The portion of a nerve cell that conducts signals is called an axon. Many of the electrical properties of axons are governed by ion channels, which are protein molecules that span the axon's cell membrane. When open, each ion channel has a pore that is filled with fluid of low resistivity and connects the interior of the cell electrically to the medium outside the cell. In contrast, the lipid-rich cell membrane in which ion channels reside has very high resistivity. Assume that a typical open ion channel spanning an axon's membrane has a resistance of 1× 10^11 Ω. We can model this ion channel, with its pore, as a 12-nm-long cylinder of radius 0.3 nm.

Question

The portion of a nerve cell that conducts signals is called an axon. Many of the electrical properties of axons are governed by ion channels, which are protein molecules that span the axon's cell membrane. When open, each ion channel has a pore that is filled with fluid of low resistivity and connects the interior of the cell electrically to the medium outside the cell. In contrast, the lipid-rich cell membrane in which ion channels reside has very high resistivity. Assume that a typical open ion channel spanning an axon's membrane has a resistance of 1× 10^11 Ω. We can model this ion channel, with its pore, as a 12-nm-long cylinder of radius 0.3 nm.
(a) What is the resistivity of the fluid in the pore?
(b) If a typical current density (current per unit cross-sectional area) in the cell membrane is 5 mA/cm^2 when the voltage across the membrane (the action potential) is 50 mV, what is the number density of open ion channels in the membrane?
(c) Cell membranes across a wide variety of organisms have a capacitance per unit area of 1 μF/cm^2. For the electrical signal in a nerve to propagate down the axon, the charge on the membrane "capacitor" must change. We still assume that a typical open ion channel spanning an axon's membrane has a resistance of 1×10^11Ω and the number density of open ion channels in the membrane is 100/μm^2. What time constant is required when the ion channels are open?