What makes a reaction network “chemical”?

Abstract

Abstract Background Reaction networks (RNs) comprise a set X of species and a set $$\mathscr {R}$$ R of reactions $$Y\rightarrow Y'$$ Y → Y ′ , each converting a multiset of educts $$Y\subseteq X$$ Y ⊆ X into a multiset $$Y'\subseteq X$$ Y ′ ⊆ X of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix $$\mathbf {S}\in \mathbb {R}^{X\times \mathscr {R}}$$ S ∈ R X × R have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for $$\mathbf {S}$$ S that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively. Results Chemically plausible RNs allow neither a perpetuum mobile, i.e., a “futile cycle” of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix $$\mathbf {S}$$ S . The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and—in a stricter sense—futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of “atoms”. In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless $$\mathbf {S}$$ S implies that they are “obligatory isomers”. In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN. Conclusions “Chemical” RNs are directed hypergraphs with a stoichiometric matrix $$\mathbf {S}$$ S whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain $$\mathbf {S}$$ S by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

Keywords

• Chemical reaction network
• Directed hypergraph
• Stoichiometric matrix
• Futile cycle
• Perpetuum mobile
• Energy conservation