Artificial chemistries are mainly used to construct virtual systems that are expected to show behavior similar to living systems. In this study, we explore possibilities of applying an artificial chemistry to modeling natural biochemical systems—or, to be specific, molecular computing systems—and show that it may be a useful modeling tool for molecular computation. We previously proposed an artificial chemistry based on string pattern matching and recombination. This article first demonstrates that this artificial chemistry is computationally universal if it has only rules that have one reactant or two reactants. We think this is a good property of an artificial chemistry that models molecular computing, because natural elementary chemical reactions, on which molecular computing is based, are mostly unimolecular or bimolecular. Then we give two illustrative example models for DNA computing in our artificial chemistry: one is for the type of computation called the Adleman-Lipton paradigm, and the other is for a DNA implementation of a finite automaton. Through the construction of these models we observe preferred properties of the artificial chemistry for modeling molecular computing, such as having no spatial structure and being flexible in choosing levels of abstraction.