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


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.


ODE Lyapunov functions Dissipative Global stability Controllability MicroRNA 

Mathematics Subject Classification

34D23 37B25 37C10 37C70 37C75 



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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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

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