Abstract
We construct two nonlinear dynamical systems of the functioning simplest circadian oscillator. Some conditions of the uniqueness of the equilibrium point of these systems are described as well as the conditions for the existence of cycles in their phase portraits.
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REFERENCES
U. Albrecht, “Timing to Perfection: The Biology of Central and Peripheral Circadian Clocks,” Neuron 74 (2), 246–260 (2012).
S. A. Newman and G. Forgacs, “Complexity and Self-Organization in Biological Development and Evolution,” Complexity in Chemistry, Biology, and Ecology (Springer, 2005), pp. 49–96.
B. C. Goodwin, Temporal Organization in Cells: A Dynamic Theory of Cellular Control Processes (Acad. Press, London, 1963).
B. C. Goodwin, “Oscillatory Behavior in Enzymatic Control Processes,” Adv. Enzyme Regul. 3, 425–438 (1965).
O. A. Podkolodnaya, N. N. Tverdokhleb, and N. L. Podkolodnyy, “Computational Modeling of the Cell Autonomous Mammalian Circadian Oscillator,” BMC Syst. Biol. 11, 27–42 (2017).
S. Almeida, M. Chaves, and F. Delaunay, “Transcription-Based Circadian Mechanism Controls the Duration of Molecular Clock States in Response to Signaling Inputs,” J. Theor. Biol. 484, 110015 (2020).
T. K. Sato, S. Panda, L. J. Miraglia, T. M. Reyes, R. D. Rudic, P. Mcnamara, K. A. Naik, G. A. Fitzgerald, S. A. Kay, and J. B. Hogenesch, “A Functional Genomics Strategy Reveals Rora as a Component of the Mammalian Circadian Clock,” Neuron 43 (4), 527–537 (2004).
S. Hastings, J. Tyson, and D. Webster, “Existence of Periodic Solutions for Negative Feedback Cellular Control System,” J. Differential Equations 25, 39–64 (1977).
J. Hofbauer, J. Mallet-Paret, and H. L. Smith, “Stable Periodic Solutions for the Hypercycle System,” J. Dyn. Diff. Equat. 3 (3), 423–436 (1991).
A. A. Akinshin, T. A. Bukharina, V. P. Golubyatnikov, and D. P. Furman, “Mathematical Modeling the Interaction of Two Cells in a Proneuralizing Cluster of Wing Imaginal Bud of D. melanogaster,” Sibir. Zh. Chist. Prikl. Mat. 14 (4), 3–10 (2014).
M. B. Elowitz and S. Leibler, “A Synthetic Oscillatory Network of Transcriptional Regulators,” Nature 403, 335–338 (2000).
System Computer Biology, Ed. by N. A. Kolchanov, S. S. Goncharov, V. A. Ivanisenko, and V. A. Likhoshvai (Izd. Sib. Otd. Ross. Akal. Nauk, Novosibirsk, 2008) [in Russian].
T. A. Bukharina, A. A. Akinshin, V. P. Golubyatnikov, and D. P. Furman, “Mathematical and Numerical Models of the Central Regulatory Circuit of the Morphogenesis System of Drosophila,” Sibir. Zh. Ind. Mat. 23 (2), 41–50 (2020) [J. Appl. Ind. Math. 14 (2), 249–255 (2020)].
L. Glass and J. S. Pasternack, “Stable Oscillations in Mathematical Models of Biological Control Systems,” J. Math. Biology 6, 207–223 (1978).
R. Smith, “Orbital Stability of Ordinary Differential Equations,” J. Differential Equations 69, 265–287 (1987).
Yu. A. Gaidov, V. P. Golubyatnikov, “About Some Nonlinear Dynamic Systems That Model the Asymmetrical Gene Networks,” Sibir. Zh. Chist. Prikl. Mat. 7 (2), 8–17 (2007).
Yu. A. Gaidov, V. P. Golubyatnikov, and E. Mjolsness, “Topological Index of a Model of p53-Mdm2 Circuit,” Inform. Vestnik Vavilov. Obshch. Genet. i Selekts. 13 (1), 160–162 (2009).
N. E. Kirillova, “On Invariant Surfaces in Gene Network Models,” Sibir. Zh. Ind. Mat. 23 (4), 69–76 (2020) [J. Appl. Ind. Math. 14 (4), 666–671 (2020)].
D. V. Anosov, Mappings of a Circle, Vector Fields, and Their Applications (Izd. MTsNMO, Moscow, 2003) [in Russian].
R. Abraham and J. Robbins, Transversal Mappings and Flows (W. A. Benjamin Inc., New York, 1967).
D. V. Anosov, “Remarks Concerning Hyperbolic Sets,” J. Math. Sci. 78 (5), 497–529 (1996).
J. S. Griffith, “Mathematics of Cellular Control Processes. I. Negative Feedback to One Gene,” J. Theor. Biol. 20 (2), 202–208 (1968).
D. Gonze and W. Abou-Jaoudé, “The Goodwin Model: Behind the Hill Function,” PLoS ONE 8 (8), e69573 (2013).
J. K. Kim, “Protein Sequestration Versus Hill-Type Repression in Circadian Clock Models,” IET Syst. Biol. 10 (4), 125–135 (2016).
A. C. Liu, H. G. Tran, E.E. Zhang, A. A. Priest, D. K. Welsh, and S. A. Kay, “Redundant Function of REV-ERB-Alpha and Beta and Nonessential Role for Bmal1 Cycling in Transcriptional Regulation of Intracellular Circadian Rhythms,” PLoS Genet. 4 (2), e1000023 (2008).
P. T. Foteinou, A. Venkataraman, L. J. Francey, R. C. Anafi, J. B. Hogenesch, and F. J. Doyle 3rd., “Computational and Experimental Insights into the Circadian Effects of SIRT1,” Proc. Nat. Acad. Sci. U.S.A. 115 (45), 11643–11648 (2018).
V. P. Golubyatnikov and N. E. Kirillova, “Phase Portraits of Models of Two Gene Networks,” Mat. Zametki Sev.-Vost. Feder. Univ. 28 (1), 3–11 (2021).
S. Smale, “A Mathematical Model of Two Cells via Turing’s Equation,” in Lecture in Applied Mathematics, Vol. 6 (AMS, 1974), pp. 15–26.
Funding
The authors were supported within the framework of the State Contract of Institute of Cytology and Genetics (project no. FWNR–2022–0020), the State Contract of Institute of Computational Mathematics and Mathematical Geophysics (project no. 0251–2021–0004), the State Contract of Sobolev Institute of Mathematics (project no. FWNF–2022–0009), and partially by the Russian Foundation for Basic Research (project no. 20–31–90011).
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Golubyatnikov, V.P., Podkolodnaya, O.A., Podkolodnyy, N.L. et al. On Conditions for the Existence of Cycles in Two Models of a Circadian Oscillator of Mammals. J. Appl. Ind. Math. 15, 597–608 (2021). https://doi.org/10.1134/S1990478921040037
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DOI: https://doi.org/10.1134/S1990478921040037