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Plasmid-derived DNA Strand Displacement Gates for Implementing Chemical Reaction Networks

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Published on: November 25, 2015

Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks.

Maya Mincheva¹, Gheorghe Craciun

¹Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115, United States. mincheva@math.niu.edu

Mathematical Biosciences and Engineering : MBE

|August 3, 2013

Summary

We identified a graph condition that prevents Turing instability in chemical reaction networks. If this condition is met, zero-eigenvalue Turing instability is impossible, aiding in the design of stable reaction-diffusion systems.

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Area of Science:

Chemical kinetics
Mathematical biology
Systems chemistry

Background:

Turing instability drives pattern formation in reaction-diffusion systems.
Zero-eigenvalue Turing instability is a specific type of instability crucial for pattern formation.
Understanding conditions that prevent this instability is vital for predicting and controlling pattern formation.

Purpose of the Study:

To establish a necessary graph-theoretic condition for zero-eigenvalue Turing instability in mass-action kinetics reaction networks.
To provide a method for determining if a given reaction network can exhibit this specific type of Turing instability.
To offer insights into the design principles for chemical reaction networks resistant to Turing instability.

Main Methods:

Utilizing the species-reaction (SR) graph representation for chemical reaction networks.
Analyzing graph-theoretic properties of the SR graph.
Relating graph properties to the eigenvalues of the Jacobian matrix in the corresponding reaction-diffusion system.
Applying the derived condition to a bifunctional enzyme model.

Main Results:

A necessary condition based on the structure of the species-reaction graph is presented for zero-eigenvalue Turing instability.
If the SR graph satisfies specific conditions, zero-eigenvalue Turing instability is ruled out for all parameter values.
If the graph-theoretic condition is not met, the system may exhibit zero-eigenvalue Turing instability for certain parameters.
The condition was successfully illustrated using a bifunctional enzyme model.

Conclusions:

The study provides a powerful graph-theoretic tool to predict the absence of zero-eigenvalue Turing instability in chemical reaction networks.
This condition simplifies the analysis of pattern formation in complex chemical systems.
The findings contribute to the rational design of chemical systems with desired stability properties.