Skip to main content
SpringerLink
Log in

Dynamical Features of the MAP Kinase Cascade

  • Chapter
  • First Online:
Modeling Cellular Systems

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 11))

Abstract

The MAP kinase cascade is an important signal transduction system in molecular biology for which a lot of mathematical modelling has been done. This paper surveys what has been proved mathematically about the qualitative properties of solutions of the ordinary differential equations arising as models for this biological system. It focuses, in particular, on the issues of multistability and the existence of sustained oscillations. It also gives a concise introduction to the mathematical techniques used in this context, bifurcation theory and geometric singular perturbation theory, as they relate to these specific examples. In addition further directions are presented in which the applications of these techniques could be extended in the future.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Altan-Bonnet, G. and Germain, R.N. 2005 Modelling T cell antigen discrimination based on feedback control of digital ERK responses. PloS Biol. 3 (11): e356.

    Google Scholar 

  2. Amin, M., Porter, S. L. and Soyer, O. S. 2013 Split histidine kinases enable ultrasensitivity and bistability in two-component signalling networks. PloS Comp. Biol. 9 (3): e1002949.

    Google Scholar 

  3. Angeli, D., Ferrell, J. E. Jr. and Sontag, E. D. 2004 Detection of multistability, bifurcation and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. USA 101, 1822–1827.

    Google Scholar 

  4. Angeli, D., Hirsch, M. and Sontag, E. 2009 Attractors in coherent systems of differential equations. J. Diff. Eq. 246, 3058–3076.

    Google Scholar 

  5. Angeli, D. and Sontag, E. D. 2006 Translation-invariant monotone systems and a global convergence result for enzymatic futile cycles. Nonlin. Anal. RWA 9, 128–140.

    Google Scholar 

  6. Conley, C. 1978 Isolated invariant sets and the Morse index. CBMS Regional Conference Series in Mathematics 38.

    Google Scholar 

  7. Conradi, C. and Mincheva, M. 2014 Catalytic constants enable the emergence of bistability in dual phosphorylation. Royal Society Interface 11, 20140158.

    Google Scholar 

  8. Conradi, C., Saez-Rodriguez, J., Gilles, E.-D. and Raisch, J. 2005 Using chemical reaction network theory to discard a kinetic mechanism hypothesis. IEEE Syst. Biol. 152, 243–248.

    Google Scholar 

  9. Conradi, C. and Shiu, A. 2015 A global convergence result for processive multisite phosphorylation systems. Bull. Math. Biol. 77, 126–155.

    Google Scholar 

  10. Errami, H., Eiswith, M., Grigoriev, D., Seiler, W. M. and Weber, A. 2015 Detection of Hopf bifurcations in chemical reaction networks using convex coordinates. J. Comp. Phys. 291, 279–302.

    Google Scholar 

  11. Feinberg, M. 1980 Lectures on chemical reactions networks. Available at https://crnt.osu.edu/lectures-chemical-reaction-networks.

  12. Fenichel, N. 1979 Geometric perturbation theory for ordinary differential equations. J. Diff. Eq. 31, 53–98.

    Google Scholar 

  13. François, P., Voisinne, G., Siggia, E. D., Altan-Bonnet, G. and Vergassola, M. 2013 Phenotypic model for early T cell activation displaying sensitivity, specificity and antagonism. Proc. Natl. Acad. Sci. USA. 110 E888–E897.

    Google Scholar 

  14. Gedeon, T. 2010 Oscillations in monotone systems with a negative feedback. SIAM J. Dyn. Sys. 9, 84–112.

    Google Scholar 

  15. Gedeon, T. and Sontag, E. D. 2007 Oscillations in multi-stable monotone systems with slowly varying feedback. J. Diff. Eq. 239, 273–295.

    Google Scholar 

  16. Goldbeter, A. and Koshland, D. E., Jr. 1981 An amplified sensitivity arising from covalent modification in biological systems. Proc. Natl. Acad. Sci. USA 78, 6840–6844.

    Google Scholar 

  17. Guckenheimer, J. and Holmes, P. 1983 Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer.

    Google Scholar 

  18. Gunawardena, J. 2007 Distributivity and processivity in multisite phosphorylation can be distinguished through steady-state invariants. Biophys. J. 93, 3828–3834.

    Google Scholar 

  19. Hell, J. and Rendall, A. D. 2015 A proof of bistability for the dual futile cycle. Nonlin. Anal. RWA. 24, 175–189.

    Google Scholar 

  20. Hell, J. and Rendall, A. D. 2016 Sustained oscillations in the MAP kinase cascade. Math. Biosci. 282, 162–173.

    Google Scholar 

  21. Huang, C.-Y. F. and Ferrell, J. E., Jr. 1996 Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc. Natl. Acad. Sci. USA 93, 10078–10083.

    Google Scholar 

  22. Igoshin, O. A., Alves, R. and Savageau, M. A. 2008 Hysteretic and graded responses in bacterial two-component signal transduction. Mol. Microbiol. 68, 1196–1215.

    Google Scholar 

  23. Jolley, C. C., Ode, K. L. and Ueda, H. R. 2012 A design principle for a posttranslational biochemical oscillator. Cell Reports 2, 938–950.

    Google Scholar 

  24. Kholodenko, B. N. 2000 Negative feedback and ultrasensitivity can bring about oscillations in the mitogen activated protein kinase cascades. Eur. J. Biochem. 267, 1583–1588.

