Goldman's equation seeks to determine the
voltage Em 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
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; Formally, it is [A] multiplied by the drift velocity of the ions, with the
drift velocity expressed using the
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. This is a first-order
ODE of the form ''y' = ay + b
, with y
= [A] and y'
= d[A]/dz
; integrating both sides from z
=0 to z
=L
with the boundary conditions [A](0) = [A]in and [A](L'') = [A]out, one gets 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_{\mathrm{A}}\mu} }{1 - e^{n_{\mathrm{A}}\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}} 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
Em, the total current density is zero : 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 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
e2μ appear, which is the
square of
eμ; in this case, the formula for the Goldman equation can be solved using the
quadratic formula. ==See also==