Membrane Permeability vs Membrane Conductance

To place the discussion in the preceding section on more quantitative ground, it will be necessary to introduce a new concept that is closely related to membrane permeability: membrane conductance. The conductance of a membrane to an ion is an index of the ability of that ion to carry current across the membrane: the higher the conductance, the greater the ion current for a given driving force. Conductance is analogous to the reciprocal of the resistance of an electrical circuit to current flow: the higher the resistance of a circuit, the lower the amount of current that flows in response to a particular voltage. This behavior of electrical circuits can be conveniently summarized by Ohm's law:

i = V/R. Here, i is the current flowing through a resistor, R, in the presence of a voltage gradient, V. The equivalent form for the flow of an ion current across a membrane is, using potassium as an example:

where gK is the conductance of the membrane to potassium ions. The unit of electrical conductance is the Siemen, abbreviated S; a 1 V battery will drive 1 ampere of current through a 1 S conductance. Similar equations can be written for sodium and chloride:

ECl)

Note that for the usual values of Em (-71 mV), EK (-80 mV), and ENa (+58 mV), the potassium current is a positive number and the sodium current is a negative number, as required by the fact that the two currents flow in opposite directions across the membrane. By convention in neurophysiology, an outward membrane current (such as iK, at the steady-state Em) is positive and an inward current (such as iNa, at the steady-state Em) is negative.

The membrane conductance to an ion is closely related to the membrane permeability to that ion, but the two are not identical. The membrane current carried by a particular ion, and hence the membrane conductance to that ion, is proportional to the rate at which ions are crossing the membrane (that is, the ion flux). That rate depends not only on the permeability of the membrane to the ion, but also on the number ofavailable ions in the solution. As an example, imagine a cell membrane with many potassium channels (Figure 5-5). The permeability of this membrane to potassium is thus high. If there are few potassium ions in solution, on the one hand, the chance is small that a K+ will encounter a channel and cross the membrane. In this case, the potassium current will be low and the conductance of the membrane to K+ will be low even though the permeability is high. On the other hand, if there are many potassium ions available to cross the membrane (Figure 5-5b), the chance that

(a) High permeability + few ions = low ionic current

Cell membrane

(b) High permeability + many ions = larger ionic current

Cell membrane

Figure 5-5 Illustration of the difference between permeability and conductance. (a) A cell membrane is highly permeable to potassium, but there is little potassium in solution. Therefore, the ionic current carried by potassium ions is small and the membrane conductance to potassium is small. (b) The same cell membrane in the presence of higher potassium concentration has a larger potassium conductance because the potassium current is larger. The permeability, however, is the same as in (a).

a K+ will encounter a channel is high, and the rate of K+ flow across the membrane will be high. The permeability remains fixed but the ionic conductance increases when more potassium ions are available. The point is that the potassium conductance of the membrane depends on the concentration of potassium at the membrane. For the most part, however, a change in permeability of a membrane to an ion produces a corresponding change in the conductance of the membrane to that ion. Thus, when we are dealing with changes in membrane conductance as in the next chapter we can treat a conductance change as a direct index of the underlying permeability change.

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