  The emergence of self-sustaining autocatalytic networks in chemical reaction
systems has been studied as a possible mechanism for modelling how living
systems first arose. It has been known for several decades that such networks
will form within systems of polymers (under cleavage and ligation reactions)
under a simple process of random catalysis, and this process has since been
mathematically analysed. In this paper, we provide an exact expression for the
expected number of self-sustaining autocatalytic networks that will form in a
general chemical reaction system, and the expected number of these networks
that will also be uninhibited (by some molecule produced by the system). Using
these equations, we are able to describe the patterns of catalysis and
inhibition that maximise or minimise the expected number of such networks. We
apply our results to derive a general theorem concerning the trade-off between
catalysis and inhibition, and to provide some insight into the extent to which
the expected number of self-sustaining autocatalytic networks coincides with
the probability that at least one such system is present.
