Resting Membrane Potential Calculator (Goldman-Hodgkin-Katz)

Calculate the resting membrane potential using the Goldman-Hodgkin-Katz equation, which considers the relative permeabilities and concentrations of key ions.

Calculator Inputs

Resting Membrane Potential

— mV

Using the Goldman-Hodgkin-Katz (GHK) equation:
Vm = (RT/F) * ln( (P_K[K⁺]_e + P_Na[Na⁺]_e + P_Cl[Cl^–]_i) / (P_K[K⁺]_i + P_Na[Na⁺]_i + P_Cl[Cl^–]_e) )
Where R is the ideal gas constant, T is absolute temperature, and F is Faraday’s constant.

Permeability vs. Potential Simulation

What is Resting Membrane Potential?

Resting membrane potential refers to the electrical potential difference (voltage) across the plasma membrane of a neuron or muscle cell when it is not stimulated or excited. This potential is maintained by the selective permeability of the cell membrane to different ions and the activity of ion pumps. It’s a fundamental property that allows cells to generate electrical signals, such as action potentials, which are critical for nerve impulse transmission and muscle contraction. A typical resting potential in animal cells ranges from -40 mV to -90 mV, with the inside of the cell being negative relative to the outside.

Who should use this calculator? Biologists, neuroscientists, physiologists, medical students, researchers, and anyone studying cellular electrophysiology can use this calculator. It’s particularly useful for understanding how changes in ion concentrations or membrane permeability affect cellular excitability.

Common misconceptions about resting membrane potential include believing it’s a static value (it can fluctuate slightly) or that it’s solely determined by one ion (it’s a complex interplay of multiple ions). Another misconception is that the inside of the cell is always positive; at rest, it’s typically negative.

Resting Membrane Potential Formula and Mathematical Explanation

The most comprehensive equation for calculating resting membrane potential, especially when multiple ions contribute significantly to the potential, is the Goldman-Hodgkin-Katz (GHK) equation. It’s an extension of the simpler Nernst equation, which considers only one ion. The GHK equation accounts for the permeability of the membrane to several ions (commonly Na⁺, K⁺, and Cl^–) and their respective electrochemical gradients.

The GHK equation can be expressed as:

V_m = (RT/F) \* ln( (Σ P_i[C_i^out] + Σ P_j[C_jⁱⁿ]) / (Σ P_i[C_iⁱⁿ] + Σ P_j[C_j^out]) )

For the three main ions (K⁺, Na⁺, Cl^–), and noting that Cl^– is typically an anion and its concentration gradient is reversed relative to cations, the equation simplifies to:

V_m = (RT/F) \* ln( (P_K[K⁺]_e + P_Na[Na⁺]_e + P_Cl[Cl^–]_i) / (P_K[K⁺]_i + P_Na[Na⁺]_i + P_Cl[Cl^–]_e) )

Variable Explanations:

GHK Equation Variables

Variable	Meaning	Unit	Typical Range
Vm	Membrane Potential	Millivolts (mV)	-40 to -90 mV
R	Ideal Gas Constant	8.314 J/(mol·K)	Constant
T	Absolute Temperature	Kelvin (K)	~310 K (37°C)
F	Faraday’s Constant	96,485 C/mol	Constant
PK, PNa, PCl	Permeability of K^+, Na^+, Cl^–	Arbitrary Units (e.g., x10^-8 cm/s)	PK: 1-10; PNa: 0.01-0.1; PCl: 0.1-1
[K^+]i, [K^+]e	Intracellular / Extracellular K^+ Concentration	Millimolar (mM)	[K^+]i: 120-170; [K^+]e: 3-7
[Na^+]i, [Na^+]e	Intracellular / Extracellular Na^+ Concentration	Millimolar (mM)	[Na^+]i: 10-20; [Na^+]e: 135-155
[Cl^–]i, [Cl^–]e	Intracellular / Extracellular Cl^– Concentration	Millimolar (mM)	[Cl^–]i: 5-15; [Cl^–]e: 100-150

The term RT/F is often combined into a single value, the “. At 37°C (310 K), RT/F ≈ 26.7 mV. This constant reflects the thermal energy available to drive ion movement. The logarithmic term represents the ratio of the contributions of the cations (Na⁺, K⁺) and the anion (Cl^–) to the membrane potential, weighted by their respective permeabilities. A higher permeability to an ion means that ion has a greater influence on the resting membrane potential.

