transport charged species across membranes

Transport of Charged Species Across Membranes

• Components of ΔG (Gibbs Free Energy)

  • The transport of charged species (ions) across biological membranes adds a new term to the Gibbs Free Energy equation.

  • The additional term is expressed as:
    $ext{ΔG = ΔG}_{ ext{concentration}} + zF ext{Δψ}$

  • Where:

    • z = Charge of the ion

    • F = Faraday's constant = 96.5 kJ/V·mol

    • Δψ = Membrane potential

• Example with Sodium Ions (Na⁺)

  • Sodium ion (Na⁺) has a charge of +1 (z = +1).

  • When moving from an area of negative charge to an area of positive charge:

  • Membrane potential (Δψ) is positive.

  • Hence, both components of the equation (zFΔψ and ΔG concentration) contribute positively to ΔG.

• Concentration Gradient Effects

  • Moving sodium from low concentration (inside cell) to high concentration (outside cell), leads to:

  • $ext{ln} rac{c_2}{c_1}$ > 0

  • Thus, the ΔG concentration term is positive.

  • Summary:

  • Both components of ΔG (concentration and charge) yield a large positive ΔG:

    • Indicating that movement is unfavorable (against both concentration gradient and charge gradient).

• Numerical Examples

  • If $rac{c_2}{c_1} = 10$ (tenfold increase outside):

  • $ext{ΔG} = 10.8 ext{ kJ/mol}$

  • If $rac{c_2}{c_1} = 100$ (hundredfold increase):

  • $ext{ΔG} = 16.7 ext{ kJ/mol}$

  • For $rac{c_2}{c_1} = 1000$ (thousandfold increase):

  • $ext{ΔG} = 22.6 ext{ kJ/mol}$

  • Increasing Δψ from 50 mV to 70 mV results in:

  • $ext{ΔG} = 24.6 ext{ kJ/mol}$ based on higher charge differential.

• Reversing Ion Movement

  • If sodium (Na⁺) is transported with the concentration gradient but still against the charge gradient (from negative to positive):

  • Positive ΔG contribution from charge (Δψ is still positive).

  • Negative ΔG contribution from concentration (spontaneous movement due to gradient).

  • Examples lead to:

    • For $rac{c_2}{c_1} = 10$ — $ext{ΔG} = -1.1 ext{ kJ/mol}$

    • For $rac{c_2}{c_1} = 100$ — $ext{ΔG} = -7 ext{ kJ/mol}$

    • For $rac{c_2}{c_1} = 1000$ — $ext{ΔG} = -13 ext{ kJ/mol}$

  • Potential increase from 50 mV to 70 mV changes ΔG from -13 kJ/mol to -11 kJ/mol (less spontaneous).

• Understanding Membrane Potential (Δψ)

  • Typical Cell Polarization:

  • Inside cell is often negative compared to outside which is positive.

  • Movement from negative to positive results in:

  • Positive Δψ.

  • When moving from positive to negative:

  • Negative Δψ occurs.

• Movement of Anions

  • Example with Chloride ion (Cl⁻):

  • Moving from positive to negative involves:

    • Negative Δψ, positive charge (z = -1), thus giving a positive net charge contribution: (z F Δψ is positive).

• Energetics of Sodium-Potassium ATPase

  • This enzyme:

  • Hydrolyzes ATP to move three sodium ions out of the cell and two potassium ions in.

  • Sodium Movement:

  • Against concentration gradient (12 mM inside, 145 mM outside).

  • Against charge gradient (inside negative, outside positive).

  • Potassium Movement:

  • Still against concentration (4 mM outside, 140 mM inside).

  • With charge gradient (inside negative).

  • The function involves enzyme phosphorylation during ATP hydrolysis, promoting ion transport.

  • Calculating ΔG for Sodium Transport:

  • Δψ = +60 mV, $c_2/c_1 = 145/12$ → use for calculations:

    • Results in ΔG = +12.2 kJ/mol for sodium.

  • Transport of 3 sodium ions: $3 imes 12.2 = 36.6 ext{ kJ}$

  • Calculating ΔG for Potassium Transport:

  • Δψ = -60 mV, $c_2/c_1 = 140/4$:

    • Results in ΔG = +3.37 kJ/mol for potassium.

  • Transport of 2 potassium ions: $2 imes 3.37 = 6.74 ext{ kJ}$

  • Total Energy Requirement:

  • Total ΔG = 36.6 + 6.74 = 43.3 kJ.

  • Hydrolysis of ATP = approx. -50 kJ/mol.

  • Overall ΔG effectiveness = -6.7 kJ, indicating highly efficient mechanism (87% efficiency).