American Institute of Mathematical Sciences

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March  2021, 26(3): 1337-1357. doi: 10.3934/dcdsb.2020165

Bistability of sequestration networks

Xiaoxian Tang 1, and  Jie Wang 2,,

1. 

School of Mathematical Sciences, Beihang University, Beijing, China
2. 

Laboratoire d'Analyse et d'Architecture des Systèmes (LAAS-CNRS), Toulouse, France

* Corresponding author: jwang@laas.fr

Received  June 2019 Revised  February 2020 Published  March 2021 Early access  May 2020

Fund Project: XT was partially funded by NSF (DMS-1752672). JW was supported by China Postdoctoral Science Foundation under grants 2018M641055

We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically stable. In addition, we provide a non-empty open set in the parameter space where a sequestration network admits bistability, and we present a procedure for computing a witness for bistability.

Keywords: Multistationarity, bistability, chemical reaction networks, mass-action kinetics, sequestration networks.
Mathematics Subject Classification: 92C40, 92C45.
Citation: Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165
References:
[1]

C. Bagowski and J. Ferrell, Bistability in the JNK cascade, Curr. Biol., 11 (2001), 1176-1182.  doi: 10.1016/S0960-9822(01)00330-X.  Google Scholar

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M. Banaji and C. Pantea, Some results on injectivity and multistationarity in chemical reaction networks, SIAM J. Appl. Dyn. Syst., 15 (2016), 807-869.  doi: 10.1137/15M1034441.  Google Scholar

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C. Conradi and A. Shiu, Dynamics of post-translational modification systems: Recent progress and future challenges, Biophys. J., 114 (2018), 507-515.   Google Scholar

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G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546.  doi: 10.1137/S0036139904440278.  Google Scholar

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G. Craciun, Y. Z. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, Proc. Natl. Acad. Sci. USA, 103 (2006), 8697-8702.  doi: 10.1073/pnas.0602767103.  Google Scholar

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B. N. Datta, An elementary proof of the stability criterion of Liénard and Chipart, Linear Algebra Appl., 22 (1978), 89-96.  doi: 10.1016/0024-3795(78)90060-5.  Google Scholar

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A. Dickenstein, M. P. Millan, A. Shiu and X. X. Tang, Multistationarity in structured reaction networks, Bull. Math. Biol., 81 (2019), 1527-1581.  doi: 10.1007/s11538-019-00572-6.  Google Scholar

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E. Feliu, Injectivity, multiple zeros and multistationarity in reaction networks, Proc. A, 471 (2015), 20140530, 18 pp. doi: 10.1098/rspa.2014.0530.  Google Scholar

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B. Félix, A. Shiu and Z. Woodstock, Analyzing multistationarity in chemical reaction networks using the determinant optimization method, Appl. Math. Comput., 287/288 (2016), 60-73.  doi: 10.1016/j.amc.2016.04.030.  Google Scholar

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J. E. Ferrell, Jr . and E. M. Machleder, The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes, Science, 280 (1998), 895-898.  doi: 10.1126/science.280.5365.895.  Google Scholar

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S. A. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk (in German), 6 (1931), 749-754.   Google Scholar

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B. Joshi and A. Shiu, Which small reaction networks are multistationary?, SIAM J. Appl. Dyn. Syst., 16 (2017), 802-833.  doi: 10.1137/16M1069705.  Google Scholar

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S. Müller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu and A. Dickenstein, Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry, Found. Comput. Math., 16 (2016), 69-97.  doi: 10.1007/s10208-014-9239-3.  Google Scholar

[21]

N. Obatake, A. Shiu, X. X. Tang and A. Torres, Oscillations and bistability in a model of ERK regulation, Journal of Mathematical Biology, 79 (2019), 1515-1549.  doi: 10.1007/s00285-019-01402-y.  Google Scholar

[22]

M. Pérez Millán and A. Dickenstein, The structure of MESSI biological systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 1650-1682.  doi: 10.1137/17M1113722.  Google Scholar

[23]

