Question

6. Bacteriorhodopsin is an integral membrane protein that uses solar energy (absorbed by its retinal chromophore) to pump protons out of the cell against an electrochemical gradient. Re-entry of the protons into the cell (down the gradient through its ATPase) is then used to drive the synthesis of ATP from ADP and Pi. Consider the following conditions: intracellular pH is 9.4 and extracellular pH is 6.4, and there is a transmembrane electrical gradient of 120 mV (negative inside). The concentration of intracellular [ATP] = 1 mM, [ADP] = 50 µM, and [Pi] = 2 mM, and G° for ATP hydrolysis is -30.5 kJ/mol. Temperature is 27 °C. a. What is the energy cost of pumping one mole of H+ out of the cell? b. What is the minimal number of H+ that must re-enter the cell through the ATPase to provide sufficient energy to synthesize each ATP? Explain how you arrived at this answer quantitatively, of course.

Answer 1

Step-by-step explanation:

Hello!

a)

You have an integral membrane protein (Bacteriorhodopsin) that uses solar energy to pump protons out of the cell against its electrochemical gradient.

To calculate the energy needed by the protein to pump one molecule out of the cell you need to apply the free energy difference equation for a charged molecule:

ΔG= 2.303RTlog₁₀
`(([H^+]out)/([H^+]in))`+zFΔV

Where

ΔG= free energy difference per molecule transferred from the inside to the outside

R= gas constant 8.315*10⁻³ kJ/mol

T= Temperature in Kelvin (273.15 + TºC)

z= Valency of the ion (ej Na⁺ is z= +1 and Cl⁻ is z= -1)

F= Faraday constant 96.5 kJ/V*mol

ΔV= membrane potential (Volts)

The information you have is:

Intracellular pH= 9.4

Extracellular pH= 6.4

ΔV= 120 mV

Temperature 27ºC

Using the values of pH you can calculate the intracellular and extracellular proton concentrations:

Remember the pH of a solution is calculated as pH=-log[H⁺] so if you reverse the logarithm, you'll find the concentration of protons:

b= logₐ C ⇒ C= aᵇ

[H⁺]intracelular= 10⁻⁹'⁴

[H⁺]extracelular= 10⁻⁶'⁴

Now you need to match the units of the temperature and electrical gradient to the ones in the equation:

The temperature needs to ve in Kelvin: T= 273.15 + 27ºC= 300.15K

The electrical gradients need to be in Volts: ΔV= 120 mV= 0.12V

Now you can replace the data in the formula and calculate how much energy is needed to pump one proton outside the cell:

ΔG= 2.303RTlog₁₀
`(([H^+]out)/([H^+]in))`+zFΔV= 2.303*( 8.315*10⁻³ kJ/mol)*300.15K*log
`((10^(-6.4))/( 10^(-9.4)) )`+(+1)*96.5 kJ/V*mol*0.12V= 28.82 kJ/mol

It costs 28.82 kJ/mol to pump one mole of H⁺ outside the cell.

b)

To synthesize one molecule of ATP the ATP synthase uses the electrical gradient produced by the spontaneous pass of protons through the membrane. This process is called the proton-motive force.

The ATP synthase consists of two subunits F₀ and F₁ with a rotational motor mechanism that allows it to produce and release ATP.

F₁ has the subunits α, β, σ, γ, ε in charge of synthesizing ATP

F₀ is a proton pore, made of 3 subunits a, b and c. 6 subunits c make up the transmembrane ring in charge of rotation, this ring is connected to the subunits γ and ε. Each time a proton passes, it makes the ring rotate which makes the subunits γ rotate.

Each time the subunit γ rotates 360º, 3 ATP molecules are synthesized and freed. A full rotation of γ means that the c-ring also made a full rotation (for this you need one proton per subunit)

If the ring has 6 c-subunits and, then each ATP molecule needs 6/3= 2 protons to need to pass for each ATP molecule.

I hope this helps!