Goldman equation

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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 $M$ monovalent positive ionic species and $A$ negative:

$E_{m} = \frac{RT}{F} \ln{ \left( \frac{ \sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\mathrm{out} + \sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\mathrm{in}}{ \sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\mathrm{in} + \sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\mathrm{out}} \right) }$

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

$E_{m, \mathrm{K}_{x}\mathrm{Na}_{1-x}\mathrm{Cl} } = \frac{RT}{F} \ln{ \left( \frac{ P_{Na^{+}}[Na^{+}]_\mathrm{out} + P_{K^{+}}[K^{+}]_\mathrm{out} + P_{Cl^{-}}[Cl^{-}]_\mathrm{in} }{ P_{Na^{+}}[Na^{+}]_\mathrm{in} + P_{K^{+}}[K^{+}]_{\mathrm{in}} + P_{Cl^{-}}[Cl^{-}]_\mathrm{out} } \right) }$

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

$E_{m,Na} = \frac{RT}{F} \ln{ \left( \frac{ P_{Na^{+}}[Na^{+}]_\mathrm{out}}{ P_{Na^{+}}[Na^{+}]_\mathrm{in}} \right) }=\frac{RT}{F} \ln{ \left( \frac{ [Na^{+}]_\mathrm{out}}{ [Na^{+}]_\mathrm{in}} \right) }$
• $E_{m}$ = The membrane potential (in volts, equivalent to joules per coulomb)
• $P_\mathrm{ion}$ = the permeability for that ion (in meters per second)
• $[ion]_\mathrm{out}$ = the extracellular concentration of that ion (in moles per cubic meter, to match the other SI units)
• $[ion]_\mathrm{in}$ = the intracellular concentration of that ion (in moles per cubic meter)
• $R$ = The ideal gas constant (joules per kelvin per mole)
• $T$ = The temperature in kelvins
• $F$ = 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)

$E_{X} = 61.5 \ \mathrm{mV} \log{ \left( \frac{ [X^{+}]_\mathrm{out}}{ [X^{+}]_\mathrm{in}} \right) } = -61.5 \ \mathrm{mV} \log{ \left( \frac{ [X^{-}]_\mathrm{out}}{ [X^{-}]_\mathrm{in}} \right) }$

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

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

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 Em 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 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

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

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 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 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 PA is the ionic permeability, defined here as

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

The electric current density JA equals the charge qA of the ion multiplied by the flux jA

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

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

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

If all the ions are monovalent—that is, if all the nA 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 appear, which is the square of eμ; in this case, the formula for the Goldman equation can be solved using the quadratic formula.