Goldman–Hodgkin–Katz Voltage Equation
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The Goldman–Hodgkin–Katz voltage equation, more commonly known as the Goldman equation, is used in cell membrane
physiology Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
to determine the
reversal potential In a biological membrane, the reversal potential is the membrane potential at which the direction of ionic current reverses. At the reversal potential, there is no net flow of ions from one side of the membrane to the other. For channels that are pe ...
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
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, and the Medicine Nobel laureates
Alan Lloyd Hodgkin Sir Alan Lloyd Hodgkin (5 February 1914 – 20 December 1998) was an English physiologist and biophysicist who shared the 1963 Nobel Prize in Physiology or Medicine with Andrew Huxley and John Eccles. Early life and education Hodgkin was ...
and Bernard Katz.


Equation for monovalent ions

The GHK voltage equation for M monovalent positive
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
ic species and A negative: :E_ = \frac \ln This results in the following if we consider a membrane separating two \mathrm_\mathrm_\mathrm-solutions: :E_ = \frac \ln It is "
Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the w ...
-like" but has a term for each permeant ion: :E_ = \frac \ln=\frac \ln *E_ = the membrane potential (in
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defi ...
s, equivalent to
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
s per
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
) *P_\mathrm = the selectivity for that ion (in meters per second) * mathrm\mathrm = the extracellular concentration of that ion (in
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
per cubic meter, to match the other SI units) * mathrm\mathrm = the intracellular concentration of that ion (in moles per cubic meter) *R = the ideal gas constant (joules per
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
per mole) *T = the temperature in kelvins *F =
Faraday's constant In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of ...
(coulombs per mole) \frac is approximately 26.7 mV at human body temperature (37 °C); when factoring in the change-of-base formula between the natural logarithm, ln, and logarithm with base 10 ( log_\exp(1)=\ln(10)=2.30258...), it becomes 26.7\,\mathrm\cdot2.303=61.5\,\mathrm, a value often used in neuroscience. :E_ = 61.5 \, \mathrm\cdot \log = -61.5 \, \mathrm\cdot \log The ionic charge determines the sign of the membrane potential contribution. During an action potential, although the membrane potential changes about 100mV, the concentrations of ions inside and outside the cell do not change significantly. They are always very close to their respective concentrations when the membrane is at their resting potential.


Calculating the first term

Using R \approx \frac, F \approx \frac, (assuming body temperature) T=37 \ ^\circ \mathrm=310 \ \mathrm and the fact that one volt is equal to one joule of energy per coulomb of charge, the equation :E_X = \frac \ln \frac can be reduced to : \begin E_X & \approx \frac \ln \frac \\ & = \frac \ln \frac \\ & \approx \frac \log \frac & \text \ln 10 \approx 2.303 \end which is the
Nernst equation In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction ( half-cell or full cell reaction) from the standard electrode potential, absolute tempe ...
.


