Journal of Mathematical Biology

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

Necessary and sufficient conditions for protocell growth

  • Erwan BiganEmail author
  • Loïc Paulevé
  • Jean-Marc Steyaert
  • Stéphane Douady


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.


Protocell Chemical reaction network Persistence Siphons 

Mathematics Subject Classification

34A12 34C11 80A30 92B05 92C42 



The authors would like to thank Pierre Legrain, Laurent Schwartz and Pierre Plateau for stimulating discussions.


  1. Angeli D, De Leenheer P, Sontag ED (2007) A Petri net approach to the study of persistence in chemical reaction networks. Math. Biosci. 210(2):598–618MathSciNetzbMATHCrossRefGoogle Scholar
  2. Angeli D, De Leenheer P, Sontag ED (2011) Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws. SIAM J. Appl. Math. 71(1):128–146MathSciNetzbMATHCrossRefGoogle Scholar
  3. Basener W, Brooks BP, Ross D (2006) The Brouwer fixed point theorem applied to rumour transmission. Appl. Math. Lett. 19(8):841–842MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bigan E, Steyaert JM, Douady S (2015a) Minimal conditions for protocell stationary growth. Artif. Life 21(2):166–192CrossRefGoogle Scholar
  5. Bigan E, Steyaert JM, Douady S (2015b) On necessary and sufficient conditions for proto-cell stationary growth. Electron Notes Theor. Comput. Sci. 316:3–15CrossRefGoogle Scholar
  6. Bigan E, Steyaert JM, Douady S (2015c) Chemical schemes for maintaining different compositions across a semi-permeable membrane with application to proto-cells. Orig. Life Evol. Biosph. 45(4):439–454CrossRefGoogle Scholar
  7. Bigan E, Steyaert JM, Douady S (2015d) Filamentation as a primitive growth mode? Phys. Biol. 12(6):066024CrossRefGoogle Scholar
  8. Božič B, Svetina S (2004) A relationship between membrane properties forms the basis of a selectivity mechanism for vesicle self-reproduction. Eur. Biophys. J. 33(7):565–571CrossRefGoogle Scholar
  9. Busa W, Nuccitelli R (1984) Metabolic regulation via intracellular pH. Am. J. Physiol. Regul. Integr. Comp. Physiol. 246(4):R409–R438Google Scholar
  10. Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. 65(5):1526–1546MathSciNetzbMATHCrossRefGoogle Scholar
  11. Craciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J. Appl. Math. 66(4):1321–1338MathSciNetzbMATHCrossRefGoogle Scholar
  12. Dickinson J (2008) Filament formation in Saccharomyces cerevisiae—a review. Folia Microbiol. 53(1):3–14CrossRefGoogle Scholar
  13. Ederer M, Gilles E (2007) Thermodynamically feasible kinetic models of reaction networks. Biophys. J. 92(6):1846–1857CrossRefGoogle Scholar
  14. Érdi P, Tóth J (1989) Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models. Manchester University Press, Manchester, UKGoogle Scholar
  15. Feinberg M (1972) Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49(3):187–194MathSciNetCrossRefGoogle Scholar
  16. Feinberg, M.: Lectures on chemical reaction networks. Notes of lectures given at the Mathematics Research Center, University of Wisconsin, Madison, WI. (1979)
  17. Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Ration. Mech. Anal. 132(4):311–370MathSciNetzbMATHCrossRefGoogle Scholar
  18. Gunawardena, J.: Chemical reaction network theory for in-silico biologists. Notes available for download at (2003)
  19. Himeoka Y, Kaneko K (2014) Entropy production of a steady-growth cell with catalytic reactions. Phys. Rev. E 90(4):042,714CrossRefGoogle Scholar
  20. Hohmann S (2002) Osmotic stress signaling and osmoadaptation in yeasts. Microbiol. Mol. Biol. Rev. 66(2):300–372CrossRefGoogle Scholar
  21. Horn F (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49(3):172–186MathSciNetCrossRefGoogle Scholar
  22. Jahreis K, Pimentel-Schmitt EF, Brückner R, Titgemeyer F (2008) Ins and outs of glucose transport systems in eubacteria. FEMS Microbiol. Rev. 32(6):891–907CrossRefGoogle Scholar
