Finding the capacitance of a cell membrane BIO EXAMPLE 23.14 What is the capacitance of the membrane of the spherical cell specified in Example 23.13? The dielectric constant of a cell membrane is approximately 9.0. times that of a parallel-plate capacitor without a dielectric, or Cmembrane = K€0A/d. Inserting the dimensions from Example 23.13, we find STRATEGIZE If we imagine opening up a cell membrane and flattening it out, we would get something that looks like a parallel-plate capacitor with the plates separated by a dielectric, as illustrated in FIGURE 23.45. K€,A 9.0(8.85 × 10-12 c²¾N•m²) 4#(2.5 × 10-5 m)² Cmembrane d 7.0× 10-9 m = 8.9 × 10-1! F PREPARE The relevant dimensions are the same as those in Example 23.13. ASSESS Though the cell is small, the cell membrane has a rea- SOLVE The capacitance of the membrane is that of a parallel- sonably large capacitance of 90 pF. This makes sense because plate capacitor filled with a dielectric, so its capacitance is k the membrane is quite thin. FIGURE 23.45 The cell membrane can also be modeled as a capacitor. The cross-section area of the capacitor Cmembrane, One plate of the capacitor is the surface area of the membrane. + Insulator between + the plates - The other plate of the The separation between the сараcitor plates is the thickness of the membrane. + EXAMPLE 23.15 Counting ions moving through a channel BIO Investigators can measure the ion flow through a single ion channel with the patch clamp technique, as illustrated in FIG- URE 23.47. A micropipette, a glass tube 1 µm in diameter, makes a seal on a patch of cell membrane that includes one sodium channel. This tube is filled with a conducting saltwater solution, and a very sensitive ammeter measures the current as sodium ions flow into the cell. A sodium channel passes an average current of 4.0 pA during the 0.40 ms that the channel is open during an action potential. How many sodium ions pass through the channel? FIGURE 23.47 Measuring the current in a single sodium channel. of ions can then be found from this total charge and the charge per ion. PREPARE Each Na* ion has a charge q= +e. SOLVE In «SECTION 22.2, we saw that the charge delivered by a steady current in time At is Q=1At. The amount of charge flowing through the channel in At=4.0 × 104 s is Micropipette Na+ Q = I At = (4.0 × 10-12 A)(4.0 × 10-4 s) = 1.6 × 10-15 C Sodium Cell This charge is due to N ions, each with q = e, so the number of channel membrane ions is 1.6 × 10-15 C 1.6 X 10-19 C N = = 10,000 e ASSESS The number of ions flowing through one channel is not large, but a cell has a great many channels. The patch clamp tech- nique and other similar procedures have allowed investigators to elucidate the details of the response of the cell membrane to a STRATEGIZE The current is a measure of how much charge flows through the ion channel per second; we can thus use the values of the current and the time to find this charge. The number stimulus.