An Application of
Bendixson-Boincare
Theorem
MAT 574- Fall 2003
Arizona State University
Math & Stat Dept
What is a gene network and its
mathematical model?
What is a gene network?


A network which consists of
mutual interactions between
products of gene and other
chemicals in a cell.
Nonlinearity, delay and
noise.
Why mathematical model?


Understanding dynamical
aspect of inner cellular
phenomena.
Modeling some gene
networks realistically as a
huge amount of
experimental data has been
accumulate
Central Dogma of Molecular Biology
Poly A Site
Termination Site
Exon
DNA
Exon
Intron
Transcription
5’Cap
RNA
processing (Cleavage &
Polyadenylation)
Poly A
Primary transcript
Splicing
Functional
Translation
Degradation
Protein
Function
Degradation
mRNA
Two gene of genetic regulatory
system.
Protein P
Promoter
Gene Q
Protein q
Promoter
Gene P
A Limit cycle generated by a two-gene model.
What does that mean?

The periodic jumping dynamics may exist robustly,
provided that there are two different time scale
dynamics in gene-protein network.

This model act as a genetic oscillator or switch is
robust for parameter variations or environment
variations.
References

Chen,L and Aihara .K. A Model of Periodic Oscillation
for Genetic Regulatory Systems, IEEE Trans. On
Circuits and Systems, Part-I, Vol.49, 2002.

Chen,L and Aihara .K. Stability of Genetic Regulatory
Networks with Time Delay, IEEE Trans. On Circuits
and Systems, Part-I, Vol.49, 2002.