  23. Jensen RH, Woolfolk CA (1985) Formation of filaments by Pseudomonas putida. Appl. Environ. Microbiol. 50(2):364–372Google Scholar
  24. Kondo Y, Kaneko K (2011) Growth states of catalytic reaction networks exhibiting energy metabolism. Phys. Rev. E 84(1):011,927CrossRefGoogle Scholar
  25. Mavelli F, Ruiz-Mirazo K (2013) Theoretical conditions for the stationary reproduction of model protocells. Integr. Biol. 5(2):324–341CrossRefGoogle Scholar
  26. 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
  27. Morgan JJ, Surovtsev IV, Lindahl PA (2004) A framework for whole-cell mathematical modeling. J. Theor. Biol. 231(4):581–596MathSciNetCrossRefGoogle Scholar
  28. Murata T (1989) Petri nets: properties, analysis and applications. Proc. IEEE 77(4):541–580CrossRefGoogle Scholar
  29. Olson AL, Pessin JE (1996) Structure, function, and regulation of the mammalian facilitative glucose transporter gene family. Annu. Rev. Nutr. 16(1):235–256CrossRefGoogle Scholar
  30. Orth JD, Thiele I, Palsson BØ (2010) What is flux balance analysis? Nat. Biotechnol. 28(3):245–248CrossRefGoogle Scholar
  31. Pawłowski PH, Zielenkiewicz P (2004) Biochemical kinetics in changing volumes. Acta Biochim. Pol. 51:231–243Google Scholar
  32. Richeson D, Wiseman J et al (2002) A fixed point theorem for bounded dynamical systems. Ill. J. Math. 46(2):491–495MathSciNetzbMATHGoogle Scholar
  33. 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
  34. Schilling CH, Letscher D, Palsson BØ (2000) Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J. Theor. Biol. 203(3):229–248CrossRefGoogle Scholar
  35. Schlosser PM, Feinberg M (1994) A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions. Chem. Eng. Sci. 49(11):1749–1767CrossRefGoogle Scholar
  36. Schuster S, Hilgetag C (1994) On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst. 2(02):165–182CrossRefGoogle Scholar
  37. Schuster S, Höfer T (1991) Determining all extreme semi-positive conservation relations in chemical reaction systems: a test criterion for conservativity. J. Chem. Soc. Faraday Trans. 87(16):2561–2566CrossRefGoogle Scholar
  38. Scott M, Gunderson CW, Mateescu EM, Zhang Z, Hwa T (2010) Interdependence of cell growth and gene expression: origins and consequences. Science 330(6007):1099–1102CrossRefGoogle Scholar
  39. Stano P, Luisi PL (2010) Achievements and open questions in the self-reproduction of vesicles and synthetic minimal cells. Chem. Commun. 46(21):3639–3653CrossRefGoogle Scholar
  40. Surovstev IV, Morgan JJ, Lindahl PA (2007) Whole-cell modeling framework in which biochemical dynamics impact aspects of cellular geometry. J. Theor. Biol. 244(1):154–166MathSciNetCrossRefGoogle Scholar
  41. Surovtsev IV, Zhang Z, Lindahl PA, Morgan JJ (2009) Mathematical modeling of a minimal protocell with coordinated growth and division. J. Theor. Biol. 260(3):422–429MathSciNetCrossRefGoogle Scholar
  42. Tadmor AD, Tlusty T (2008) A coarse-grained biophysical model of E. coli and its application to perturbation of the rRNA operon copy number. PLoS Comput. Biol. 4(4):e1000,038MathSciNetCrossRefGoogle Scholar
  43. Wei J (1962) Axiomatic treatment of chemical reaction systems. J. Chem. Phys. 36(6):1578–1584CrossRefGoogle Scholar
  44. Weiße AY, Oyarzún DA, Danos V, Swain PS (2015) Mechanistic links between cellular trade-offs, gene expression, and growth. Proc. Natl. Acad. Sci. 112(9):E1038–E1047CrossRefGoogle Scholar
  45. Wittmann C, Hans M, Van Winden WA, Ras C, Heijnen JJ (2005) Dynamics of intracellular metabolites of glycolysis and TCA cycle during cell-cycle-related oscillation in Saccharomyces cerevisiae. Biotechnol. Bioeng. 89(7):839–847CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Erwan Bigan
    • 1
    • 2
    Email author
  • 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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