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Chemical Systems with Limit Cycles.

Radek Erban¹, Hye-Won Kang²

¹Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK. erban@maths.ox.ac.uk.

Bulletin of Mathematical Biology

|July 4, 2023

Summary

This summary is machine-generated.

Researchers demonstrate the existence of chemical reaction networks (CRNs) capable of generating an arbitrary number of stable limit cycles. These CRNs can be constructed using low-order reactions, offering insights into complex chemical dynamics.

Keywords:

Chemical reaction networks Limit cycles Mass action kinetics

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Area of Science:

Chemical Kinetics
Systems Biology
Nonlinear Dynamics

Background:

Chemical reaction networks (CRNs) are frequently modeled using systems of ordinary differential equations (ODEs) under mass action kinetics.
These ODE models describe the temporal evolution of chemical species concentrations via polynomial rate laws.
Understanding the oscillatory behavior, specifically stable limit cycles, in CRNs is crucial for comprehending complex biological and chemical systems.

Purpose of the Study:

To investigate the theoretical possibility of constructing CRNs that exhibit an arbitrarily large number of stable limit cycles.
To determine the constraints on reaction order and the number of chemical species required for generating multiple stable limit cycles.

Main Methods:

Theoretical analysis of chemical reaction network dynamics.
Construction of specific CRN models to demonstrate the existence of multiple stable limit cycles.
Mathematical derivation of bounds for the number of species and reactions based on kinetics order and desired limit cycles.

Main Results:

Existence proof for a CRN whose ODE model possesses at least K stable limit cycles for any integer K.
Demonstration that such CRNs can be achieved with at most second-order reactions if the number of species scales linearly with K.
Presentation of bounds on minimal species and reaction counts for CRNs with K stable limit cycles, considering up to seventh-order kinetics.
Finding that CRNs with only two species can exhibit K stable limit cycles if reaction order scales linearly with K.

Conclusions:

It is theoretically possible to design chemical reaction networks capable of generating an arbitrary number of stable oscillations.
The complexity (number of species and reactions) required to achieve multiple limit cycles is dependent on the order of the reactions.
This work provides fundamental insights into the rich dynamical behaviors achievable within the framework of mass action kinetics.