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Plants do cellular respiration and release CO2 into the air, but does the plant also use some of this CO2 to do the Calvin cycle? Or does all the CO2 required for the Calvin cycle come in through the stomata?

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Gases like CO2 move quite freely in biological systems, approximating the laws of diffusion (note: there are of course some exceptions, especially underwater, as well as with proteins that assist in bulk transport of CO2 like hemoglobin).

If you were to measure the net amount of CO2 coming into the stomata, it would indeed approximate the net CO2 use by the plant. From this, you could make a claim like "the CO2 a plant needs comes in through the stomata": this is true for the net, and that's usually what's most important to a biologist.

However, if you want to get picky about what really happens at a molecular level, the flux of CO2 coming through the stomata is not the same as the net: no one is keeping track of the individual gas molecules, they just float around back and forth. The only reason there is a net flow at all is due to a concentration gradient: because the plant uses some CO2, there is less inside than outside. When molecules move around randomly, that means that a few more end up coming inside than are leaving. When we talk about net diffusion we are talking about the averages of many many molecules over time.

Included in that random motion are not just the CO2 molecules present inside and outside the plant, but also those that the plant is producing through cellular respiration. By chance, certainly, some of these molecules could end up back in the Calvin cycle. Some could even exit through the stomata and come back in (my intuition is that this is quite improbable for an individual molecule but I haven't done the math...the important concept is that it's all one "pool").

If you really wanted to know, you could set up a Monte Carlo simulation (or use differential equations) and calculate probabilities of all sorts of event sequences for individual molecules. You could also tilt the scales a bit if you, say, put the equivalent of a 'plastic bag' over a leaf, either physically or in a model: if there's a limited amount of CO2 in the "outside" space it would be much more likely for individual molecules to be "reused".

From a biologists' perspective, though, and especially for ubiquitous, freely moving molecules like CO2, it's usually only important to think about net flow.

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    $\begingroup$ Great answer — thanks for fixing the egregious mistake I introduced into the title! $\endgroup$ – tyersome Jan 9 at 3:30
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    $\begingroup$ @tyersome Haha no problem. Shh don't tell anyone but I edited my answer before the time edits register to fix the place I referred to CO2 as an "ion" - maybe a bit too much neuroscience for me. Happens to us all sometimes :) $\endgroup$ – Bryan Krause Jan 9 at 3:32

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