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An Application of Bendixson-Boincare Theorem

This article discusses the concept of gene networks and their mathematical modeling, focusing on nonlinearity, delay, and noise. It explores the importance of understanding the dynamical aspects of cellular phenomena and how gene networks can be modeled using experimental data. The article also introduces the Central Dogma of Molecular Biology and describes the various processes involved in gene expression. The role of genetic regulatory systems and the generation of limit cycles in a two-gene model are explored. The robustness of the model as a genetic oscillator or switch is discussed in relation to parameter and environmental variations. References to relevant research papers are provided.

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An Application of Bendixson-Boincare Theorem

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

  2. 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 What is a gene network and its mathematical model?

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

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

  5. A Limit cycle generated by a two-gene model. q A Slow manifold B fast manifold E fast manifold Equilibruim locus of slow system fast manifold c D Slow manifold p

  6. 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.

  7. 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.

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