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# GHK current equation

The Goldman-Hodgkin-Katz current equation (or GHK current equation) describes the current carried by an ionic species 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 called electrodiffusion.

## The eponyms of the equation

The American David E. Goldman of Columbia University, and the English Nobel laureates Alan Lloyd Hodgkin and Bernard Katz derived this equation.

## Assumptions underlying the validity of the equation

Several assumptions are made in deriving the GHK current equation:

• The membrane is a homogeneous substance
• The electrical field is constant so that the transmembrane potential varies linearly across the membrane
• The ions access the membrane instantaneously from the intra- and extracellular solutions
• The permeant ions do not interact
• The movement of ions is affected by both concentration and voltage differences

## The equation

The GHK current equation for an ion S: $I_{S} = P_{S}z_{S}^2\frac{V_{m}F^{2}}{RT}\frac{[\mbox{S}]_{i} - [\mbox{S}]_{o}\exp(-z_{S}V_{m}F/RT)}{1 - \exp(-z_{S}V_{m}F/RT)}$

where

• IS is the current across the membrane carried by ion S, measured in amperes (A = C·s-1)
• PS is the permeability of ion S measured in m3·s-1
• zS is the charge of ion S in elementary charges
• Vm is the transmembrane potential in volts
• F is the Faraday constant, equal to 96,485 C·mol-1 or J·V-1·mol-1
• R is the gas constant, equal to 8.314 J·K-1·mol-1
• T is the absolute temperature, measured in kelvins (= degrees Celsius + 273.15)
• [S]i is the intracellular concentration of ion S, measured in mol·m-3 or mmol·l-1
• [S]o is the extracellular concentration of ion S, measured in mol·m-3

## Rectification and the GHK current equation

Since one of the assumptions of the GHK current 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 (or current) approaches an asymptotic value as the membrane potential diverges from zero. These asymptotes are $I_{S|i\to o} = P_{S}z_{S}^2 \frac{V_{m}F^{2}}{RT}[\mbox{S}]_{i}\ \mbox{for}\ V_{m} \gg \; 0$

and $I_{S|o\to i} = P_{S}z_{S}^2 \frac{V_{m}F^{2}}{RT}[\mbox{S}]_{o}\ \mbox{for}\ V_{m} \ll \; 0$

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 IS against Vm. It is evident that the ratio between the two asymptotes is merely the ratio between the two concentrations of S, [S]i and [S]o. Thus, if the two concentrations are identical, the slope will be identical (and constant) throughout the voltage range (corresponding to Ohm's law). 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, and the effect is called rectification.

The GHK current 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.

## References

• Bertil Hille Ion channels of excitable membranes, 3rd ed., Sinauer Associates, Sunderland, MA (2001). ISBN 0-88214-320-2