Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system

Magalí Giaroli; Rick Rischter; Mercedes P. Millán; Alicia Dickenstein

doi:10.3934/mbe.2019381

Mathematical Biosciences and Engineering, 2019, 16(6): 7589-7615. doi: 10.3934/mbe.2019381. Research article Special Issues

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Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system

Magalí Giaroli, Rick Rischter, Mercedes P. Millán, Alicia Dickenstein

1 Dto. de Matemática, FCEN, Universidad de Buenos Aires, and IMAS (UBA-CONICET), Ciudad Universitaria, Pab. I, C1428EGA Buenos Aires, Argentina
2 Instituto de Matemática e Computação, IMC, Universidade Federal de Itajubá (UNIFEI), Av. BPS 1303, Bairro Pinheirinho, 37500-903, Itajubá, Minas Gerais, Brazil

† The authors contributed equally to this work.

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Special Issues: Mathematical analysis of reaction networks: theoretical advances and applications

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The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the reaction rate constants and total conservation constants, the system can have 2[n/2] +1 positive steady states (that is, n+1 positive steady states for n even and n positive steady states for n odd). In this paper we give open parameter regions in the space of reaction rate constants and total conservation constants that ensure these number of positive steady states, while assuming in the modeling that roughly only 1/4 of the intermediates occur in the reaction mechanism. This result is based on the general framework developed by Bihan, Dickenstein, and Giaroli (2018), which can be applied to other networks. We also describe how to implement these tools to search for multistationarity regions in a computer algebra system and present some computer aided results.

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Keywords: multistationarity; distributive sequential n-site phosphorylation/dephosphorylation system; steady states; regions of multistationarity; intermediate species

Citation: Magalí Giaroli, Rick Rischter, Mercedes P. Millán, Alicia Dickenstein. Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system. Mathematical Biosciences and Engineering, 2019, 16(6): 7589-7615. doi: 10.3934/mbe.2019381

References:

1. W. Lim, B. Meyer and T. Pawson, Cellular signaling: principles and mechanisms, Garland Science, 2014.
2. A. Dickenstein, Biochemical reaction networks: an invitation for algebraic geometers, MCA 2013, Contemp. Math., 656 (2016), 65–83.
3. N. I. Markevich, J. B. Hoek and B. N. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell Biol., 164 (2004), 353–359.
4. L. Wang and E. Sontag, On the number of steady states in a multiple futile cycle, J. Math. Biol., 57 (2008), 29–52.
5. J. Hell and A. D. Rendall, A proof of bistability for the dual futile cycle, Nonlinear Anal-Real., 24 (2015), 175–189.
6. C. Conradi, E. Feliu, M. Mincheva, et al., Identifying parameter regions for multistationarity, PLOS Computat. Biol., 13 (2017), e1005751.
7. C. Conradi and M. Mincheva, Catalytic constants enable the emergence of bistability in dual phosphorylation, J. R. Soc. Interface, 11 (2014), 20140158.
8. D. Flockerzi, K. Holstein and C. Conradi, N-site phosphorylation systems with 2N-1 steady states, Bull. Math. Biol., 76 (2014), 1892–1916.
9. K. Holstein, D. Flockerzi and C. Conradi, Multistationarity in Sequential Distributed Multisite Phosphorylation Networks, Bull. Math. Biol., 75 (2013), 2028–2058.
10. C. Conradi, A. Iosif and T. Kahle, Multistationarity in the space of total concentrations for systems that admit a monomial parametrization, T. Bull. Math. Biol., 2019, 1–36.
11. M. Thomson and J. Gunawardena, Unlimited multistability in multisite phosphorylation systems, Nature, 460 (2009), 274.
12. E. Feliu, A. D. Rendall and C. Wiuf, A proof of unlimited multistability for phosphorylation cycles, preprint, arXiv:1904.02983.
13. F. Bihan, A. Dickenstein and M. Giaroli, Lower bounds for positive roots and regions of multistationarity in chemical reaction networks, J. Algebra, (2019), to appear.
14. E. Feliu and C. Wiuf, Simplifying biochemical models with intermediate species, J. R. Soc. Interface, 10 (2013), 20130484.
15. E. Feliu and A. Sadeghimanesh, The multistationarity structure of networks with intermediates and a binomial core network, B. Math. Biol., 81 (2019), 2428–2462.
16. M. P. Millán and A. Dickenstein, The structure of MESSI biochemical networks, SIAM J. Appl. Dyn. Syst., 17 (2018), 1650–1682.
17. A. G. Kušnirenko, Newton polyhedra and Bezout's theorem, Funkcional. Anal. i Priložen, 10 (1976), 82–83. English translation: Funct. Anal. Appl., 10 (1977), 233–235.
18. A. Dickenstein, M. Giaroli, M. P. Millán, et al., Detecting the multistationarity structure in enzymatic networks, work in progress.
19. F. Bihan, F. Santos and P-J. Spaenlehauer, A polyhedral method for sparse systems with many positive solutions, SIAM J. Appl. Algebra Geom., 2 (2018), 620–645.
20. M. Giaroli, F. Bihan and A. Dickenstein, Regions of multistationarity in cascades of Goldbeter-Koshland loops, J. Math. Biol., 78 (2019), 1115–1145.
21. J. De Loera, J. Rambau and F. Santos, Triangulations: Structures for Algorithms and Applications, 2010, Series Algorithms and Computation in Mathematics, Springer, Berlin.
22. W.A. Stein, et al., Sage Mathematics Software (Version 8.4), The Sage Development Team, 2018. Available from: http://www.sagemath.org.
23. Maple 18 (2014) Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.
24. E. Feliu, On the reaction rate constants that enable multistationarity in the two-site phosphorylation cycle, preprint, arXiv:1809.07275.
25. A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, 4th Ed., Wiley-Blackwell, 2012.

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This article has been cited by:

1. Frédéric Bihan, Alicia Dickenstein, Magalí Giaroli, Lower bounds for positive roots and regions of multistationarity in chemical reaction networks, Journal of Algebra, 2019, 10.1016/j.jalgebra.2019.10.002

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