Nets with Mana: A Framework for Chemical Reaction Modelling

TL;DR

This paper introduces a categorical framework for modeling chemical reactions using Petri nets with limited transition firing, called mana, to better represent enzyme-dependent biochemical processes.

Contribution

It develops a new categorical Petri net model with mana, proves their equivalence through categorical constructions, and generalizes to include regeneration of mana, capturing catalytic and biochemical interactions.

Findings

Categorical Petri nets with mana model enzyme-dependent reactions.

Equivalence of two categorical constructions for these nets.

Generalization to mana regeneration representing catalysts.

Abstract

We use categorical methods to define a new flavor of Petri nets where transitions can only fire a limited number of times, specified by a quantity that we call mana. We do so with chemistry in mind, looking at ways of modelling the behavior of chemical reactions that depend on enzymes to work. We prove that such nets can be either obtained as a result of a comonadic construction, or by enriching them with extra information encoded into a functor. We then use a well-established categorical result to prove that the two constructions are equivalent, and generalize them to the case where the firing of some transitions can "regenerate" the mana of others. This allows us to represent the action of catalysts and also of biochemical processes where the byproducts of some chemical reaction are exactly the enzymes that another reaction needs to work.