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Stream: theory: category theory

Topic: category theory in chemistry

John Baez (Feb 01 2021 at 16:26):

Seeing all these great discussions of category theory in computer science, I feel a bit sad because what I'm really interested in is category theory in chemistry.

It's definitely there, because chemical reaction networks - a widely used framework in chemistry - are equivalent to Petri nets where every transition is labelled with a nonnegative real numbers, called its "rate constant". I call these Petri nets with rates.

Petri nets are presentations of free commutative monoidal categories: strict symmetric monoidal categories where the symmetry morphisms are identities. Thus, a Petri net with rates presents a free commutative monoidal category $C$ equipped with a functor $r: C \to [0,\infty)$.

John Baez (Feb 01 2021 at 16:30):

It is remarkable that chemists have proved a large number of extremely deep theorems about the objects - and that there exist some profound open conjectures about them. Five are listed on page 4 here:

• Gheorghe Craciun and Abhishek Deshpande, Endotactic networks and toric differential inclusions.

(You don't need to understand what an "endotactic network" or a "toric differential inclusion" is to understand these conjectures.)

John Baez (Feb 01 2021 at 16:36):

Many theorems and conjectures are independent of the rate constants and are thus theorems about Petri nets!

Today I'm interested in this paper:

• Alex Blokhuisa, David Lacostea and Philippe Nghe, Universal motifs and the diversity of autocatalytic systems.

A "motif" is a small Petri net that is often found as a subobject in the big Petri nets that describe all the chemical reactions in cells. It's believed that motifs describe certain biological functions that transcend the specific chemicals involved in the chemical reactions. That is, they're important patterns in biochemistry.

John Baez (Feb 01 2021 at 16:38):

This paper studies "autocatalytic" motifs - that is, Petri nets describing chemical reactions where chemicals catalyze the production of other chemicals involved.

John Baez (Feb 01 2021 at 16:38):

To understand the origin of life we need to understand autocatalytic reactions... but they're also fundamental to how life continues to create the chemicals it relies on.

John Baez (Feb 01 2021 at 16:39):

I think it's cool that this study of autocatalytic motifs is "really" (in the sense category theorists use this word) the study of free commutative monoidal categories! But so far category theorists haven't help this study at all.

John Baez (Feb 01 2021 at 16:45):

Here are two examples, drawn in a number of styles:

two motifs

John Baez (Feb 01 2021 at 16:48):

The first one is the free commutative monoidal category generated by morphisms

$A \otimes E \to EA , \quad EA \to A \otimes E$
$EA \to E \otimes B, \quad E \otimes B \to EA$

(Chemists traditionally write the tensor product as $+$ even though it's not coproduct.)

John Baez (Feb 01 2021 at 16:50):

Here $EA$ is just a funny name for an object, which chemists use because it's $E$ and $A$ "stuck together" (not the tensor product though). We could just as well call it $C$ and write the generating morphisms of our free commutative monoidal category as follows:

$A \otimes E \to C , \quad C \to A \otimes E$
$C \to E \otimes B, \quad E \otimes B \to C$

John Baez (Feb 01 2021 at 16:52):

Once you get used to this the chemist's notation is helpful, since it says what's going on: $A$ and $E$ can stick together, and break apart forming $B$ and $E$. It's probably clearer if we only include the "forwards" morphisms:

$A \otimes E \to EA$
$EA \to E \otimes B$

John Baez (Feb 01 2021 at 16:53):

$E$ stands for "enzyme". It's an enzyme catalyzing a process where $A$ turns into $B$.

Spencer Breiner (Feb 01 2021 at 22:40):

John Baez said:

Here $EA$ is just a funny name for an object, which chemists use because it's $E$ and $A$ "stuck together" (not the tensor product though). ```quote

Spencer Breiner (Feb 01 2021 at 22:42):

Any idea whether "stuck together" can be interpreted in terms functorial boxes? If the rate constant in one direction is much greater than the other, that seems somewhat similar to (co)laxity.

John Baez (Feb 01 2021 at 22:49):

I don't remember what "functorial boxes" are. The big problem is that given two molecules E and A, there are often many different ways to stick them together and get a bigger molecule. "EA" is a sloppy, ambiguous notation for some particular molecule that we can get by sticking together E and A. In other words, there's no way to guess what "EA" means knowing only E and A: you have to ask the chemist which molecule is meant.

John Baez (Feb 01 2021 at 22:50):

So, a mathematician or (worse) a computer scientist would not be happy with this notation "EA".

Spencer Breiner (Feb 01 2021 at 23:19):

I learned about them from the gorgeous diagrams in Fritz & Perrone, though there is a longer lineage.

The basic idea is that you can represent co/lax monoidal functors in string diagrams by putting shaded boxes around the objects/morphisms coming from the domain category.

Spencer Breiner (Feb 01 2021 at 23:22):

Presumably different ways of sticking together would correspond to different functors.

