Theoretical conditions for the stationary reproduction of model protocells

In previous works we have explored the dynamics of chemically reacting proto-cellular systems, under different experimental conditions and kinetic parameters, by means of our stochastic simulation platform 'ENVIRONMENT'. In this paper we, somehow, turn the question around: accepting some broad modeling assumptions, we investigate the conditions under which simple protocells will spontaneously settle into a stationary reproducing regime, characterized by a regular growth/division cycle and the maintenance of a certain standard size and chemical composition across generations. In the first part, starting from purely geometric considerations, the condition for stationary reproduction of a protocell will be expressed in terms of a growth control coefficient (gamma). Then, an explicit relationship, the osmotic synchronization condition, will be analytically derived under a set of kinetic simplifications and taking into account the osmotic pressure balance operating across the protocell membrane. In the second part of the paper, this general condition that constrains different molecular/kinetic parameters and features of the system (reaction rates, permeability coefficients, metabolite concentrations, system volume) will be applied to different cases of self-producing vesicles, predicting the stationary protocell size or lifetime. Finally, in order to test the validity of our analytic results and predictions, the case study is contrasted with data obtained through both stochastic and deterministic computational algorithms.