M. Pérez Millán, A. Dickenstein, A. Shiu and C. Conradi, Chemical reaction systems with toric steady states, Bull. Math. Biol., 74 (2012), 1027-1065.  doi: 10.1007/s11538-011-9685-x.  Google Scholar

[24]

A. Sadeghimanesh and E. Feliu, The multistationarity structure of networks with intermediates and a binomial core network, Bulletin of Mathematical Biology, 81 (2019), 2428-2462.  doi: 10.1007/s11538-019-00612-1.  Google Scholar

[25]

P. M. Schlosser and M. Feinberg, A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions, Chem. Eng. Sci., 49 (1994), 1749-1767.  doi: 10.1016/0009-2509(94)80061-8.  Google Scholar

[26]

G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92-113.  doi: 10.1016/j.mbs.2012.05.004.  Google Scholar

[27]

A. Shiu and T. de Wolff, Nondegenerate multistationarity in small reaction networks, Discrete Contin. Dyn. Syst. B, 24 (2019), 2683-2700.  doi: 10.3934/dcdsb.2018270.  Google Scholar

[28]

M. Thomson and J. Gunawardena, The rational parameterisation theorem for multisite post-translational modification systems, J. Theoret. Biol., 261 (2009), 626-636.  doi: 10.1016/j.jtbi.2009.09.003.  Google Scholar

[29]

C. Wiuf and E. Feliu, Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species, SIAM J. Appl. Dyn. Syst., 12 (2013), 1685-1721.  doi: 10.1137/120873388.  Google Scholar

[30]

W. Xiong and J. E. Ferrell, A positive-feedback-based bistable 'memory module' that governs a cell fate decision, Nature, 426 (2003), 460-465.  doi: 10.1038/nature02089.  Google Scholar

show all references

References:
[1]

C. Bagowski and J. Ferrell, Bistability in the JNK cascade, Curr. Biol., 11 (2001), 1176-1182.  doi: 10.1016/S0960-9822(01)00330-X.  Google Scholar

[2]

M. Banaji and C. Pantea, Some results on injectivity and multistationarity in chemical reaction networks, SIAM J. Appl. Dyn. Syst., 15 (2016), 807-869.  doi: 10.1137/15M1034441.  Google Scholar

[3]

C. Conradi, E. Feliu, M. Mincheva and C. Wiuf, Identifying parameter regions for multistationarity, PLoS Comput. Biol., 13 (2017). doi: 10.1371/journal.pcbi.1005751.  Google Scholar

[4]

C. Conradi and A. Shiu, Dynamics of post-translational modification systems: Recent progress and future challenges, Biophys. J., 114 (2018), 507-515.   Google Scholar

[5]

G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546.  doi: 10.1137/S0036139904440278.  Google Scholar

[6]

G. Craciun, Y. Z. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, Proc. Natl. Acad. Sci. USA, 103 (2006), 8697-8702.  doi: 10.1073/pnas.0602767103.  Google Scholar

[7]

G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: Semiopen mass action systems, SIAM J. Appl. Math., 70 (2010), 1859-1877.  doi: 10.1137/090756387.  Google Scholar

[8]

B. N. Datta, An elementary proof of the stability criterion of Liénard and Chipart, Linear Algebra Appl., 22 (1978), 89-96.  doi: 10.1016/0024-3795(78)90060-5.  Google Scholar

[9]

A. Dickenstein, M. P. Millan, A. Shiu and X. X. Tang, Multistationarity in structured reaction networks, Bull. Math. Biol., 81 (2019), 1527-1581.  doi: 10.1007/s11538-019-00572-6.  Google Scholar

[10]

M. Domijan and M. Kirkilionis, Bistability and oscillations in chemical reaction networks, J. Math. Biol., 59 (2009), 467-501.  doi: 10.1007/s00285-008-0234-7.  Google Scholar

[11]

E. Feliu, Injectivity, multiple zeros and multistationarity in reaction networks, Proc. A, 471 (2015), 20140530, 18 pp. doi: 10.1098/rspa.2014.0530.  Google Scholar

