Discrepancies between extinction events and boundary equilibria in reaction networks

Area of Science:

• Biochemistry
• Mathematical Biology
• Chemical Kinetics

Background:

• Reaction networks model chemical species interactions using deterministic (differential equations) and stochastic (Markov chains) regimes.
• Connections between deterministic and stochastic models exist, with deterministic models as limits of rescaled stochastic models.
• Long-term behaviors of both models are linked for networks with specific properties like weak reversibility and zero deficiency.

Purpose of the Study:

• To investigate the conjectured links between the long-term behaviors of deterministic and stochastic reaction network models.
• To determine if positive recurrence in stochastic models implies positive equilibria in deterministic models.
• To ascertain if boundary equilibria in deterministic models imply extinction events in stochastic models.

Main Methods:

• Analysis of autonomous systems of differential equations for deterministic models.
• Application of continuous-time Markov chains for stochastic models.
• Examination of specific network properties, including bimolecular reactions, mass conservation, and absolute concentration robustness.

Main Results:

• The study proves that implications between stochastic positive recurrence and deterministic positive equilibria do not hold in general.
• It is demonstrated that boundary equilibria in deterministic models do not necessarily imply extinction events in stochastic models.
• These disproven implications hold even for mass-conserving bimolecular reaction networks.

Conclusions:

• The conjecture linking the long-term behaviors of deterministic and stochastic reaction network models is disproven.
• The findings answer negatively a 2014 literature conjecture regarding these model connections.
• The results highlight limitations in generalizing behaviors between deterministic and stochastic modeling regimes for chemical kinetics.