Instability of the steady state solution in cell cycle population structure models with feedback

  • Balázs Bárány
  • Gregory Moses
  • Todd YoungEmail author


We show that when cell–cell feedback is added to a model of the cell cycle for a large population of cells, then instability of the steady state solution occurs in many cases. We show this in the context of a generic agent-based ODE model. If the feedback is positive, then instability of the steady state solution is proved for all parameter values except for a small set on the boundary of parameter space. For negative feedback we prove instability for half the parameter space. We also show by example that instability in the other half may be proved on a case by case basis.


Yeast metabolic oscillations Temporal clustering Phase synchronization 

Mathematics Subject Classification

34C25 37N25 92D25 



B.B. acknowledges support from the Grants EP/J013560/1 and OTKA K104745. T.Y. was partially supported by the National Science Foundation Grant 1418787. B.B. and T.Y. thank the staff of the Warwick Mathematics Institute for their hospitality while this paper was written.


  1. Bell GI (1968) Cell growth and division III. Conditions for balanced exponential growth in a mathematical model. Biophys. J. 8:431–444CrossRefGoogle Scholar
  2. Bell GI, Anderson EC (1967) Cell growth and division I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures. Biophys. J. 7:329–351CrossRefGoogle Scholar
  3. Boczko EM, Stowers CC, Gedeon T, Young T (2010) ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast. J. Biolog. Dyn. 4:328–345MathSciNetCrossRefGoogle Scholar
  4. Breeden LL (2014) \(\alpha \)-Factor synchronization of budding yeast. Methods Enzymol. 283:332–342CrossRefGoogle Scholar
  5. Breitsch N, Moses G, Young TR, Boczko EM (2015) Cell cycle dynamics: clustering is universal in negative feedback systems. J. Math. Biol. 70(5):1151–1175. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Buckalew R (2014) Cell cycle clustering in a nonlinear mediated feedback model. Discrete Cont. Dyn. Syst. B 19(4):867–881MathSciNetCrossRefGoogle Scholar
  7. Buckalew R, Finley K, Tanda S, Young T (2015) Evidence for internuclear signaling in Drosophila embryogenesis. Dev. Dyn. 244:1014–1021. CrossRefGoogle Scholar
  8. Burnetti AJ, Aydin M, Buchler NE (2016) Cell cycle start is coupled to entry into the yeast metabolic cycle across diverse strains and growth rates. MBoC 27:64–74CrossRefGoogle Scholar
  9. Cohn A (1922) Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Zeit 14:110–148CrossRefGoogle Scholar
  10. Danø S, Madsen MF, Sørensen PG (2007) Quantitative characterization of cell synchronization in yeast. PNAS 104:12732–12736CrossRefGoogle Scholar
  11. Diekmann O, Gyllenberg M, Thieme H, Verduyn Lunel SM (1993a) A cell-cycle model revisited. Centrum for Wiskunde en Informatica, Report AM-R9305, pp 1–18Google Scholar
  12. Diekmann O, Gyllenberg M, Thieme H (1993b) Perturbing semigroups by solving Stieltjes renewal equations. J. Differ. Integral Equ. 6:155–181MathSciNetzbMATHGoogle Scholar
  13. Diekmann O, Heijmans H, Thieme H (1984) On the stability of the cell size distribution. J. Math. Biol. 19:227–248MathSciNetCrossRefGoogle Scholar
  14. Duboc P, Marison L, von Stockar U (1996) Physiology of Saccharomyces cerevisiae during cell cycle oscillations. J. Biotechnol. 51:57–72CrossRefGoogle Scholar
  15. Futcher B (2006) Metabolic cycle, cell cycle and the finishing kick to start. Genome Biol. 7:107–111CrossRefGoogle Scholar
  16. Gong X, Buckalew R, Young T, Boczko E (2014a) Cell cycle dynamics in a response/signaling feedback system with a gap. J. Biol. Dyn. 8:79–98. MathSciNetCrossRefGoogle Scholar
  17. Gong X, Moses G, Neiman A, Young T (2014) Noise-induced dispersion and breakup of clusters in cell cycle dynamics. J. Theor. Biol. 335:160–169. MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hannsgen KB, Tyson JJ (1985) Stability of the steady-state size distribution in a model of cell growth and division. J. Math. Biol. 22:293–301MathSciNetCrossRefGoogle Scholar
  19. Hannsgen KB, Tyson JJ, Watson LT (1985) Steady-state size distributions in probabilistic models of the cell division cycle. SIAM J. Appl. Math. 45(4):523–540MathSciNetCrossRefGoogle Scholar
  20. Heijmans HJAM (1984) On the stable size distribution of populations reproducing by fission into two unequal parts. Math. Biosci. 72:19–50MathSciNetCrossRefGoogle Scholar
  21. Heijmans HJAM (1985) An eigenvalue problem related to cell growth. J. Math. Anal. Appl. 111:253–280MathSciNetCrossRefGoogle Scholar
  22. Kuenzi MT, Fiechter A (1969) Changes in carbohydrate composition and trehalose activity during the budding cycle of Saccharomyces cerevisiae. Arch Microbiol. 64:396–407Google Scholar
  23. Lasota A, Mackey MC (1984) Globally asymptotic properties of proliferating cell populations. J. Math. Biol. 19:43–62MathSciNetCrossRefGoogle Scholar
  24. Moses G (2015) Dynamical systems in biological modeling: clustering in the cell division cycle of yeast. Dissertation, Ohio University, July 2015Google Scholar
  25. Munch T, Sonnleitner B, Fiechter A (1992) The decisive role of the Saccharomyces cervisiae cell cycle behavior for dynamic growth characterization. J. Biotechnol. 22:329–352CrossRefGoogle Scholar
  26. Murray D, Klevecz R, Lloyd D (2003) Generation and maintenance of synchrony in Saccharomyces cerevisiae continuous culture. Exp. Cell. Res. 287:10–15CrossRefGoogle Scholar
  27. Robertson JB, Stowers CC, Boczko EM, Johnson CH (2008) Real-time luminescence monitoring of cell-cycle and respiratory oscillations in yeast. PNAS 105:17988–17993CrossRefGoogle Scholar
  28. Stowers C, Young T, Boczko E (2011) The structure of populations of budding yeast in response to feedback. Hypoth. Life Sci. 1:71–84Google Scholar
  29. Tyson JJ, Hannsgen KB (1985a) The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions. J. Theor. Biol. 113:29–62MathSciNetCrossRefGoogle Scholar
  30. Tyson JJ, Hannsgen KB (1985b) Global asymptotic stability of the size distribution in probabilistic models of the cell cycle. J. Math. Biol. 22:61–68MathSciNetCrossRefGoogle Scholar
  31. Uchiyama K, Morimoto M, Yokoyama Y, Shioya S (1996) Cell cycle dependency of rice \(\alpha \)-amylase production in a recombinant yeast. Biotechnol. Bioeng. 54:262–271CrossRefGoogle Scholar
  32. Young T, Fernandez B, Buckalew R, Moses G, Boczko E (2012) Clustering in cell cycle dynamics with general responsive/signaling feedback. J. Theor. Biol. 292:103–115CrossRefGoogle Scholar
  33. Zietz S (1977) Mathematical modeling of cellular kinetics and optimal control theory in the service of cancer chemotherapy. Dissertation, Department of Mathematics, University of California, BerkeleyGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics InstituteWarwick UniversityCoventryUK
  2. 2.Department of StochasticsBudapest University of Technology and EconomicsBudapestHungary
  3. 3.Mathematics, Ohio UniversityAthensUSA

Personalised recommendations