The free energy generated by ion movement in the cell is used by the cell to do work. Compute the minimum number of millimoles of sodium that would be needed to pump one millimole of calcium ions if the following concentrations were present in the cell. Assume the temperature to be 370C and recall that R=0.008314 kJ/mol-K.
Na+(aq, 31 mM) -> Na+(aq, 9 mM)
Ca2+(aq, 12 mM) -> Ca2+(aq, 800 mM)
To compute the minimum number of millimoles of sodium required to pump one millimole of calcium ions, we can use the Nernst equation. The Nernst equation relates the concentration gradient of an ion to the electrical potential required to maintain equilibrium across the cell membrane.
The Nernst equation is given by:
E = (RT / zF) * ln ([Cout] / [Cin])
Where:
E is the electrical potential
R is the gas constant (0.008314 kJ/mol-K)
T is the temperature (in Kelvin)
z is the charge of the ion
F is the Faraday constant (96,485 C/mol)
[Cout] is the concentration outside the cell
[Cin] is the concentration inside the cell
ln is the natural logarithm
For pumping calcium ions (Ca2+), we need to find the electrical potential required to maintain equilibrium. The charge of calcium ions (z) is 2.
Let's calculate the electrical potential required separately for sodium (Na+) and calcium (Ca2+) ions using the given concentrations.
For sodium ions:
E_Na = (RT / zF) * ln ([Cout_Na] / [Cin_Na])
Substituting the values:
R = 0.008314 kJ/mol-K
T = 37 + 273 = 310 K (converting from degrees Celsius to Kelvin)
z = 1 (since sodium ions have a charge of +1)
F = 96,485 C/mol
[Cout_Na] = 9 mM
[Cin_Na] = 31 mM
E_Na = (0.008314 * 310 / (1 * 96485)) * ln (9 / 31)
Using a calculator, we can find that E_Na ≈ -0.109 V (rounded to 3 decimal places).
For calcium ions:
E_Ca = (RT / zF) * ln ([Cout_Ca] / [Cin_Ca])
Substituting the values:
R = 0.008314 kJ/mol-K
T = 310 K
z = 2 (since calcium ions have a charge of +2)
F = 96,485 C/mol
[Cout_Ca] = 800 mM
[Cin_Ca] = 12 mM
E_Ca = (0.008314 * 310 / (2 * 96485)) * ln (800 / 12)
Using a calculator, we can find that E_Ca ≈ 0.451 V (rounded to 3 decimal places).
Now, we can calculate the difference in the electrical potential between sodium and calcium ions:
ΔE = E_Ca - E_Na
ΔE = 0.451 - (-0.109) ≈ 0.560 V (rounded to 3 decimal places).
The free energy change (ΔG) associated with moving one mole of ions across the membrane is given by the equation:
ΔG = -zFΔE
Where z is the charge of the ion and F is the Faraday constant. Since we have millimoles of ions in this case, we need to convert ΔG to kJ/mol by dividing by 1000.
ΔG = -2 * 96,485 * 0.560 / 1000 ≈ -107.76 kJ/mol (rounded to 2 decimal places)
To compute the minimum number of millimoles of sodium required to pump one millimole of calcium ions, we can use the equation:
ΔG = -RT * ln ([Cout_Na] / [Cin_Na])
Rearranging the equation to solve for [Cout_Na]:
[Cout_Na] = [Cin_Na] * e^(-ΔG / (RT))
Substituting the values:
R = 0.008314 kJ/mol-K
T = 310 K
ΔG = -107.76 kJ/mol
[Cin_Na] = 31 mM
[Cout_Na] = 31 * e^(-(-107.76 / (0.008314 * 310)))
Using a calculator, we can find that [Cout_Na] ≈ 682 mM (rounded to the nearest whole number).
Therefore, the minimum number of millimoles of sodium ions needed to pump one millimole of calcium ions is approximately 682 millimoles.