Goldman equation

The Goldman equation or Goldman-Hodgkin-Katz equation (GHK equation for short) according to David Eliot Goldman (1910–1998), Alan Lloyd Hodgkin and Bernard Katz is a way of calculating the membrane potential , taking into account several permeating ions .

Explanation

It allows the calculation of a membrane potential for a membrane that is permeable to various ions, such as sodium , potassium and chloride ions, and can be interpreted as a generalization of the Nernst equation , which is often used as a first approximation for the description of the Equilibrium potential difference ( resting membrane potential ) across a cell membrane is used.

In contrast to the Nernst equation , the Goldman equation is not based on a state of equilibrium, but on the principle of a steady state . This means here that the sum of all ion currents must be zero (while for example a potassium current can exist on the extracellular side). Further assumptions of the Goldman equation are the independence of the ions from one another, and a linear drop in potential across the membrane thickness - because of the resulting constant field, one often speaks of a "constant field equation". In particular, no consideration is given to the fact that at rest membrane potentials the currents do not go across the entire membrane, but rather through individual channels (the equation was established - in 1943 by Goldman and in 1949 by Hodgkin and Katz - before ion channels were known).

The ion current amplitudes depend in a complicated way on the membrane voltage and the ion concentration and can therefore only be calculated approximately. In the Goldman equation, the ion current is approximated as a function of the ion concentration and a coefficient, the so-called permeability P. The permeability is derived from Fick's law of diffusion . It is the quotient of the diffusion constant and the membrane thickness.
Note that the concentration ratios, from inside and outside for each ion, over the entire course of an action potential, remain practically constant at their resting potential ratios. Even if the membrane voltage changes by approx. 100mV, the currents that occur for the change in the ion concentrations are negligible.

formula

For a membrane potential mediated by potassium ( ), chloride ( ), and sodium ( ) currents, the Goldman equation is ${\ displaystyle {\ text {K}} ^ {+}}$ ${\ displaystyle {\ text {Cl}} ^ {-}}$ ${\ displaystyle {\ text {Na}} ^ {+}}$

${\ displaystyle U _ {\ text {M}} = {\ frac {RT} {F}} \ cdot \ ln {\ frac {P _ {\ text {Na}} \ cdot [{\ text {Na}} ^ { +}] _ {a} + P _ {\ text {K}} \ cdot [{\ text {K}} ^ {+}] _ {a} + P _ {\ text {Cl}} \ cdot [{\ text {Cl}} ^ {-}] _ {i}} {P _ {\ text {Na}} \ cdot [{\ text {Na}} ^ {+}] _ {i} + P _ {\ text {K} } \ cdot [{\ text {K}} ^ {+}] _ {i} + P _ {\ text {Cl}} \ cdot [{\ text {Cl}} ^ {-}] _ {a}}} }$ ,

wherein the universal gas constant , the absolute temperature (in Kelvin ), the Faraday constant and the permeability (see text) referred to; stands for outside, for inside. ${\ displaystyle R}$ ${\ displaystyle T}$ ${\ displaystyle F}$ ${\ displaystyle P}$ ${\ displaystyle a}$ ${\ displaystyle i}$

example

Example of the resting membrane potential of the giant axon in cuttlefish :

	c (outside) in mmol	c (inside) in mmol	Nernst potential in mV
Na ⁺	460	50	+56
K ⁺	10	410	−94
Cl ^-	540	60	+55

ΔΨ = −56 mV

Individual evidence

↑ Sadava et al .: Purve's Biology . Spektrum Akademie Verlag, 2011, p. 1261.

literature

Goldman, DE (1943): Potential, impedance, and rectification in membranes . In: Journal of General Physiology , 27: 37-60