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Stream: deprecated: chemistry

Topic: problems to work on

John Baez (Apr 29 2022 at 05:06):

We've started talking about some problems to work on. Here are some I suggested:

1) coding - features that we could add to the AlgebraicPetri package for dealing with open Petri nets.

2) studying "motifs" - basically, Petri nets that commonly show up in chemical reaction networks because they have good properties. Maybe we could use AlgebraicPetri to create a library of motifs, and study their behavior, alone and composed with each other. For more on motifs, see:

• John J. Tyson and Béla Novák, Functional motifs in biochemical reaction networks.

3) studying "absolute concentration robustness" - certain Petri nets with rates have equilibria where the amount of certain chemicals is independent of initial conditions. There are theorems about this, so far only for "closed" Petri nets. For example:

• Martin Feinberg and Guy Shinar, Design principles for robust biochemical reaction networks: What works, what cannot work, and what might almost work.

• Badal Joshi and Gheorghe Craciun, Reaction network motifs for static and dynamic absolute concentration robustness.

4) studying "detailed balanced" open Petri nets with rates. This is a constraint from thermodynamics that holds for "realistic" systems, but is violated by most open Petri nets with rates. I wrote a paper touching on this with the students at ACT2018, but we never published it. There's more to do!

*John C. Baez, Blake S. Pollard, Jonathan Lorand and Maru Sarazola, Biochemical coupling through emergent conservation laws.

5) studying the relation between open Petri nets with rates and stock-flow diagrams. Evan, Sophie and I have already been thinking about this. This could easily pull us into topics like this:

• John Baez, Kenny Courser and Christina Vasilakopoulou, Structured versus decorated cospans.

Sam Tenka (Apr 29 2022 at 15:52):

The idea of "motifs" especially interests me! Might there be space to work on "motif discovery" given a bunch of nets? I'm imagining something in the style of my academic sibling Kevin Ellis et al's dreamcoder --- its main contribution is taking a bunch of (typed lambda calculus) programs, then extracting common subprograms to "compress the input" and more importantly to discover subroutines potentially useful for future programming tasks. Their notion of "subprogram" is broader and more semantically sensible than you might expect, since two programs related by some functional equation and with no shared syntactic subtrees will often (but not always, to avoid rice theorem difficulties) be recognized as having a common subprogram. Perhaps a similar method can be used for motif discovery?

John Baez (Apr 29 2022 at 17:45):

We have some software relevant to this as part of AlgebraicJulia, which is a bunch of software for applied category theory, including Petri nets. James Fairbanks writes:

We have Petri net hom finding implemented with a constraint satisfaction problem solver. It can be used to match the motif patterns in a net or compute a motif distribution of a Petri net if you have a set of motifs you want to match. That would give you a tool for doing some experimental mathematics and checking conjectures on some examples rapidly. The CSP solver isn’t really fast, but it is way faster than doing it by hand.

John Baez (Apr 29 2022 at 17:49):

I don't know how this software works myself, but we've got people who do. So one kind of project would be to enhance this software, and another kind it would be to use it to do something interesting in chemistry or biochemistry. It might be fun to do these hand in hand - try to do some mathematical chemistry, and build software as part of that project.

Sam Tenka (May 03 2022 at 21:08):

@Sophie Libkind you mentioned something about fast/slow decompositions in chemistry... I'd love to hear what you had in mind!

Sam Tenka (May 03 2022 at 21:50):

Re problem (3) above [on robustness / fixedpoint existence and uniqueness], I'll copy one of my zoom chat comments from earlier today: it would be really cool to see in detail how this monoidal functor from reaction nets to dynamics models *oscillating* reactions (e.g. BriggsRauscher) --- or more generally the sorts of cool far-from-equilibrium phenomena (as clarified by the great chemist Prigogine) that happen when we fix one side of our bath as "hot" (or high concentration of source X) and the other "side" as "cold" (or low concentration of target Y). Unlike in the heat equation, in chemistry the flow allows v interesting behaviors within the bath that aren't just linear interpolation. Sort of an analyzable microcosm of how the [sun]->(earth)->[space] pair of heat baths supports very non-equilibrium-looking life!

John Baez (May 03 2022 at 22:03):

Oscillating reactions are cool. For example this chemical reaction network

$A \rightarrow X$

$2X + Y \rightarrow 3X$

$B + X \rightarrow Y + D$

$X \rightarrow E$

is called the Brusselator, and with suitable rate constants it has oscillating solutions (click the link for graphs).

Now that we have AlgebraicPetri we could easily simulate this... at least if @Sophie Libkind tells us how!