Practical Examples (Real-World Use Cases)

Example 1: Typical Neuron at Rest

Let’s calculate the resting membrane potential for a typical neuron under physiological conditions:

• P_K = 1.0 (arbitrary units)
• P_Na = 0.04 (arbitrary units)
• P_Cl = 0.45 (arbitrary units)
• [K⁺]_i = 150 mM, [K⁺]_e = 5 mM
• [Na⁺]_i = 15 mM, [Na⁺]_e = 145 mM
• [Cl^–]_i = 10 mM, [Cl^–]_e = 120 mM
• Temperature = 37°C (RT/F ≈ 26.7 mV)

Calculation:
Numerator = (1.0 * 5) + (0.04 * 145) + (0.45 * 10) = 5 + 5.8 + 4.5 = 15.3
Denominator = (1.0 * 150) + (0.04 * 15) + (0.45 * 120) = 150 + 0.6 + 54 = 204.6
V_m = 26.7 mV \* ln(15.3 / 204.6)
V_m = 26.7 mV \* ln(0.07478)
V_m = 26.7 mV \* (-2.60) ≈ -69.4 mV

Interpretation: This result of -69.4 mV is close to the typical resting membrane potential for neurons, indicating that the high resting permeability to K⁺ dominates the potential, making the inside of the cell negative. The contribution of Na⁺ and Cl^– slightly shifts the potential from the pure K⁺ equilibrium potential.

Example 2: Effect of Increased Sodium Permeability

Consider a scenario where a toxin increases the membrane’s permeability to sodium ions:

• P_K = 1.0
• P_Na = 0.4 (10x increase from typical)
• P_Cl = 0.45
• Ion concentrations remain the same as Example 1.
• Temperature = 37°C (RT/F ≈ 26.7 mV)

Calculation:
Numerator = (1.0 * 5) + (0.4 * 145) + (0.45 * 10) = 5 + 58 + 4.5 = 67.5
Denominator = (1.0 * 150) + (0.4 * 15) + (0.45 * 120) = 150 + 6 + 54 = 210
V_m = 26.7 mV \* ln(67.5 / 210)
V_m = 26.7 mV \* ln(0.3214)
V_m = 26.7 mV \* (-1.135) ≈ -30.3 mV

Interpretation: The significant increase in sodium permeability causes a substantial depolarization (less negative) of the membrane potential, shifting it from -69.4 mV to -30.3 mV. This is because the membrane potential is now more influenced by the electrochemical gradient for sodium. This level of depolarization might bring the cell closer to its threshold for firing an action potential, or if the increase is severe, it could lead to excitotoxicity. This highlights the critical role of selective ion permeability in maintaining cellular function and excitability.

How to Use This Resting Membrane Potential Calculator

1. Input Ion Permeabilities: Enter the relative permeabilities for Potassium (P_K), Sodium (P_Na), and Chloride (P_Cl). These values can be in arbitrary units, but consistency is key. Higher numbers mean greater permeability. For typical resting conditions, P_K is much higher than P_Na and P_Cl.
2. Input Ion Concentrations: Provide the intracellular ([Ion]_i) and extracellular ([Ion]_e) concentrations for Potassium (K⁺), Sodium (Na⁺), and Chloride (Cl^–) in millimolar (mM).
3. Specify Temperature: Input the temperature in degrees Celsius (°C). The default is 37°C, representing body temperature.
4. Calculate: Click the “Calculate Potential” button.

How to Read Results:

• Primary Result (V_m): This is the calculated resting membrane potential in millivolts (mV). A negative value indicates the inside of the cell is negatively charged relative to the outside.
• Intermediate Values: These display the calculated contribution ratio of the ions to the potential, showing the impact of each ion’s concentration gradient and permeability.
• Formula Explanation: Provides context on the Goldman-Hodgkin-Katz equation used.
• Simulation Chart: Visually demonstrates how changes in P_K and P_Na affect the calculated membrane potential when other factors are held constant.

Decision-Making Guidance: Use the calculator to explore “what-if” scenarios. For example, how does a condition like hyperkalemia (high extracellular K⁺) affect the resting potential? Or how might a drug that blocks sodium channels (reducing P_Na) alter cellular excitability? The results can help predict cellular responses and understand disease mechanisms.

