The
Goldman-Hodgkin-Katz voltage equation, more commonly known as the
Goldman equation is used in cell membrane physiology to determine the equilibrium potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.
The discoverers of this are
David E. GoldmanDavid E. Goldman was a scientist famous for the Goldman equation which he derived for his doctorate degree at Columbia University....
of
Columbia UniversityColumbia University in the City of New York is a private university in the United States and a member of the Ivy League. Columbia's main campus lies in the Morningside Heights neighborhood in the borough of Manhattan, in New York City...
, and the English Nobel laureates
Alan Lloyd HodgkinSir Alan Lloyd Hodgkin, OM, KBE, FRS was a British physiologist and biophysicist, who won the 1963 Nobel Prize in Physiology or Medicine.-Early life:...
and
Bernard KatzSir 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...
.
The GHK voltage equation for monovalent positive ionic species and negative:
This results in the following if we consider a membrane separating two -solutions:
It is "Nernst-like" but has a term for each permeant ion.
The
Goldman-Hodgkin-Katz voltage equation, more commonly known as the
Goldman equation is used in cell membrane physiology to determine the equilibrium potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.
The discoverers of this are
David E. GoldmanDavid E. Goldman was a scientist famous for the Goldman equation which he derived for his doctorate degree at Columbia University....
of
Columbia UniversityColumbia University in the City of New York is a private university in the United States and a member of the Ivy League. Columbia's main campus lies in the Morningside Heights neighborhood in the borough of Manhattan, in New York City...
, and the English Nobel laureates
Alan Lloyd HodgkinSir Alan Lloyd Hodgkin, OM, KBE, FRS was a British physiologist and biophysicist, who won the 1963 Nobel Prize in Physiology or Medicine.-Early life:...
and
Bernard KatzSir 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...
.
Equation for monovalent ions
The GHK voltage equation for monovalent positive ionic species and negative:
This results in the following if we consider a membrane separating two -solutions:
It is "Nernst-like" but has a term for each permeant ion. The
Nernst equationIn electrochemistry, the Nernst equation is an equation which 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...
can be considered a special case of the Goldman equation for only one ion:
- = The membrane potential (in Volt
The volt is the SI derived unit of electromotive force, commonly called "voltage". It is also the unit for the related but slightly different quantity electric potential difference...
s, equivalent to JouleThe joule , named for James Prescott Joule, is the derived unit of energy in the International System of Units. It is the energy exerted by a force of one newton acting to move an object through a distance of one metre...
s per CoulombThe coulomb is the SI derived unit of electric charge. It is named after Charles-Augustin de Coulomb.- Definition :If 2 like point charges of equal magnitudes are placed in a vacuum at a distance of 1 metre away from each other and if they repel each other with a force of 9*1000000000 Newton, then...
)
- = the permeability
Permeability, permeable and semipermeable have several meanings:*Permeability , the degree of magnetization of a material in response to a magnetic field...
for that ion (in meters per second)
- = the extracellular concentration of that ion (in moles
The mole is a unit of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity. The name "mole" was coined in German by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier...
per cubic meter, to match the other SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
units)
- = the intracellular concentration of that ion (in moles
The mole is a unit of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity. The name "mole" was coined in German by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier...
per cubic meter)
- = The ideal gas constant (Joules per Kelvin
The kelvin is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero kelvin...
per moleThe mole is a unit of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity. The name "mole" was coined in German by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier...
)
- = The temperature in Kelvin
The kelvin is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero kelvin...
- = Faraday's constant
In physics and chemistry, the Faraday constant is the magnitude of electric charge per mole of electrons. While most uses of the Faraday constant, denoted F, have been replaced by the standard SI unit, the coulomb, the Faraday is still widely used in calculations in electrochemistry...
(CoulombThe coulomb is the SI derived unit of electric charge. It is named after Charles-Augustin de Coulomb.- Definition :If 2 like point charges of equal magnitudes are placed in a vacuum at a distance of 1 metre away from each other and if they repel each other with a force of 9*1000000000 Newton, then...
s per moleThe mole is a unit of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity. The name "mole" was coined in German by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier...
)
The first term, before the parenthesis, can be reduced to 61.5 log for calculations at human body temperature (37°C)
Note that the ionic charge determines the sign of the membrane potential contribution.
The usefulness of the GHK equation to
electrophysiologistsElectrophysiology 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...
is that it allows one to calculate the predicted membrane potential for any set of specified permeabilities. For example, if one wanted to calculate the resting potential of a cell, they would use the values of ion permeability that are present at rest (e.g. ). If one wanted to calculate the peak voltage of an
action potentialAn action potential is a transient alteration of the transmembrane voltage across an excitable membrane generated by the activity of voltage-gated ion channels embedded in the membrane. Action potentials play multiple roles in several types of excitable cells such as neurons, myocytes, and...
