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Revisiting a synthetic intracellular regulatory network that exhibits oscillations

  • Jonathan TylerEmail author
  • Anne Shiu
  • Jay Walton
Article
  • 52 Downloads

Abstract

In 2000, Elowitz and Leibler introduced the repressilator—a synthetic gene circuit with three genes that cyclically repress transcription of the next gene—as well as a corresponding mathematical model. Experimental data and model simulations exhibited oscillations in the protein concentrations across generations. Müller et al. (J Math Biol 53(6):905–937, 2006) generalized the model to an arbitrary number of genes and analyzed the resulting dynamics. Their new model arose from five key assumptions, two of which are restrictive given current biological knowledge. Accordingly, we propose a new repressilator system that allows for general functions to model transcription, degradation, and translation. We prove that, with an odd number of genes, the new model has a unique steady state and the system converges to this steady state or to a periodic orbit. We also give a necessary and sufficient condition for stability of steady states when the number of genes is even and conjecture a condition for stability for an odd number. Finally, we derive a new rate function describing transcription that arises under more reasonable biological assumptions than the widely used single-step binding assumption. With this new transcription-rate function, we compare the model’s amplitude and period with that of a model with the conventional transcription-rate function. Taken together, our results enhance our understanding of genetic regulation by repression.

Keywords

Repressilator Biological clock Gene regulatory network Hurwitz matrix Hopf bifurcation 

Notes

Acknowledgements

AS thanks Mariano Beguerisse Díaz and Heather A. Harrington for helpful discussions. The authors thank Jake A. Pitt, Ruben Perez-Carrasco, and 2 conscientious referees for their helpful comments and suggestions that helped us improve the work.

References

  1. Allen LJS (2006) An introduction to mathematical biology. Pearson, Upper Saddle RiverGoogle Scholar
  2. Berget SM, Moore C, Sharp PA (1977) Spliced segments at the 5’ terminus of adenovirus 2 late mRNA. Proc Natl Acad Sci USA 74(8):3171–3175CrossRefGoogle Scholar
  3. Cooper GM (2000) The cell: a molecular approach, 2nd edn. Sinauer Associates, SunderlandGoogle Scholar
  4. Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335CrossRefGoogle Scholar
  5. Forger DB (2017) Biological clocks, rhythms, and oscillations: the theory of biological timekeeping. The MIT Press, CambridgezbMATHGoogle Scholar
  6. He Q, Liu Y (2005) Degradation of the neurospora circadian clock protein frequency through the ubiquitin–proteasome pathway. Biochem Soc Trans 33(5):953–956CrossRefGoogle Scholar
  7. Kim JK (2016) Protein sequestration versus Hill-type repression in circadian clock models. IET Syst Biol 10(4):125–135CrossRefGoogle Scholar
  8. Kulaeva OI, Hsieh F-K, Chang H-W, Luse DS, Studitsky VM (2013) Mechanism of transcription through a nucleosome by RNA polymerase II. Biochim Biophys Acta 1829(1):76–83Google Scholar
  9. Lillacci G, Khammash M (2010) Parameter estimation and model selection in computational biology. PLOS Comput Biol 6(3):1–17MathSciNetCrossRefzbMATHGoogle Scholar
  10. Mallet-Paret J, Smith HL (1990) The Poincare–Bendixson theorem for monotone cyclic feedback systems. J Dyn Differ Equ 2(4):367–421MathSciNetCrossRefzbMATHGoogle Scholar
  11. MATLAB version 9.2.0. (2017) Matlab optimization toolbox, R2017a. The MathWorks Inc, Natick, MassachusettsGoogle Scholar
  12. Müller S, Hofbauer J, Endler L, Flamm C, Widder S, Schuster P (2006) A generalized model of the repressilator. J Math Biol 53(6):905–937MathSciNetCrossRefzbMATHGoogle Scholar
  13. Page KM, Perez-Carrasco R (2018) Degradation rate uniformity determines success of oscillations in repressive feedback regulatory networks. J R Soc Interface 15(142):20180157CrossRefGoogle Scholar
  14. Ullrich D (2008) Complex made simple. American Mathematical Society, ProvidenceCrossRefzbMATHGoogle Scholar
  15. Yang X (2002) Generalized form of Hurwitz–Routh criterion and Hopf bifurcation of higher order. Appl Math Lett 15:615–621MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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