Stoichiometric Reaction Network: Representability and Autocatalysis

Reaction networks (RN) comprise a set $X$ of species and a set $R$ of reactions $Y\to Y'$, each converting a multiset of educts into a a multiset 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. 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. These conditions are 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 the RN implies that they are "obligatory isomers", in which case they always have the same sum formula. 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.

Autocatalysis is a deceptively simple concept, referring to the situation that a chemical species $X$ catalyzes its own formation. Given a RN it is not at all straightforward to identify species that are autocatalytic in the sense that there is a sub-network that takes $X$ as input and produces more than one copy of $X$ as output. The difficulty arises from the need to distinguish autocatalysis e.g. from the superposition of a cycle that consumes and produces equal amounts of $X$ and a pathway that produces $X$. A number of competing notions, such as exclusive autocatalysis and autocatalytic cycles, have been introduced and studied to some extent. However, many questions remain open for future research.