, one would simply substitute the permeabilities that are present at that time (e.g. ).
Calculating the first term
Using , , (assuming room temperature) and the fact that one
voltThe volt is the SI derived unit of electromotive force, commonly called "voltage". It is also the unit for the related but slightly different quantity electric potential difference...
is equal to one joule of energy per coulomb of charge, the equation
can be reduced to
Derivation
Goldman's equation seeks to determine the
voltageVoltage is commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt...
Em across a membrane. A
Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
is used to describe the system, with the
z direction being perpendicular to the membrane. Assuming that the system is symmetrical in the
x and
y directions (around and along the axon, respectively), only the
z direction need be considered; thus, the voltage
Em is the
integralIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
of the
z component of the
electric fieldIn physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. This electric field exerts a force on other electrically charged objects...
across the membrane.
According to Goldman's model, only two factors influence the motion of ions across a permeable membrane: the average electric field and the difference in ionic
concentrationIn chemistry, concentration is the measure of how much of a given substance there is mixed with another substance. This can apply to any sort of chemical mixture, but most frequently the concept is limited to homogeneous solutions, where it refers to the amount of solute in the solvent.To...
from one side of the membrane to the other. The electric field is assumed to be constant across the membrane, so that it can be set equal to
Em/
L, where
L is the thickness of the membrane. For a given ion denoted A with valence
nA, its
fluxIn 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 the amount that flows through a unit area per unit time...
jA—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula
The first term corresponds to
Fick's law of diffusionFick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855.- Fick's First Law :...
, which gives the flux due to
diffusionMolecular diffusion, often called simply diffusion, is a net transport of molecules from a region of higher concentration to one of lower concentration by random molecular motion. The result of diffusion is a gradual mixing of material...
down the
concentrationIn chemistry, concentration is the measure of how much of a given substance there is mixed with another substance. This can apply to any sort of chemical mixture, but most frequently the concept is limited to homogeneous solutions, where it refers to the amount of solute in the solvent.To...
gradient, i.e., from high to low concentration. The constant
DA is the diffusion constant of the ion A. The second term reflects the
fluxIn 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 the amount that flows through a unit area per unit time...
due to the electric field, which increases linearly with the electric field; this is a
Stokes-Einstein relationIn physics the Einstein relation is a previously unexpected connection revealed independently by Albert Einstein in 1905 and by Marian Smoluchowski in their papers on Brownian motion:linking D, the diffusion constant, and μp, the mobility of the particles; where is...
applied to
electrophoretic mobilityElectrophoresis is the motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric field. This electrokinetic phenomenon was observed for the first time in 1807 by Reuss , who noticed that the application of a constant electric field caused clay particles...
. The constants here are the
chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
valence-In molecules, atoms, or subatomic particles:* Valence * Valence bond theory in chemistry* Valence shell in chemistry* Valence band in physics* Valence quarks in particle physics-In other sciences:* Valency...
nA of the ion A (e.g., +1 for K
+, +2 for Ca
2+ and −1 for Cl
−), the
temperatureIn physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...
T (in
KelvinThe kelvin is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero kelvin...
s), the molar
gas constantThe gas constant is a physical constant which is featured in a large number of fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation...
R , and the Faraday
F, which is the total charge of a
moleThe mole is a unit of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity. The name "mole" was coined in German by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier...
of
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s.
Using the mathematical technique of
separation of variablesIn mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
, the equation may be separated
Integrating both sides from
z=0 (inside the membrane) to
z=
L (outside the membrane) yields the solution
where μ is a dimensionless number
and
PA is the ionic permeability, defined here as
The
electric currentElectric current can mean, depending on the context, a flow of electric charge or the rate of flow of electric charge ....
densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
JA equals the charge
qA of the ion multiplied by the flux
jA
There is such a current associated with every type of ion that can cross the membrane. By assumption, at the Goldman voltage
Em, the total current density is zero
If all the ions are monovalent—that is, if all the
nA equal either +1 or -1—this equation can be written
whose solution is the Goldman equation
where
If divalent ions such as
calciumCalcium 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 grey alkaline earth metal, and is the fifth most abundant element by mass in the Earth's crust...
are considered, terms such as
e2μ appear, which is the
squareIn algebra, the square of a number is that number multiplied by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x²...
of
eμ; in this case, the formula for the Goldman equation can be solved using the quadratic formula.
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