A renewal model for the superconcentration effect

Many efforts in the origin of life research focus on the under- standing of the very first steps that led to the emergence of organic compounds and metabolic pathways in the pre-biotic solution. A less investigated issue regards those crucial evolutionary steps leading to the formation of membrane compartments, which came into play as hosts for the first forms of cellular metabolism. Luisi et al. tackled this issue experimentally, studying the behaviour of the solutes in a water solution of ferritine proteins and lipids. In this setting, the lipid molecules, due to hydrophobic forces, tend to organize spontaneously in quasi-spherical structures, called liposomes or vesicles. These structures, while are forming but still open, allow the random passage of ferritine molecules. Surprisingly, the authors found that, when lipid surfaces close up in a protein-containing solution to form vesicles, the entrapment frequency does not follow the expected Poisson distribution, but tends to assume a power-law behaviour, characterized by many empty vesicles (no or very little entrapped ferritine solute), and a long decreasing tail with extremely crowded vesicles. This is referred to as "Supercon- centration Effect". Consequently, the experimental finding of Luisi et al. proves that liposomes can spontaneously capture a very high number of macromolecular solutes, even when forming in diluted solutions. This observation overcomes one of the major problems in prebiotic chemistry, i.e., how intravesicular solutes can spontaneously reach the relatively high concentrations needed for the metabolic processes to occur. As a result of this intriguing behavior, these solute-rich vesicles could be able to facilitate and support the emergence of a cellular metabolism, due to the high molecular concentration. As the Poisson assumption does not apply, in this work we propose a stochastic model based on renewal theory [Cox], describing independent critical events randomly occurring in time. Waiting Times (WTs) among events are then mutually independent and they are only characterized by the WT distribution. We are here interested in describing the Gibbs ensemble of liposomes, with the critical event corresponding to the closure of the liposome membrane, after which no flux of ferritine proteins can occur on the liposome surface. We show that simple assumptions about the liposome-ferritine interactions, can explain the emergence of a power-law decay in the distribution of ferritine molecules trapped inside the liposomes, thus sheding light on the role of renewal theory in the emergence of self-organized macro-structures in pre-biotic systems. [l] Luisi et al, ChemBioChem 11, 1989 (2010). [2] D.R. Cox, Renewal Theory Ed.: Methuen, London (1962). [3] Paradisi et al, AIP Conf. Proc. 1510, 151 (2013).