Synchronization phenomena in surface-reaction models of protocells

A class of generic models of protocells is introduced, which are inspired by the Los Alamos bug hypothesis but which, due to their abstraction level, can be applied to a wider set of detailed protocell hypotheses. These models describe the coupled growth of the lipid container and of the self-replicating molecules. A technique to analyze the dynamics of populations of such protocells is described, which couples a continuous-time formalism for the growth between two successive cell divisions, and a discrete map that relates the quantity of self-replicating molecules in successive generations. This technique allows one to derive several properties in an analytical way. It is shown that, under fairly general assumptions, the two growth rates synchronize, so that the lipid container doubles its size when the number 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; here it is an emergent property. We also compare the rate of duplication of two populations generated by two different protocells with different kinds of self-replicating molecules, considering the interesting case where the rate of self-replication of one kind is higher than that of the other, but its contribution to the container growth rate is smaller. It is shown that in this case the population of offspring of the protocell with the faster-replicating molecule will eventually grow faster than the other. The case where two different types of self-replicating monomers are present in the same protocell is also analyzed, and it is shown that, if the replication follows a first-order kinetic equation, then the faster replicator eventually displaces the slower one, whereas if the growth is sublinear the two coexist. It is also proven by an appropriate rescaling of time that the results that concern the system asymptotic dynamics hold both for micelles and vesicles.