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An all-encompassing global convergence result for processive multisite phosphorylation systems. (English) Zbl 1379.92013
Summary: Phosphorylation, the enzyme-mediated addition of a phosphate group to a molecule, is a ubiquitous chemical mechanism in biology. Multisite phosphorylation, the addition of phosphate groups to multiple sites of a single molecule, may be distributive or processive. Distributive systems, which require an enzyme and substrate to bind several times in order to add multiple phosphate groups, can be bistable. Processive systems, in contrast, require only one binding to add all phosphate groups, and were recently shown to be globally stable. However, this global convergence result was proven only for a specific mechanism of processive phosphorylation/dephosphorylation (namely, all catalytic reactions are reversible). Accordingly, we generalize this result to allow for processive phosphorylation networks in which each reaction may be irreversible, and also to account for possible product inhibition. We accomplish this by first defining an all-encompassing processive network that encapsulates all of these schemes, and then appealing to recent results of M. M. de Freitas et al. [J. Math. Biol. 74, No. 4, 887–932 (2017; Zbl 1362.34083)] that assert global convergence by way of monotone systems theory and network/graph reductions (corresponding to removing intermediate complexes). Our results form a case study into the question of when global convergence is preserved when reactions and/or intermediate complexes are added to or removed from a network.

MSC:
92C40 Biochemistry, molecular biology
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
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References:
[1] Gunawardena, J., Multisite protein phosphorylation makes a good threshold but can be a poor switch, Proc. Natl. Acad. Sci., 102, 41, 14617-14622, (2005)
[2] Patwardhan, P.; Miller, W. T., Processive phosphorylation: mechanism and biological importance, Cell. Signal., 19, 11, 2218-2226, (2007)
[3] Perez Millán, M.; Turjanski, A. G., MAPK’s networks and their capacity for multistationarity due to toric steady states, Math. Biosci., 262, 125-137, (2015) · Zbl 1315.92029
[4] Conradi, C.; Mincheva, M., Catalytic constants enable the emergence of bistability in dual phosphorylation, J. R. Soc. Interface, 11, 95, (2014)
[5] Holstein, K.; Flockerzi, D.; Conradi, C., Multistationarity in sequential distributed multisite phosphorylation networks, Bull. Math. Biol., 75, 11, 2028-2058, (2013) · Zbl 1283.92030
[6] Manrai, A. K.; Gunawardena, J., The geometry of multisite phosphorylation, Biophys. J., 95, 12, 5533-5543, (2008)
[7] Wang, L.; Sontag, E., On the number of steady states in a multiple futile cycle, J. Math. Biol., 57, 1, 29-52, (2008) · Zbl 1141.92022
[8] Hell, J.; Rendall, A. D., A proof of bistability for the dual futile cycle, Nonlinear Anal. - Real World Appl., 24, 175-189, (2015) · Zbl 1331.34089
[9] Markevich, N. I.; Hoek, J. B.; Kholodenko, B. N., Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell. Biol., 164, 353-359, (2004)
[10] Johnston, M. D., Translated chemical reaction networks, Bull. Math. Biol., 76, 6, 1081-1116, (2014) · Zbl 1297.92096
[11] Thomson, M.; Gunawardena, J., The rational parameterisation theorem for multisite post-translational modification systems, J. Theoret. Biol., 261, 4, 626-636, (2009) · Zbl 1403.92085
[12] Pérez Millán, M.; Dickenstein, A.; Shiu, A.; Conradi, C., Chemical reaction systems with toric steady states, Bull. Math. Biol., 74, 5, 1027-1065, (2012) · Zbl 1251.92016
[13] Conradi, C.; Shiu, A., A global convergence result for processive multisite phosphorylation systems, Bull. Math. Biol., 77, 1, 126-155, (2015) · Zbl 1334.92169
[14] Angeli, D.; Sontag, E. D., Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles, Nonlinear Anal. Real World Appl., 9, 1, 128-140, (2008) · Zbl 1401.92086
[15] Ali, M., New Approach to the Stability and Control of Reaction Networks, Ph.D. thesis, (2015), Imperial College London London
[16] Rao, S., Global stability of a class of futile cycles, J. Math. Biol., 74, 709-726, (2017) · Zbl 1361.92027
[17] M. Marcondes de Freitas, C. Wiuf, E. Feliu, Intermediates and Generic Convergence to Equilibria, arXiv preprint arXiv:1606.09480 (2016). · Zbl 1372.92036
[18] Suwanmajo, T.; Krishnan, J., Mixed mechanisms of multi-site phosphorylation, J. R. Soc. Interface, 12, 107, (2015)
[19] Donnell, P.; Banaji, M., Local and global stability of equilibria for a class of chemical reaction networks, SIAM J. Appl. Dyn. Syst., 12, 2, 899-920, (2013) · Zbl 1284.92120
[20] Angeli, D.; De Leenheer, P.; Sontag, E., Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates, J. Math. Biol., 61, 4, 581-616, (2010) · Zbl 1204.92038
[21] Ali Al-Radhawi, M.; Angeli, D., New approach to the stability of chemical reaction networks: piecewise linear in rates Lyapunov functions, IEEE Trans. Autom. Control, 61, 1, 76-89, (2016) · Zbl 1359.93440
[22] Gunawardena, J., Distributivity and processivity in multisite phosphorylation can be distinguished through steady-state invariants, Biophys. J., 93, 11, 3828-3834, (2007)
[23] Feliu, E.; Wiuf, C., Enzyme-sharing as a cause of multi-stationarity in signalling systems, J. R. Soc. Interface, 9, 71, 1224-1232, (2012)
[24] Aoki, K.; Yamada, M.; Kunida, K.; Yasuda, S.; Matsuda, M., Processive phosphorylation of ERK MAP kinase in Mammalian cells, Proc. Natl. Acad. Sci. USA, 108, 31, 12675-12680, (2011)
[25] Marcondes de Freitas, M.; Feliu, E.; Wiuf, C., Intermediates, catalysts, persistence, and boundary steady states, J. Math. Biol., 1-46, (2016)
[26] Angeli, D.; De Leenheer, P.; Sontag, E. D., A Petri net approach to the study of persistence in chemical reaction networks, Math. Biosci., 210, 2, 598-618, (2007) · Zbl 1133.92322
[27] Shiu, A.; Sturmfels, B., Siphons in chemical reaction networks, Bull. Math. Biol., 72, 6, 1448-1463, (2010) · Zbl 1198.92020
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