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Stream: deprecated: chemistry

Topic: the hyperfield of signs

John Baez (May 27 2022 at 03:18):

It might be fun to interpret motifs as giving matrices in the hyperfield of signs. This is a number system with 3 elements,

$\mathbb{S} = \{+, 0, - \}$

where you add, multiply, subtract and divide by nonzero elements in the "obvious way", taking into account the fact that addition can be multivalued, namely $\mathbf{+}$ plus $\mathbf{-}$ takes all 3 values, $\mathbf{+}, 0, \mathbf{-}$. This is the only case where we need the multivaluedness.

Benjamin Merlin Bumpus (he/him) (May 27 2022 at 09:55):

Interesting! Just to check I understand, say we have the motif $m := x \vdash y \leftarrow x$ which says that y inhibits x and x promotes y. Would $A_m := \begin{pmatrix} 0 & + \\ - & 0 \end{pmatrix}$ be the $\mathbb{S}$-matrix of $m$ (rows and columns indexed by $(x,y)$)?
I wonder what matrix powers end up meaning here; for example, what does $A_m^2 = \begin{pmatrix} \{0,1-1\} & 0 \\ 0 & \{0,1-1\} \end{pmatrix}$ encode?
Either way, I like the idea :smile:

By the way, I'm assuming that by "motifs" you mean something in the style of the Tyson and Novák paper we looked at yesterday?

EDIT: i just noticed that you liked to matrix representations of the motifs and that these differ from what I wrote above. I'll leave what I wrote unchanged since I think that it's nice to have "non-examples". Anyway, looking at your link, it seems like the correct matrix should have been $A_m := \begin{pmatrix} - & + \\ - & - \end{pmatrix}$. However, the diagonal entries being non-zero is confusing since I'd only expect this to be the case if the motif had self-loop inhibition arrows...

John Baez (May 28 2022 at 00:02):

I linked to the matrix representations of motifs shown in Tyson and Novák paper.

John Baez (May 28 2022 at 00:03):

They say that the diagonal entries are negative - except for chemicals that catalyze their own production - because the chemicals get "used up" as time goes on.

John Baez (May 28 2022 at 00:05):

I don't think taking powers of these matrices is extremely useful in chemistry except insofar as

$\frac{d}{dt} v(t) = A v(t)$

has solution

$v(t) = (1 + tA + \frac{t^2A^2}{2!} + \cdots ) v(0)$

John Baez (May 28 2022 at 00:06):

But maybe I'm just missing some uses of matrix multiplication here.

There should be lots of uses for linear algebra over $\mathbb{S}$, and thus matrix multiplication.

Sam Tenka (Aug 14 2022 at 22:45):

@Evan Patterson could i ask you a bit about the functor between regnets and stockflows you constructed? i think it'd be cool to consider codomains other than Z/2Z to slice over (hence generalizing reg nets and potentially the
stockflow functors). might it be that this holds interest not just because it's a fun knob to turn (tho this is justification enough for me) but because it would allow one to gradually incorporate more and more detail in one's analysis of an actual chemical system? For example, one might tags each arrow with some rate; then one encounters the hyperfield ambiguity John mentions above when trying to mod out back to signs. I believe that an interesting avenue for relating such different levels of detail is to look at forced conclusions rather than possible conclusions --- so {Plus} minus {Plus} evaluates to {} --- as one does in the programming languages topic of refinement types / abstract interpretation. This is includy rather than quotienty.

there are also questions of modularity: to what extent does gluing of two regulatory nets induce a gluing of stockflow diagrams (one can think of this as: to what extent it's a 2 functor or something) --- I believe that this form of exact compostiionality fails (at least in the naive setup I'm using to imagine examples) but that its failure tells us something interesting and isn't necessarily something to fix.