GHK current equation
Encyclopedia
The Goldman–Hodgkin–Katz flux equation (or GHK flux equation) describes the ionic flux
carried by an ion
ic 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 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.
of Columbia University
, and the English Nobel laureates Alan Lloyd Hodgkin
and Bernard Katz
derived this equation.
where
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
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 unknown!
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
:
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, [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
scaled by the surface area). 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.
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
carried by an ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
ic species across a cell membrane
Cell membrane
The cell membrane or plasma membrane is a biological membrane that separates the interior of all cells from the outside environment. The cell membrane is selectively permeable to ions and organic molecules and controls the movement of substances in and out of cells. It basically protects the cell...
as a function of the transmembrane potential
Membrane potential
Membrane potential is the difference in electrical potential between the interior and exterior of a biological cell. All animal cells are surrounded by a plasma membrane composed of a lipid bilayer with a variety of types of proteins embedded in it...
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
Nernst-Planck equation
The Nernst–Planck equation is a conservation of mass equation used to describe the motion of chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient \nabla c and an electric field E=-\nabla \phi...
and the GHK flux equation is a solution to the Nernst-Planck equation
Nernst-Planck equation
The Nernst–Planck equation is a conservation of mass equation used to describe the motion of chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient \nabla c and an electric field E=-\nabla \phi...
with the assumptions listed below.
Origin
The American David E. GoldmanDavid E. Goldman
David E. Goldman was a scientist famous for the Goldman equation which he derived for his doctorate degree at Columbia University....
of Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
, and the English Nobel laureates Alan Lloyd Hodgkin
Alan Lloyd Hodgkin
Sir Alan Lloyd Hodgkin, OM, KBE, PRS was a British physiologist and biophysicist, who shared the 1963 Nobel Prize in Physiology or Medicine with Andrew Huxley and John Eccles....
and Bernard Katz
Bernard Katz
Sir Bernard Katz, FRS was a German-born biophysicist, noted for his work on nerve biochemistry. He shared the Nobel Prize in physiology or medicine in 1970 with Julius Axelrod and Ulf von Euler...
derived this equation.
Assumptions
Several assumptions are made in deriving the GHK flux equation (Hille 2001) :- 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):where
S is the current across the membrane carried by ion S, measured in ampereAmpereThe ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
s per centimeter squared (A = C·s−1cm−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 voltVoltThe volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
s - F is the Faraday constant, equal to 96,485 C·mol−1 or J·V−1·mol−1
- R is the gas constantGas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
, equal to 8.314 J·K−1·mol−1 - T is the absolute temperature, measured in KelvinKelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
(= 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 potentialReversal potential
In a biological membrane, the reversal potential of an ion is the membrane potential at which there is no net flow of that particular ion from one side of the membrane to the other...
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
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 unknown!We turn to l'Hôpital's rule
L'Hôpital's rule
In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
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 thuswhich is the definition of the reversal potential.
By setting
we can also obtain the reversal potential :which reduces to :
and produces the Nernst equation
Nernst equation
In electrochemistry, the Nernst equation is an equation that can be used to determine the equilibrium reduction potential of a half-cell in an electrochemical cell. It can also be used to determine the total voltage for a full electrochemical cell...
:
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 valueAsymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
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, [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 lawOhm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
scaled by the surface area). 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
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
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
Rectifier
A rectifier is an electrical device that converts alternating current , which periodically reverses direction, to direct current , which flows in only one direction. The process is known as rectification...
.
The GHK flux equation is mostly used by electrophysiologists
Electrophysiology
Electrophysiology is the study of the electrical properties of biological cells and tissues. It involves measurements of voltage change or electric current on a wide variety of scales from single ion channel proteins to whole organs like the heart...
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
Action potential
In physiology, an action potential is a short-lasting event in which the electrical membrane potential of a cell rapidly rises and falls, following a consistent trajectory. Action potentials occur in several types of animal cells, called excitable cells, which include neurons, muscle cells, and...
. The most common example is probably intracellular calcium
Calcium
Calcium is the chemical element with the symbol Ca and atomic number 20. It has an atomic mass of 40.078 amu. Calcium is a soft gray alkaline earth metal, and is the fifth-most-abundant element by mass in the Earth's crust...
, [Ca2+]i, which during a cardiac action potential
Cardiac action potential
In electrocardiography, the cardiac action potential is a specialized action potential in the heart, necessary for the electrical conduction system of the heart....
cycle can change 100-fold or more, and the ratio between [Ca2+]o and [Ca2+]i can reach 20,000 or more.