Respiration, photosynthesis and electrochemical gradients

An ion crosses both chemical and electrical differences

For transfer of an ion from out to in under the same reference convention, its electrochemical potential difference is Δμ̃=RT ln(a_in/a_out)+zF(ψ_in−ψ_out). The activity ratio is dimensionless, z the signed charge number and ψ electrical potential. Both terms are energies per mole of transferred ions.

A concentration difference alone is insufficient for a charged solute. A voltage can reinforce or oppose the chemical contribution, and reversing the direction reverses both terms. This thermodynamic expression gives a driving-energy difference, not a complete transport-flux equation or membrane rate.

Electron transport in respiratory systems and light-driven processes in oxygenic photosynthesis can participate in creating electrochemical gradients used by coupled machinery. The actual membranes, carriers, coupling and conditions determine the account; there is no universal gradient or ATP-per-oxygen number supplied by this statement.

Calculate the two signed contributions

Invent maintained compartments at T=310 K with pH_out=7.00, pH_in=8.00 and Δψ=ψ_in−ψ_out=−0.120 V. For proton transfer out→in, z=+1 and a_in/a_out=10⁻⁸/10⁻⁷=0.100 using the stated pH activity definition.

The chemical contribution is RT ln(0.100)=−5934.87488 J mol⁻¹. The electrical contribution is F×(−0.120)=−11578.2399 J mol⁻¹. Their sum is −17513.1147 J mol⁻¹, equivalent to Δμ̃/F=−0.181510644 V for these monovalent proton transfers.

Keep the pH difference but change Δψ to +0.120 V. The sum becomes +5643.36497 J mol⁻¹: the electrical opposition outweighs the favorable chemical term. The plotted line varies only the assigned voltage. A negative total is thermodynamic driving energy in the stated direction, not proof of a particular transport rate.

An energetic minimum is not a motor measurement

Assign an ATP-synthesis event energy +48000 J mol⁻¹ under compatible maintained inner-compartment conditions at the same temperature. For a stipulated link to n of the preceding inward proton transfers, the model sums ΔG_net=48000+n(−17513.1147) J mol⁻¹.

With n=3 the net is −4539.34421 J mol⁻¹. The smallest integer satisfying this assigned ideal energy budget is three, since 48000/17513.1147≈2.7408. This is an energetic bound for the invented event, not a measured coupling stoichiometry or a guarantee that the event occurs.

If only a stipulated 85.0% of the proton driving energy reaches the event, three transfers give +3341.55742 J mol⁻¹ and are insufficient in this energy model; four give −11544.5901 J mol⁻¹. Mechanistic geometry, other transport costs and kinetic constraints remain separate. The assigned synthesis energy is not silently imported from the earlier ATP example at a different temperature and composition.

A supplied flux budget needs the actual coupling model

For a separate synthetic F-type motor model, stipulate twelve ion-carrying ring sites, one proton per site per complete turn and three ATP events per turn. The geometric count is twelve/three=4 protons per ATP event. It differs from an energetic minimum and is not a universal rotor geometry.

Supply steady pump flux 100 protons s⁻¹ and leak flux 20 protons s⁻¹, with sufficient maintained driving energy and tight productive coupling assumed. Productive flux is 80 protons s⁻¹. This supports 80/12=6.66666667 turns s⁻¹ and 20.0 ATP events s⁻¹ in the assigned motor budget.

A real pump need not hold the same flux when gradients or leaks change. Rotor stoichiometries vary among studied systems, and ATP production also depends on transport and other processes. This toy cannot establish a complete cellular ATP/oxygen ratio.

Water oxidation and absorbed-photon events are different inventories

In an oxygenic context, the formal half-reaction 2H₂O → O₂+4H⁺+4e⁻ preserves atoms and charge. It counts four electron-transfer equivalents per O₂ event. This is the oxidation direction; the full reverse oxygen reduction also requires four electrons, rather than two per whole O₂ molecule.

For a deliberately ideal excitation ledger, stipulate two excitation stages per transferred electron and one absorbed photon per stage. One assigned O₂ event with four electron transfers therefore corresponds to eight absorbed-photon events. This is not a measured quantum yield or a sufficient complete budget for making one hexose.

Cyclic electron flow returns electrons through a circuit without adding the same water-oxidation extent. Four stipulated extra circuits therefore do not automatically add an O₂ event. Carbon fixation also needs carbon input, ATP, reducing equivalents and regenerated intermediates. “Dark reaction” does not mean independent of the supply created by light-driven processes, and anoxygenic photosynthetic systems require a different donor account.

A complete yield needs more than one local count

The gradient-energy calculation, energetic coupling bound, ring-site budget and ideal photon ledger answer different questions. Combining them into a claimed cellular yield would require compatible conditions and a full account of electron donors, acceptors, carrier shuttles, transport, leaks and other uses.

Use the biological scope actually examined. Higher-plant chloroplast and mitochondrial accounts do not specify every organism’s machinery. These invented examples contain no measured membrane gradient, universal photosynthetic efficiency or actual ATP yield.

Worked examples

Try it yourself

1. Retain the pH difference but set Δψ=+0.120 V. Find inward-transfer energy; then reverse the original −0.120 V transfer direction.

   Show the answer and reasoning

   Opposed inward value +5643.36497 J mol⁻¹; reversing the original direction gives +17513.1147 J mol⁻¹.

   Recombine the chemical term with the new electrical sign. Reversing the original direction negates both original contributions.

2. Only 85.0% of proton driving energy is available to the +48000 J mol⁻¹ event. Compare three and four transfers.

   Show the answer and reasoning

   Three give +3341.55742 J mol⁻¹ and are energetically insufficient; four give −11544.5901 J mol⁻¹.

   Multiply available driving energy by 0.850 before counting transfers. Energetic sufficiency here is not a proof of a motor mechanism.

3. Raise leak to forty protons s⁻¹ at the same stipulated hundred-proton pump flux. Find ATP event flux for the assigned motor.

   Show the answer and reasoning

   15.0 ATP events s⁻¹.

   Sixty productive protons per second divided by four per ATP gives fifteen. A real pump’s response to a changed gradient has not been supplied.

4. Add four stipulated cyclic electron circuits without additional water oxidation. What additional O₂ is implied, and is hexose production fixed?

   Show the answer and reasoning

   No additional O₂ is implied by those circuits; hexose production is not determined.

   No extra water-oxidation extent or carbon-fixation/ATP/reductant budget was added. Do not turn an electron circuit count into a complete biosynthetic yield.

What to carry forward

• Define transfer direction before assigning gradient signs.
• Chemical and electrical terms can reinforce or oppose.
• Energetic bounds differ from actual coupling stoichiometry.
• Productive flux must exclude specified leaks.
• Oxygenic photon/electron counting is only part of a complete yield account.