[12]

B. Félix, A. Shiu and Z. Woodstock, Analyzing multistationarity in chemical reaction networks using the determinant optimization method, Appl. Math. Comput., 287/288 (2016), 60-73.  doi: 10.1016/j.amc.2016.04.030.  Google Scholar

[13]

J. E. Ferrell, Jr . and E. M. Machleder, The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes, Science, 280 (1998), 895-898.  doi: 10.1126/science.280.5365.895.  Google Scholar

[14]

S. A. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk (in German), 6 (1931), 749-754.   Google Scholar

[15]

H. Hong, X. X. Tang and B. C. Xia, Special algorithm for stability analysis of multistable biological regulatory systems, J. Symbolic Comput., 70 (2015), 112-135.  doi: 10.1016/j.jsc.2014.09.039.  Google Scholar

[16]

M. D. Johnston, S. Müller and C. Pantea, A deficiency-based approach to parametrizing positive equilibria of biochemical reaction systems, Bull. Math. Biol., 81 (2019), 1143-1172.  doi: 10.1007/s11538-018-00562-0.  Google Scholar

[17]

B. Joshi and A. Shiu, A survey of methods for deciding whether a reaction network is multistationary, Math. Model. Nat. Phenom., 10 (2015), 47-67.  doi: 10.1051/mmnp/201510504.  Google Scholar

[18]

B. Joshi and A. Shiu, Which small reaction networks are multistationary?, SIAM J. Appl. Dyn. Syst., 16 (2017), 802-833.  doi: 10.1137/16M1069705.  Google Scholar

[19] R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York-London, 1982.   Google Scholar
[20]

S. Müller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu and A. Dickenstein, Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry, Found. Comput. Math., 16 (2016), 69-97.  doi: 10.1007/s10208-014-9239-3.  Google Scholar

[21]

N. Obatake, A. Shiu, X. X. Tang and A. Torres, Oscillations and bistability in a model of ERK regulation, Journal of Mathematical Biology, 79 (2019), 1515-1549.  doi: 10.1007/s00285-019-01402-y.  Google Scholar

[22]

M. Pérez Millán and A. Dickenstein, The structure of MESSI biological systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 1650-1682.  doi: 10.1137/17M1113722.  Google Scholar

[23]

M. Pérez Millán, A. Dickenstein, A. Shiu and C. Conradi, Chemical reaction systems with toric steady states, Bull. Math. Biol., 74 (2012), 1027-1065.  doi: 10.1007/s11538-011-9685-x.  Google Scholar

[24]

A. Sadeghimanesh and E. Feliu, The multistationarity structure of networks with intermediates and a binomial core network, Bulletin of Mathematical Biology, 81 (2019), 2428-2462.  doi: 10.1007/s11538-019-00612-1.  Google Scholar

[25]

P. M. Schlosser and M. Feinberg, A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions, Chem. Eng. Sci., 49 (1994), 1749-1767.  doi: 10.1016/0009-2509(94)80061-8.  Google Scholar

[26]

G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92-113.  doi: 10.1016/j.mbs.2012.05.004.  Google Scholar

[27]

A. Shiu and T. de Wolff, Nondegenerate multistationarity in small reaction networks, Discrete Contin. Dyn. Syst. B, 24 (2019), 2683-2700.  doi: 10.3934/dcdsb.2018270.  Google Scholar

[28]

M. Thomson and J. Gunawardena, The rational parameterisation theorem for multisite post-translational modification systems, J. Theoret. Biol., 261 (2009), 626-636.  doi: 10.1016/j.jtbi.2009.09.003.  Google Scholar

[29]

C. Wiuf and E. Feliu, Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species, SIAM J. Appl. Dyn. Syst., 12 (2013), 1685-1721.  doi: 10.1137/120873388.  Google Scholar

[30]

W. Xiong and J. E. Ferrell, A positive-feedback-based bistable 'memory module' that governs a cell fate decision, Nature, 426 (2003), 460-465.  doi: 10.1038/nature02089.  Google Scholar

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