1. An Application of Bendixson-Boincare Theorem MAT 574- Fall 2003 Arizona State University Math & Stat Dept

2. 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

3. Central Dogma of Molecular Biology Exon Transcription Poly A SiteTermination Site 5’Cap ExonIntron RNA processing (Cleavage & Polyadenylation) Poly A Splicing Functional mRNA Primary transcript Translation Protein Degradation DNA Function Degradation

4. Two gene of genetic regulatory system. Promoter Gene Q Protein q Promoter Protein P Gene P

6. A Limit cycle generated by a two-gene model.

9. 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.

10. 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.