Cole Comfort (Feb 01 2021 at 23:23):

They go back much earlier, like in this paper of Cockett and Seely. But I don't know who invented them, or if they are just folklore.

John Baez (Feb 01 2021 at 23:28):

Spencer Breiner said:

Presumably different ways of sticking together would correspond to different functors.

Molecules are quirky and individualistic enough that I'm having trouble seeing how to describe the ways of sticking them together using a functor, though sometimes there are semi-systematic recipes like "attach the ketyl group of molecule A to the methyl group of molecule B". (I don't know if that particular recipe makes sense, but it sounds like things chemists say.)

John Baez (Feb 01 2021 at 23:28):

But then if B has 17 methyl groups you say "huh?"

John Baez (Feb 01 2021 at 23:29):

Anyway, all this seems wide-open to me... though there's been some great work on chemistry via graph rewriting, that comes within range of this.

Matteo Capucci (he/him) (Feb 02 2021 at 09:17):

Shall we move this to #practice: applied ct?

Jules Hedges (Feb 02 2021 at 10:32):

I'd be in favour of that... partly because I wish that stream was more lively than it is

Ben Sprott (Feb 02 2021 at 18:57):

I had a fun thought about wires that bend around and back into the box they came out of in a monoidal category string diagram. In chemistry terms, aren't these catalysts?

John Baez (Feb 02 2021 at 18:57):

Yes, those are catalysts.

Joe Moeller (Feb 02 2021 at 19:00):

We wrote a paper about that:

• John Baez, John Foley, Joe Moeller. Network models from Petri nets with Catalysts, Compositionality, Volume 1, Issue 4, 2019.

Ben Sprott (Feb 02 2021 at 19:15):

And an autocatlytic system is just a box with wires coming out and back in (I think). This is related to life itself. Can anyone discuss this?

John Baez (Feb 02 2021 at 19:31):

Here's a picture from that paper, showing the sort of thing @Ben Sprott asked about:

Petri net with a catalyst

This is a Petri net, not a string diagram in a monoidal category - but any Petri net gives a commutative monoidal category, and the string diagrams in that category locally resemble the Petri net itself, so it's indeed connected to what Ben was thinking.

Ben Sprott (Feb 02 2021 at 19:37):

My intuition was just that atoms have to be Hilbert spaces. I guess that is a simplification, you have to tensor all the electrons together and tensor that with the etc etc..perhaps I just decided that an atom is a string in Hilb and a box is a unitary transformation. It is fairly safe since it's all just quantum mechanics.

John Baez (Feb 02 2021 at 19:54):

In particle physics strings bending back up describe antiparticles, which are - roughly, though not exactly - the dual objects of particles. As Feynman put it, "antiparticles are particles going backward in time". (That too is a rough statement, not precise.)

In chemistry, strings bending back up just mean that a molecule that leaves a reaction is able to come around back and partake in another reaction of the same kind.

John Baez (Feb 02 2021 at 19:59):

I told Gheorghe Craciun, an expert on reaction networks who seems to have proved the Global Attractor Conjecture, about this paper:

• Alex Blokhuisa, David Lacostea and Philippe Nghe, Universal motifs and the diversity of autocatalytic systems.

He replied:

I was not aware of the Blokhuisa-Lacostea-Nghe paper, but it does look very interesting!

I have been very interested in autocatalytic networks recently, and we posted some work on arXiv recently:

• G. Craciun, A. Deshpande, B. Joshi and P. Y. Yu, Autocatalytic systems and recombination: a reaction network perspective https://arxiv.org/abs/2012.06033

• B. Joshi and G. Craciun, Autocatalytic Networks: An Intimate Relation between Network Topology and Dynamics https://arxiv.org/abs/2006.01384

Ben Sprott (Feb 02 2021 at 20:14):

Thanks @John Baez !!

John Baez (Feb 02 2021 at 20:42):

Somehow my interest in the strange properties of nitrogen-14 and my interest in autocatalysis seem to have collided! The carbon-nitrogen-oxygen cycle is the main way hydrogen gets fused into helium in stars > 1.3 times as massive as the Sun - though it also plays a role in the Sun. And it's a clearly a case of catalysis:

the CNO cycle

John Baez (Feb 02 2021 at 20:44):

Abstractly it looks a lot like the Krebs cycle or Calvin cycle in biochemistry: little things come in, little things go out, but the big things go round and round.

John Baez (Feb 02 2021 at 21:26):

I'd like to prove general theorems about reaction networks ( $\approx$ Petri nets) of this sort: there's a loop, and at each reaction (= transition) there's at most one thing going in and at most one thing going out. (These are called bimolecular reactions in chemistry.)

John Baez (Feb 02 2021 at 21:27):

It follows that the rate at which each reaction in the loop proceeds is linearly proportional to the amount of things that go in at that stage.