Emergence of protocellular growth laws

Template-directed replication is known to obey a parabolic growth law due to product inhibition (Sievers & Von Kiedrowski 1994 Nature 369 , 221; Lee et al . 1996 Nature 382 , 525; Varga & Szathmáry 1997 Bull. Math. Biol . 59 , 1145). We investigate a template-directed replication with a coupled template catalysed lipid aggregate production as a model of a minimal protocell and show analytically that the autocatalytic template–container feedback ensures balanced exponential replication kinetics; both the genes and the container grow exponentially with the same exponent. The parabolic gene replication does not limit the protocellular growth, and a detailed stoichiometric control of the individual protocell components is not necessary to ensure a balanced gene–container growth as conjectured by various authors (Gánti 2004 Chemoton theory ). Our analysis also suggests that the exponential growth of most modern biological systems emerges from the inherent spatial quality of the container replication process as we show analytically how the internal gene and metabolic kinetics determine the cell population's generation time and not the growth law (Burdett & Kirkwood 1983 J. Theor. Biol . 103 , 11–20; Novak et al . 1998 Biophys. Chem . 72 , 185–200; Tyson et al . 2003 Curr. Opin. Cell Biol . 15 , 221–231). Previous extensive replication reaction kinetic studies have mainly focused on template replication and have not included a coupling to metabolic container dynamics (Stadler et al . 2000 Bull. Math. Biol . 62 , 1061–1086; Stadler & Stadler 2003 Adv. Comp. Syst . 6 , 47). The reported results extend these investigations. Finally, the coordinated exponential gene–container growth law stemming from catalysis is an encouraging circumstance for the many experimental groups currently engaged in assembling self-replicating minimal artificial cells (Szostak 2001 et al . Nature 409 , 387–390; Pohorille & Deamer 2002 Trends Biotech . 20 123–128; Rasmussen et al . 2004 Science 303 , 963–965; Szathmáry 2005 Nature 433 , 469–470; Luisi et al . 2006 Naturwissenschaften 93 , 1–13). 1