A spherical neuron-sized cell has radius

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Molecular machines, transport and bioenergetics · Molecular machines, transport and bioenergetics

Life at Thermal Energy, Ch The membrane: barrier, capacitor and pump

Problem

a = 15\,\mathrm{\mu m}

, resting potential

V_m = -60\,\mathrm{mV}

, membrane capacitance

C_m = 1\,\mathrm{\mu F\,cm^{-2}}

, and intracellular

\mathrm{K^+}

concentration

140\,\mathrm{mM}

. Take body temperature,

310\,\mathrm{K}

, where

k_BT/e = 26.7\,\mathrm{mV}

, with

\mathrm{Na^+}

12\,\mathrm{mM}

inside and

145\,\mathrm{mM}

outside, and

\mathrm{K^+}

140\,\mathrm{mM}

inside and

5\,\mathrm{mM}

outside. ((a)) Find the number of elementary charges that must sit on the membrane to hold this cell at

-60\,\mathrm{mV}

, and express it as a fraction of the cell's total

\mathrm{K^+}

ions. ((b)) A rival pump hypothesis proposes a stoichiometry of four

\mathrm{Na^+}

pumped out for every three

\mathrm{K^+}

pumped in, per ATP. Using the free-energy cost of moving one ion at this cell's resting potential, decide whether this stoichiometry could be driven by one ATP (which supplies about

20k_BT

in a cell, and no more than the

21k_BT

found in Chapter 3).

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