A key step in the origin of life is the emergence of a primitive metabolism. This requires the formation of a subset of chemical reactions that is both self-sustaining and collectively autocatalytic. A generic theory to study such processes (called RAF theory) has provided a precise and computationally effective way to address these questions, both on simulated data and in laboratory studies. One of the classic applications of this theory (arising from Stuart Kauffmans pioneering work in the 1980s) involves networks of polymers under cleavage and ligation reactions; in the first part of this paper, we provide the first exact description of the number of such reactions under various model assumptions. Conclusions from earlier studies relied on either approximations or asymptotic counting, and we show that the exact counts lead to similar (though not always identical) asymptotic results. In the second part of the paper, we solve some questions posed in more recent papers concerning the computational complexity of some key questions in RAF theory. In particular, although there is a fast algorithm to determine whether or not a catalytic reaction network contains a subset that is both self-sustaining and autocatalytic (and, if so, find one), determining whether or not sets exist that satisfy certain additional constraints exist turns out to be NP-complete.
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