We determine conditions under which a random biochemical system is likely to contain a subsystem that is both autocatalytic and able to survive on some ambient 'food' source. Such systems have previously been investigated for their relevance to origin-of-life models. In this paper we extend earlier work, by finding precisely the order of catalysation required for the emergence of such self-sustaining autocatalytic networks. This answers questions raised in earlier papers, yet also allows for a more general class of models. We also show that a recently described polynomial-time algorithm for determining whether a catalytic reaction system contains an autocatalytic, self-sustaining subsystem is unlikely to adapt to allow inhibitory catalysation--in this case we show that the associated decision problem is NP-complete.