Goldman–Hodgkin–Katz flux equation

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. 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.

Origin

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

Assumptions

Several assumptions are made in deriving the GHK flux equation (Hille 2001, p. 445) :

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

Equation

The GHK flux equation for an ion S (Hille 2001, p. 445):

Φ

S

=

P

S

z

S

2

V

m

F

2

R

T

[

S

]

i

−

[

S

]

o

exp

⁡

(

−

z

S

V

m

F

/

R

T

)

1

−

exp

⁡

(

−

z

S

V

m

F

/

R

T

)

{\displaystyle \Phi _{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

Φ

{\displaystyle \Phi }

S is the current density (flux) outward through the membrane carried by ion S, measured in amperes per square meter (A·m−2)

PS is the permeability of the membrane for ion S measured in m·s−1

zS is the valence of ion S

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

Implicit definition of reversal potential

The reversal potential is shown to be contained in the GHK flux equation (Flax 2008). The proof is replicated from the reference (Flax 2008) here.

We wish to show that when the flux is zero, the transmembrane potential is not zero. Formally it is written

lim

Φ

S

→

0

V

m

≠

0

{\displaystyle \lim _{\Phi _{S}\rightarrow 0}V_{m}\neq 0}

which is equivalent to writing

lim

V

m

→

0

Φ

S

≠

0

{\displaystyle \lim _{V_{m}\rightarrow 0}\Phi _{S}\neq 0}

, which states that when the transmembrane potential is zero, the flux is not zero.