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Non-explosivity of Stochastically Modeled Reaction Networks that are Complex Balanced.

David F Anderson1, Daniele Cappelletti2, Masanori Koyama3

  • 1Department of Mathematics, University of Wisconsin-Madison, Madison, USA.

Bulletin of Mathematical Biology
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PubMed
Summary

Stochastic reaction models are non-explosive if their governing equations decay sufficiently fast relative to reaction rates. This finding is particularly relevant for complex-balanced reaction networks, ensuring their stability and predictable behavior.

Keywords:
Continuous-time Markov chainExplosivityReaction networkStationary distributionStochastic reaction networks

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Area of Science:

  • * Mathematical biology
  • * Computational chemistry
  • * Chemical kinetics

Background:

  • * Stochastic reaction networks are used to model biochemical systems.
  • * Explosivity in these models can lead to computational challenges and unpredictable behavior.
  • * Understanding conditions for non-explosivity is crucial for reliable simulations.

Purpose of the Study:

  • * To establish criteria for non-explosivity in stochastically modeled reaction networks.
  • * To investigate the relationship between the decay rate of the Kolmogorov forward equation and model stability.
  • * To demonstrate that complex-balanced reaction networks are a specific case of non-explosive models.

Main Methods:

  • * Analysis of the Kolmogorov forward equation for stochastic reaction networks.
  • * Derivation of conditions based on the decay rate of the equation's solution.
  • * Application of these conditions to complex-balanced reaction networks.

Main Results:

  • * A sufficient condition for non-explosivity is identified: the decay rate of the Kolmogorov forward equation must be sufficiently fast relative to the transition rates.
  • * This condition guarantees that the system does not explode in finite time.
  • * Complex-balanced reaction networks are proven to satisfy this non-explosivity condition.

Conclusions:

  • * The study provides a clear mathematical criterion for ensuring the non-explosivity of stochastic reaction models.
  • * The findings simplify the analysis of complex-balanced reaction networks, confirming their inherent stability.
  • * This work contributes to the development of more robust and reliable computational tools for systems biology and chemical kinetics.