This article is cited in 5 papers

Differentical equations, dynamical systems and optimal control

Mathematical and numerical models of two asymmetric gene networks

V. P. Golubyatnikov^a, M. V. Kazantsev^b, N. E. Kirillova^c, T. A. Bukharina<sup>d

a</sup> Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
^b Polzunov Altai State Technical University, Lenin avenue, 46, 656038, Barnaul, Russia
^c Novosibirsk State University, Pirogova street, 1, 630090, Novosibirsk, Russia
^d The Federal reaearch center Institute of Cytology and Genetics SB RAS, Lavrent'ev avenue, 10, 630090, Novosibirsk, Russia

Abstract: We construct and study mathematical models of two gene networks: a circular gene network of molecular repressilator, and a natural gene network which does not have circular structure. For the first model, we consider discretization of phase portrait of corresponding nonlinear dynamical system and find conditions of existence of an oscillating trajectory (cycle) in this phase portrait. The second model describes the central regulatory circuit of one gene network which acts on early stage of the fruit fly Drosophila melanogaster mechanoreceptors morphogenesis. For both models we give biological interpretations of our numerical simulations and give a short description of software elaborated specially for these experiments.

Keywords: nonlinear dynamical systems, cycles, phase portraits, gene networks models, hyperbolic equilibrium points, Grobman-Hartman theorem, Brouwer fixed point theorem, numerical analysis.

UDC: 517.93

MSC: 37N25

Received March 26, 2018, published October 25, 2018

Language: English

DOI: 10.17377/semi.2018.15.103

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