Goldman equation

The Goldman-Hodgkin-Katz voltage equation, more commonly known as the Goldman equation is used in cell membrane physiology to determine the potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.

The discoverers of the formula

GHK was the work of three men - the American David E. Goldman of Columbia University, and the English Nobel laureates Alan Lloyd Hodgkin and Bernard Katz.

A variation on Nernst

"GHK" as it is most often referred to by electrophysiologists, is a variation on the Nernst equation. The Nernst equation can essentially calculate the membrane potential of a cell when only one ion is permeant, as long as the concentrations of that ion both inside and outside the cell are known. The Nernst equation cannot, however, deal with cells having permeability to more than one ion.

One notable distinction, however, is that the Nernst equation deals with a thermodynamic equilibrium, whereas the GHK equation deals with a non-equilibrium steady state. Therefore, the Nernst equation is exact and independent of the type of boundary between the electrochemical cells. The GHK voltage equation is not exact and makes assumptions with regards to the mechanism of diffusion, which influences the final result.

The equation

The GHK equation for $N$ positive ionic species and $M$ negative:

$E_{m}={\frac {RT}{F}}\ln {\left({\frac {\sum _{i}^{N}P_{M_{i}^{+}}[M_{i}^{+}]_{out}+\sum _{j}^{M}P_{A_{j}^{-}}[A_{j}^{-}]_{in}}{\sum _{i}^{N}P_{M_{i}^{+}}[M_{i}^{+}]_{in}+\sum _{j}^{M}P_{A_{j}^{-}}[A_{j}^{-}]_{out}}}\right)}$

This results in the following if we consider a membrane separating two $K_{x}Na_{1-x}Cl$-solutions:

$E_{m,K_{x}Na_{1-x}Cl}={\frac {RT}{F}}\ln {\left({\frac {P_{Na^{+}}[Na^{+}]_{out}+P_{K^{+}}[K^{+}]_{out}+P_{Cl^{-}}[Cl^{-}]_{in}}{P_{Na^{+}}[Na^{+}]_{in}+P_{K^{+}}[K^{+}]_{in}+P_{Cl^{-}}[Cl^{-}]_{out}}}\right)}$

It is "Nernst-like" but has a term for each permeant ion.

• $E_{m}$ = The membrane potential
• $P_{ion}$ = the permeability for that ion
• $[ion]_{out}$ = the extracellular concentration of that ion
• $[ion]_{in}$ = the intracellular concentration of that ion
• $R$ = The ideal gas constant
• $T$ = The temperature in kelvins
• $F$ = Faraday's constant

The first term, before the parenthesis, can be reduced to 59 log for calculations under standard physiological conditions. Note that the ionic charge determines the sign of the membrane potential contribution.

The usefulness of the GHK equation to electrophysiologists is that it allows one to calculate the predicted membrane potential for any set of specified permeabilities. For example, if one wanted to calculate the resting potential of a cell, they would use the values of ion permeability that are present at rest (e.g. $P_{K^{+}}>>P_{Na^{+}}$). If one wanted to calculate the peak voltage of an action potential, one would simply substitute the permeabilities that are present at that time (e.g. $P_{Na^{+}}>>P_{K^{+}}$).

See Also

GHK current equation

External links

• Nernst/Goldman Equation Simulator