  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.

Molecular Networks;Combinatorics;Populations and Evolution