    Google Scholar 

  25. Kothamachu, V.B., Feliu, E., Wiuf, C., Cardelli, L., Soyer, O.S. 2013 Phosphorelays provide tunable signal processing capabilities for the cell. PLoS Comp. Biol. 9(11):e1003322.

    Google Scholar 

  26. Kuznetsov, Y. 1995 Elements of Bifurcation Theory. Springer, Berlin.

    Google Scholar 

  27. Legewie, S., Schoeberl, B., Blüthgen, N, and Herzel, H. 2007 Competing docking interactions can bring about bistability in the MAPK cascade. Biophys. J. 93, 2279–2288.

    Google Scholar 

  28. Liu, P., Kevrekidis, I. G. and Shvartsman, S. V. 2011 Substrate-dependent control of ERK phosphorylation can lead to oscillations. Biophys. J. 101, 2572–2581.

    Google Scholar 

  29. Markevich, N. I., Hoek, J. B. and Kholodenko, B. N. 2004 Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J. Cell Biol. 164, 353–359.

    Google Scholar 

  30. Nakajima, T., Satomi, Y., Kitayama, Y., Terauchi, K., Kiyohara, R., Takao, T. and Kondo, T. 2005 Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro. Science 308, 414–415.

    Google Scholar 

  31. Ortega, F., Garcés, J. L., Mas, F., Kholodenko, B. N. and Cascante, M. 2006 Bistability from double phosphorylation in signal transduction. FEBS J. 273, 3915–3926.

    Google Scholar 

  32. Prabakaran, S., Gunawardena, J. and Sontag, E. D. 2014 Paradoxical results in perturbation-based network reconstruction. Biophys. J. 106, 2720–2728.

    Google Scholar 

  33. Qiao, L., Nachbar, R. B., Kevrekidis, I. G. and Shvartsman, S. Y. 2007 Bistability and oscillations in the Huang-Ferrell model of MAPK signaling. PloS Comp. Biol. 3, 1819–1826.

    Google Scholar 

  34. Rendall, A. D. 2012 Mathematics of the NFAT signalling pathway. SIAM J. Appl. Dyn. Sys. 11, 988–1006.

    Google Scholar 

  35. Richard, A. and Comet, J. P. 2010 Stable periodicity and negative circuits in differential systems. J. Math. Biol. 63, 593–600.

    Google Scholar 

  36. Shankaran, H., Ippolito, D. L., Chrisler, W. B., Resat, H., Bollinger, N., Opresko, L. K. and Wiley, H. S. 2009 Rapid and sustained nuclear-cytoplasmic ERK oscillations induced by epidermal growth factor. Mol. Sys. Biol. 5, 332.

    Google Scholar 

  37. Shinar, G. and Feinberg, M. 2010 Structural sources of robustness in biochemical reaction networks. Science 327, 1389–1391.

    Google Scholar 

  38. Sontag E. D. 2001 Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automat. Contr. 46, 1028–1047.

    Google Scholar 

  39. Stock, A. M., Robinson, V. L. and Goudreau, P. N. 2000 Two-component signal transduction. Ann. Rev. Biochem. 69, 183–215.

    Google Scholar 

  40. Vanderbauwhede, A. 1989 Centre Manifolds, Normal Forms and Elementary Bifurcations. Dynamics Reported 2, 89–169.

    Google Scholar 

  41. Ventura, A. C., Sepulchre, J.-A. Merajver, S. D. 2008 A hidden feedback in signalling cascades is revealed. PLoS Comp. Biol. 4(3):e1000041.

    Google Scholar 

  42. Ventura, A. C. and Sepulchre, J.-A. 2013 Intrinsic feedbacks in MAPK signalling cascades lead to bistability and oscillations. Acta Biotheor. 61, 59–78.

    Google Scholar 

  43. Wang, L. and Sontag, E. D. 2008 On the number of steady states in a multiple futile cycle. J. Math. Biol. 57, 29–52.

    Google Scholar 

  44. Wang, L. and Sontag, E. D. 2008 Singularly perturbed monotone systems and an application to double phosphorylation cycles. J. Nonlin. Sci. 18, 527–550.

    Google Scholar 

  45. Widmann, C., Gibson, G., Jarpe, M. B. and Johnson, G. L. 1999 Mitogen-activated protein kinase: conservation of a three-kinase module from yeast to human. Physiol. Rev. 79, 143–180.

    Google Scholar 

  46. Zumsande, M. and Gross, T. 2010 Bifurcations and chaos in the MAPK signalling cascade. J. Theor. Biol. 265, 481–491.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Für Mathematik, Freie Universität Berlin, Arnimallee 7, 14195, Berlin, Germany

    Juliette Hell

  2. Institut Für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099, Mainz, Germany

    Alan D. Rendall

Authors
  1. Juliette Hell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alan D. Rendall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Juliette Hell or Alan D. Rendall .

Editor information

Editors and Affiliations

  1. Heidelberg University, Heidelberg, Germany

    Frederik Graw

  2. Heidelberg University, Heidelberg, Germany

    Franziska Matthäus

  3. Heidelberg University, Heidelberg, Germany

    Jürgen Pahle

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hell, J., Rendall, A.D. (2017). Dynamical Features of the MAP Kinase Cascade. In: Graw, F., Matthäus, F., Pahle, J. (eds) Modeling Cellular Systems. Contributions in Mathematical and Computational Sciences, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-45833-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45833-5_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45831-1

  • Online ISBN: 978-3-319-45833-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation