- Problems[[Student version, January 17, 2003]] 485
yield? Comment on the continuous appearance of Hodgkin and Huxley’s conductance curves in the
light of your estimate.
12.8 T 2 Extracellular resistance
Repeat our derivation of the nonlinear cable equation, but this time don’t set the external fluid’s
resistivity equal to zero. Instead, let Γ 1 denote the electrical resistance per unit length of the
extracellular fluid (we found that the axoplasm’s resistance per length is Γ 2 =(πa^2 κ)−^1 ). Get a
new estimate of the propagation speedθand see how it depends on Γ 1. Compare your answer
qualitatively to Hodgkin’s 1938 results (Section 12.2.3 on page 456).
12.9Estimate for channel conductivity
a. Imagine a sodium channel as a cylindrical tube about 0. 5 nmin diameter (the diameter of a hy-
drated ion) and 4nmlong (the thickness of a bilayer membrane). Use the discussion of Section 4.6.1
on page 121 to estimate the permeability of a membrane studded with such channels at an area
densityσchan.
b. Use your result from Your Turn 11c on page 419 to estimate the corresponding conductance per
area. Take the concentration of ions to bec= 250 mM.
c. Convert your result to conductance per channel;σchanwill drop out of your answer. Get a nu-
merical answer and compare to the experimental value quoted above.
[Remark: Certainly the result you obtained is very rough: We cannot expect the results of macro-
scopic diffusion theory to apply to a channel so narrow that ions must pass through it single-file!
Nevertheless, you’ll see that the idea of a water-filled channel can give the magnitude of real con-
ductances observed in experiments.]
12.10Mechanotransduction
Review Problem 6.7 on page 212. How could the arrangement shown in panel (b) help your ear to
transduce sound (mechanical stimulation) into electrical signals (action potentials)?