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I've been looking at the light-dependent reactions on Wikipedia and Khan Academy.

This is the net reaction:

$$2H_2O + 2NADP^+ + 3ADP + 3P_i \to O_2 + 2NADPH + 3ATP$$

So for every $2H_2O$ I expect $2NADPH$ and $3ATP$. I tried to count:

Diagram of light-dependent reactions, with counts.

These are my assumptions:

  • Cytochrome b6 complex pumps $1H^+$ for each $1e^-$.
  • FNR makes $1NADPH$ for each $2e^-$.
  • ATP synthase makes $1ATP$ for each $1H^+$.

If we want to end up with $3ATP$, then $3H^+$ need to diffuse through. The thing is, there's too many $H^+$. $4H^+$ is contributed by the cytochrome b6 complex pumping alone, and another potential $4H^+$ is contributed by the water splitting.

Clearly I'm missing something here that would lead to the correct count.

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I remember being confused by the numbers when I first studies it primarily because most sources seem to differ on the stoichiometry. I have tried to explain why is it difficult to comment upon such large equations, hope it helps!

Short Answer

There are two faults in your assumptions.

  1. The plastoquinone-cytochrome cycle actually pumps $2H^+$ per electron (terms and conditions apply)
  2. ATP-synthase requires $\approx 4$ protons per ATP produced

So, 12 protons per 2 water molecules is what is pumped, which makes 3 ATP.

Full Answer

As you might notice, the above scheme does not account for many things. For starters, since the 4$H^+$ from water splitting are produced and not pumped, only 2 of those need to be pumped back via ATP-Synthase to restore equilibrium. So even if you account for the $2H^+$ consumed by $2NADP^+$, there is still something that doesn't quite fit. Furthermore, some sources also try to account for the acidity of the inorganic $Pi$ in the ATP-synthesis which will further allow for changes in the $H^+/e^-$ ratio.

Why the confusion?

Most of what we know of the stoichiometry is based on a conglomerate of experiments, beginning with experiments on isolated chloroplasts. As a result, some experiments give us some parts of the stoichiometry (we know that the $NADPH/ATP$ ratio is around $2/3$) and others give us the reactions. Often times, many possible reactions are proposed, from which, the ones which can best be integrated with results from different experiments are thought to represent the actual truth. But this process is often mangled with complications, unexplained patches in the theory and the like. Furthermore, biological processes often are stochastic and do not have perfect stoichiometries. This is all overlooking the actual disagreements between experiments (the $H^+/ATP$ ratio for the synthase had significant variations in different experiments).

Okay, so now what?

Don't loose hope. We do know quite a bit about photosynthesis. What we currently understand about photosynthesis (light reactions) is thus:

  1. There are three distinct processes that can produce a chemiosmotic gradient of $H^+$.

    a. Non-cyclic Photophosphorylation is the main method. This constitutes the popular Z-scheme involving a linear electron transport from water to the final acceptor $NADPH$, and this process pumps about $12H^+$ per $2H_2O$

    b. Pseudo-cyclic Photophosphorylation which is basically the same as above, but the final acceptor is an $O_2$ molecule, with similar ratios.

    c. Cyclic Photophosphorylation which allows for recycling an electron, that is, proton pumping without there being a final acceptor of electron solely by internal shuffling of the electron between different states (primarily involved Plastoquinone (Q) cycle. This can pump $2H^+$ extra for 1 electron.

  2. The ATP-synthase is slightly different from that found in animal mitochondria. The number of protons per ATP depends on the rotational symmetry of the synthase. (details in the references below). Basically, initially, it was thought to have 12-fold symmetry (needing $12H^+$ per rotation, in which $3ATP$ is produced, hence $4H^+$ per $ATP$) but was later found to have 14-fold symmetry actually needing $14H^+$ for the observed $3/2$ ratio of $ATP/NADPH$.

  3. The number of Photosystem 1 units(where the cyclic version operates) is more than that of Photosystem 2 units. This means, that cyclic photophosphorylation goes on simultaneously at these centres, and will account for the extra $2H^+$ required to meet the experimentally observed ratio (1 in every 5 electron is "recycled").

References

  1. A good overall paper addressing all the main issues. Photosynthesis of ATP—Electrons, Proton Pumps, Rotors, and Poise; John F.Alle; Cell Volume 110, Issue 3, 9 August 2002, Pages 273-276

  2. General resources on these reactions. Here, here and here.

  3. Studies on the $H^+/ATP$ ratio.

    a. Petersen, Jan et al. “Comparison of the H+/ATP ratios of the H+-ATP synthases from yeast and from chloroplast” Proceedings of the National Academy of Sciences of the United States of America vol. 109,28 (2012): 11150-5.

    b. Watt IN, Montgomery MG, Runswick MJ, Leslie AG, Walker JE. Bioenergetic cost of making an adenosine triphosphate molecule in animal mitochondria. Proc Natl Acad Sci U S A. 2010;107(39):16823-7.

  4. Old studies to illustrate the difficulty of explaining the results of many studies:

    a. Function of the Two Cytochrome Components in Chloroplasts: A Working Hypothesis; R. HILL & FAY BENDALL; Nature - volume 186, pages136–137 (1960)

    b. Photosynthesis by Isolated Chloroplasts; DANIEL I. ARNON, M. B. ALLEN & F. R. WHATLEY; Nature - volume 174, pages394–396 (1954)

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