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

Some Results on Global Stability of Messenger RNA–MicroRNA Interaction

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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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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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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Keywords

  • ODE
  • Lyapunov functions
  • Dissipative
  • Global stability
  • Controllability
  • MicroRNA

Mathematics Subject Classification

  • 34D23
  • 37B25
  • 37C10
  • 37C70
  • 37C75