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On attractors in dynamical systems modeling genetic networks

Year 2023, Volume: 7 Issue: 2, 486 - 498, 23.07.2023
https://doi.org/10.31197/atnaa.1248853

Abstract

The system of ordinary differential equations that arises in the theory of genetic networks is studied. Attracting sets of a special kind is the focus of the study. These attractors appear as combinations of attractors of lower dimensions,
which are stable limit cycles. The properties of attractors are studied. Visualizations and examples are provided.

References

  • [1] R. Edwards and L. Ironi. Periodic solutions of gene networks with steep sigmoidal regulatory functions. Physica D, 282 (2014), 1 - 15. https://doi.org/10.1016/j.physd.2014.04.013
  • [2] C. Furusawa, K. Kaneko. A generic mechanism for adaptive growth rate regulation. PLoS Comput Biol 4(1), 2008: e3. doi:10.1371/journal.pcbi.0040003
  • [3] G. van der Heijden. Hopf bifurcation. www.ucl.ac.uk/~ucesgvd/hopf.pdf
  • [4] Le-Zhi Wang, Ri-Qi Su, Zi-Gang Huang, Xiao Wang, Wen-Xu Wang, Grebogi Celso and Ying-Cheng Lai: A geometrical approach to control and controllability of nonlinear dynamical networks. Nature Communications, Volume 7, Article number: 11323 (2016), DOI: 10.1038/ncomms11323
  • [5] S.P. Cornelius, W.L. Kath, A.E. Motter. Realistic control of network dynamic. Nature Communications, Volume 4, Article number: 1942 (2013), 1–9.
  • [6] H.D. Jong. Modeling and Simulation of Genetic Regulatory Systems: A Literature Review, J. Comput Biol. 2002;9(1):67-103, DOI:10.1089/10665270252833208
  • [7] V.W. Noonburg Differential Equations: From Calculus to Dynamical Systems, Providence, Rhode Island: MAA Press, 2019, 2nd edition. [8] L. Perko. Differential Equations and Dynamical Systems, 3rd Edition. [9] F. Sadyrbaev, S. Atslega, E. Brokan. (2020) Dynamical Models of Interrelation in a Class of Artificial Networks. In: Pinelas S., Graef J.R., Hilger S., Kloeden P., Schinas C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham.
  • [10] F. Sadyrbaev, V. Sengileyev, A. Silvans. On Coexistence of Inhibition and Activation in Genetic Regulatory Networks. 19 Intern. Confer. Numer. Analys and Appl. Mathematics, Rhodes, Greece, 20-26 September 2021, To appear in AIP Conference Proceedings.
  • [11] F. Sadyrbaev, I. Samuilik. V. Sengileyev. On Modelling of Genetic Regulatory Networks. WSEAS Trans. Electronics, vol. 12, 2021, 73-80.
  • [12] I. Samuilik, F. Sadyrbaev. Mathematical Modelling of Leukemia Treatment. WSEAS Trans. of Computers, vol. 20, 2021, 274-281.
  • [13] I. Samuilik, F. Sadyrbaev, D. Ogorelova. Mathematical modeling of three-dimensional genetic regulatory networks using logistic and Gompertz functions, WSEAS Trans. on Systems and Control, vol. 17, 2022, 101-107.
  • [14] I. Samuilik, F. Sadyrbaev, V. Sengileyev. Examples of periodic biological oscillators: transition to a six-dimensional system, WSEAS Trans. Computer Research, vol. 10, 2022, 50-54.
  • [15] H.R. Wilson, J.D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J., vol 12 (1), 1972, pp. 1-24.
Year 2023, Volume: 7 Issue: 2, 486 - 498, 23.07.2023
https://doi.org/10.31197/atnaa.1248853

Abstract

References

  • [1] R. Edwards and L. Ironi. Periodic solutions of gene networks with steep sigmoidal regulatory functions. Physica D, 282 (2014), 1 - 15. https://doi.org/10.1016/j.physd.2014.04.013
  • [2] C. Furusawa, K. Kaneko. A generic mechanism for adaptive growth rate regulation. PLoS Comput Biol 4(1), 2008: e3. doi:10.1371/journal.pcbi.0040003
  • [3] G. van der Heijden. Hopf bifurcation. www.ucl.ac.uk/~ucesgvd/hopf.pdf
  • [4] Le-Zhi Wang, Ri-Qi Su, Zi-Gang Huang, Xiao Wang, Wen-Xu Wang, Grebogi Celso and Ying-Cheng Lai: A geometrical approach to control and controllability of nonlinear dynamical networks. Nature Communications, Volume 7, Article number: 11323 (2016), DOI: 10.1038/ncomms11323
  • [5] S.P. Cornelius, W.L. Kath, A.E. Motter. Realistic control of network dynamic. Nature Communications, Volume 4, Article number: 1942 (2013), 1–9.
  • [6] H.D. Jong. Modeling and Simulation of Genetic Regulatory Systems: A Literature Review, J. Comput Biol. 2002;9(1):67-103, DOI:10.1089/10665270252833208
  • [7] V.W. Noonburg Differential Equations: From Calculus to Dynamical Systems, Providence, Rhode Island: MAA Press, 2019, 2nd edition. [8] L. Perko. Differential Equations and Dynamical Systems, 3rd Edition. [9] F. Sadyrbaev, S. Atslega, E. Brokan. (2020) Dynamical Models of Interrelation in a Class of Artificial Networks. In: Pinelas S., Graef J.R., Hilger S., Kloeden P., Schinas C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham.
  • [10] F. Sadyrbaev, V. Sengileyev, A. Silvans. On Coexistence of Inhibition and Activation in Genetic Regulatory Networks. 19 Intern. Confer. Numer. Analys and Appl. Mathematics, Rhodes, Greece, 20-26 September 2021, To appear in AIP Conference Proceedings.
  • [11] F. Sadyrbaev, I. Samuilik. V. Sengileyev. On Modelling of Genetic Regulatory Networks. WSEAS Trans. Electronics, vol. 12, 2021, 73-80.
  • [12] I. Samuilik, F. Sadyrbaev. Mathematical Modelling of Leukemia Treatment. WSEAS Trans. of Computers, vol. 20, 2021, 274-281.
  • [13] I. Samuilik, F. Sadyrbaev, D. Ogorelova. Mathematical modeling of three-dimensional genetic regulatory networks using logistic and Gompertz functions, WSEAS Trans. on Systems and Control, vol. 17, 2022, 101-107.
  • [14] I. Samuilik, F. Sadyrbaev, V. Sengileyev. Examples of periodic biological oscillators: transition to a six-dimensional system, WSEAS Trans. Computer Research, vol. 10, 2022, 50-54.
  • [15] H.R. Wilson, J.D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J., vol 12 (1), 1972, pp. 1-24.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Diana Ogorelova

Felix Sadyrbaev 0000-0001-5074-804X

Inna Samuılık

Early Pub Date August 7, 2023
Publication Date July 23, 2023
Published in Issue Year 2023 Volume: 7 Issue: 2

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