Only a tiny fraction of the diffusible ions move across a cell membrane in
establishing a Nernst potential (see Focus On 20: Membrane Potentials), so
there is no detectable concentration change. Consider a typical cell with a
volume of \(10^{-8} \mathrm{cm}^{3},\) a surface area \((A)\) of \(10^{-6}
\mathrm{cm}^{2},\) and a membrane thickness \((l)\) of \(10^{-6} \mathrm{cm}\)
Suppose that \(\left[\mathrm{K}^{+}\right]=155 \mathrm{mM}\) inside the cell and
\(\left[\mathrm{K}^{+}\right]=4 \mathrm{mM}\) outside the cell and that the
observed Nernst potential across the cell wall is \(0.085 \mathrm{V}\). The
membrane acts as a charge-storing device called a capacitor, with a
capacitance, \(C,\) given by
$$C=\frac{\varepsilon_{0} \varepsilon A}{l}$$
where \(\varepsilon_{0}\) is the dielectric constant of a vacuum and the product
\(\varepsilon_{0} \varepsilon\) is the dielectric constant of the membrane,
having a typical value of \(3 \times 8.854 \times 10^{-12}\) \(\mathrm{C}^{2}
\mathrm{N}^{-1} \mathrm{m}^{-2}\) for a biological membrane. The SI unit of
capacitance is the firad, \(1 \mathrm{F}=1\) coulomb per volt \(=1
\mathrm{CV}^{-1}=1 \times \mathrm{C}^{2} \mathrm{N}^{-1} \mathrm{m}^{-1}\)
(a) Determine the capacitance of the membrane for the typical cell described.
(b) What is the net charge required to maintain the observed membrane
potential?
(c) How many \(\mathrm{K}^{+}\) ions must flow through the cell membrane to
produce the membrane potential?
(d) How many \(\mathrm{K}^{+}\) ions are in the typical cell?
(e) Show that the fraction of the intracellular \(K^{+}\) ions transferred
through the cell membrane to produce the membrane potential is so small that
it does not change \(\left[\mathrm{K}^{+}\right]\) within the cell.
