  Over the last years, analyses performed on a stochastic model of catalytic
reaction networks have provided some indications about the reasons why wet-lab
experiments hardly ever comply with the phase transition typically predicted by
theoretical models with regard to the emergence of collectively
self-replicating sets of molecule (also defined as autocatalytic sets, ACSs), a
phenomenon that is often observed in nature and that is supposed to have played
a major role in the emergence of the primitive forms of life. The model at
issue has allowed to reveal that the emerging ACSs are characterized by a
general dynamical fragility, which might explain the difficulty to observe them
in lab experiments. In this work, the main results of the various analyses are
reviewed, with particular regard to the factors able to affect the generic
properties of catalytic reactions network, for what concerns, not only the
probability of ACSs to be observed, but also the overall activity of the
system, in terms of production of new species, reactions and matter.