Sam Tenka (May 03 2022 at 22:24):

@John Baez Yes! key there, though, is that A's (and for symmetry or to enable reversibility let's say E's) concentrations are fixed (perhaps so large as to be so). That's part of the "open system"ness and part of the nonequilibrium I mentioned in that sun/earth/space metaphor

Spencer Breiner (May 03 2022 at 22:31):

$B$ and $D$ also? It seems like otherwise $D$ will grow monotonically until $B$ depletes.

John Baez (May 03 2022 at 22:38):

Good point - I imagine that some concentrations oscillate until B runs out. You can make up "artificial" reaction networks that oscillate forever, but reality they never do. So it looks like Wikipedia chose to use a "realistic" one. If you just scratched out B and D from the reaction network it would be less realistic... but presumably the realistic one would behave similarly as long as there was enough B around

John Baez (May 03 2022 at 22:38):

And yes, @Sam Tenka, it might be wiser to treat this particular oscillator as an open reaction network, where B flows in and D flows out.

John Baez (May 03 2022 at 22:47):

Getting back to another of Sam's point: I would really like to study the black-boxing of some of the open reaction networks in Prigonine and Kondepudi's book Modern Thermodynamics. An interesting one is in section 18.4.1 here.

Sam Tenka (May 03 2022 at 22:58):

@John Baez neat! do you have specific questions in mind? (e.g. what semialgebraic relations this induces? or perhaps more qualitative questions about, e.g., bounding the number of steady flow states using Morse theory etc...) (btw, this is probably obvious to the conneuseiur (and probably the spelling of conneseiur is, too!), but what is a semialgebraic relation? Are they closed under "or"? In algebra, as long as we are in a domain we can insist that f vanish or that g vanish by asking that f*g vanish. But idk how to do this for inequalities (perhaps one can make progress if one uses that in an ordered ring all squares are nonnegative??) [edit: I had forgotten that in an ordered ring squares are always nonnegative!]

John Baez (May 03 2022 at 23:01):

As for the question "what is a semialgebraic relation", you can't beat my paper with Blake.

Sam Tenka (May 03 2022 at 23:03):

Ah, cool, so on page ~30 we see that we put in closure under finitary "or" and "and" by hand.

John Baez (May 03 2022 at 23:03):

The main good thing about them is that they're closed under composition yet still quite manageable - this relies on a nontrivial result, the Tarski-Seidenberg theorem (explained in my paper).

They're also closed under "and" and "or", but those operations aren't so easy to achieve by combining chemical reactions.

Sam Tenka (May 03 2022 at 23:04):

Cool! that sounds like one of the lemmas that says that 1st order theory of the reals is complete...?

Sam Tenka (May 03 2022 at 23:05):

ah, TarskiSeidenberg is indeed satisfying to learn! https://en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem

Sam Tenka (May 03 2022 at 23:06):

@John Baez you said that "and" and "or" aren't so easy to achieve in chemistry... are they always achievable with enough blood sweat and tears? (maybe the answer is in that paper and I should just read it instead of skimming?)

John Baez (May 03 2022 at 23:40):

I meant I have no idea how to do it. You should definitely read my paper - that's one of the main things we'll be using in our work, I imagine - but the answer is not in there.

Sophie Libkind (May 04 2022 at 03:29):

Now that we have AlgebraicPetri we could easily simulate this... at least if @Sophie Libkind tells us how!

Maybe we could do this at one of our meetings!

John Baez (May 04 2022 at 15:26):

Sophie Libkind said:

Now that we have AlgebraicPetri we could easily simulate this... at least if @Sophie Libkind tells us how!

Maybe we could do this at one of our meetings!

Sure! How would this go? Would it be "we" doing it, or you doing it while we watch? The more you let us help, the longer it will take - like having your kid help you make breakfast. :upside_down: But it would be useful. Maybe we should try to schedule out a time where people can take the time to do this? (I've been avoiding "scheduling" because it's so hard for 14 people.)

Sam Tenka (May 04 2022 at 15:28):

I'm interested in this! I'm pretty good at programming but I haven't played around much with Julia!

Sophie Libkind (May 04 2022 at 22:23):

John Baez said:

Sophie Libkind said:

Now that we have AlgebraicPetri we could easily simulate this... at least if @Sophie Libkind tells us how!

Maybe we could do this at one of our meetings!

Sure! How would this go? Would it be "we" doing it, or you doing it while we watch? The more you let us help, the longer it will take - like having your kid help you make breakfast. :upside_down: But it would be useful. Maybe we should try to schedule out a time where people can take the time to do this? (I've been avoiding "scheduling" because it's so hard for 14 people.)

Haha, that's a great point! Maybe the easiest thing to do is for me to make a notebook with the example and share it. We can walk through it during a meeting if there's interest. That way people who aren't able attend the meetings can still enjoy!

John Baez (May 04 2022 at 22:32):

That sounds great. I'm thinking of having the next meeting on Thursday, just for a change of pace. I don't know if that works for you (or other people).

