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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 494-513. doi: 10.3934/mbe.2020027
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On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle

  • 1.
    Life Science Engineering, HTW Berlin, Wilhelminenhoftstr. 75, 10459 Berlin, Germany
  • 2.
    Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
  • 3.
    Department of Mathematical Sciences, Northern Illinois University, 1425 W. Lincoln Hwy., DeKalb IL 60115, USA
  • Received: 15 May 2019 Accepted: 23 September 2019 Published: 17 October 2019

  • Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parametrize the steady state and Hurwitz matrices.

    Citation: Carsten Conradi, Elisenda Feliu, Maya Mincheva. On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 494-513. doi: 10.3934/mbe.2020027

    Related Papers:

  • Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parametrize the steady state and Hurwitz matrices.


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    [1] C. Conradi and A. Shiu, Dynamics of post-translational modification systems: recent progress and future directions, Biophys. J., 114 (2018), 507-515.
    [2] T. Suwanmajo and J. Krishnan, Exploring the intrinsic behaviour of multisite phosphorylation systems as part of signalling pathways, J. R. Soc. Interface, 15 (2018), 20180109.
    [3] J. Gunawardena, Multisite protein phosphorylation makes a good threshold but can be a poor switch, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 14617-14622.
    [4] C. Salazar and T. Hofer, Multisite protein phosphorylation-from molecular mechanisms to kinetic models, FEBS J., 276 (2009), 3177-3198.
    [5] M. Thomson and J. Gunawardena, Unlimited multistability in multisite phosphorylation systems, Nature, 460 (2009), 274-277.
    [6] C. Conradi, D. Flockerzi and J. Raisch, Multistationarity in the activation of an MAPK: parametrizing the relevant region in parameter space, Math. Biosci., 211 (2008), 105-131.
    [7] C. Conradi and M. Mincheva, Catalytic constants enable the emergence of bistability in dual phosphorylation, J. R. S. Interface, 11.
    [8] J. Hell and A. D. Rendall, A proof of bistability for the dual futile cycle, Nonlinear Anal. Real World Appl., 24 (2015), 175-189.
    [9] D. Flockerzi, K. Holstein and C. Conradi, N-site Phosphorylation Systems with 2N-1 Steady States, Bull. Math. Biol., 76 (2014), 1892-1916.
    [10] L. Wang and E. D. Sontag, On the number of steady states in a multiple futile cycle, J. Math. Biol., 57 (2008), 29-52.
    [11] C. Conradi and A. Shiu, A global convergence result for processive multisite phosphorylation systems, Bull. Math. Biol., 77 (2015), 126-155.
    [12] C. Conradi, M. Mincheva and A. Shiu, Emergence of oscillations in a mixed-mechanism phosphorylation system, Bull. Math. Biol., 81 (2019), 1829-1852.
    [13] T. Suwanmajo and J. Krishnan, Mixed mechanisms of multi-site phosphorylation, J. R. Soc. Interface, 12 (2015), 20141405.
    [14] N. Obatake, A. Shiu, X. Tang and A. Torres, Oscillations and bistability in a model of ERK regulation, J. Math. Biol., (2019), https://doi.org/10.1007/s00285-019-01402-y.
    [15] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, 1995.
    [16] H. Errami, M. Eiswirth, D. Grigoriev, et al., Detection of hopf bifurcations in chemical reaction networks using convex coordinates, J. Comput. Phys., 291 (2015), 279-302.
    [17] M. El Kahoui and A. Weber, Deciding Hopf bifurcations by quantifier elimination in a softwarecomponent architecture, J. Symbolic Comput., 30 (2000), 161-179.
    [18] W. M. Liu, Criterion of hopf bifurcations without using eigenvalues, Journal of Mathematical Analysis and Applications, 182 (1994), 250-256.
    [19] X. Yang, Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order, Appl. Math. Lett., 15 (2002), 615-621.
    [20] B. L. Clarke, Stability of Complex Reaction Networks, 1-215, John Wiley & Sons, Ltd, 2007. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1002/9780470142622.ch1.
    [21] B. L. Clarke, Stoichiometric network analysis, Cell Biochem. Biophys., 12 (1988), 237-253.
    [22] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
    [23] B. D. Aguda and B. L. Clarke, Bistability in chemical reaction networks: Theory and application to the peroxidase-oxidase reaction, J. Chem. Phys., 87 (1987), 3461-3470.
    [24] K. Gatermann, M. Eiswirth and A. Sensse, Toric ideals and graph theory to analyze hopf bifurcations in mass action systems, J. Symbol. Comput., 40 (2005), 1361-1382.
    [25] C. Conradi and D. Flockerzi, Switching in mass action networks based on linear inequalities, SIAM J. Appl. Dyn. Syst., 11 (2012), 110-134.
    [26] E. Feliu, C. Lax, S. Walcher, et al., Quasi-steady state and singular perturbation reduction for reaction networks with non-interacting species, arXiv, 1908.11270.
    [27] A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, 3rd edition, Portland Press, London, 2004.
    [28] A. Goeke, S. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differ. Equations, 259 (2015), 1149-1180.
    [29] A. Goeke, S. Walcher and E. Zerz, Classical quasi-steady state reduction-a mathematical characterization, Phys. D, 345 (2017), 11-26.
    [30] H.-R. Tung, Precluding oscillations in Michaelis-Menten approximations of dual-site phosphorylation systems, Math. Biosci., 306 (2018), 56-59.
    [31] M. Banaji, Inheritance of oscillation in chemical reaction networks, Appl. Math. Comp., 325 (2018), 191-209.
    [32] C. Conradi, E. Feliu, M. Mincheva, et al., Identifying parameter regions for multistationarity, PLoS Comput. Biol., 13 (2017), e1005751.
    [33] A. Sadeghimanesh and E. Feliu, The multistationarity structure of networks with intermediates and a binomial core network, Bull. Math. Biol., 81 (2019), 2428-2462.
    [34] A. von Kamp, S. Thiele, O. Hädicke, et al., Use of CellNetAnalyzer in biotechnology and metabolic engineering, J. Biotechnol., 261 (2017), 221-228.
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