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Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria
Bistability of sequestration networks
1. | School of Mathematical Sciences, Beihang University, Beijing, China |
2. | Laboratoire d'Analyse et d'Architecture des Systèmes (LAAS-CNRS), Toulouse, France |
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.
References:
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C. Bagowski and J. Ferrell,
Bistability in the JNK cascade, Curr. Biol., 11 (2001), 1176-1182.
doi: 10.1016/S0960-9822(01)00330-X. |
[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. |
[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. |
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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 |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
E. Feliu, Injectivity, multiple zeros and multistationarity in reaction networks, Proc. A, 471 (2015), 20140530, 18 pp.
doi: 10.1098/rspa.2014.0530. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York-London, 1982.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
G. Shinar and M. Feinberg,
Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92-113.
doi: 10.1016/j.mbs.2012.05.004. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
E. Feliu, Injectivity, multiple zeros and multistationarity in reaction networks, Proc. A, 471 (2015), 20140530, 18 pp.
doi: 10.1098/rspa.2014.0530. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York-London, 1982.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
G. Shinar and M. Feinberg,
Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92-113.
doi: 10.1016/j.mbs.2012.05.004. |
[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. |
[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. |
[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. |
[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. |
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