Phase portraits of two gene networks models

  • Golubyatnikov Vladimir P., vladimir.golubyatnikov1@fulbrightmail.org Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
  • Kirillova Nataliya E., kne@math.nsc.ru Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia
Keywords: gene networks models, non-linear dynamical systems, phase portrait, equilibrium point, stability, Vyshnegradskii criterion

Abstract

We construct mathematical models of functioning of two few-components gene networks which regulate circadian rhythmes in organisms by means of combinations of positive and negative feedbacks between components of these networks. Both models are represented in the form of non-linear dynamical systems of biochemical kinetics. It is shown that the phase portraits of both models contain exactly one equilibrium point each and in both cases for all values of parameters of these dynamical systems, eigenvalues of their linearization matrices at their equilibrium points are either negative or have negative real parts. Thus, these equilibrium points are stable. We construct their invariant neighborhoods and describe the behavior of trajectories of these systems. Biological interpretations of these results are given as well.

References


[1]
Banks H. T. and Mahaffy J. M., “Stability of cyclic gene models for systems involving repression,” J. Theor. Biology, 74, 323–334 (1978).

[2]
Bass J., “Circadian topology of metabolism,” Nature, 491, No. 7424, 348–356 (2012).

[3]
Podkolodnaya О. А., “Molecular genetic aspects of the interaction of the circadian clocks and metabolism of energy substrates of mammals [in Russian],” Genetics, 50, No. 2, 1–13 (2014).

[4]
Likhoshvai V., Golubyatnikov V., Demidenko G., Evdokimov A., and Fadeev S., “Gene networks theory,” in: Computational Systems Biology [in Russian], pp. 395–480, Izdat. SO RAN, Novosibirsk (2008).

[5]
Bukharina T. A., Golubyatnikov V. P., Kazantsev M. V., Kirillova N. E., and Furman D. P., “Mathematical and numerical models of two asymmetric gene networks,” Sib. Electron. Math. Rep., 15, 1271–1283 (2018).

[6]
Golubyatnikov V. P., Mjolsness E., and Gaidov Yu. A., “Topological index of a model of p53 −Mdm2 circuit,” Inform. Vestn. Vavilov. Obshch. Genetikov i Seleksts., 13, No. 1, 160–162 (2009).

[7]
Chumakov G. A. and Chumakova N. A., “Homoclinic cycles in one gene network model [in Russian],” Mat. Zamet. SVFU, 21, No. 4, 97–106 (2014).

[8]
Ayupova N. B. and Golubyatnikov V. P., “A three-cells model of the initial stage of development of one proneural cluster,” J. Appl. Ind. Math., 11, No. 2, 1–7 (2017).

[9]
Golubyatnikov V. P. and Gradov V. S., “Non-uniqueness of cycles in piecewise-linear models of circular gene networks,” Sib. Adv. Math., 31, No. 1, 1–12 (2021).

[10]
Vyshnegradskiy I. A. “ On regulators of direct action [in Russian],” Izv. Tekhnolog. Inst., Imper. Akad. Nauk, Saint-Petersburg, 21–62 (1877).

[11]
Postnikov M. M., Stable Polynomials [in Russian], URSS, Moscow (2004).

[12]
Smale S., “A mathematical model of two cells via Turing’s equation,” in: Lecture Notes in Applied Mathematics, vol. 16, pp. 15–26, Amer. Math. Soc., Providence, RI (1974).

[13]
Akinshin A. A., Bukharina T. A., Golubyatnikov V. P., and Furman D. P., “Mathematical modeling of the interaction of two cells in the proneural cluster of the wing imaginal disc D. melanogaster [in Russian],” Sib. Zhurn. Chistoy Prikl. Mat., 14, No. 4, 3–10 (2014).

[14]
Ayupova N. B. and Golubyatnikov V. P., “The structure of the phase portrait of one piecewise linear dynamic system,” J. Appl. Ind. Math., 13, No. 4, 1–8 (2019).
How to Cite
Golubyatnikov, V. and Kirillova, N. (2021) “Phase portraits of two gene networks models”, Mathematical notes of NEFU, 28(1), pp. 3-11. doi: https://doi.org/10.25587/SVFU.2021.68.70.001.
Section
Mathematics