Surface-reaction models of protocells

We present a class of models aiming to describe generic protocells hypotheses, improving a model introduced elsewhere[13]. These models, inspired by the 'Los Alamos bug' hypothesis, are composed by two coupled subsystems: a self-replicating molecule-SRM- and a lipid container. The latter grows thanks to the replication of the former, which in turn can produce copies of itself thanks to the very existence of the lipid container, as it is assumed that SRMs are preferentially found in the lipid phase. Nevertheless, due to abstraction level of our models, they can be applied to a wider set of detailed protocell hypotheses. It can thus be shown that, under fairly general assumptions of generic non-linear growth law for the container and replication for the SRM, the two growth rates synchronize, so that the lipid container doubles its size when the quantity of self-replicating molecules has also doubled ' thus giving rise to exponential growth of the population of protocells. Such synchronization had been postulated a priori in previous models of protocells, while it is here an emergent property. Our technique, combining a continuous-time formalism, for the growth between two successive protocell divisions, and a discrete map, relating the quantity of self-replicating molecules in successive generations, allows one to derive several properties in an analytical way.