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The kinetic space of multistationarity in dual phosphorylation. (English) Zbl 1496.92028

In biology and biochemistry, phosphorylation (as well as its inverse, dephosphorylation) is a fundamental modification process consisting in the attachment (or removal, resp.) of a phosphate group. It is important in cell signaling and is a special case of post-translational modification (PTM) in that it, e.g., changes the behavior of a compound w.r.t. a membrane. It is managed by an enzyme reaction open to quantitative description by the ODE system of a mass action reaction network. The standard machinery is applied to the enzyme reaction of a dual phosphorylation, involving a substrate with two phosphorylation sites and two enzymes, resulting in a polynomial ODE system with nine species concentrations and 12 rate constants (parameters). The objective is to study the parameter space w.r.t. points of multistationarity. In the situation investigated here, real algebraic geometry allows rather precise statements on the parameter areas of monostationarity and multistationarity, their boundaries and their connectedness. Suitable polynomials, their signs, Newton polytopes and cylindrical algebraic decompositions play a decisive role, as well as application of symbolic algorithms. The approach explored here is relevant not only for the system itself, but lends itself also to test the examination of similar models.

MSC:

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92C42 Systems biology, networks
92C40 Biochemistry, molecular biology
92C37 Cell biology
14P10 Semialgebraic sets and related spaces
37N25 Dynamical systems in biology
92E20 Classical flows, reactions, etc. in chemistry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
34A34 Nonlinear ordinary differential equations and systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Software:

CoNtRol; crntwin; MESSI
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References:

