Derivation of the Goldman Equation

The Goldman equation, or constant-field equation, is important to an understanding of the factors that govern the steady-state membrane potential. As discussed in Chapter 4, the Goldman equation describes the nonequilibrium membrane potential reached when two or more ions with unequal equilibrium potentials are free to move across the membrane. The basic strategy in this derivation is to use the flux equations derived in Appendix A to solve separately for the ionic current carried by each permeant ion and then to set the sum of all ionic currents equal to zero. The derivation is somewhat more complex than that of the Nernst equation in Appendix A, and it requires some knowledge of differential and integral calculus to follow in detail. Nevertheless, it should be possible for those without the necessary mathematics to follow the logic of the steps and thus to gain some insight into the physical mechanisms described by the equation.

When several ions are moving across the membrane simultaneously, a steady value of membrane potential will be reached when the sum of the ionic currents carried by the individual ions is zero; that is, for permeant ions A, B, and C

The first step in arriving at a value of membrane potential that satisfies this condition is to solve for the net ionic flux, 0, for each ion separately. The total flux for a particular ion will be the sum of the flux due to the concentration gradient and the flux due to the electrical gradient:

The expressions for 0C and Ov are given by Equations (A-10) and (A-12) in Appendix A. Thus, Equation (B-2) becomes

If it is assumed that the electric field across the membrane is constant (this is the constant-field assumption from which the equation draws its alternative name) and that the thickness of the membrane is a, then dV/d x = V/a In that case, Equation (B-3) can be written as

uRT dx RTa This is a differential equation of the form

C exp (JP(x) d x) = Jq exp (JP(x) d x) d x + constant (B-6)

In this instance, Q = ®T/(uRT) and P(x) = (ZFV)/(RTa). Making these substitutions and integrating Equation (B-6) across the membrane of thickness a (that is, from 0 to a) gives

C exp

yRTay

0T Ca uRT

exp f ZFVx"-

This becomes

Ca exP

Or uRT

ZFVx

UZFV^

yRTaj or

Ca exP

rZFV^

0T RTa uRT ZFV Rearranging and combining terms yields exp f ZFVaa-

Ca exp

Or a uZFV

This can be solved for 0T to yield

uZFV

Now, Ca and C0 are the concentrations of the ion just within the membrane. These concentrations are related to the concentrations in the fluids inside and outside the cell by Ca = ¡Cin and C0 = ¡Cout,where ¡3 is the oil-water partition coefficient for the ion in question. Substituting these in Equation (B-8) gives

The permeability constant, pi, for a particular ion is given by pi = fkiRT/a, or p/RT=fiu/a. Making this substitution in Equation (B-9) gives

Pi ZFV RT

The flux, 0T, for an ion can be converted to a flow of electrical current, as required in Equation (B-1), by multiplying by ZF (the number of coulombs per mole of ion); therefore

This is the expression we need for each ion in Equation (B-1). For instance, if the three permeant ions are Na, K, and Cl with permeabilities pNa, pK, and pCl, then Equation (B-1) becomes (keeping in mind that the valence of chloride is —1)

F 2V

PkPLeFV/RT - [K]out) + pNa([Na]ineFV/RT - [Na]o exp (FV/RT ) - 1

Multiplying through by -exp (FV/RT)/-exp (FV/RT) and rearranging yields F 2V

RT(exp (FV/RT) - 1) - (Pk[K] out + PNa[Na] out + Pcl[Cl]in)] = 0

This requires that

(Pk[K] out + PNa[Na] out + Pcl[Cl] i] (PjK]in + PNa[Na]in + Pcl[Cl]ou

Taking the natural logarithm of both sides and solving for V yields the usual form of the Goldman equation

0 0