In Search of Chaos in Genetic Systems

Abstract

A three-dimensional multiparametric system of ordinary differential equations, arising in the theory of genetic networks, is considered. The examples of chaotic behavior are constructed using the methodology by Shilnikov. This methodology requires the existence of a saddle-focus points satisfying some additional conditions. As the result, reach dynamical behavior of solutions can be observed, including chaotic behavior of solutions.

Keywords

• Mathematical model
• Dynamical system
• Attractors
• Genetic regulatory networks

References

1. Barbuti, R., Gori, Milazzo, and Nasti, 2020 A survey of gene regulatory networks modelling methods: from differential equations, to Boolean and qualitative bioinspired models. Journal of Membrane Computing 2: 207–226.
2. Brokan, E. and F. Sadyrbaev, 2016 On attractors in gene regulatory systems. AIP Conf. Proc. 1809, 020010 .
3. Brokan, E. and F. Sadyrbaev, 2018 Attraction in n-dimensional differential systems from network regulation theory. Mathematical Methods in the Applied Sciences 41: 7498–7509.
4. Das, A., P. Das, and A. Roy, 2000 Chaos In A Three-Dimensional General Model Of Neural Network. Applied Mathematical Modelling pp. 511–522.
5. Das, A., P. Das, and A. Roy, 2002 Chaos In A Three-Dimensional General Model Of Neural Network. International Journal of Bifurcation and Chaos 12: 2271–2281.
6. Deng, B., M. Han, and S.-B. Hsu, 2017 Numerical Proof For Chemostat Chaos of Shilnikov Type. Chaos 27.
7. Gonchenko, S., A.Gonchenko, Kazakov, Kozlov, and Bakhanova, 2019 Spiral chaos of three-dimensional flows. Izvestija vuzov 27: 7–52.
8. Ibraheem and K. Raied, 2022 Generating a Novel Chaotic System by Coupling (Rossler-Chen) Systems. Research Square . Jong, D., 2002 Modeling and simulation of genetic regulatory systems. Journal of Computational Biology 9: 67–103.

9. Kardynska, M., D. Kogut, M. Pacholczyk, and J. Smieja, 2023 Mathematical modeling of regulatory networks of intracellular processes. Computational and Structural Biotechnology Journal 21: 1523–1532.
10. Kozlovska, O. and F. Sadyrbaev, 2022 Models of genetic networks with given properties. Transactions on Computer Research 10: 43–49.
11. Magnitskii, N. and Sidorov, 2006 New methods for chaotic dynamics. World Scientific . Ogorelova, D., F. Sadyrbaev, and V. Sengileyev, 2020 Control in Inhibitory Genetic Regulatory Network Models. Contemporary Mathematics 1: 393–400.
12. Peter, I., 2020 The function of architecture and logic in developmental gene regulatory networks. Current Topics in Developmental Biology, Academic Press 139: 267–295.
13. Saeed, N. A., H. A. Saleh, W. A. El-Ganaini, M. Kamel, and M. S. Mohamed, 2023 On a new three-dimensional chaotic system with adaptive control and chaos synchronization. Shock and Vibration pp. 1–19.
14. Samuilik, I., 2022 Genetic engineering construction of a network of four dimensions with a chaotic attractor. Vibroengineering Procedia 44: 66–70.
15. Samuilik, I. and F. Sadyrbaev, 2023 On trajectories of a system modeling evolution of genetic networks. Mathematical Biosciences and Engineering 20: 2232–2242.
16. Sandri, M., 1996 Numerical calculation of Lyapunov exponents. Mathematica Journal 6: 78–84.
17. Santillan, M., 2008 On the Use of the Hill Functions in Mathematical Models of Gene Regulatory Networks. Mathematical Modelling of Natural Phenomena 3: 85–97.
18. Schlitt, T., 2013 Approaches to Modeling Gene Regulatory Networks: A Gentle Introduction. In Silico Systems Biology p. 13–35.
19. Shilnikov, L., 1965 A case of the existence of a denumerable set of periodic motions. Doklady Akademii Nauk SSSR, 1965 160: 558–561.
20. Sprott, J., 2010 Elegant Chaos.World Scientific . Vijesh, N., S. K.Chakrabarti, and J. Sreekumar, 2013 Modeling of gene regulatory networks. Biomedical Science and Engineering pp. 223–231.
21. Zhang, Z., Y.Weiming, and et al., 2012 Chaotic motifs in gene regulatory networks. PLOS ONE 7: 1–11.