Computational Complexity of Atomic Chemical Reaction Networks

Abstract

Informally, a chemical reaction network is “atomic” if each reaction may be interpreted as the rearrangement of indivisible units of matter. There are several reasonable definitions formalizing this idea. We investigate the computational complexity of deciding whether a given network is atomic according to each of these definitions.

Primitive atomic, which requires each reaction to preserve the total number of atoms, is shown to be equivalent to mass conservation. Since it is known that it can be decided in polynomial time whether a given chemical reaction network is mass-conserving [28], the equivalence we show gives an efficient algorithm to decide primitive atomicity.

Subset atomic further requires all atoms be species. We show that deciding if a network is subset atomic is in $\mathsf {NP}$, and “whether a network is subset atomic with respect to a given atom set” is strongly $\mathsf {NP}$-$\mathsf {complete}$.

Reachably atomic, studied by Adleman, Gopalkrishnan et al. [1, 22], further requires that each species has a sequence of reactions splitting it into its constituent atoms. Using a combinatorial argument, we show that there is a polynomial-time algorithm to decide whether a given network is reachably atomic, improving upon the result of Adleman et al. that the problem is decidable. We show that the reachability problem for reachably atomic networks is $\mathsf {PSPACE}$-$\mathsf {complete}$.

Finally, we demonstrate equivalence relationships between our definitions and some cases of an existing definition of atomicity due to Gnacadja [21].

This work was supported by NSF grant 1619343.