Sufficient conditions for emergent synchronization in protocell models

In this paper, we study general protocell models aiming to understand the synchronization phenomenon of genetic material and container productions, a necessary condition to ensure sustainable growth in protocells and eventually leading to Darwinian evolution when applied to a population of protocells. Synchronization has been proved to be an emergent property in many relevant protocell models in the class of the so-called surface reaction models, assuming both linear- and non-linear dynamics for the involved chemical reactions. We here extend this analysis by introducing and studying a new class of models where the relevant chemical reactions are assumed to occur inside the protocell, in contrast with the former model where the reaction site was the external surface. While in our previous studies, the replicators were assumed to compete for resources, with out any direct interaction among them, we here improve both models by allowing linear interaction between replicators: catalysis and/ or inhibition. Extending some techniques previously introduced, we are able to give a quite general analytical answer about the synchronization phenomenon in this more general context. We also report on results of numerical simulations to support the theory, where applicable, and allow the investigation of cases which are not amenable to analytical calculations.