Goldman-Hodgkin-Katz Equation Calculator

When excitable cells, like neurons and muscle cells, are at rest, they concentrate sodium (Na⁺) and chloride (Cl⁻) ions outside the cell membrane. In contrast, potassium (K⁺) ions and large anions, such as phosphates and proteins, are concentrated inside the cell.

Na⁺/K⁺ ATPase pumps in the membrane actively maintain the concentration gradients of the two cations. In contrast, the concentration gradient for Cl⁻ ions develops passively. Intracellular anions accumulate within the cell because they are too large to pass through the membrane.

This distribution of anions and cations maintains the fluid compartments on each side of the membrane electrically neutral overall while allowing concentration gradients to drive ion movement across the membrane when transport mechanisms are available.

Differences in ion concentration create the driving force for ion movement, but the actual membrane potential depends on which ions are able to cross the membrane.

The number of open ion channels determines the permeability of a membrane. The cell membrane of excitable cells contains a relatively high number of K⁺ leak channels, making it most permeable to K⁺. In contrast, the cell membrane has fewer open channels for Na⁺ and Cl^–, resulting in lower permeability for these ions.

Potassium’s dominance is reflected in the permeability scale: K⁺ is assigned a reference value of 1 (P_K = 1), while Na⁺ has much lower permeability (0.01 – 0.05) and Cl^– has moderate permeability (0.4 – 0.5).

The following table shows typical ion concentrations and relative permeabilities in mammalian cells:

This difference in permeability means that not all ions contribute equally to the membrane potential—those with higher permeability have greater influence.

The GHK equation calculates the membrane potential (Vm) by considering the equilibrium potential of each ion, based on its concentration gradient, and adjusting for or weighing the membrane’s permeability to those ions.

Equilibrium Potential

An ion’s equilibrium potential (E_ion) is the opposing electrical force (voltage) that prevents an ion’s net movement across the cell membrane along its concentration gradient. An ion’s equilibrium potential can be calculated using the Nernst equation.

E_ion = 61 × log₁₀ (Ion_(out) / Ion_(in))

The opposing electrical force forms as an ion species moves across the cell membrane along its concentration gradient. As like charges accumulate on the opposing side of the membrane, they increasingly repel the movement of similarly charged ions. At equilibrium, ions still move across the membrane but in equal numbers, preventing either side of the membrane from gaining or losing net charge.

Weighting Factor

The term ‘weighted’ indicates that the GHK equation also considers the membrane permeability of each ion. In the equation, each ion’s concentration ratio is multiplied by its membrane permeability factor.

((P_ion × Ion_(out)) / (P_ion × Ion_(in)))

The greater the membrane’s permeability to a given ion, the more that ion’s equilibrium potential dominates the tug-of-war and pulls the measured membrane potential towards it.

Resulting Membrane Potential

Because the membrane is most permeable to K⁺, the resting membrane potential tends toward this ion’s equilibrium potential. However, Na⁺ permeability allows these ions to enter the cell, pulling the membrane potential in a positive direction. This balance of opposing influences results in a typical resting potential of approximately -70 mV. Chloride ions have little impact because their equilibrium potential is close to the resting membrane potential.