The Goldman–Hodgkin–Katz flux 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. Electrodiffusion is most accurately defined by the Nernst–Planck equation and the GHK flux equation is a solution to the Nernst–Planck equation with the assumptions listed below.
The reversal potential is shown to be contained in the GHK flux equation. The proof is replicated from the reference here. We wish to show that when the flux is zero, the transmembrane potential is not zero. Formally it is written which is equivalent to writing, which states that when the transmembrane potential is zero, the flux is not zero. However, due to the form of the GHK flux equation when,. This is a problem as the value of is indeterminate. 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, if and thus which is the definition of the reversal potential. By setting we can also obtain the reversal potential : which reduces to : and produces the Nernst equation :
Rectification
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 S against Vm. It is evident that the ratio between the two asymptotes is merely the ratio between the two concentrations of S, i and o. Thus, if the two concentrations are identical, the slope will be identical throughout the voltage range. 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 i and 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, i, which during a cardiac action potential cycle can change 100-fold or more, and the ratio between o and i can reach 20,000 or more.
