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 ofdetailed protocell hypotheses. These models describe the coupledgrowth of the lipid container and of the self-replicating molecules.A technique to analyze the dynamics of populations of such protocellsis described, which couples a continuous-time formalism for thegrowth between two successive cell divisions, and a discrete mapthat relates the quantity of self-replicating molecules in successivegenerations. This technique allows one to derive several properties inan analytical way. It is shown that, under fairly general assumptions,the two growth rates synchronize, so that the lipid container doublesits size when the number of self-replicating molecules has alsodoubled—thus giving rise to exponential growth of the populationof protocells. Such synchronization had been postulated a priori inprevious models of protocells; here it is an emergent property. We alsocompare the rate of duplication of two populations generated by twodifferent protocells with different kinds of self-replicating molecules,considering the interesting case where the rate of self-replicationof one kind is higher than that of the other, but its contribution tothe container growth rate is smaller. It is shown that in this case thepopulation of offspring of the protocell with the faster-replicatingmolecule will eventually grow faster than the other. The case wheretwo different types of self-replicating monomers are present inthe same protocell is also analyzed, and it is shown that, if thereplication follows a first-order kinetic equation, then the fasterreplicator eventually displaces the slower one, whereas if the growthis sublinear the two coexist. It is also proven by an appropriaterescaling of time that the results that concern the system asymptoticdynamics hold both for micelles and vesicles.