# State-dependent regulatory networks via sheaves

I was thinking about the point of regulatory networks recently, and an interesting extension presented itself to me.

Regulatory networks capture “first-order” structure in a network of interacting species. I.e., does species A upregulate, downregulate, or have indeterminate effect on species B?

However, such effects may not be constant across all populations levels if there are nonlinearities. For instance, it could be the case that at high proportions of A to B, A upregulates B, but at low proportions of A to B, A downregulates B.

In order to model this, instead of having sets of edges, we could have sheaves of edges. These sheaves would be over the site consisting of the open sets of \mathbb{R}^S_{> 0}, where S is the set of species. Then there would be source and target functions to the constant sheaf of vertices. An edge would exist between two species in a certain open U \subset \mathbb{R}^S_{> 0} if across that regime there was a consistent relationship between the species.

Of course, this does not lend itself to an easily describable model in the way that traditional reg nets do. But the data of such a model could be the output of some procedure that attempted to determine causality across a wide spectrum of different population levels. Then that data could be analyzed for patterns that persisted across many different population levels.

I don’t think I will do anything with this; I just wanted to record it in case it was interesting to anyone.

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Do you think there would be a way to extract a differential equation (or family thereof) from this input?

My idea behind this was less to be a modeling framework, and more to be a result of analysis, i.e. something like computing persistent homology.

Right, this sheaf of regulatory networks is the analysis that you want to do on the system. It’s interesting because the absence of an edge here has a very specific meaning “X and Y do not have a consistent relationship throughout this open”. In the traditional perspective, a missing edge is more like an unknown relationship than a relationship known not to exist.

You can then ask about this sheaf to understand the system. The global sections are those edges that are true in every regime. They are like iron laws of the dynamics.

If the dynamics are given by polynomials, can we actually compute these things?

I don’t really know what the algorithm would be… maybe this could be a job for an SMT solver? I.e. you could maybe phrase the query “does this vertex have a causal relation on this vertex in this regime” in terms of some first order logic problem.

Makes sense, I guess that has a similar feel to the purpose regulatory networks. It took me some time to realize when working on the paper that in some sense the part about turning a Petri net with links into a regnet was more fundamental than turning a regnet into a family of equations, since the regnet is more of a meaningful output than input. It makes me wonder how the compositionality can best be used since a regnet seems very nice to have at the small or medium scale but less so at the large scale where our theory should be most helpful since it’s beyond readability. But maybe then something like homomorphism search is the idea, to find perhaps unexpected patterns in say a large signed category.