Derivation

Goldman's equation seeks to determine the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
''E''''m'' 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 ''E''''m'' is the
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of the ''z'' component of the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
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 In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
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 ''E''''m''/''L'', where ''L'' is the thickness of the membrane. For a given ion denoted A with valence ''n''A, its
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
''j''A—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula : j_ = -D_ \left( \frac - \frac \frac \left \mathrm\right\right) The first term corresponds to
Fick's law of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion e ...
, which gives the flux due to
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
down the
concentration In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
gradient, i.e., from high to low concentration. The constant ''D''A is the
diffusion constant Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
of the ion A. The second term reflects the
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
due to the electric field, which increases linearly with the electric field; Formally, it is multiplied by the drift velocity of the ions, with the drift velocity expressed using the Stokes–Einstein relation applied to
electrophoretic mobility Electrophoresis, from Ancient Greek ἤλεκτρον (ḗlektron, "amber") and φόρησις (phórēsis, "the act of bearing"), is the motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric fi ...
. The constants here are the
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
valence ''n''A of the ion A (e.g., +1 for K+, +2 for Ca2+ and −1 for Cl), the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
''T'' (in
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
s), the molar
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
''R'', and the
faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, ...
''F'', which is the total charge of a mole of
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s. This is a first-order
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of the form ''y' = ay + b'', with ''y'' = and ''y = d d''z''; integrating both sides from ''z''=0 to ''z''=''L'' with the boundary conditions 0) = sub>in and ''L'') = sub>out, one gets the solution : j_ = \mu n_ P_ \frac where μ is a dimensionless number : \mu = \frac and ''P''A is the ionic permeability, defined here as : P_ = \frac The
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
''J''A equals the charge ''q''A of the ion multiplied by the flux ''j''A : J_ = q_ j_ Current density has units of (Amperes/m2). Molar flux has units of (mol/(s m2)). Thus, to get current density from molar flux one needs to multiply by Faraday's constant F (Coulombs/mol). F will then cancel from the equation below. Since the valence has already been accounted for above, the charge qA of each ion in the equation above, therefore, should be interpreted as +1 or -1 depending on the polarity of the ion. There is such a current associated with every type of ion that can cross the membrane; this is because each type of ion would require a distinct membrane potential to balance diffusion, but there can only be one membrane potential. By assumption, at the Goldman voltage ''E''''m'', the total current density is zero : J_ = \sum_ J_ = 0 (Although the current for each ion type considered here is nonzero, there are other pumps in the membrane, e.g. Na+/K+-ATPase, not considered here which serve to balance each individual ion's current, so that the ion concentrations on either side of the membrane do not change over time in equilibrium.) If all the ions are monovalent—that is, if all the ''n''A equal either +1 or -1—this equation can be written : w - v e^ = 0 whose solution is the Goldman equation : \frac = \mu = \ln \frac where : w = \sum_ P_ \left \mathrm^ \right + \sum_ P_ \left \mathrm^ \right : v = \sum_ P_ \left \mathrm^ \right + \sum_ P_ \left \mathrm^ \right If divalent ions such as
calcium Calcium is a chemical element with the symbol Ca and atomic number 20. As an alkaline earth metal, calcium is a reactive metal that forms a dark oxide-nitride layer when exposed to air. Its physical and chemical properties are most similar to ...
are considered, terms such as ''e'' appear, which is the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of ''e''μ; in this case, the formula for the Goldman equation can be solved using the
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, g ...
.


See also

*
Bioelectronics Bioelectronics is a field of research in the convergence of biology and electronics. Definitions At the first C.E.C. Workshop, in Brussels in November 1991, bioelectronics was defined as 'the use of biological materials and biological architectu ...
*
Cable theory Classical cable theory uses mathematical models to calculate the electric current (and accompanying voltage) along passive neurites, particularly the dendrites that receive synaptic inputs at different sites and times. Estimates are made by model ...
*
GHK current equation GHK may refer to: * Gahcho Kue Aerodrome, in the Northwest Territories, Canada * Geko Karen, a language of Burma * GHK algorithm, a regression model * Ghotki railway station, in Pakistan * Glasgow High Kelvinside, a Scottish rugby union club * Goldm ...
*
Hindmarsh–Rose model The 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''(''t''), which is ...
*
Hodgkin–Huxley model The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical charact ...
* Morris–Lecar model *
Nernst equation In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction ( half-cell or full cell reaction) from the standard electrode potential, absolute tempe ...
*
Saltatory conduction In neuroscience, saltatory 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. The uninsulated nodes of Ranvier are th ...


References


External links


Subthreshold membrane phenomena
Includes a well-explained derivation of the Goldman-Hodgkin-Katz equation
Nernst/Goldman Equation Simulator
{{Webarchive, url=https://web.archive.org/web/20100808191814/http://www.nernstgoldman.physiology.arizona.edu/ , date=2010-08-08



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 (1943)
Physical chemistry Electrochemical equations