In general, the flow of ions through the membrane is driven by concentration gradients and also by the electric field. The contribution to the flow from the electric field is given by Planck's equation z
where u is the mobility of the ion, defined as the velocity of the ion under a constant unit electric field; z is the valence of the ion, so that z/|z| is the sign of the force on the ion; c is the concentration of S; and 0 is the electrical potential, so that —V0 is the electrical field.
There is a relationship, determined by Einstein, between the ionic mobility u and Fick's diffusion constant:
When the effects of concentration gradients and electrical gradients are combined, we obtain the Nernst-Planck equation
If the flow of ions and the electric field are transverse to the membrane, we can view (2.59) as the one-dimensional relation
The Nernst equation
The Nernst equation can be derived from the Nernst-Planck electrodiffusion equation (2.60). When the flux J is zero, we find
so that
c dx RT dx
Now suppose that the cell membrane extends from x = 0 (the inside) to x = L (the outside), and let subscripts i and e denote internal and external quantities respectively. Then, integrating from x = 0 to x = L we get
1c RT
and thus the potential difference across the membrane, V = fa - fa, is given by
which is the Nernst equation.
This derivation of the Nernst equation relies on the Nernst-Planck electrodiffusion equation, and so is not a derivation from first principles. The derivation from first principles can be given, but it is beyond the scope of this text. The interested reader is referred to Levine (1978) or Denbigh (1981) for the details.
In general, the electric potential fa is determined by the local charge density, and so J must be found by solving a coupled system of equations (this is discussed in detail in Chapter 3). However, a useful result is obtained by assuming that the electric field in the membrane is constant, and thus decoupled from the effects of charges moving through the membrane. Suppose we have two reservoirs separated by a semipermeable membrane of thickness L, such that the potential difference across the membrane is V. On the left of the membrane (the inside) [S] = ci, and on the right (the outside)
[S] = ce. If the electric field is constant through the membrane, we have d^/dx = -V/L, where V = 0(0) — 0(L) is the membrane potential.
At steady state and with no production of ions, the flux must be constant. In this case, the Nernst-Planck equation (2.59) is an ordinary differential equation for the concentration c, dc zFV J „
dx RTL D
whose solution is
RTL ) DzVF
where we have used the left boundary condition c(0) = ci. To satisfy the boundary condition c(L) = ce, it must be that
where J is the flux density with units (typically) of moles per area per unit time. This flux density becomes an electrical current density (current per unit area) when multiplied by zF, the number of charges carried per mole, and thus z2F2 ci — ce exp (—RV) Is = Pszj^V1 e p( —zRT) , (2.68)
where PS = D/L is the permeability of the membrane to S. This is the famous Goldman-Hodgkin-Katz (GHK) current equation. It plays an important role in models of cellular electrical activity.
This flow is zero if the diffusively driven flow and the electrically driven flow are in balance, which occurs, provided that z = 0, if
which is, as expected, the Nernst potential.
If there are several ions that are separated by the same membrane, then the flow of each of these is governed separately by its own current-voltage relationship. In general there is no potential at which these currents are all zero. However, the potential at which the net electrical current is zero is called the Goldman-Hodgkin-Katz potential. For a collection of ions all with valence z = ±1, we can calculate the GHK potential directly. For zero net electrical current, it must be that
where Pj = Dj/L. This expression can be solved for V, to get
For example, if the membrane separates sodium (Na+,z = 1), potassium (K+,z = 1), and chloride (Cl—, z = — 1) ions, then the GHK potential is
V=_RT ln/ PNa[Na+]i + Pk[K+] + Pcl[Cl— f F n\ PNa[Na+], + Pk[K+], + Pcl[Cl—])' (-)
It is important to emphasize that neither the GHK potential nor the GHK current equation are universal expressions like the Nernst equation. Both depend on the assumption of a constant electric field, and other models give different expressions for the transmembrane current and reversal potential. In Chapter 3 we present a detailed discussion of other models of ionic current and compare them to the GHK equations. However, the importance of the GHK equations is so great, and their use so widespread, that their separate presentation here is justified.
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