Key Factors That Affect Resting Membrane Potential Results

1. Relative Ion Permeabilities (P_K, P_Na, P_Cl): This is the most direct factor. The GHK equation weights the ion concentration gradients by their permeabilities. Typically, P_K is highest at rest, making the membrane potential close to the K⁺ equilibrium potential. Any change that alters these permeabilities (e.g., ion channel activity, toxins, drugs) will significantly shift V_m.
2. Ion Concentration Gradients: The difference between the intracellular and extracellular concentrations of ions ([K⁺]_i vs [K⁺]_e, etc.) creates the electrochemical driving force for each ion. For example, high extracellular K⁺ (hyperkalemia) reduces the driving force for K⁺ efflux, causing depolarization. Conversely, low extracellular K⁺ causes hyperpolarization.
3. Temperature: Temperature affects the kinetic energy of ions and the rate of ion transport. Higher temperatures increase ion movement and can slightly alter the membrane potential, primarily by affecting the RT/F term in the GHK equation. Standard physiological temperature is 37°C.
4. Presence of Other Ions: While this calculator focuses on K⁺, Na⁺, and Cl^–, other ions like Ca²⁺ and organic anions also contribute to the cell’s electrical properties, especially under certain conditions or in specific cell types. The GHK equation can be extended to include more ions if their permeabilities and concentrations are known.
5. Activity of Ion Pumps: The Na⁺/K⁺-ATPase pump actively transports ions against their concentration gradients, maintaining the steep concentration gradients that the GHK equation relies upon. While the pump’s direct electrogenic effect (net charge movement) is usually small, its role in preserving the gradients is crucial for establishing and maintaining the resting potential over time. Disrupting pump function indirectly affects V_m.
6. Cell Type and Membrane Composition: Different cell types have unique sets of ion channels and transporters, leading to variations in their resting membrane potentials. For instance, glial cells often have a resting potential primarily determined by K⁺, making them even more negative than typical neurons. Muscle cells have their own characteristic resting potentials influencing their excitability.
7. Changes in pH: Alterations in extracellular or intracellular pH can affect the function of ion channels and transporters, indirectly influencing ion permeabilities and concentrations, and thus the membrane potential.

Frequently Asked Questions (FAQ)

What is the typical resting membrane potential of a cell?

The typical resting membrane potential for animal cells, particularly neurons and muscle cells, ranges from -40 mV to -90 mV. Neurons are often around -70 mV.

Why is the resting membrane potential negative?

It’s negative primarily because the cell membrane is much more permeable to potassium ions (K⁺) than to sodium ions (Na⁺) at rest. K⁺ tends to flow out of the cell down its concentration gradient, leaving behind unbalanced negative charges inside the cell.

What happens if extracellular potassium (K⁺) increases?

An increase in extracellular K⁺ (hyperkalemia) reduces the concentration gradient for K⁺ efflux. This makes the membrane potential less negative (depolarization), moving it closer to zero. This can lead to increased cell excitability or, in severe cases, arrhythmias.

How does the sodium-potassium pump relate to resting potential?

The Na⁺/K⁺-ATPase pump actively transports 3 Na⁺ ions out of the cell for every 2 K⁺ ions it brings in. While this pump has a small direct electrogenic effect, its primary role is maintaining the steep Na⁺ and K⁺ concentration gradients across the membrane, which are essential for establishing the resting potential calculated by the GHK equation.

Is the GHK equation always accurate?

The GHK equation is a powerful model but relies on assumptions, such as constant field theory and uniform ion distribution. It provides a good approximation for many physiological conditions but may not perfectly capture complex scenarios involving highly localized ion concentration changes, complex channel gating, or the influence of less abundant ions.

What does it mean for a cell to be depolarized or hyperpolarized?

Depolarization means the membrane potential becomes less negative (moves closer to zero). Hyperpolarization means the membrane potential becomes more negative (moves further away from zero). Both are critical for cellular signaling.

Can Chloride ions (Cl^–) affect resting potential?

Yes, Chloride ions can influence the resting membrane potential, especially in cells where the chloride equilibrium potential is close to the resting potential or when chloride permeability is significant. In many neurons, the resting potential is close to the chloride equilibrium potential, meaning changes in chloride permeability might not drastically shift the resting potential but are crucial for inhibitory signaling.

What is the significance of the P_Na/P_K ratio?

The ratio of sodium to potassium permeability (P_Na/P_K) is a key determinant of how close the resting membrane potential is to the equilibrium potentials of Na⁺ and K⁺ (E_Na and E_K). A low P_Na/P_K ratio (typical at rest) means V_m is closer to E_K. An increased P_Na/P_K ratio leads to depolarization and increased excitability.