Rules, hypergraphs, and probabilities: The three-level analysis of chemical reaction systems and other stochastic stoichiometric population processes
Fig 1
The three levels (upper labels) in a Stochastic Stoichiometric Population Process realized as a rule-based system, with chemical reactions as an example.
Rules correspond to reaction mechanisms, in which a context K comprising the active atoms can support two bond configurations that we generically term patterns: a reactant pattern L is converted to a product pattern R by the reaction. The reactant pattern is embedded as a sub-graph in one or more literal molecules G by a map m, and the conversion maps l and r on patterns are used to generate embeddings d and m′, and conversions l′ and r′ to new literal molecules H, so that the remainders of the literal molecules outside the reacting bonds are “carried along” by the mechanism in a structure-preserving way. The second level formed by the action of rules on molecules is, in our treatment, the generator of a stochastic process in the form of a chemical reaction network (CRN) connecting literal molecule types by literal reaction types. The third level is a state space in which collections of the molecules evolve stochastically under the generator. Middle labels give the mathematical structures that express each level. For rules, they are morphisms from category theory; the commutative diagram is known as a double pushout. For reaction networks, the representation is called a multi-hypergraph (because the inputs and outputs may have multiple copies of the same molecule type, and so are termed “multisets”). For state spaces in a population process, the states are points in a lattice where the coordinates count molecule copy-numbers. Each level is connected to the next by a (generally) one-to-many generative relation: mechanisms generate CRNs (both molecules and reactions) through network expansion, in which the same rule may be instantiated in many different reactions. Sets of transitions from the CRN as a generator are embedded in the state space as paths of population states; the same reaction sequence may have indefinitely or infinitely many images through states with different numbers of molecules. The bottom labels give result-types at each level. For rules, they consist of the algebra of dependencies for activation of a rule on patterns created or eliminated by other rules. For reaction networks, they may be integer-flow solutions to a conversion problem. For state spaces, they are stochastically evolving population states, or distributions over states and their transitions that may evolve deterministically under a master equation. Left and middle panels are reproduced from [11], Fig 6.6 and Fig 12.3 respectively, and other terms used here are explained at length in that dissertation.