## A Biochemistry Question

Does anyone know a real-world example of a cycle like this:

or in other words, this:

$\begin{array}{ccc} \mathrm{A} + \mathrm{C}_1 \longrightarrow \mathrm{C}_2 \\ \mathrm{X} + \mathrm{C}_2 \longrightarrow \mathrm{C}_3 \\ \mathrm{C}_3 \longrightarrow \mathrm{B} + \mathrm{C}_4 \\ \mathrm{C}_4 \longrightarrow \mathrm{Y} + \mathrm{C}_1 \end{array}$

where the reaction

$\mathrm{A} \to \mathrm{B}$

is exergonic (i.e., involves a decrease in free energy) while

$\mathrm{X} \to \mathrm{Y}$

is endergonic (i.e., involves a free energy increase)?

The idea is that the above cycle, presumably catalyzed so that all the reactions go fairly fast under normal conditions, ‘couples’ the exergonic reaction, which ‘wants to happen’, to the endergonic reaction, which doesn’t… thus driving the endergonic one.

I would love an example from biochemistry. This is like a baby version of much more elaborate cycles such as the citric acid cycle, shown here:

in a picture from Stryer’s Biochemistry. I’m writing a paper on this stuff with Jonathan Lorand, Blake Pollard and Maru Sarazola, and we have—presumably obvious—reasons to want to discuss a simpler cycle!

### 11 Responses to A Biochemistry Question

1. Wolfgang says:

Just a quick guess:

https://en.m.wikipedia.org/wiki/Urea_cycle

• John Baez says:

Hmm, that’s a bit more complicated, but it’s still a lot simpler than the citric acid cycle, and it may be better for our own purposes in some other ways:

Thanks!

• John Baez says:

Oh, so there’s a 4-cycle embedded in a larger reaction network here, namely:

$\begin{array}{rcl} \mathrm{A}_1 + \textrm{carbamoyl phosphate} &\longrightarrow & \mathrm{A}_2 + \mathrm{P}_{\mathrm{i}} \\ \mathrm{A}_2 + \mathrm{Asp} + \mathrm{ATP} &\longrightarrow& \mathrm{A}_3 + \mathrm{AMP} + \mathrm{PP}_{\mathrm{i}} \\ \mathrm{A}_3 &\longrightarrow& \mathrm{A}_4 + \mathrm{fumarate} \\ \mathrm{A}_4 + \mathrm{H}_2\mathrm{O} &\longrightarrow& \mathrm{A}_1 + \mathrm{urea} \end{array}$

• Wolfgang says:

Yes, I noted the four cycle, but did not check for the energetic conditions you would like to impose. From a faint memory of my biochemistry course maybe fatty acid synthesis/degradation could be another thing to look for? But I guess these are more complicated again.

• Wolfgang says:

You might find this instructive, too (though complex and with mostly acyclic pathways included):

http://biochemical-pathways.com/#/map/1

• Wolfgang says:

Another thought. If you are not fixed to biochemical cycles you could look at catalytic cycles of organic chemistry coupling reactions, here is one example (others exist, which are quite similar):

https://en.m.wikipedia.org/wiki/Suzuki_reaction

• John Baez says:

I think we’ll go with the urea cycle—it does a lot of what we want. Thanks, Wolfgang!

• LambertD says:

Does it HAVE to be from biochemistry? Because that could look a lot like a Feynman diagram :D

• John Baez says:

It does look like a Feynman diagram, and I wrote a whole book on the connection between chemistry and Feynman diagrams:

• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics, World Scientific, 2018.

But, I need an example from biochemistry…. and I’m using the urea cycle, which comes close. If Wolfgang tells me his full name, we’ll credit him.