- Problems[[Student version, January 17, 2003]] 483
Problems
12.1Conduction velocity
The Chippendale Mupp is a mythical creature who bites his tail before going to sleep. As the poets
have sung, his tail is so long that he doesn’t feel the pain until it’s time to wake up, eight hours
after going to sleep. Suppose that a single unmyelinated axon connects the Mupp’s tail to its spinal
cord and use axon parameters appropriate to squid. Given that the range of axon diameters in real
organisms is 0.2–1000μm,estimate how long the Mupp’s tail must be.
12.2Discharging the battery
Imagine the resting axon membrane as a capacitor, an insulating layer that separates charge,
creating the resting potential drop.
a. How much charge per unit area must pass through the membrane to discharge the capacitor
(that is, bringV fromV^0 =− 60 mVto zero)?
b. Reexpress your answer to (a) by giving the surface area per excess proton charge needed to
maintainV^0 =− 60 mV.Then express it a third time, as the charge per unit length of axon, taking
the squid giant axon to be a cylinder of radius 0. 5 mm.
c. We saw that depolarization is largely the result of the passage of sodium ions. Estimate the effect
on the interior ion concentration of a charge transfer of the sort just described, as follows. Again
imagine the giant axon as a cylinder filled with salt solution, with ion concentrations given by the
data in Table 11.1 on page 416. Find the total number of interior sodium ions per length. Find the
corresponding number if the interior sodium concentration matched the exterior value. Subtract
these two numbers and compare to the total number of sodium ions passing through the membrane
as estimated in (b).
d. Comment on your answer in the light of the observation an axon can continue to transmit many
action potentials after its ion pumps have been shut down.
12.3Contributions to capacitance
a. Estimate the capacitance per area of a lipid bilayer. Consider only the electrically insulating
part of the bilayer, the lipid tails, as a layer of oil about 2nmthick. The dielectric constant of oil
isεoil/ε 0 ≈2.
b. T 2 As mentioned in Section 12.1.2, the charge-screening layers in the water on either side of the
membrane also contribute to its capacitance (see Section 7.4.3′on page 250). In physiological salt
concentrations these layers are each roughly 0. 7 nmthick. Use the formula for capacitors in series
from first-year physics, and your result in (a), to estimate the contribution to the total capacitance
from these layers.
12.4Afterhyperpolarization
a. The Example on page 447 showed how the resting membrane conductances (Equation 11.9 on page
420) give a quasi-steady membrane potential in rough agreement with the actual resting potential.
Repeat the calculation using the conductances measured during an action potential (Equation 12.14
on page 457), and interpret in the light of Figure 12.6b on page 457.
b. Hodgkin and Katz also found that the membrane conductances immediatelyafterthe passage
of an action potential did not return immediately to their resting values. Instead they found that
gNa+fell to essentially zero, whilegK+≈ 4 gCl−. Repeat your calculation using these values, and
again interpret in the light of Figure 12.6b.