Goldman equation
Encyclopedia
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. Goldman
of Columbia University
, and the English Nobel laureates Alan Lloyd Hodgkin
and Bernard Katz
.
monovalent positive ion
ic 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 equation
can be considered a special case of the Goldman equation for only one ion:
The first term, before the parenthesis, can be reduced to 61.5 mV 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 electrophysiologists
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 potential
, one would simply substitute the permeabilities that are present at that time (e.g.
).
, 
, (assuming body temperature) 
and the fact that one volt is equal to one joule of energy per coulomb of charge, the equation
can be reduced to
Em across a membrane. A Cartesian coordinate system
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 integral
of the z component of the electric field
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 concentration
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 flux
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 diffusion
, which gives the flux due to diffusion
down the concentration
gradient, i.e., from high to low concentration. The constant DA is the diffusion constant of the ion A. The second term reflects the flux
due to the electric field, which increases linearly with the electric field; this is a Stokes-Einstein relation
applied to electrophoretic mobility
. The constants here are the charge
valence
nA of the ion A (e.g., +1 for K+, +2 for Ca2+ and −1 for Cl−), the temperature
T (in kelvin
s), the molar gas constant
R, and the faraday F, which is the total charge of a mole of electron
s.
Using the mathematical technique of separation of variables
, 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 current
density
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 calcium
are considered, terms such as e2μ appear, which is the square of eμ; in this case, the formula for the Goldman equation can be solved using the quadratic formula.
Physiology
Physiology is the science of the function of living systems. This includes how organisms, organ systems, organs, cells, and bio-molecules carry out the chemical or physical functions that exist in a living system. The highest honor awarded in physiology is the Nobel Prize in Physiology or...
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. Goldman
David 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...
.
Equation for monovalent ions
The GHK voltage equation for monovalent positive ionIon
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 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 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...
can be considered a special case of the Goldman equation for only one ion:
= The membrane 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, equivalent to jouleJouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s per coulomb)
= the permeability for that ion (in meters per second)
= the extracellular concentration of that ion (in molesMole (unit)The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
per cubic meter, to match the other SISiSi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units)
= the intracellular concentration of that ion (in moles per cubic meter)
= The ideal gas constant (joules per 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...
per mole)
= The temperature in kelvins
= Faraday's constant (coulombs per mole)
The first term, before the parenthesis, can be reduced to 61.5 mV 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 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...
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 potentialAction 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...
, one would simply substitute the permeabilities that are present at that time (e.g.
).Calculating the first term
Using, , (assuming body temperature) and the fact that one volt is equal to one joule of energy per coulomb of charge, the equationcan be reduced to
Derivation
Goldman's equation seeks to determine the voltageVoltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
Em across a membrane. A Cartesian coordinate system
Cartesian coordinate system
A 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 integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the z component of the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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 concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
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 flux
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...
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 diffusion
Fick's law of diffusion
Fick'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 diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
down the concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
gradient, i.e., from high to low concentration. The constant DA is the diffusion constant of the ion A. The second term reflects the flux
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...
due to the electric field, which increases linearly with the electric field; this is a Stokes-Einstein relation
Einstein relation (kinetic theory)
In 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...
applied to electrophoretic mobility
Electrophoresis
Electrophoresis, also called cataphoresis, 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...
. The constants here are the charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
valence
Valence (chemistry)
In chemistry, valence, also known as valency or valence number, is a measure of the number of bonds formed by an atom of a given element. "Valence" can be defined as the number of valence bonds...
nA of the ion A (e.g., +1 for K+, +2 for Ca2+ and −1 for Cl−), the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
T (in kelvin
Kelvin
The 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...
s), the molar gas constant
Gas constant
The 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,...
R, and the faraday F, which is the total charge of a mole of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s.
Using the mathematical technique of separation of variables
Separation of variables
In 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 current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
density
Current density
Current 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 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...
are considered, terms such as e2μ appear, which is the square of eμ; in this case, the formula for the Goldman equation can be solved using the quadratic formula.
See also
- GHK current equationGHK current equationThe Goldman–Hodgkin–Katz flux equation describes the ionic flux 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...
- Nernst equationNernst equationIn 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...
- Hodgkin-Huxley modelHodgkin-Huxley modelThe Hodgkin–Huxley model is a mathematical model that describes how action potentials in neurons are initiated and propagated....
- Saltatory conductionSaltatory conductionSaltatory conduction is the propagation of action potentials along myelinated axons from one node of Ranvier to the next node, increasing the conduction velocity of action potentials without needing to increase the diameter of an axon.-Mechanism:Because the cytoplasm of the axon is electrically...
- BioelectronicsBioelectronicsBioelectronics is a recently coined term for a field of research that works to establish a synergy between electronics and biology. One of the main forums for information about the field is the Elsevier journal Biosensors and Bioelectronics, published since 1990...
- Cable theoryCable theoryClassical cable theory uses mathematical models to calculate the flow of electric current along passive neuronal fibers particularly dendrites that receive synaptic inputs at different sites and times...
- Morris-Lecar model
- Hindmarsh-Rose modelHindmarsh-Rose modelThe Hindmarsh–Rose model of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments made with a single neuron. The relevant variable is the membrane potential, x, which is written in dimensionless units...
External links
- Nernst/Goldman Equation Simulator
- Goldman-Hodgkin-Katz Equation Calculator
- Nernst/Goldman interactive Java applet The membrane voltage is calculated interactively as the number of ions are changed between the inside and outside of the cell.
- Potential, Impedance, and Rectification in Membranes by Goldman (1949)