Piecewise linear dynamical system modeling gene network with variable feedback. (Russian, English) Zbl 1413.37061
Sib. Zh. Chist. Prikl. Mat. 16, No. 4, 28-37 (2016); translation in J. Math. Sci., New York 230, No. 1, 46-54 (2018).
Summary: We construct a discretization of the phase portrait of a 3-dimensional dynamical system of biochemical kinetics with piecewise linear right-hand sides. We describe geometry of the phase portrait and construct an invariant piecewise linear surface bounded by a stable cycle of this system composed of eight linear segments.
References:
| [1] | A. A. Akinshin and V. P. Golubyatnikov, “Geometric characteristics of cycles in some symmetric dynamical systems” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform.12, No. 2, 3-12 (2012). · Zbl 1289.37048 |
| [2] | V. P. Golubyatnikov and I. V. Golubyatnikov, “On periodic trajectories in odd-dimensional gene network models,” Rus. J. Numer. Anal. Math. Model.26, No. 4, 397-412 (2011). · Zbl 1238.34053 |
| [3] | V. A. Likhoshvai, V. P. Golubyatnikov, G. V. Demidenko, S. I. Fadeev, and A. A. Evdokimov, “Theory of gene networks” [in Russian], In: Computational Systems Biology, pp. 395-480, SB RAS Press, Novosibirsk (2008). |
| [4] | Yu. A. Gaidov and V. P. Golubyatnikov, “On the existence and stability of cycles in gene networks with variable feedbacks,” Contemp. Math.553, 61-74 (2011). · Zbl 1238.37036 · doi:10.1090/conm/553/10932 |
| [5] | N. B. Ayupova and V. P. Golubyatnikov, “On the uniqueness of a cycle in an asymmetric three-dimensional model of a molecular repressilator,” J. Appl. Ind. Math.8, No. 2. 153-157 (2014). · Zbl 1340.34164 · doi:10.1134/S199047891402001X |
| [6] | N. B. Ayupova and V. P. Golubyatnikov, “On two classes of nonlinear dynamical systems: The four-dimensional case,” Sib. Math. J.56, No. 2, 231-236 (2015). · Zbl 1357.37043 · doi:10.1134/S0037446615020044 |
| [7] | Yu. A. Gaidov and V. P. Golubyatnikov, “On cycles and other geometric phenomena in phase portraits of some nonlinear dynamical systems,” pp. 225-233, Springer, New York etc. (2014). · Zbl 1348.34063 |
| [8] | M. B. Elowitz and S. Leibler, “A synthetic oscillatory network of transcriptional regulators,” Nature403, 335-338 (2000). · doi:10.1038/35002125 |
| [9] | G. Yu. Riznichenko, Lectures on Mathematical Models in Biology [in Russian], RCD, Moscow etc. (2011). |
| [10] | Yu. I. Gilderman, “On the limit cycles of piecewise affine systems,” Sov. Math., Dokl.17, 1328-1332 (1977). · Zbl 0382.34016 |
| [11] | V. G. Demidenko, “Reconstruction of the parameters of the homogeneous linear models of the gene network dynamics” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform.8, No. 3, 51-59 (2008). · Zbl 1249.65011 |
| [12] | I. I. Matveeva and A. M. Popov, “On properties of solutions to one system modeling a multistage synthesis of a substance” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform.9, No. 3, 86-94 (2009). · Zbl 1249.34137 |
| [13] | L. Glass and J. S. Pasternack, “Stable oscillations in mathematical models of biological control systems,” J. Math. Biol.6, 207-223 (1978). · Zbl 0391.92001 · doi:10.1007/BF02547797 |
| [14] | M. V. Kazantsev, “On some properties of the domain graphs of dynamical systems” [in Russian], Sib. Zh. Ind. Mat.18, No. 4, 42-48 (2015). · Zbl 1349.92061 |
| [15] | B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov,, Modern Geometry [in Russian], Nauka, Moscow (1980) · Zbl 0433.53001 |
| [16] | V. I. Arnold, Mathematical Methods of Classical Mechanics Springer, New York etc. (1978). · Zbl 0386.70001 |
| [17] | V. A. Likhoshvai, V. V. Kogai, S. I. Fadeev, and T. M. Khlebodarova, “On the chaos in gene networks,” J. Bioinform. Comput. Biol.11, No. 1, 1340009 (2013). · doi:10.1142/S021972001340009X |
| [18] | V. A. Likhoshvai, V. V. Kogai, S. I. Fadeev, and T. M. Khlebodarova, “Alternative splicing can lead to chaos,” J. Bioinform. Comput. Biol.13, No. 1, 1540003 (2015). · doi:10.1142/S021972001540003X |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
