What makes a reaction network "chemical"?
Abstract
Reaction networks (RNs) comprise a set $X$ of species and a set $\mathscr{R}$ of reactions $Y\to Y'$, each converting a multiset of educts $Y\subseteq X$ into a multiset $Y'\subseteq X$ 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 $\mathbf{S} \in \mathbb{R}^{X\times\mathscr{R}}$ have been discussed extensively in the literature. Here, we provide sufficient conditions for $\mathbf{S}$ that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively. Chemically plausible RNs allow neither a perpetuum mobile, i.e., a "futile cycle" of reactions with nonvanishing 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 $\mathbf{S}$. 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. Furthermore, it is 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 $\mathbf{S}$ 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.01646
 Bibcode:
 2022arXiv220101646M
 Keywords:

 Quantitative Biology  Molecular Networks;
 Mathematics  Combinatorics;
 Mathematics  Metric Geometry;
 Physics  Chemical Physics