484 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
12.5Conduction as diffusion
Section 4.6.4 argued that the conduction of electricity by a salt solution was just another diffusion
problem. Mobile charged objects (sodium and chloride ions) get pulled by an external force (from
the imposed electrostatic potential gradient) and drift at a net speedvdriftmuchsmaller than their
thermal velocity. Let’s rederive the result of that section and see how successful this claim is.
We’ll study fully dissociated salts with singly charged ions, like table salt, NaCl. In this problem
take all diffusion constants to have the approximate value for a generic small molecule in water,
D≈ 1 μm^2 /ms.
The resistance of a column of conductor with cross-sectional areaAand lengthLis not intrinsic
to the material in the column, but a related quantity, the electrical conductivity, is. We defined
conductivity^14 byκ=L/(AR).
Not surprisingly, the conductivity of a salt solution goes up when we add more salt. For low
concentrations, one finds experimentally at room temperature
κ=(12. 8 Ω−^1 ·m−^1 )·(c/ 1 M), (12.28)wherec/ 1 Mis the salt concentration expressed in mole/L.Wewant to understand the magnitude
of the numerical prefactor.
Apotential differenceV across the cell gives an electric fieldE=V/L.Each mole of salt gives
2 Nmoleions (one Na+and one Cl−perNaCl molecule). Each ion drifts under the applied electric
field. Suppose the solution is sufficiently dilute that our ideal-solution formulas are applicable.
a. Write the force on each ion in terms ofV,L,and known constants.
b. Write the resulting drift velocityvdriftin terms ofV,L,and known constants.
c. Get a formula for the number of Na+ions crossing the centerline of the cell in time dt.
d. Write the resulting currentIin the cell in terms ofV, A,L, c,and known constants.
e. Write a formula forκin terms ofcand known constants. Discuss every factor in this formula
and its physical meaning.
f. Put in the numbers and compare to experiment (Equation 12.28).
g. Now evaluate the conductivity for the ion concentrations characteristic of squid axoplasm (see
Table 11.1 on page 416; pretend that you can use the dilute-solution formulas and ignore ions not
listed on the table). Compare your answer to the measured value ofκ≈ 3 Ω−^1 m−^1.
h. What would you expect for a solution of magnesium chloride? You can suppose that (c/ 1 M)
moles of MgCl 2 dissociates completely into Mg2+and Cl−in one liter of water.
12.6Analytical solution for simplified action potential
Show that the function ̄v(y)=(1+eαy)−^1 solves Equation 12.23, if we take the parameterQto
begiven by
√
2 /s(s
2 −^1)
.Hence derive the speed of the action potential (Equation 12.24). αis
another constant, which you are to find.
12.7Discrete versus continuous
a. Use the overall membrane conductance per areagtot^0 of the resting squid axon membrane, the
500-fold increase in total conductance during the action potential, and the conductance of a single
open sodium channel to estimate the density of sodium channels in the squid axon membrane.
Compare to the accepted value of roughly 300μm−^2 in squid.
b. For a cylindrical axon of diameter 1mm,howmany channels per unit length does your estimate
(^14) Youmaybemore familiar with theresistivity,which is 1/κ. Because the resistanceRhas the units of ohms
(denotedΩ), and an ohm is aJ·s/coul^2 ,κhas the SI unitscoul^2 /(J·m·s).