Goldman equation

The Goldman-Hodgkin-Katz voltage equation, more commonly known as the Goldman equation is used in cell membrane physiology to determine the 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 $N$ monovalent positive ionic species and $M$ negative:

:$E\_\{m\}\; =\; frac\{RT\}\{F\}\; ln\{\; left(\; frac\{\; sum\_\{i\}^\{N\}\; P\_\{M^\{+\}\_\{i\; [M^\{+\}\_\{i\}]\; \_mathrm\{out\}\; +\; sum\_\{j\}^\{M\}\; P\_\{A^\{-\}\_\{j\; [A^\{-\}\_\{j\}]\; \_mathrm\{in\{\; sum\_\{i\}^\{N\}\; P\_\{M^\{+\}\_\{i\; [M^\{+\}\_\{i\}]\; \_mathrm\{in\}\; +\; sum\_\{j\}^\{M\}\; P\_\{A^\{-\}\_\{j\; [A^\{-\}\_\{j\}]\; \_mathrm\{out\; ight)\; \}$

This results in the following if we consider a membrane separating two $K\_\{x\}Na\_\{1-x\}Cl$-solutions:

:$E\_\{m,\; K\_\{x\}Na\_\{1-x\}Cl\; \}\; =\; frac\{RT\}\{F\}\; ln\{\; left(\; frac\{\; P\_\{Na^\{+\; [Na^\{+\}]\; \_mathrm\{out\}\; +\; P\_\{K^\{+\; [K^\{+\}]\; \_mathrm\{out\}\; +\; P\_\{Cl^\{-\; [Cl^\{-\}]\; \_mathrm\{in\}\; \}\{\; P\_\{Na^\{+\; [Na^\{+\}]\; \_mathrm\{in\}\; +\; P\_\{K^\{+\; [K^\{+\}]\; \_\{mathrm\{in\; +\; P\_\{Cl^\{-\; [Cl^\{-\}]\; \_mathrm\{out\}\; \}\; ight)\; \}$

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::$E\_\{m,Na\}\; =\; frac\{RT\}\{F\}\; ln\{\; left(\; frac\{\; P\_\{Na^\{+\; [Na^\{+\}]\; \_mathrm\{out\{\; P\_\{Na^\{+\; [Na^\{+\}]\; \_mathrm\{in\; ight)\; \}=frac\{RT\}\{F\}\; ln\{\; left(\; frac\{\; [Na^\{+\}]\; \_mathrm\{out\{\; [Na^\{+\}]\; \_mathrm\{in\; ight)\; \}$

*$E\_\{m\}$ = The membrane potential
*$P\_mathrm\{ion\}$ = the permeability for that ion
*$[ion]\; \_mathrm\{out\}$ = the extracellular concentration of that ion
*$[ion]\; \_mathrm\{in\}$ = the intracellular concentration of that ion
*$R$ = The ideal gas constant
*$T$ = The temperature in kelvins
*$F$ = Faraday's constant

The first term, before the parenthesis, can be reduced to 61.5 log for calculations at human body temperature (37° C)

:$E\_\{X\}\; =\; 61.5\; log\{\; left(\; frac\{\; [X^\{+\}]\; \_mathrm\{out\{\; [X^\{+\}]\; \_mathrm\{in\; ight)\; \}\; =\; -61.5\; log\{\; left(\; frac\{\; [X^\{-\}]\; \_mathrm\{out\{\; [X^\{-\}]\; \_mathrm\{in\; ight)\; \}$

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. $P\_\{K^\{+\; gg\; P\_\{Na^\{+$). 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. $P\_\{Na^\{+\; gg\; P\_\{K^\{+$).

Derivation

Goldman's equation seeks to determine the voltage "E"_"m" across a membrane. [cite book | author = Junge D | date = 1981 | title = Nerve and Muscle Excitation | edition = 2nd edition | publisher = Sinauer Associates | location = Sunderland, MA | isbn = 0-87893-410-3 | pages = pp. 33–37] 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

:$j\_\{mathrm\{A\; =\; -D\_\{mathrm\{A\; left(\; frac\{dleft\; [\; mathrm\{A\}\; ight]\; \}\{dz\}\; +\; frac\{n\_\{mathrm\{AF\}\{RT\}\; frac\{E\_\{m\{L\}\; left\; [\; mathrm\{A\}\; ight]\; ight)$

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

:$frac\{dleft\; [\; mathrm\{A\}\; ight]\; \}\{-frac\{j\_\{mathrm\{A\}\{D\_\{mathrm\{A\}\; +\; frac\{n\_\{mathrm\{AFE\_\{m\{RTL\}\; left\; [\; mathrm\{A\}\; ight]\; \}\; =\; dz$

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

:$j\_\{mathrm\{A\; =\; mu\; n\_\{mathrm\{A\; P\_\{mathrm\{Afrac\{left\; [\; mathrm\{A\}\; ight]\; \_\{mathrm\{out\; -\; left\; [\; mathrm\{A\}\; ight]\; \_\{mathrm\{in\; e^\{n\_\{mathrm\{Amu\}\; \}\{1\; -\; e^\{n\_\{mathrm\{Amu$

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

:$w\; =\; sum\_\{mathrm\{cations\; C\; P\_\{mathrm\{C\; left\; [\; mathrm\{C\}^\{+\}\; ight]\; \_\{mathrm\{out\; +\; sum\_\{mathrm\{anions\; A\; P\_\{mathrm\{A\; left\; [\; mathrm\{A\}^\{-\}\; ight]\; \_\{mathrm\{in$

:$v\; =\; sum\_\{mathrm\{cations\; C\; P\_\{mathrm\{C\; left\; [\; mathrm\{C\}^\{+\}\; ight]\; \_\{mathrm\{in\; +\; sum\_\{mathrm\{anions\; A\; P\_\{mathrm\{A\; left\; [\; mathrm\{A\}^\{-\}\; ight]\; \_\{mathrm\{out$

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.

ee also

*GHK current equation
*Nernst equation

References

External links

* [http://www.nernstgoldman.physiology.arizona.edu/ Nernst/Goldman Equation Simulator]
* [http://thevirtualheart.org/GHKindex.html 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.