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TheGoldman–Hodgkin–Katz voltage equation, sometimes called theGoldman equation, is used in cell membranephysiology to determine theresting potential across a cell's membrane, taking into account all of the ions that arepermeant through that membrane.

The discoverers of this areDavid E. Goldman ofColumbia University, and the Medicine Nobel laureatesAlan Lloyd Hodgkin andBernard Katz.

The GHK voltage equation for${\displaystyle M}$ monovalent positiveionic species and${\displaystyle A}$ negative:

${\displaystyle E_m=\frac{RT}{F}\ln \left(\frac{\sum _i^n{P}_{M_i^{+}}[M_i^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}+\sum _j^m{P}_{A_j^{-}}[A_j^{-}{]}_{\mathrm{i}\mathrm{n}}}{\sum _i^n{P}_{M_i^{+}}[M_i^{+}{]}_{\mathrm{i}\mathrm{n}}+\sum _j^m{P}_{A_j^{-}}[A_j^{-}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\right)}$

This results in the following if we consider a membrane separating two${\displaystyle {\mathrm{K}}_x{\mathrm{N}\mathrm{a}}_{1-x}\mathrm{C}\mathrm{l}}$-solutions:^[1][2][3]

${\displaystyle E_{m,{\mathrm{K}}_x{\text{Na}}_{1-x}\mathrm{C}\mathrm{l}}=\frac{RT}{F}\ln \left(\frac{P_{\text{Na}}[{\text{Na}}^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}+P_{\text{K}}[{\text{K}}^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}+P_{\text{Cl}}[{\text{Cl}}^{-}{]}_{\mathrm{i}\mathrm{n}}}{P_{\text{Na}}[{\text{Na}}^{+}{]}_{\mathrm{i}\mathrm{n}}+P_{\text{K}}[{\text{K}}^{+}{]}_{\mathrm{i}\mathrm{n}}+P_{\text{Cl}}[{\text{Cl}}^{-}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}\right)}$

It is "Nernst-like" but has a term for each permeant ion:

${\displaystyle E_{m,\text{Na}}=\frac{RT}{F}\ln \left(\frac{P_{\text{Na}}[{\text{Na}}^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}{P_{\text{Na}}[{\text{Na}}^{+}{]}_{\mathrm{i}\mathrm{n}}}\right)=\frac{RT}{F}\ln \left(\frac{[{\text{Na}}^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}{[{\text{Na}}^{+}{]}_{\mathrm{i}\mathrm{n}}}\right)}$

• ${\displaystyle E_m}$ = the membrane potential (involts, equivalent tojoules percoulomb)
• ${\displaystyle P_{\mathrm{i}\mathrm{o}\mathrm{n}}}$ = the selectivity for that ion (in meters per second)
• ${\displaystyle [\mathrm{i}\mathrm{o}\mathrm{n}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}$ = the extracellular concentration of that ion (inmoles per cubic meter, to match the otherSI units)^[4]
• ${\displaystyle [\mathrm{i}\mathrm{o}\mathrm{n}{]}_{\mathrm{i}\mathrm{n}}}$ = the intracellular concentration of that ion (in moles per cubic meter)^[4]
• ${\displaystyle R}$ = theideal gas constant (joules perkelvin per mole)^[4]
• ${\displaystyle T}$ = the temperature in kelvins^[4]
• ${\displaystyle F}$ =Faraday's constant (coulombs per mole)

${\displaystyle \frac{RT}{F}}$ 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${\displaystyle ([{\log }_{10}\exp (1){]}^{-1}=\ln (10)=\mathrm{2.30258...})}$, it becomes${\displaystyle 26.7\mathrm{m}\mathrm{V}\cdot 2.303=61.5\mathrm{m}\mathrm{V}}$, a value often used in neuroscience.

${\displaystyle E_X=61.5\mathrm{m}\mathrm{V}\cdot \log \left(\frac{[X^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}{[X^{+}{]}_{\mathrm{i}\mathrm{n}}}\right)=-61.5\mathrm{m}\mathrm{V}\cdot \log \left(\frac{[X^{-}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}{[X^{-}{]}_{\mathrm{i}\mathrm{n}}}\right)}$

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.

