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October  2021, 26(10): 5407-5419. doi: 10.3934/dcdsb.2020349

Effective reduction of a three-dimensional circadian oscillator model

1. 

School of Mathematics and Statistics & Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China

2. 

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

3. 

School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Shuang Chen

Received  July 2020 Published  November 2020

Fund Project: This work was partly supported by the NSFC grants 11531006, 11771449, 11771161, and the Hubei provincial postdoctoral science and technology activity project

We investigate the dynamics of a three-dimensional system modeling a molecular mechanism for the circadian rhythm in Drosophila. We first prove the existence of a compact attractor in the region with biological meaning. Under the assumption that the dimerization reactions are fast, in this attractor we reduce the three-dimensional system to a simpler two-dimensional system on the persistent normally hyperbolic slow manifold.

Citation: Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5407-5419. doi: 10.3934/dcdsb.2020349
References:
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S. BoieV. KirkJ. Sneyd and M. Wechselberger, Effects of quasi-steady-state reduction on biophysical models with oscillations, J. Theoret. Biol., 393 (2016), 16-31.  doi: 10.1016/j.jtbi.2015.12.011.  Google Scholar

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[3]

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F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.  Google Scholar

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J. Dunlap, Molecular bases for circadian clocks, Cell, 96 (1999), 271-290.  doi: 10.1016/S0092-8674(00)80566-8.  Google Scholar

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N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

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N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.  Google Scholar

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N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

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A. GoekeS. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.  doi: 10.1016/j.jde.2015.02.038.  Google Scholar

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D. A. Goussis and H. N. Najm, Model reduction and physical understanding of slowly oscillating processes: The circadian cycle, Multiscale Model. Simul., 5 (2006), 1297-1332.  doi: 10.1137/060649768.  Google Scholar

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G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

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I. Kosiuk and P. Szmolyan, Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle, J. Math. Biol., 72 (2016), 1337-1368.  doi: 10.1007/s00285-015-0905-0.  Google Scholar

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M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.  Google Scholar

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C. Kuehn, Multiple Time Scale Dynamics, Appl. Math. Sci., 191, Springer, Swizerland, 2015. doi: 10.1007/978-3-319-12316-5.  Google Scholar

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C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.  Google Scholar

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W. Liu, Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 67 (2000), 134-180.  doi: 10.1006/jdeq.2000.3778.  Google Scholar

[28]

U. Maas and S. Pope, Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239-264.  doi: 10.1016/0010-2180(92)90034-M.  Google Scholar

[29]

D. McMillenN. KopellJ. Hasty and J. Collins, Synchronizing genetic relaxation oscillators by intercell signaling, Proc. Natl. Acad. Sci. USA, 99 (2002), 679-684.   Google Scholar

[30]

J. E. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 93–146. doi: 10.1016/S1874-575X(02)80024-8.  Google Scholar

[31]

S. Schecter, Exchange lemmas 1: Deng's lemma, J. Differenital Equations, 245 (2008), 392-410.  doi: 10.1016/j.jde.2007.08.011.  Google Scholar

[32]

S. Schecter, Exchange lemmas 2: General exchange lemma, J. Differenital Equations, 245 (2008), 411-441.  doi: 10.1016/j.jde.2007.10.021.  Google Scholar

[33]

J. TysonC. HongC. Thron and B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417.  doi: 10.1016/S0006-3495(99)77078-5.  Google Scholar

[34]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci., 105, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4312-0.  Google Scholar

[35]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992. doi: 10.1007/978-1-4757-4969-4_4.  Google Scholar

show all references

References:
[1]

S. BoieV. KirkJ. Sneyd and M. Wechselberger, Effects of quasi-steady-state reduction on biophysical models with oscillations, J. Theoret. Biol., 393 (2016), 16-31.  doi: 10.1016/j.jtbi.2015.12.011.  Google Scholar

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.  Google Scholar

[3]

B. Deng, The Šil'nikov problem, exponential expansion, strong $\lambda$-lemma, $C^{1}$ linearization, and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.  doi: 10.1016/0022-0396(89)90100-9.  Google Scholar

[4]

B. Deng and G. Hines, Food chain chaos due to transcritical point, Chaos, 13 (2003), 578-585.  doi: 10.1063/1.1576531.  Google Scholar

[5]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[6]

F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Amer. Math. Soc., Vol. 577, Providence, 1996. doi: 10.1090/memo/0577.  Google Scholar

[7]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.  Google Scholar

[8]

J. Dunlap, Molecular bases for circadian clocks, Cell, 96 (1999), 271-290.  doi: 10.1016/S0092-8674(00)80566-8.  Google Scholar

[9]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar

[10]

N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109-1137.  doi: 10.1512/iumj.1974.23.23090.  Google Scholar

