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 axonâs 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.
(a) Assume that a typical open ion channel spanning an axonâs membrane has a resistance of 1 x 10^11 Ω. We can model this ion channel, with its pore, as a 12-nm-long cylinder of radius 0.3 nm. What is the resistivity of the fluid in the pore?
(b) In a simple model of an axon conducting a nerve signal, ions move across the cell membrane through open ion channels, which act as purely resistive elements. If a typical current density 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?