Question

The sodium potassium pump transports 5 ions for every molecule of ATP that is hydrolyzed: 3 Na+ out of the cell and 2 K+ into the cell. The pump maintains internal Na+ at 13 mM and external Na+ at 144 mM; it maintains internal K+ at 138 mM and external K+ at 9 mM. Assuming that the membrane potential is –66 mV (inside negative relative to outside), the temperature is 37 °C, and F is 23.1 kcal/V·mol, how much energy does it take to run the pump (i.e., how many kcal of energy are required per 5 mol of ions pumped)? Report your answer to the nearest tenth of a unit.

Answer 1

First of all the parameters. Ion concentration are:

internal Na+ = 13 mM; external Na+ = 144 mM; internal K+ = 138 mM; external K+ = 9 mM; and membrane potential = -66mV (negative inside the cell)

I apply these values for constants:

gas stable R = 2 cal mol-1 K-1

Faraday stable F= 23,100 cal mol-1 V-1

and I’ll deduce that the hotness, T, is 37? = 310 K

The Na+/K+ power uses 1 bit of ATP to go 3 Na+ absent of the cell and 2 K+ into the cell.

The ability of the drive is so power obligatory to move these ions/strength on large by ATP hydrolysis.

How much energy is needed to move the ions? In each case present is a payment from the information that the ions are enthused along a attention gradient (the substance part) and from the pressure group along a gradient of electrical potential (since of the casing potential). In allowing for these mechanism we boast to think on the subject of whether force needs to be used whilst the ions stir, or whether it is unconfined.

For the substance part we can use

?G = -RTln(relative amount of attention, inside as well as outer surface), where ln = normal logarithm

The symbol of this ?G will depend upon how the ratio is spoken (in/out or out/in) but the absolute value determination be the same. So I calculate an total value and then infer the sign of the free power change.

For Na+ ?G = -(2 x 310 x ln(11/144)) = 1594.5891766 cal mol-1 = 1595 cal mol-1

Since Na+ is life form enthused up a attention incline, labor have got to be done so the free energy modify is constructive.

apply the identical formula and logic for the pressure group of K+:

For K+ ?G = -(2 x 310 x ln(139/7)) = 1852.90954613 cal mol-1 = 1853 cal mol-1

(This worth is better than that for Na+ since the attentiveness incline is steeper.)

Next I compute the electrical machinery by:

?G = zF?, anywhere z is the accuse on the ion = 1 in both luggage, and ? is the covering latent (in volts).