WordNet

English physiologist who, with Andrew Huxley, discovered the role of potassium and sodium atoms in the transmission of the nerve impulse (1914-1998) (同)Alan Hodgkin, Sir Alan Hodgkin, Alan Lloyd Hodgkin
English chemist (born in Egypt) who used crystallography to study the structure of organic compounds (1910-1994) (同)Dorothy Hodgkin, Dorothy Mary Crowfoot Hodgkin
English physician who first described Hodgkins disease (1798-1866) (同)Thomas Hodgkin
the act of regarding as equal (同)equating
a mathematical statement that two expressions are equal
United States anarchist (born in Russia) who opposed conscription; was deported to the Soviet Union in 1919 (1869-1940) (同)Emma Goldman
unknown god; an epithet of Prajapati and Brahma

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等式; 方程式

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/09/05 17:33:12」(JST)

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The Goldman–Hodgkin–Katz voltage equation, more commonly known as the Goldman equation, is used in cell membrane physiology to determine the reversal 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.

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 membrane potential (in volts, equivalent to joules per coulomb)
= the permeability for that ion (in meters per second)
= the extracellular concentration of that ion (in moles per cubic meter, to match the other SI units)
= the intracellular concentration of that ion (in moles per cubic meter)
= The ideal gas constant (joules per kelvin per mole)
= The temperature in kelvins
= Faraday's constant (coulombs per mole)

The first term, before the parentheses, 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. Note also that 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 then membrane is at their resting potential.

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 equation

can be reduced to

\begin{align} E_X & \approx \frac{.0267 \ \mathrm{ V}}{z} \ln \frac {X_o}{X_i} \\ & = \frac{26.7 \ \mathrm{ mV}}{z} \ln \frac {X_o}{X_i} \\ & \approx \frac{61.5 \ \mathrm{ mV} }{z} \log \frac {X_o}{X_i} & \text{ since } \ln 10 \approx 2.30 \end{align}

Derivation

Goldman's equation seeks to determine the voltage E_m across a membrane.^[1] 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 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 E_m/L, where L is the thickness of the membrane. For a given ion denoted A with valence n_A, its flux j_A—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula

j_{\mathrm{A}} = -D_{\mathrm{A}} \left( \frac{d\left[ \mathrm{A}\right]}{dz} - \frac{n_{\mathrm{A}}F}{RT} \frac{E_{m}}{L} \left[ \mathrm{A}\right] \right)

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 D_A 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 n_A of the ion A (e.g., +1 for K⁺, +2 for Ca²⁺ and −1 for Cl⁻), the temperature T (in kelvins), the molar gas constant R, and the faraday F, which is the total charge of a mole of electrons.

Using the mathematical technique of separation of variables, the equation may be separated

\frac{d\left[ \mathrm{A}\right]}{-\frac{j_{\mathrm{A}}}{D_{\mathrm{A}}} + \frac{n_{\mathrm{A}}FE_{m}}{RTL} \left[ \mathrm{A}\right]} = dz

Integrating both sides from z=0 (inside the membrane) to z=L (outside the membrane) yields the solution

j_{\mathrm{A}} = \mu n_{\mathrm{A}} P_{\mathrm{A}} \frac{\left[ \mathrm{A}\right]_{\mathrm{out}} - \left[ \mathrm{A}\right]_{\mathrm{in}} e^{n_{}\mu} }{1 - e^{n_{}\mu }}

where μ is a dimensionless number

\mu = \frac{F E_{m}}{RT}

and P_A is the ionic permeability, defined here as

P_{\mathrm{A}} = \frac{D_{\mathrm{A}}}{L}

The electric current density J_A equals the charge q_A of the ion multiplied by the flux j_A

J_{A} = q_{\mathrm{A}} j_{\mathrm{A}}

There is such a current associated with every type of ion that can cross the membrane. By assumption, at the Goldman voltage E_m, the total current density is zero

J_{tot} = \sum_{A} J_{A} = 0

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^{\mu} = 0

whose solution is the Goldman equation

\frac{F E_{m}}{RT} = \mu = \ln \frac{w}{v}

where

w = \sum_{\mathrm{cations\ C}} P_{\mathrm{C}} \left[ \mathrm{C}^{+} \right]_{\mathrm{out}} + \sum_{\mathrm{anions\ A}} P_{\mathrm{A}} \left[ \mathrm{A}^{-} \right]_{\mathrm{in}}

v = \sum_{\mathrm{cations\ C}} P_{\mathrm{C}} \left[ \mathrm{C}^{+} \right]_{\mathrm{in}} + \sum_{\mathrm{anions\ A}} P_{\mathrm{A}} \left[ \mathrm{A}^{-} \right]_{\mathrm{out}}

If divalent ions such as calcium are considered, terms such as e^2μ appear, which is the square of e^μ; in this case, the formula for the Goldman equation can be solved using the quadratic formula.

References

^ Junge D (1981). Nerve and Muscle Excitation (2nd edition ed.). Sunderland, Massachusetts: Sinauer Associates. pp. 33–37. ISBN 0-87893-410-3.

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)

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English Journal

Extracellular potassium inhibits Kv7.1 potassium channels by stabilizing an inactivated state.

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Kv7.1 (KCNQ1) channels are regulators of several physiological processes including vasodilatation, repolarization of cardiomyocytes, and control of secretory processes. A number of Kv7.1 pore mutants are sensitive to extracellular potassium. We hypothesized that extracellular potassium also modulate
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Functional identification of an outwardly rectifying pH- and anesthetic-sensitive leak K(+) conductance in hippocampal astrocytes.

Chu KC, Chiu CD, Hsu TT, Hsieh YM, Huang YY, Lien CC.SourceInstitute of Neuroscience and Brain Research Center, National Yang-Ming University, Section 2, Li-Nong St., Taipei, Taiwan.
The European journal of neuroscience.Eur J Neurosci.2010 Sep;32(5):725-35. doi: 10.1111/j.1460-9568.2010.07323.x. Epub 2010 Jul 28.
Astrocytes function as spatial K(+) buffers by expressing a rich repertoire of K(+) channels. Earlier studies suggest that acid-sensitive tandem-pore K(+) channels, mainly TWIK-related acid-sensitive K(+) (TASK) channels, mediate part of the passive astroglial membrane conductance. Here, using a com
PMID 20673311

Japanese Journal

人工膜と生体膜におけるチャネルとゲート

谷岡明彦
膜 20(6), 377-385, 1995-11-01
… Nernst-Planck equation, Donnan equilibrium, Goldman-Hodgkin-Katz equation, phenomenological equation based on nonequilibrium thermodynamics had played an important role for the explanation of transport phenomena through biological membrane and the development of ion-exchange membranes. …
NAID 10009855562

Studies on membrane potentials of excitable membranes using a sublayer model.

Nakagaki Masayuki,Takagi Ryosuke
膜 8(5), 301-310, 1983
… One of them is the change of permeability coefficient of Hodgkin & Katz. … If these two theories are to be compatible, the permeability coefficient of Hodgkin & Katz ought to be correlated with the opening and closing of ionic channels. …
NAID 130001440400

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★リンクテーブル★

リンク元	「ゴールドマン・ホジキン・カッツの式」「ゴールドマンの式」「Goldman-Hodgkin-Katz式」
関連記事	「equation」「Goldman」