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In the practical analysis of chemical reaction networks, it is often assumed that no explicit catalysts exist, i.e., that species appear both as a reactant and a product in the same reaction.
In this case, the stoichiometric matrix uniquely identifies the corresponding reaction network (RN), and methods from matrix theory naturally apply. However, catalysis plays an important role, in particular for biochemical reactions. We, therefore, adapt here the concepts derived for autocatalytic cores, the minimal units accounting for the emergence of autocatalysis, to RNs with explicitly catalyzed reactions. In this setting, we confirm that an inspection of the stoichiometric matrix alone is inconclusive concerning the presence and number of autocatalytic cores, and that a more delicate algebraic analysis is required. Nevertheless, this generalization demonstrates that, up to certain subtleties, both the graph and matrix representations of autocatalytic cores are preserved. We additionally show that in the common case of unit stoichiometries (0 and 1), autocatalytic cores containing explicitly catalyzed reactions always have a spanning subgraph that is composed of a single loop with a simple metabolite-to-reaction chord.
