  Autocatalysis underlies the ability of chemical and biochemical systems to
replicate. Recently, Blokhuis et al. gave a stoechiometric definition of
autocatalysis for reaction networks, stating the existence of a combination of
reactions such that the balance for all autocatalytic species is strictly
positive, and investigated minimal autocatalytic networks, called {\em
autocatalytic cores}. By contrast, spontaneous autocatalysis -- namely,
exponential amplification of all species internal to a reaction network,
starting from a diluted regime, i.e. low concentrations -- is a dynamical
property.
  We introduce here a topological condition (Top) for autocatalysis, namely:
restricting the reaction network description to highly diluted species, we
assume existence of a strongly connected component possessing at least one
reaction with multiple products (including multiple copies of a single
species). We find this condition to be necessary and sufficient for
stoechiometric autocatalysis. When degradation reactions have small enough
rates, the topological condition further ensures dynamical autocatalysis,
characterized by a strictly positive Lyapunov exponent giving the instantaneous
exponential growth rate of the system.
  The proof is generally based on the study of auxiliary Markov chains. We
provide as examples general autocatalytic cores of Type I and Type III in the
typology of Blokhuis et al. In a companion article, Lyapunov exponents and the
behavior in the growth regime are studied quantitatively beyond the present
diluted regime .

Molecular Networks