Advertisement

Journal of Mathematical Biology

, Volume 47, Issue 3, pp 249–269 | Cite as

Wave bifurcation and propagation failure in a model of Ca2+ release

  • Y. Timofeeva
  • S. Coombes
Article

Abstract.

The De Young Keizer model for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP3) receptors. Systematic reductions of the kinetic schemes for IP3 dynamics have proved especially fruitful in understanding the transition from excitable to oscillatory behaviour. With the inclusion of diffusive transport of calcium ions the model also supports wave propagation. The analysis of waves, even in reduced models, is typically only possible with the use of numerical bifurcation techniques. In this paper we review the travelling wave properties of the biophysical De Young Keizer model and show that much of its behaviour can be reproduced by a much simpler Fire-Diffuse-Fire (FDF) type model. The FDF model includes both a refractory process and an IP3 dependent threshold. Parameters of the FDF model are constrained using a comprehensive numerical bifurcation analysis of solitary pulses and periodic waves in the De Young Keizer model. The linear stability of numerically constructed solution branches is calculated using pseudospectral techniques. The combination of numerical bifurcation and stability analysis also allows us to highlight the mechanisms that give rise to propagation failure. Moreover, a kinematic theory of wave propagation, based around numerically computed dispersion curves is used to predict waves which connect periodic orbits. Direct numerical simulations of the De Young Keizer model confirm this prediction. Corresponding travelling wave solutions of the FDF model are obtained analytically and are shown to be in good qualitative agreement with those of the De Young Keizer model. Moreover, the FDF model may be naturally extended to include the discrete nature of calcium stores within a cell, without the loss of analytical tractability. By considering calcium stores as idealised point sources we are able to explicitly construct solutions of the FDF model that correspond to saltatory periodic travelling waves.

Keywords

 Calcium waves Stability Lattice models 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allbritton, N., Meyer, T.: Localized calcium spikes and propagating calcium waves. Cell Calcium 14, 691–697 (1993)Google Scholar
  2. 2.
    Balmforth, N.J., Ierley, G.R., Spiegel, E.A.: Chaotic pulse trains. SIAM J. Appl. Math. 54, 1291–1334 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bär, M., Falcke, M., Levine, H., Tsimring, L.S.: Discrete stochastic modeling of calcium channel dynamics. Phys. Rev. Lett. 84, 5664–5667 (2000)CrossRefGoogle Scholar
  4. 4.
    Berridge, M.J.: Elementary and global aspects of Calcium signalling. J. Phys. 499, 291–306 (1997)Google Scholar
  5. 5.
    Bugrim, E.A., Zhabotinsky, Epstein, I.R.: Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites. Biophys. J. 73, 2897–2906 (1997)Google Scholar
  6. 6.
    Callamaras, N., Marchant, J.S., Sun, X.P., Parker, I.: Activation and co-ordination of InsP(3)-mediated elementary Ca2+ events during global Ca2+ signals in Xenopus oocytes. J. Phys. 509, 81–91 (1998)Google Scholar
  7. 7.
    Coombes, S.: The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of Ca2+ release. Bull. Math. Biol. 63, 1–20 (2001a)Google Scholar
  8. 8.
    Coombes, S.: From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves. Math. Biosci. 170, 155–172 (2001b)Google Scholar
  9. 9.
    De Young, G.W., Keizer, J.: A single pool IP3-receptor based model for agonist stimulated Ca2+ oscillations. Proc. National Acad. Sci. USA 89, 9895–9899 (1992)Google Scholar
  10. 10.
    Falcke, M., Tsimring, L., Levine, H.: Stochastic spreading of intracellular Ca2+ release. Phys. Rev. E 62, 2636–2643 (2000)CrossRefGoogle Scholar
  11. 11.
    Glendinning, P., Sparrow, C.: T-point: a codimension two heteroclinic bifurcation. J. Statis. Phys. 43, 479–488 (1986)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Izu, L.T., Wier, W.G., Balke, C.W.: Evolution of cardiac waves from stochastic calcium sparks. Biophys. J. 80, 103–120 (2001)Google Scholar
  13. 13.
    Keener, J., Sneyd, J.: Mathematical Physiology. Springer, 1998Google Scholar
  14. 14.
    Keizer, J., De Young, G.W.: Simplification of a realistic model of IP3-induced Ca2+ oscillations. J. Theoretical Biology 166, 431–442 (1994)CrossRefGoogle Scholar
  15. 15.
    Keizer, J.E., Smith, G.D., Ponce Dawson, S., Pearson, J.: Saltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75, 595–600 (1998)Google Scholar
  16. 16.
    Li, Y., Rinzel, J.: Equatons for InsP3 receptor mediated [Ca2+]i oscillations derived from a detailed kinetic model: A Hodgkin-Huxley like formalism. J. Theor. Biol. 166, 461–473 (1994)CrossRefGoogle Scholar
  17. 17.
    Lipp, P., Bootman, M.D.: To quark or to spark, that is the question. J. Phys. 502, 1–1 (1997)Google Scholar
  18. 18.
    Pencea, C.S., Hentschel, H.G.E.: Excitable calcium wave propagation in the presence of localized stores. Phys. Rev. E 62, 8420–8426 (2000)CrossRefGoogle Scholar
  19. 19.
    Rinzel, J., Maginu, K.: Non-equilibrium Dynamics in Chemical Systems. Kinematic analysis of wave pattern formation in excitable media. Springer-Verlag, 1984, pp. 107–113Google Scholar
  20. 20.
    Sneyd, J., LeBeau, A., Yule, D.: Traveling waves of calcium in pancreatic acinar cells: model construction and bifurcation analysis. Physica D 145, 158–179 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Sneyd, J., Keizer, J., Sanderson, M.J.: Mechanisms of calcium oscillations and waves: a quantitative analysis. FASEB 9, 1463–1472 (1995)Google Scholar
  22. 22.
    Tang, Y., Stephenson, J.L., Othmer, H.G.: Simplification and analysis of models of calcium dynamics based on IP3 – sensitive calcium channel kinetics. Biophys. J. 70, 246–263 (1996)Google Scholar
  23. 23.
    Trefethen, L.N.: Spectal Methods in MATLAB. SIAM, 2000Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK

Personalised recommendations