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June  2019, 24(6): 2683-2700. doi: 10.3934/dcdsb.2018270

Nondegenerate multistationarity in small reaction networks

1. 

Texas A&M University, Department of Mathematics, Mailstop 3368, College Station, TX 77843-3368, USA

2. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author: Anne Shiu

Received  January 2018 Revised  May 2018 Published  October 2018

Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques may point the way forward for larger networks.

Citation: Anne Shiu, Timo de Wolff. Nondegenerate multistationarity in small reaction networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2683-2700. doi: 10.3934/dcdsb.2018270
References:
[1]

M. Banaji and C. Pantea, The inheritance of nondegenerate multistationarity in chemical reaction networks, SIAM J. Appl. Math., 78 (2018), 1105-1130, arXiv: 1608.08400. doi: 10.1137/16M1103506. Google Scholar

[2]

J. L. Cherry and F. R. Adler, How to make a biological switch, J. Theoret. Biol., 203 (2000), 117-133. doi: 10.1006/jtbi.2000.1068. Google Scholar

[3]

C. Conradi, E. Feliu, M. Mincheva and C. Wiuf, Identifying parameter regions for multistationarity, PLoS Comput. Biol., 13(2017), e1005751. 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. doi: 10.1016/j.bpj.2017.11.3787. Google Scholar

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J. P. DexterT. Dasgupta and J. Gunawardena, Invariants reveal multiple forms of robustness in bifunctional enzyme systems, Integr. Biol., 7 (2015), 883-894. doi: 10.1039/C5IB00009B. Google Scholar

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M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors Ⅰ. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268. doi: 10.1016/0009-2509(87)80099-4. Google Scholar

[8]

B. FélixA. 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

[9]

K. Gatermann and B. Huber, A family of sparse polynomial systems arising in chemical reaction systems, J. Symb. Comput., 33 (2002), 275-305. doi: 10.1006/jsco.2001.0512. Google Scholar

[10]

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994. doi: 10.1007/978-0-8176-4771-1. Google Scholar

[11]

M. GopalkrishnanE. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797. doi: 10.1137/130928170. Google Scholar

[12]

D. J. Grabiner, Descartes' rule of signs: Another construction, Amer. Math. Monthly, 106 (1999), 854-856. doi: 10.1080/00029890.1999.12005131. Google Scholar

[13]

E. GrossH. A. HarringtonZ. Rosen and B. Sturmfels, Algebraic systems biology: A case study for the {Wnt} pathway, Bull. Math. Biol., 78 (2016), 21-51. doi: 10.1007/s11538-015-0125-1. Google Scholar

[14]

K. L. Ho and H. A. Harrington, Bistability in apoptosis by receptor clustering, PLoS Comput. Biol., 6 (2010), e1000956, 9pp. doi: 10.1371/journal.pcbi.1000956. Google Scholar

[15]

B. Joshi and A. Shiu, Atoms of multistationarity in chemical reaction networks, J. Math. Chem., 51 (2013), 153-178. doi: 10.1007/s10910-012-0072-0. Google Scholar

[16]

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

[17]

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

[18]

M. P. 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

[19]

Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, vol. 26 of London Mathematical Society Monographs. New Series, The Clarendon Press, Oxford University Press, Oxford, 2002. Google Scholar

[20]

M. A. Sweeney, Conditions for solvability in chemical reaction networks at quasi-steady-state, Preprint, arXiv: 1712.05533.Google Scholar

[21]

T. Theobald and T. de Wolff, Norms of roots of trinomials, Math. Ann., 366 (2016), 219-247. doi: 10.1007/s00208-015-1323-8. Google Scholar

[22]

V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications, vol. 98 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992, Translated from the Russian by B. Goldfarb. Google Scholar

show all references

References:
[1]

M. Banaji and C. Pantea, The inheritance of nondegenerate multistationarity in chemical reaction networks, SIAM J. Appl. Math., 78 (2018), 1105-1130, arXiv: 1608.08400. doi: 10.1137/16M1103506. Google Scholar

[2]

J. L. Cherry and F. R. Adler, How to make a biological switch, J. Theoret. Biol., 203 (2000), 117-133. doi: 10.1006/jtbi.2000.1068. Google Scholar

[3]

