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A new necessary condition on interaction graphs for multistationarity.

M Kaufman¹, C Soulé, R Thomas

¹Unit of Theoretical and Computational Biology, Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), Campus Plaine, C.P. 231, B-1050 Brussels, Belgium. Marcelle.Kaufman@ulb.ac.be

Journal of Theoretical Biology

|August 8, 2007

Summary

We explore interaction graphs derived from ordinary differential equations to understand system dynamics. Our findings provide new conditions for system instability and multiple steady states in biological networks.

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Area of Science:

Dynamical systems theory
Graph theory
Mathematical biology

Background:

Dynamical systems are modeled by ordinary differential equations.
Interaction graphs represent system dynamics using Jacobian matrices.
Understanding system behavior, like stability and multiple steady states, is crucial.

Purpose of the Study:

To investigate the relationship between circuits in interaction graphs and the dynamic behavior of dynamical systems.
To establish new theoretical conditions for qualitative unstability and the existence of multiple stationary states.
To illustrate these findings with examples from biological regulatory networks.

Main Methods:

Defining interaction graphs from the sign matrix of the Jacobian of ordinary differential equations.
Formulating and proving theorems relating graph circuits to system dynamics.
Analyzing two-variable regulatory modules as case studies.

Main Results:

A sufficient condition for qualitative unstability in dynamical systems was proven.
A necessary condition for the existence of several stationary states was established.
The theoretical results were demonstrated using examples from biological networks.

Conclusions:

Circuits in interaction graphs play a significant role in the dynamic behavior of systems.
The derived conditions offer valuable insights into system stability and multiplicity of states.
This work provides a framework for analyzing complex biological regulatory networks.