Necessary and sufficient conditions for protocell growth
- 241 Downloads
We consider a generic protocell model consisting of any conservative chemical reaction network embedded within a membrane. The membrane results from the self-assembly of a membrane precursor and is semi-permeable to some nutrients. Nutrients are metabolized into all other species including the membrane precursor, and the membrane grows in area and the protocell in volume. Faithful replication through cell growth and division requires a doubling of both cell volume and surface area every division time (thus leading to a periodic surface area-to-volume ratio) and also requires periodic concentrations of the cell constituents. Building upon these basic considerations, we prove necessary and sufficient conditions pertaining to the chemical reaction network for such a regime to be met. A simple necessary condition is that every moiety must be fed. A stronger necessary condition implies that every siphon must be either fed, or connected to species outside the siphon through a pass reaction capable of transferring net positive mass into the siphon. And in the case of nutrient uptake through passive diffusion and of constant surface area-to-volume ratio, a sufficient condition for the existence of a fixed point is that every siphon be fed. These necessary and sufficient conditions hold for any chemical reaction kinetics, membrane parameters or nutrient flux diffusion constants.
KeywordsProtocell Chemical reaction network Persistence Siphons
Mathematics Subject Classification34A12 34C11 80A30 92B05 92C42
The authors would like to thank Pierre Legrain, Laurent Schwartz and Pierre Plateau for stimulating discussions.
- Busa W, Nuccitelli R (1984) Metabolic regulation via intracellular pH. Am. J. Physiol. Regul. Integr. Comp. Physiol. 246(4):R409–R438Google Scholar
- Érdi P, Tóth J (1989) Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models. Manchester University Press, Manchester, UKGoogle Scholar
- Feinberg, M.: Lectures on chemical reaction networks. Notes of lectures given at the Mathematics Research Center, University of Wisconsin, Madison, WI. https://crnt.osu.edu/LecturesOnReactionNetworks (1979)
- Gunawardena, J.: Chemical reaction network theory for in-silico biologists. Notes available for download at http://vcp.med.harvard.edu/papers/crnt (2003)
- Jensen RH, Woolfolk CA (1985) Formation of filaments by Pseudomonas putida. Appl. Environ. Microbiol. 50(2):364–372Google Scholar
- Molenaar D, van Berlo R, de Ridder D, Teusink B (2009) Shifts in growth strategies reflect tradeoffs in cellular economics. Mol. Syst. Biol. 5(1):323Google Scholar
- Pawłowski PH, Zielenkiewicz P (2004) Biochemical kinetics in changing volumes. Acta Biochim. Pol. 51:231–243Google Scholar
- Schaechter M, Maaløe O, Kjeldgaard NO (1958) Dependency on medium and temperature of cell size and chemical composition during balanced growth of Salmonella typhimurium. Microbiology 19(3):592–606Google Scholar