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 .
Can prelife proceed without cell division? A recently proposed mechanism suggests that transient compartmentalization could have preceded cell division in prebiotic scenarios. Here, we study transient compartmentalization dynamics in the presence of
mutations and noise in replication, as both can be detrimental the survival of compartments. Our study comprises situations where compartments contain uncoupled autocatalytic reactions feeding on a common resource, and systems based on RNA molecules copied by replicases, following a recent experimental study. Using the theory of branching processes, we show analytically that two regimes are possible. In the diffusion-limited regime, replication is asynchronous which leads to a large variability in the composition of compartments. In contrast, in a replication-limited regime, the growth is synchronous and thus the compositional variability is low. Typically, simple autocatalysts are in the former regime, while polymeric replicators can access the latter. For deterministic growth dynamics, we introduce mutations that turn functional replicators into parasites. We derive the phase boundary separating coexistence or parasite dominance as a function of relative growth, inoculation size and mutation rate. We show that transient compartmentalization allows coexistence beyond the classical error threshold, above which the parasite dominates. Our findings invite to revisit major prebiotic transitions, notably the transitions towards cooperation, complex polymers and cell division.
