Journal of Mathematical Biology

, Volume 73, Issue 6–7, pp 1627–1664 | Cite as

Necessary and sufficient conditions for protocell growth

  • Erwan Bigan
  • Loïc Paulevé
  • Jean-Marc Steyaert
  • Stéphane Douady
Article

Abstract

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.

Keywords

Protocell Chemical reaction network Persistence Siphons 

Mathematics Subject Classification

34A12 34C11 80A30 92B05 92C42 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Erwan Bigan
    • 1
    • 2
  • Loïc Paulevé
    • 3
  • Jean-Marc Steyaert
    • 1
  • Stéphane Douady
    • 2
  1. 1.Laboratoire d’InformatiqueÉcole PolytechniquePalaiseauFrance
  2. 2.Laboratoire Matière et Systèmes ComplexesUniversité Paris DiderotParisFrance
  3. 3.Laboratoire de Recherche en InformatiqueCNRS and Université Paris SudOrsayFrance

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