Using${\displaystyle R\approx \frac{8.3\mathrm{J}}{\mathrm{K}\cdot \mathrm{m}\mathrm{o}\mathrm{l}}}$,${\displaystyle F\approx \frac{9.6\times {10}^4\mathrm{J}}{\mathrm{m}\mathrm{o}\mathrm{l}\cdot \mathrm{V}}}$, (assuming body temperature)${\displaystyle T=37{}^{\circ }\mathrm{C}=310\mathrm{K}}$ and the fact that one volt is equal to one joule of energy per coulomb of charge, the equation

${\displaystyle E_X=\frac{RT}{zF}\ln \frac{X_o}{X_i}}$

can be reduced to

which is theNernst equation.

Goldman's equation seeks to determine thevoltageE_m across a membrane.^[5] ACartesian coordinate system is used to describe the system, with thez direction being perpendicular to the membrane. Assuming that the system is symmetrical in thex andy directions (around and along the axon, respectively), only thez direction need be considered; thus, the voltageE_m is theintegral of thez component of theelectric 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 ionicconcentration 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 toE_m/L, whereL is the thickness of the membrane. For a given ion denoted A with valencen_A, itsfluxj_A—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula

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

The first term corresponds toFick's law of diffusion, which gives the flux due todiffusion down theconcentration gradient, i.e., from high to low concentration. The constantD_A is thediffusion constant of the ion A. The second term reflects theflux due to the electric field, which increases linearly with the electric field; Formally, it is [A] multiplied by the drift velocity of the ions, with thedrift velocity expressed using theStokes–Einstein relation applied toelectrophoretic mobility. The constants here are thechargevalencen_A of the ion A (e.g., +1 for K⁺, +2 for Ca²⁺ and −1 for Cl⁻), thetemperatureT (inkelvins), the molargas constantR, and thefaradayF, which is the total charge of a mole ofelectrons.

This is a first-orderODE of the formy' = ay + b, withy = [A] andy' = d[A]/dz; integrating both sides fromz=0 toz=L with the boundary conditions [A](0) = [A]_in and [A](L) = [A]_out, one gets the solution

${\displaystyle j_{\mathrm{A}}=\mu n_{\mathrm{A}}{P}_{\mathrm{A}}\frac{{\left[\mathrm{A}\right]}_{\mathrm{o}\mathrm{u}\mathrm{t}}-{\left[\mathrm{A}\right]}_{\mathrm{i}\mathrm{n}}{e}^{n_{\mathrm{A}}\mu }}{1-e^{n_{\mathrm{A}}\mu }}}$

where μ is a dimensionless number

${\displaystyle \mu =\frac{F{E}_m}{RT}}$

andP_A is the ionic permeability, defined here as

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

Theelectric currentdensityJ_A equals the chargeq_A of the ion multiplied by the fluxj_A

${\displaystyle J_A=q_{\mathrm{A}}{j}_{\mathrm{A}}}$

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 q_A 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 voltageE_m, the total current density is zero

${\displaystyle J_{tot}=\sum _A{J}_A=0}$

(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 then_A equal either +1 or -1—this equation can be written

${\displaystyle w-v{e}^{\mu }=0}$

whose solution is the Goldman equation

${\displaystyle \frac{F{E}_m}{RT}=\mu =\ln \frac{w}{v}}$

where

${\displaystyle w=\sum _{\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{C}}{P}_{\mathrm{C}}{\left[{\mathrm{C}}^{+}\right]}_{\mathrm{o}\mathrm{u}\mathrm{t}}+\sum _{\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{A}}{P}_{\mathrm{A}}{\left[{\mathrm{A}}^{-}\right]}_{\mathrm{i}\mathrm{n}}}$

${\displaystyle v=\sum _{\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{C}}{P}_{\mathrm{C}}{\left[{\mathrm{C}}^{+}\right]}_{\mathrm{i}\mathrm{n}}+\sum _{\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{A}}{P}_{\mathrm{A}}{\left[{\mathrm{A}}^{-}\right]}_{\mathrm{o}\mathrm{u}\mathrm{t}}}$

If divalent ions such ascalcium are considered, terms such ase^2μ appear, which is thesquare ofe^μ; in this case, the formula for the Goldman equation can be solved using thequadratic formula.

• Bioelectronics
• Cable theory
• GHK current equation
• Hindmarsh–Rose model
• Hodgkin–Huxley model
• Morris–Lecar model
• Nernst equation
• Saltatory conduction

• Subthreshold membrane phenomena Includes a well-explained derivation of the Goldman-Hodgkin-Katz equation
• Nernst/Goldman Equation SimulatorArchived 2010-08-08 at theWayback Machine
• 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 (1943)

• Physical chemistry
• Electrochemical equations

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