The Goldman–Hodgkin–Katz flux equation (or GHK flux equation or GHK current density equation) describes the ionic flux across a cell membrane as a function of the transmembrane potential and the concentrations of the ion inside and outside of the cell.
Since both the voltage and the concentration gradients influence the movement of ions, this process is a simplified version of electrodiffusion.
The American David E. Goldman of Columbia University, and the English Nobel laureates Alan Lloyd Hodgkin and Bernard Katz derived this equation.
Several assumptions are made in deriving the GHK flux equation (Hille 2001, p. 445) : The GHK flux equation for an ion S (Hille 2001, p. 445): where The reversal potential is shown to be contained in the GHK flux equation (Flax 2008).
We turn to l'Hôpital's rule to find the solution for the limit: where
represents the differential of f and the result is : It is evident from the previous equation that when
we can also obtain the reversal potential : which reduces to : and produces the Nernst equation : Since one of the assumptions of the GHK flux equation is that the ions move independently of each other, the total flow of ions across the membrane is simply equal to the sum of two oppositely directed fluxes.
Each flux approaches an asymptotic value as the membrane potential diverges from zero.
These asymptotes are and where subscripts 'i' and 'o' denote the intra- and extracellular compartments, respectively.
Keeping all terms except Vm constant, the equation yields a straight line when plotting
As the ratio between the two concentrations increases, so does the difference between the two slopes, meaning that the current is larger in one direction than the other, given an equal driving force of opposite signs.
This is contrary to the result obtained if using Ohm's law scaled by the surface area, and the effect is called rectification.
The GHK flux equation is mostly used by electrophysiologists when the ratio between [S]i and [S]o is large and/or when one or both of the concentrations change considerably during an action potential.
The most common example is probably intracellular calcium, [Ca2+]i, which during a cardiac action potential cycle can change 100-fold or more, and the ratio between [Ca2+]o and [Ca2+]i can reach 20,000 or more.