C. Conradi, E. Feliu, M. Mincheva and C. Wiuf, Identifying parameter regions for multistationarity, PLoS Comput. Biol., 13(2017), e1005751. 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. doi: 10.1016/j.bpj.2017.11.3787. Google Scholar

[5]

J. P. DexterT. Dasgupta and J. Gunawardena, Invariants reveal multiple forms of robustness in bifunctional enzyme systems, Integr. Biol., 7 (2015), 883-894. doi: 10.1039/C5IB00009B. Google Scholar

[6]

A. Dickenstein, Biochemical reaction networks: An invitation for algebraic geometers, in Mathematical Congress of the Americas, vol. 656 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2016, 65-83. doi: 10.1090/conm/656/13076. Google Scholar

[7]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors Ⅰ. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268. doi: 10.1016/0009-2509(87)80099-4. Google Scholar

[8]

B. FélixA. 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

[9]

K. Gatermann and B. Huber, A family of sparse polynomial systems arising in chemical reaction systems, J. Symb. Comput., 33 (2002), 275-305. doi: 10.1006/jsco.2001.0512. Google Scholar

[10]

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994. doi: 10.1007/978-0-8176-4771-1. Google Scholar

[11]

M. GopalkrishnanE. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797. doi: 10.1137/130928170. Google Scholar

[12]

D. J. Grabiner, Descartes' rule of signs: Another construction, Amer. Math. Monthly, 106 (1999), 854-856. doi: 10.1080/00029890.1999.12005131. Google Scholar

[13]

E. GrossH. A. HarringtonZ. Rosen and B. Sturmfels, Algebraic systems biology: A case study for the {Wnt} pathway, Bull. Math. Biol., 78 (2016), 21-51. doi: 10.1007/s11538-015-0125-1. Google Scholar

[14]

K. L. Ho and H. A. Harrington, Bistability in apoptosis by receptor clustering, PLoS Comput. Biol., 6 (2010), e1000956, 9pp. doi: 10.1371/journal.pcbi.1000956. Google Scholar

[15]

B. Joshi and A. Shiu, Atoms of multistationarity in chemical reaction networks, J. Math. Chem., 51 (2013), 153-178. doi: 10.1007/s10910-012-0072-0. Google Scholar

[16]

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

[17]

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

[18]

M. P. 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

[19]

Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, vol. 26 of London Mathematical Society Monographs. New Series, The Clarendon Press, Oxford University Press, Oxford, 2002. Google Scholar

[20]

M. A. Sweeney, Conditions for solvability in chemical reaction networks at quasi-steady-state, Preprint, arXiv: 1712.05533.Google Scholar

[21]

T. Theobald and T. de Wolff, Norms of roots of trinomials, Math. Ann., 366 (2016), 219-247. doi: 10.1007/s00208-015-1323-8. Google Scholar

[22]

V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications, vol. 98 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992, Translated from the Russian by B. Goldfarb. Google Scholar

Figure 1.  Stoichiometric compatibility classes for the network in Example 2.2.
Table 1.  Summary of results on nondegenerate multistationarity for small reactions. Here r denotes the number of reactions and s the number of species. See Section 2.
Network propertyNondegenerately multistationary?
Network with only 1 species ($s=1$)If and only if some subnetwork is 2-alternating (Proposition 1.1) [17]
Network consists of 1 reaction ($r=1$) or 1 reversible-reaction pairNo (Proposition 1.2) [17]
Network consists of 2 reactions ($r=2$)See Proposition 1.3 [17]
$r+s \leq 3$No ([17,Corollary 3.8])
$s=2$ and 1 irreversible reaction and 1 reversible-reaction pairSee Theorem 3.5
$s=2$ and 2 reversible-reaction pairsSee Theorem 3.6
Network propertyNondegenerately multistationary?
Network with only 1 species ($s=1$)If and only if some subnetwork is 2-alternating (Proposition 1.1) [17]
Network consists of 1 reaction ($r=1$) or 1 reversible-reaction pairNo (Proposition 1.2) [17]
Network consists of 2 reactions ($r=2$)See Proposition 1.3 [17]
$r+s \leq 3$No ([17,Corollary 3.8])
$s=2$ and 1 irreversible reaction and 1 reversible-reaction pairSee Theorem 3.5
$s=2$ and 2 reversible-reaction pairsSee Theorem 3.6
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