446 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
3.The electrical potential is the same at either end of a wire, and among any set of joined wires.
4.The potential changes by a fixed amount across a battery symbol.
5.The potential changes by the variable amountIRacross a resistor symbol.We’ll refer to the first two of these statements as “Kirchoff’s first law.” We prohibit charge buildup
in ordinary circuits because of the prohibitive potential energy cost usually associated with it. In
the cellular context, too, the separation of charge across micrometer-size regions is energetically
very costly (see the Example on page 229 and Problem 12.2). Point (4) summarizes the discussion
of Section 11.1.2, where we found the potential jump across a membrane. Point (5) restates the
Ohmic hypothesis for membrane conductance (Equation 11.8 on page 418).
In the rest of this subsection we will adapt and extend Figure 12.3a to get a more realistic
description of the resting cell membrane (Figure 12.3b).
Conductances as pipes The only “wire” in Figure 12.3a is the one joining the resistor to the
battery. Thus items (3–5) in the list above amount to the statement that the total potential jump
∆Vis the sum of two contributions, ∆V=IR+VNernst.This statement just reproduces the Ohmic
hypothesis (Equation 11.8 on page 418).
Thus, the electrical circuit analogy appears to be useful for describing membranes. But Fig-
ure 12.3a describes the behavior of justonespecies of ion, just as in first-year physics you studied
circuits with only one kind of charge carrier, the electron. Our situation is slightly different: We
have at leastthreeimportant kinds of charge carriers (Na+,K+,and Cl−), each of whose numbers
is separately fixed. Moreover, the conductance of a membrane will be different for different species
(see for example Equation 11.9 on page 420). It might seem as though we would need to write
circuit diagrams with three different kinds of wires, like the separate hot and cold plumbing in your
house!
Fortunately we don’t need to go to this extreme. First note that there is only one kind of electric
potentialV.Anycharged particle feels the same force per charge,−ddVx.Second, not only do all
kinds of charged particles feel a common potential, they also allcontributeto that potential in the
same way. Thus the total electrostatic energy cost of a charge arrangement reflects only the spatial
separation of net charge, without distinguishing between the types of charge. For example, pushing
some sodium into a cell while at the same time pushing an equal number of potassium ions out (or
an equal number of chloride ions in) creates no net separation of charge, and carries no electrostatic
energy cost.
Thus when writing circuit diagrams we can combine the various types of wires when dealing
with elements, like the cell’s cytoplasm, that do not discriminate between ion types.^2 Wecan
think of these wires as depicting one kind of pipe, with amixtureof different “fluids” (analogous
to the various ion species) flowing at a common “pressure” (the potentialV), and a constraint
that the total “volume of fluid” flowing in (total current) should equal that flowing out. Our wires
must branch into different types when we describe the membrane, which has different resistances
to different ion species in the mixture. In addition, each fluid will have a different entropic force
driving it (the various Nernst potentials). We accommodate these facts by drawing the membrane
as a compound object, with one resistor/battery pair in parallel for each ion species (Figure 12.3b).
Notice that this figure doesnotimply that all three Nernst potentials are equal. Instead, the
(^2) T 2 Weare neglecting possible differences in bulk resistivity among the ion species.