2022/01/05 by Stefan C. Müller, Müller, Stefan, Christoph Flamm +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #Chemical Physics (physics.chem-ph) #Combinatorics (math.CO) #Computational Drug Discovery Methods #FOS: Biological sciences #FOS: Mathematics #FOS: Physical sciences #Gene Regulatory Network Analysis #Metric Geometry (math.MG) #Molecular Networks (q-bio.MN) #Origins and Evolution of Life
paper · pdf · doi:10.48550/arxiv.2201.01646
openalex publication_date 2022/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Reaction networks (RNs) comprise a set X of species and a set \mathscrR of reactions Y→ Y', each converting a multiset of educts Y⊆ X into a multiset 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 S ∈ ℝ^X×\mathscrR have been discussed extensively in the literature. Here, we provide sufficient conditions for 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 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 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 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.