Advertisement

Bulletin of Mathematical Biology

, Volume 80, Issue 1, pp 32–45 | Cite as

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

Some Results on Global Stability of Messenger RNA–MicroRNA Interaction
Original Article
  • 156 Downloads

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.

Keywords

ODE Lyapunov functions Dissipative Global stability Controllability MicroRNA 

Mathematics Subject Classification

34D23 37B25 37C10 37C70 37C75 

Notes

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.

References

  1. 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):76–89MathSciNetCrossRefMATHGoogle Scholar
  2. 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, SingaporeGoogle Scholar
  3. Cheng K, Hsu S, Lin S (1981) Some results on global stability of a predator-prey system. J Math Biol 12:115–126MathSciNetCrossRefMATHGoogle Scholar
  4. Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J Appl Math 65(5):1526–1546MathSciNetCrossRefMATHGoogle Scholar
  5. Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132:311–370MathSciNetCrossRefMATHGoogle Scholar
  6. Figliuzzi M, Marinari E, De Martino A (2013) MicroRNAs as a selective channel of communication between competing RNAs: a steady-state theory. Biophys J 104:1203–1213CrossRefGoogle Scholar
  7. Flach E, Schnell S (2006) Use and abuse of the quasi-steady-state approximation. IEE Proc Syst Biol 153(4):187–191CrossRefGoogle Scholar
  8. Flach E, Schnell S (2010) Stability of open pathways. Math Biosci 228(2):147–152MathSciNetCrossRefMATHGoogle Scholar
  9. Giza DE, Vasilescu C, Calin GA (2014) MicroRNAs and ceRNAs: therapeutic implications of RNA networks. Expert Opin Biol Ther 14(9):1285–93CrossRefGoogle Scholar
  10. Halanay A, Răsvan V (1993) Application of Liapunov methods in stability. Springer, DordrechtCrossRefMATHGoogle Scholar
  11. Hale J (1988) Asymptotic behavior of dissipative systems, vol 25. AMS, ProvidenceMATHGoogle Scholar
  12. Hartman P (1964) Ordinary differential equations. Wiley, New YorkMATHGoogle Scholar
  13. Hirsch M, Smale S, Devaney R (2004) Differential equations, dynamical systems, and an introduction to chaos. Elsevier, New YorkMATHGoogle Scholar
  14. Korobeinikov A (2004) Lyapunov functions and global properties for seir and seis epidemic models. Math Med Biol 21:75–83CrossRefMATHGoogle Scholar
  15. Ndiaye I, Gouze J (2012) Global stability of reversible enzymatic metabolic chains. Acta Biotheor 61(1):41–57CrossRefGoogle Scholar
  16. 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
  17. 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, USAGoogle Scholar
  18. Răsvan V, Ştefan R (2007) Systemes non lineaires. Hermes Science, Lavoisier, ParisGoogle Scholar
  19. Smith H (1995) Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. AMS, Providence, Rhode IslandMATHGoogle Scholar
  20. 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
  21. 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 CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of Mathematical Methods and ModelsUniversity Politehnica of BucharestBucharestRomania
  2. 2.Department of Automatic Control and Systems EngineeringUniversity Politehnica of BucharestBucharestRomania

Personalised recommendations