[11]

N. Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), 81-93.  doi: 10.1512/iumj.1977.26.26006.  Google Scholar

[12]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[13] D. B. Forger, Biological Clocks, Rhythms, and Oscillations, MIT Press, Cambridge, MA, 2017.   Google Scholar
[14]

A. GoekeS. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.  doi: 10.1016/j.jde.2015.02.038.  Google Scholar

[15]

D. Gonze, Modeling circadian clocks: From equations to oscillations, Cent. Eur. J. Bio., 6 (2011), 699-711.  doi: 10.2478/s11535-011-0061-5.  Google Scholar

[16]

D. A. Goussis and H. N. Najm, Model reduction and physical understanding of slowly oscillating processes: The circadian cycle, Multiscale Model. Simul., 5 (2006), 1297-1332.  doi: 10.1137/060649768.  Google Scholar

[17]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[18]

C. K. R. T. Jones, Geometric Singular Perturbation Theory, in Dynamical Systems (eds. R. Johnson), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[19]

C. K. R. T. JonesT. J. Kaper and N. Kopell, Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal., 27 (1996), 558-577.  doi: 10.1137/S003614109325966X.  Google Scholar

[20]

H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66-93.  doi: 10.1016/S0167-2789(02)00386-X.  Google Scholar

[21]

J. Keener and J. Sneyd, Mathematical Physiology, Int. Appl. Math., 8, Springer-Verlag, New York, 1998. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[22]

I. Kosiuk and P. Szmolyan, Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle, J. Math. Biol., 72 (2016), 1337-1368.  doi: 10.1007/s00285-015-0905-0.  Google Scholar

[23]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.  doi: 10.1137/S0036141099360919.  Google Scholar

[24]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.  Google Scholar

[25]

C. Kuehn, Multiple Time Scale Dynamics, Appl. Math. Sci., 191, Springer, Swizerland, 2015. doi: 10.1007/978-3-319-12316-5.  Google Scholar

[26]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.  Google Scholar

[27]

W. Liu, Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 67 (2000), 134-180.  doi: 10.1006/jdeq.2000.3778.  Google Scholar

[28]

U. Maas and S. Pope, Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239-264.  doi: 10.1016/0010-2180(92)90034-M.  Google Scholar

[29]

D. McMillenN. KopellJ. Hasty and J. Collins, Synchronizing genetic relaxation oscillators by intercell signaling, Proc. Natl. Acad. Sci. USA, 99 (2002), 679-684.   Google Scholar

[30]

J. E. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 93–146. doi: 10.1016/S1874-575X(02)80024-8.  Google Scholar

[31]

S. Schecter, Exchange lemmas 1: Deng's lemma, J. Differenital Equations, 245 (2008), 392-410.  doi: 10.1016/j.jde.2007.08.011.  Google Scholar

[32]

S. Schecter, Exchange lemmas 2: General exchange lemma, J. Differenital Equations, 245 (2008), 411-441.  doi: 10.1016/j.jde.2007.10.021.  Google Scholar

[33]

J. TysonC. HongC. Thron and B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417.  doi: 10.1016/S0006-3495(99)77078-5.  Google Scholar

[34]

S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci., 105, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4312-0.  Google Scholar

[35]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992. doi: 10.1007/978-1-4757-4969-4_4.  Google Scholar

33]">Figure 1.  The mechanism for the circadian oscillator model (1). Adapted from [33]
33,Table 1,p.2414], that is, $ v_{m} = 1 $, $ k_{3} = k_{m} = 0.1 $, $ v_{p} = 0.5 $, $ k_{1} = 10 $, $ k_{2} = 0.03 $, $ P_{c} = 0.1 $ and $ J_{p} = 0.05 $. System (16) with $ \widetilde{k}_{2} = 1 $ has small parameter $ \varepsilon = 0.0003 $">Figure 2.  The attraction of the slow manifold. The surface is the critical manifolds $ \mathcal{M}_{0} $, which is the zeroth-order approximation of the slow manifold, and the discrete orbits, respectively, start from $ (10,10,2) $, $ (15,15,2) $, $ (20,20,2) $, $ (10,20,2) $ and $ (20,10,2) $. Here $ k_{a} = 20000 $, $ k_{d} = 100 $ and the remaining parameters in (2) are chosen as in [33,Table 1,p.2414], that is, $ v_{m} = 1 $, $ k_{3} = k_{m} = 0.1 $, $ v_{p} = 0.5 $, $ k_{1} = 10 $, $ k_{2} = 0.03 $, $ P_{c} = 0.1 $ and $ J_{p} = 0.05 $. System (16) with $ \widetilde{k}_{2} = 1 $ has small parameter $ \varepsilon = 0.0003 $
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