin-Katz Equation Calculator: Another resource to calculate a membrane potential using the modified nernst equation.