[1] Bihan, F.; Dickenstein, A.; Giaroli, M., Lower bounds for positive roots and regions of multistationarity in chemical reaction networks, J. Algebra, 542, 367-411 (2020) · Zbl 1453.92120
[2] Chen, C., Davenport, J.H., Moreno Maza, M., Xia, B., Xiao, R.: Computing with semi-algebraic sets represented by triangular decomposition. In: Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation (ISSAC 2011), pp. 75-82. ACM Press, Boca Raton (2011) · Zbl 1323.14029
[3] Cohen, P., The structure and regulation of protein phosphatases, Annu. Rev. Biochem., 58, 453-508 (1989)
[4] Conradi, C.; Feliu, E.; Mincheva, M., On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle, Math. Biosci. Eng., 1, 17, 494-513 (2020) · Zbl 1470.92116
[5] Conradi, C.; Feliu, E.; Mincheva, M.; Wiuf, C., Identifying parameter regions for multistationarity, PLoS Comput. Biol., 13, 10, e1005751 (2017)
[6] Conradi, C.; Flockerzi, D., Multistationarity in mass action networks with applications to ERK activation, J. Math. Biol., 65, 1, 107-156 (2012) · Zbl 1278.37058
[7] Conradi, C.; Flockerzi, D.; Raisch, J.; Stelling, J., Subnetwork analysis reveals dynamic features of complex (bio)chemical networks, Proc. Nat. Acad. Sci., 104, 49, 19175-80 (2007)
[8] Conradi, C.; Mincheva, M., Catalytic constants enable the emergence of bistability in dual phosphorylation, J. R. S. Interface, 11, 20140158 (2014)
[9] Conradi, C.; Mincheva, M.; Shiu, A., Emergence of oscillations in a mixed-mechanism phosphorylation system, Bull. Math. Biol., 81, 6, 1829-1852 (2019) · Zbl 1415.92092
[10] Conradi, C.; Shiu, A., Dynamics of post-translational modification systems: recent progress and future directions, Biophys. J., 114, 3, 507-515 (2018)
[11] Craciun, G.; Helton, JW; Williams, RJ, Homotopy methods for counting reaction network equilibria, Math. Biosci., 216, 2, 140-149 (2008) · Zbl 1153.92015
[12] Donnell, P.; Banaji, M.; Marginean, A.; Pantea, C., Control: an open source framework for the analysis of chemical reaction networks, Bioinformatics, 30, 11, 1633-1634 (2014)
[13] Dressler, M.; Iliman, S.; de Wolff, T., An approach to constrained polynomial optimization via nonnegative circuit polynomials and geometric programming, J. Symb. Comput., 91, 149-172 (2019) · Zbl 1411.12003
[14] Dressler, M.; Iliman, S.; de Wolff, T., A positivstellensatz for sums of nonnegative circuit polynomials, SIAM J. Appl. Algebra Geom., 1, 1, 536-555 (2017) · Zbl 1372.14051
[15] Ellison, P., Feinberg, M., Ji, H., Knight, D.: Chemical Reaction Network Toolbox, Version 2.2. http://www.crnt.osu.edu/CRNTWin (2012)
[16] Feinberg, M., The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal., 132, 4, 311-370 (1995) · Zbl 0853.92024
[17] Feliu, E., Injectivity, multiple zeros, and multistationarity in reaction networks, Proc. R. Soc. A., 471, 20140530 (2015) · Zbl 1371.92147
[18] Feliu, E.; Wiuf, C., Enzyme-sharing as a cause of multi-stationarity in signalling systems, J. R. S. Interface, 9, 71, 1224-32 (2012)
[19] Feliu, E.; Wiuf, C., Variable elimination in post-translational modification reaction networks with mass-action kinetics, J. Math. Biol., 66, 1, 281-310 (2013) · Zbl 1256.92019
[20] Feng, S.; Sáez, M.; Wiuf, C.; Feliu, E.; Soyer, OS, Core signalling motif displaying multistability through multi-state enzymes, J. R. S. Interface, 13, 123, 20160524 (2016)
[21] Flockerzi, D.; Holstein, K.; Conradi, C., N-site phosphorylation systems with 2N-1 steady states, Bull. Math. Biol., 76, 8, 1892-1916 (2014) · Zbl 1300.92028
[22] Hell, J.; Rendall, AD, A proof of bistability for the dual futile cycle, Nonlinear Anal. Real World Appl., 24, 175-189 (2015) · Zbl 1331.34089
[23] Hell, J., Rendall, A.D.: Dynamical features of the map kinase cascade. In: Modeling Cellular Systems, volume 11. Springer, Cham (2017) · Zbl 1352.92072
[24] Huang, CY; Ferrell, JE, Ultrasensitivity in the mitogen-activated protein kinase cascade, Proc. Natl. Acad. Sci. USA, 93, 10078-10083 (1996)
[25] Iliman, S., de Wolff, T.: Amoebas, nonnegative polynomials and sums of squares supported on circuits. Res. Math. Sci. 3(9) (2016) · Zbl 1415.11071
[26] Kurpisz, A., de Wolff, T.: New dependencies of hierarchies in polynomial optimization. In: Davenport, J.H., Wang, D., Kauers, M., Bradford, R.J. (Eds.) Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation (ISSAC 2019). Beijing, China, July 15-18, 2019, pp. 251-258. ACM (2019) · Zbl 1467.90074
[27] Laurent, M.; Kellershohn, N., Multistability: a major means of differentiation and evolution in biological systems, Trends Biochem. Sci., 24, 11, 418-422 (1999)
[28] Markevich, NI; Hoek, JB; Kholodenko, BN, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell Biol., 164, 353-359 (2004)
[29] Motzkin, T.S.: The arithmetic-geometric inequality. In: Inequalities: Proceedings, Volume 1, chapter 10, pp. 203-224. Academic Press, Boca Raton (1967)
[30] Ozbudak, EM; Thattai, M.; Lim, HN; Shraiman, BI; Van Oudenaarden, A., Multistability in the lactose utilization network of Escherichia coli, Nature, 427, 6976, 737-740 (2004)
[31] Pantea, C.; Koeppl, H.; Craciun, G., Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks, Discrete Contin. Dyn. Syst. Ser. B, 17, 6, 2153-2170 (2012) · Zbl 1253.80023
[32] Pérez Millán, M.; Dickenstein, A., The structure of MESSI biological systems, SIAM J. Appl. Dyn. Syst., 17, 1650-1682 (2018) · Zbl 1395.92071
[33] Pérez Millán, M.; Dickenstein, A.; Shiu, A.; Conradi, C., Chemical reaction systems with toric steady states, Bull. Math. Biol., 74, 1027-1065 (2012) · Zbl 1251.92016
[34] Qiao, L.; Nachbar, RB; Kevrekidis, IG; Shvartsman, SY, Bistability and oscillations in the Huang-Ferrell model of MAPK signaling, PLoS Comput. Biol., 3, 9, 1819-1826 (2007)
[35] Reznick, B., Forms derived from the arithmetic-geometric inequality, Math. Ann., 283, 3, 431-464 (1989) · Zbl 0637.10015
[36] Thomson, M.; Gunawardena, J., The rational parameterization theorem for multisite post-translational modification systems, J. Theor. Biol., 261, 626-636 (2009) · Zbl 1403.92085
[37] Thomson, M.; Gunawardena, J., Unlimited multistability in multisite phosphorylation systems, Nature, 460, 274-277 (2009)
[38] Torres, A., Feliu, E.: Symbolic Proof of Bistability in Reaction Networks. SIAM J Appl. Dyn. Syst. To appear (2020) · Zbl 1465.37105
[39] Vol’pert, AI, Differential equations on graphs. Math. USSR-Sb, 17, 571-582 (1972)
[40] Wang, L.; Sontag, ED, On the number of steady states in a multiple futile cycle, J. Math. Biol., 57, 1, 29-52 (2008) · Zbl 1141.92022
[41] Wiuf, C.; Feliu, E., Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species, SIAM J. Appl. Dyn. Syst., 12, 1685-1721 (2013) · Zbl 1278.92012
[42] Xiong, W.; Ferrell, JE Jr, A positive-feedback-based bistable ‘memory module’ that governs a cell fate decision, Nature, 426, 6965, 460-465 (2003)
[43] Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, Vol. 152. Springer, New York (1995) · Zbl 0823.52002
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