11.2. Ion pumping[[Student version, January 17, 2003]] 423
the total free-energy cost to pump one sodium ion out is thus−e(∆V−VNaNernst+ )=e(60mV+54mV)=e× 114 mV.Forinward pumping of potassium, the corresponding calculation gives+e(∆V−VKNernst+ )=e(− 60 mV−(− 75 mV)) =e× 15 mV.which is also positive. The total cost of one cycle is then 3(e× 114 mV)+2(e×
15 mV)= 372 eV=15kBTr. (The uniteV,or“electron volt,” is defined in Ap-
pendix A.) ATP hydrolysis, on the other hand, liberates about 19kBTr(see Prob-
lem 10.3). The pump is fairly efficient; only 6kBTris lost as thermal energy.Let us see how the discovery of ion pumping helps make sense of the data presented in Table 11.1
on page 416. Certainly the sodium–potassium pump’s net effect of pushing one unit of positive
charge out of the cell will drive the cell’s interior potential down, away from the sodium Nernst
potential and toward that of potassium. The net effect of removing one osmotically active ion from
the cell per cycle also has the right sign to reduce the osmotic imbalance we found in Donnan
equilibrium (Equation 11.6 on page 415).
Tostudy pumping quantitatively, we first note that a living cell is in a steady state, since it
maintains its potential and ion concentrations indefinitely (as long as it remains alive). Thus there
must be no net flux of any ion; otherwise some ion would pile up somewhere, eventually changing
the concentrations. Every ion must either be impermeant (like the interior macromolecules), or in
Nernst equilibrium, or actively pumped. Those ions that are actively pumped (Na+and K+in
our simplified model) must separately have their Ohmic leakage exactly matched by their active
pumping rates. Our model assumes thatjKpump+ =(− 2 /3)jNapump+ and thatjNapump+ >0, since our
convention is thatjis the flux directed outward. Summarizing this paragraph, in steady state we
must havejNa+=jK+=0,or
jKpump+ =−jKOhmic+ =−^23 jpumpNa+ =−^23 (−jOhmicNa+ ).In this model chloride is permeant and not pumped, so its Nernst potential must agree with
the resting membrane potential. Indeed from Table 11.1, its Nernst potential really is in good
agreement with the actual membrane potential ∆V=− 60 mV.Turning to sodium and potassium,
the previous paragraph implies that the Ohmic part of the corresponding ion fluxes must be in the
ratio− 2 /3. The Ohmic hypothesis (Equation 11.8) says that
−^23(
∆V−VNernstNa+)
gNa+=(
∆V−VKNernst+)
gK+.Solving for ∆Vgives
∆V=2 gNa+VNaNernst+ +3gK+VNernstK+
2 gNa++3gK+. (11.12)
Wenow substitute the Nernst potentials appearing in Table 11.1 on page 416, and the measured
relation between conductances (Equation 11.9), finding ∆V=− 72 mV.Wecan then compare our
prediction to the actual resting potential, about− 60 mV.
Our model is thus moderately successful at explaining the observed membrane potential. In part
the inaccuracy stemmed from our use of the Ohmic (linear) hypothesis for membrane conduction,
Equation 11.8, when at least one permeant species (sodium) was far from equilibrium. Nevertheless,