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Qualitative Analysis of an ODE Model of a Class of Enzymatic Reactions

Some Results on Global Stability of Messenger RNA–MicroRNA Interaction

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Abstract

The present paper analyzes an ODE model of a certain class of (open) enzymatic reactions. This type of model is used, for instance, to describe the interactions between messenger RNAs and microRNAs. It is shown that solutions defined by positive initial conditions are well defined and bounded on \([0, \infty )\) and that the positive octant of \({\mathbb {R}}^3\) is a positively invariant set. We prove further that in this positive octant there exists a unique equilibrium point, which is asymptotically stable and a global attractor for any initial state with positive components; a controllability property is emphasized. We also investigate the qualitative behavior of the QSSA system in the phase plane \({\mathbb {R}}^2\). For this planar system we obtain similar results regarding global stability by using Lyapunov theory, invariant regions and controllability.

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References

  • Al-Radhawi M, Angeli D (2016) New approach to the stability of chemical reaction networks: piecewise linear in rates Lyapunov functions. IEEE Trans Autom Control 61(1):76–89

    Article  MathSciNet  MATH  Google Scholar 

  • Belgacem I, Gouze J (2012) Global stability of full open reversible Michaelis-Menten reactions. In: Proceedings of the 8th IFAC symposium on advanced control of chemical processes, Furama Riverfront, Singapore

  • Cheng K, Hsu S, Lin S (1981) Some results on global stability of a predator-prey system. J Math Biol 12:115–126

    Article  MathSciNet  MATH  Google Scholar 

  • Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J Appl Math 65(5):1526–1546

    Article  MathSciNet  MATH  Google Scholar 

  • Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132:311–370

    Article  MathSciNet  MATH  Google Scholar 

  • Figliuzzi M, Marinari E, De Martino A (2013) MicroRNAs as a selective channel of communication between competing RNAs: a steady-state theory. Biophys J 104:1203–1213

    Article  Google Scholar 

  • Flach E, Schnell S (2006) Use and abuse of the quasi-steady-state approximation. IEE Proc Syst Biol 153(4):187–191

    Article  Google Scholar 

  • Flach E, Schnell S (2010) Stability of open pathways. Math Biosci 228(2):147–152

    Article  MathSciNet  MATH  Google Scholar 

  • Giza DE, Vasilescu C, Calin GA (2014) MicroRNAs and ceRNAs: therapeutic implications of RNA networks. Expert Opin Biol Ther 14(9):1285–93

    Article  Google Scholar 

  • Halanay A, Răsvan V (1993) Application of Liapunov methods in stability. Springer, Dordrecht

    Book  MATH  Google Scholar 

  • Hale J (1988) Asymptotic behavior of dissipative systems, vol 25. AMS, Providence

    MATH  Google Scholar 

  • Hartman P (1964) Ordinary differential equations. Wiley, New York

    MATH  Google Scholar 

  • Hirsch M, Smale S, Devaney R (2004) Differential equations, dynamical systems, and an introduction to chaos. Elsevier, New York

    MATH  Google Scholar 

  • Korobeinikov A (2004) Lyapunov functions and global properties for seir and seis epidemic models. Math Med Biol 21:75–83

    Article  MATH  Google Scholar 

  • Ndiaye I, Gouze J (2012) Global stability of reversible enzymatic metabolic chains. Acta Biotheor 61(1):41–57

    Article  Google Scholar 

  • Rao S (2016) Global stability of a class of futile cycles. J Math Biol. https://doi.org/10.1007/s00285-016-1039-8

    Google Scholar 

  • Rao S, van der Schaft A, Jayawardhana B (2013) Stability analysis of chemical reaction networks with fixed boundary concentrations. In: Proceedings of 52nd IEEE Conference on Decision and Control (CDC), Florence, USA

  • Răsvan V, Ştefan R (2007) Systemes non lineaires. Hermes Science, Lavoisier, Paris

    Google Scholar 

  • Smith H (1995) Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. AMS, Providence, Rhode Island

    MATH  Google Scholar 

  • Vasilescu C, Olteanu M, Flondor P, Calin G (2013) Fractal-like kinetics of intracellular enzymatic reactions: a chemical framework of endotoxin tolerance and a possible non-specific contribution of macromolecular crowding to cross-tolerance. Theor Biol Med Model. https://doi.org/10.1186/1742-4682-10-55

    Google Scholar 

  • Vasilescu C, Dragomir M, Tanase M, Giza DE, Purnichescu-Purtan R, Chen M, Yeung SC, Calin GA (2017) Circulating miRNAs in sepsis–a network under attack: an in-silico prediction of the potential existence of miRNA sponges in sepsis. PLoS ONE 12(8):e0183334. https://doi.org/10.1371/journal.pone.0183334

    Article  Google Scholar 

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Acknowledgements

We thank Dr. Catalin Vasilescu, Professor at the Carol Davila University for Medicine and Pharmacy Bucharest, for introducing us into the subject and for valuable discussions.

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Author notes

  1. Paul Flondor, Mircea Olteanu and Radu Stefan hereby state that they have equally contributed to this work.

Authors and Affiliations

  1. Department of Mathematical Methods and Models, University Politehnica of Bucharest, Bucharest, Romania

    Paul Flondor & Mircea Olteanu

  2. Department of Automatic Control and Systems Engineering, University Politehnica of Bucharest, Bucharest, Romania

    Radu Ştefan

Authors
  1. Paul Flondor
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  2. Mircea Olteanu
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  3. Radu Ştefan
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Correspondence to Radu Ştefan.

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Flondor, P., Olteanu, M. & Ştefan, R. Qualitative Analysis of an ODE Model of a Class of Enzymatic Reactions. Bull Math Biol 80, 32–45 (2018). https://doi.org/10.1007/s11538-017-0360-8

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  • DOI: https://doi.org/10.1007/s11538-017-0360-8

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