Reid Barton (May 05 2022 at 18:32):

I'm not sure what this is for but it seems to me that roughly half of these times are in the past

Sam Tenka (May 05 2022 at 18:34):

Reid Barton said:

I'm not sure what this is for but it seems to me that roughly half of these times are in the past

i think John and Sophie above meant a meeting this May 12th

Todd Trimble (May 06 2022 at 02:35):

John Baez said:

Sophie Libkind said:

Now that we have AlgebraicPetri we could easily simulate this... at least if @Sophie Libkind tells us how!

Maybe we could do this at one of our meetings!

Sure! How would this go? Would it be "we" doing it, or you doing it while we watch? The more you let us help, the longer it will take - like having your kid help you make breakfast. :upside_down: But it would be useful. Maybe we should try to schedule out a time where people can take the time to do this? (I've been avoiding "scheduling" because it's so hard for 14 people.)

Happily, the kids come of age and no longer do they need your help, they do it all by themselves. My daughter, aged 18, is an instinctual cook and makes a mean breakfast. Quite delightful! :smile:

Benjamin Merlin Bumpus (he/him) (May 09 2022 at 10:12):

John Baez said:

That sounds great. I'm thinking of having the next meeting on Thursday, just for a change of pace. I don't know if that works for you (or other people).

Thursday the 12th? That would work for me :+1:

John Baez (May 09 2022 at 19:01):

Yes, the 12th.

John Baez (May 20 2022 at 05:58):

Here's an article about now motifs can act like digital circuits:

• John J. Tyson and Béla Novák, Functional motifs in biochemical reaction networks.

But it'd take some work to turn these motifs into Petri nets! I think it should be possible....

Sam Tenka (May 26 2022 at 20:28):

@Sophie Libkind 's very useful collaborative doc on today's "What to Work on" meeting is here:
https://docs.google.com/document/d/1O40zx0dfDVY1lxpW9YK6HwSs7RliQ7NwLBGIeWupMN4/edit

Sam Tenka (Jun 13 2022 at 04:55):

@John Baez --- this was perhaps implicit from the beginning, but I only just now caught on : the category of monically open undirected graphs is a bit like the category of cobordisms between (say, smooth unoriented compact boundaryless dimension-k) manifolds. (In fact, they agree when we restrict focus to only chain graphs, i.e. all nodes either have degree zero and are in X image intersect Y image or degree one and in X image symmetric difference Y image or have degree two and are in neither image) Moreover, functors from monically open undirected graphs are, if we squint, a bit like tqfts --- the domain is cobordism-like, the codomain is number-like (I don't mean that N is a monoid, which algebraists like; that would be level mixing. I mean that N, like the category of finite dimensional hilbert spaces, has a good dimension theory that gives a complete invariant of morphisms).

Is there an analogous cobordism-y view of monically open petri nets? I'd love to think of them with the same "arm" of my brain :laughing:

((btw I like the phrase "monically open petrinet" because it allows the acronym mo.op.pe to condense to "mope". :tongue: ))

John Baez (Jun 13 2022 at 17:48):

Mope nets.

John Baez (Jun 13 2022 at 17:49):

As to your question: there's been a lot of work on the category of open graphs, not using the "monic" condition on the cospan legs of the open graph.

John Baez (Jun 13 2022 at 17:53):

I wrote about it here:

• John Baez, Brandon Coya and Fransciscus Rebro, Props in network theory.

especially near Proposition 27.

John Baez (Jun 13 2022 at 17:56):

Here we note that the category of open graphs (category theorist's graphs: directed multigraphs) with edges labelled by elements of some set $\mathcal{L}$ is the free symmetric monoidal category on a special commutative Frobenius monoid $x$ whose underlying object $x$ has a morphisms $\ell : x \to x$ for each $\ell \in \mathcal{L}$

John Baez (Jun 13 2022 at 17:56):

That's a mouthful, but getting around to your point: the category $2 \mathsf{Cob}$ of 2-dimensional oriented cobordisms is the free symmetric monoidal category on a commutative Frobenius monoid.

John Baez (Jun 13 2022 at 17:57):

So there's a strong similarity!

John Baez (Jun 13 2022 at 17:59):

I could heighten the similarity by ignoring the labels on edges in our open graphs. This is the same as having just one label.

John Baez (Jun 13 2022 at 17:59):

So, the category of open graphs is the free symmetric monoidal category on a special commutative Frobenius monoid $x$ whose underlying object $x$ is equipped with a morphism $\ell : x \to x$.

John Baez (Jun 13 2022 at 18:00):

This morphism is "the edge", and the fact that $\ell^2 \ne \ell$ means that

---o---

is not the same as

---

where o means "vertex".