Rule-based modeling using wildcards

Many biological molecules exist in multiple variants, such as proteins with different post-translational modifications, DNAs with different sequences, and phospholipids with different chain lengths. Representing these variants as distinct species, as most biochemical simulators do, leads to the problem that the number of species, and chemical reactions that interconvert them, typically increase combinatorially with the number of ways that the molecules can vary. This can be alleviated by \"rule-based modeling methods,\" in which software generates the chemical reaction network from relatively simple \"rules.\" This article presents a new approach to rule-based modeling. It is based on wildcards that match to species names, much as wildcards can match to file names in computer operating systems. It is much simpler to use than the formal rule-based modeling approaches developed previously but can also lead to unintended consequences if not used carefully. This article demonstrates rule-based modeling with wildcards through examples for: signaling systems, protein complexation, polymerization, nucleic acid sequence copying and mutation, the \"SMILES\" chemical notation, and others. The method is implemented in Smoldyn, a spatial and stochastic biochemical simulator, for both the generate-first and on-the-fly expansion, meaning whether the reaction network is generated before or during the simulation.