Proton-motive force (a) Use Equation 18.61 to make an...

Proton-motive force (a) Use Equation 18.61 to make an estimate of the contribution of the $\mathrm{H}^{+}$ concentration gradient to the proton-motive force in a cell. The typical pH difference is roughly 1 (about 7.7 inside and 7.0 outside). (b) It takes about four protons to make one ATP from ADP. If the free energy needed is in the range $20-23 k_{\mathrm{B}} T,$ how large must $V$ be in order to generate an ATP assuming that the energy of the proton-motive force can be harvested with $100 \%$ efficiency? Recall that the definition of 1 eV $\approx 160 \mathrm{zJ} \approx 40 \mathrm{k}_{\mathrm{B}} T$ is the energy gained by moving a charge of one electron through a potential difference of one volt. (Problem courtesy of Daniel Fisher.)

Proton-motive force
(a) Use Equation 18.61 to make an estimate of the contribution of the $\mathrm{H}^{+}$ concentration gradient to the proton-motive force in a cell. The typical pH difference is roughly 1 (about 7.7 inside and 7.0 outside).
(b) It takes about four protons to make one ATP from ADP. If the free energy needed is in the range $20-23 k_{\mathrm{B}} T,$ how large must $V$ be in order to generate an ATP assuming that the energy of the proton-motive force can be harvested with $100 \%$ efficiency? Recall that the definition of 1 eV $\approx 160 \mathrm{zJ} \approx 40 \mathrm{k}_{\mathrm{B}} T$ is the energy gained by moving a charge of one electron through a potential difference of one volt. (Problem courtesy of Daniel Fisher.)

Key Concepts

Proton-Motive Force

Proton-motive force (PMF) is the electrochemical potential generated across a biological membrane due to a combination of a pH (chemical) gradient and an electrical potential difference. This gradient results from differences in proton concentrations on either side of the membrane, and its energy can be harnessed to perform cellular work, most notably the synthesis of ATP via chemiosmosis.

pH Gradient and Concentration Gradients

A pH gradient is a specific type of concentration gradient where there is a difference in hydrogen ion concentration across a membrane. Because pH is a logarithmic measure of hydrogen ion concentration, even a unit difference in pH implies roughly a tenfold difference in H+ concentration. This gradient contributes significantly to the PMF by establishing a potential difference that drives protons across the membrane.

ATP Synthesis via Chemiosmosis

ATP synthesis in cells often relies on the energy stored in the proton-motive force. In the process of chemiosmosis, the flow of protons back across the membrane through ATP synthase allows the enzyme to harness the energy from the proton gradient to phosphorylate ADP, forming ATP. The stoichiometry, such as the coupling of multiple protons for each ATP produced, is a key factor in understanding the efficiency and energetics of this process.

Energy Conversion Efficiency and Free Energy

The generation of ATP from ADP involves a specific free energy change that must be met by the energy provided through proton movement. The relationship between the potential difference (voltage) and the energy change per proton is critical. When considering the efficiency of the system (even at 100% efficiency as in a theoretical scenario), understanding the required potential necessary to shift enough energy per proton to meet or exceed the free energy requirement for ATP synthesis is essential.

Relationship Between Electrical Potential and Energy per Charge

The energy gained or lost when a charge is moved across a potential difference is given by the product of the charge and the voltage. This connection is fundamental to the operation of proton-motive force, linking the chemical gradient (and its associated energy) to a measurable electrical potential difference. In biological systems, converting between energy units (such as electronvolts, joules, or multiples of thermal energy kT) is important to relate the physical process of proton movement to the chemical work of ATP synthesis.

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