Skip to main content
SpringerLink
Log in

Wave speeds of density dependent Nagumo diffusion equations – inspired by oscillating gap-junction conductance in the islets of Langerhans

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract.

The equation u t =(D(u)u x ) x +f(u) arises in several biological examples and is known to have wave solutions for appropriate D and f. We give here a new formula for finding an approximation to the wave speed, relevant for comparing experiments with model simulations. This is done in details for the simple example D(u)=u+k and an N-shaped f, derived from a model of coupled pancreatic β-cells, where the coupling conductance follows the electrical activity as it is found in experiments. On the way, we claim that the wave speed does not depend on the parameter g K , ATP , mimicking the glucose concentration in the islet, in sharp contrast to the claim set forth in the article by Aslanidi et al. [4].

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andreu, E., Bernat, S., Sanchez-Andres, J.V.: Oscillation of gap junction electrical coupling in the mouse pancreatic islets of Langerhans, J. Physiol. 498, 753–761 (1997)

    CAS  Google Scholar 

  2. Aronson, D.G.: Density-Dependent Interaction-Diffusion Systems, in W.E. Stewart, W.H. Ray and C.C. Conley (eds.): Dynamics and Modelling of Reactive Systems, (Academic Press, New York 1980) pp. 161–176

  3. Ashcroft, F.M., Rorsman, P.: Electrophysiology of the pancreatic β-cell, Prog. Biophys. Mol. Biol. 54, 87–143 (1989)

    CAS  Google Scholar 

  4. Aslanidi, O.V., Mornev, O.A., Skyggebjerg, O., Arkhammar, P., Thastrup, O., Sørensen, M.P., Christiansen, P.L., Conradsen, K., Scott, A.C.: Excitation Wave Propagation as a Possible Mechanism for Signal Transmission in Pancreatic Islets of Langerhans. Biophys. J. 80, 1195–1209 (2001)

    CAS  PubMed  Google Scholar 

  5. Aslanidi, O.V., Mornev, O.A., Vesterager, M., Sørensen, M.P., Christiansen, P.L.: A Model for Glucose-induced Wave Propagation in Pancreatic Islets of Langerhans. J. Theor. Biol. 215, 273–286 (2002)

    CAS  PubMed  MathSciNet  Google Scholar 

  6. Charollais, A., Gjinovci, A., Huarte, J., Bauquis, J., Nadal, A., Martín, F., Andreu, E., Sáfnchez-Andrés, J.V., Calabrese, A., Bosco, D., Soria, B., Wollheim, C.B., Herrera, P.L., Meda, P.: Junctional communication of pancreatic β cells contributes to the control of insulin secretion and glucose tolerance, J. Clin. Invest. 106, 235–243 (2000)

    CAS  Google Scholar 

  7. Eddlestone, G.T., Goncalves, A., Bangham, J.A.: Electrical coupling between cells in islets of Langerhans from mouse. J. Membr. Biol. 77, 1–14 (1984)

    CAS  PubMed  Google Scholar 

  8. Engler, H.: Relations Between Travelling Wave Solutions of Quasilinear Parabolic Equations, Proc. Am. Math. Soc. 93, 297–302 (1985)

    Google Scholar 

  9. Gilding, B.H., Kersner, R.: Travelling waves in nonlinear diffusion-convection- reaction. (Memorandum No. 1585, Faculty of Mathematical Sciences, University of Twente 2001)

  10. Gilon, P., Henquin, J.-C.: Influence of Membrane Potential Changes on Cytoplasmic Ca2+ Concentration in an Electrically Excitable Cell, the Insulin-secreting Pancreatic B-cell, J. Biol. Chem. 267, 20713–20720 (1992)

    CAS  Google Scholar 

  11. Goforth, P.B., Bertram, R., Khan, F.A., Zhang, M., Sherman, A., Satin, L.S.: Calcium-activated K+ Channels of Mouse β-cells are controlled by Both Store and Cytoplasmic Ca2+: Experimental ad Theoretical Studies, J. Gen. Physiol. 120, 307–322 (2002)

    CAS  Google Scholar 

  12. Hadeler, K.P.: Travelling fronts and free boundary value problems, in Albrecht, J., Collatz, L., and Hoffmann, K.-H.: Numerical Treatment of Free Boundary Value Problems (Birkhauser Verlag, Basel, 1982) pp. 90–107

  13. Hadeler, K.P.: Free boundary problems in biological models, in Fasano, A., and Primicerio, M. (eds.): Free Boundary Problems: Theory and Applications, Volume II. (Pitman Advanced Publishing Program, Boston, 1983) pp. 664–671

  14. Keizer, J., Magnus, G.: ATP-sensitive potassium channel and bursting in the pancreatic beta cell. A theoretical study. Biophys. J. 56, 229–242 (1989)

    CAS  Google Scholar 

  15. Mears, D., Sheppard, N., Atwater, I., Rojas, E.: Magnitude and Modulation of Pancreatic β-Cell Gap Junction Electrical Conductance In Situ, J. Membr. Biol. 146, 163–176 (1995)

    CAS  Google Scholar 

  16. Misler, S., Falke, L.C., Gillis, K., McDaniel, M.L.: A metabolite-regulated potassium channel in rat pancreatic B cells, PNAS 83, 7119–7123 (1986)

    Google Scholar 

  17. Mornev, O.A.: Modification of the Biot method on the basis of the principle of minimum dissipation (with an application to the problem of propagation of nonlinear concentration waves in an autocatalytic medium), Russian J. Phys. Chem. 72, 112–118 (1998)

    Google Scholar 

  18. Pedersen, M.G.: Homogenization of heterogeneously coupled bistable ODE’s - applied to excitation waves in pancreatic isles of Langerhans, J. Biol. Phys. In press.

  19. Perez-Armendariz, M., Roy, C., Spray, D.C., Bennett, M.V.: Biophysical properties of gap junctions between freshly dispersed pairs of mouse pancreatic beta cells, Biophys. J. 59, 76–92 (1991)

    CAS  Google Scholar 

  20. Prentki, M., Matschinsky, F.M.: Ca2+, cAMP, and Phospholipid-Derived Messengers in Coupling Mechanisms of Insulin Secretion, Physiol. Rev. 67, 1185–1248 (1987)

    CAS  Google Scholar 

  21. Rutter, G.A.: Nutrient secretion coupling in the pancreatic islet β-cell: recent advances, Mol Aspects Med. 22, 247–284 (2001)

    Google Scholar 

  22. Sánchez-Garduño, F., Maini, P.K.: Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations, J. Math. Biol. 35, 713–728 (1997)

    Google Scholar 

  23. Scott, A.C.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. (Oxford University Press, Oxford 1999)

  24. Sherman, A.: Calcium and membrane potential oscillations in pancreatic β-cells, in Othmer, H.G., Adler, F.R., Lewis, M.A., and Dallon, J.C. (eds.): Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology. (Prentice-Hall, New York 1997) pp. 199–217

  25. Valdeolmillos, M., Santos, R.M., Contreras, D. Soria, B., Rosario, L.M.: Glucose-induced oscillations of intracellular Ca2+ concentration resembling bursting electrical activity in single mouse islets of Langerhans, FEBS Lett. 259, 19–23 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Morten Gram Pedersen

Authors
  1. Morten Gram Pedersen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Morten Gram Pedersen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pedersen, M. Wave speeds of density dependent Nagumo diffusion equations – inspired by oscillating gap-junction conductance in the islets of Langerhans. J. Math. Biol. 50, 683–698 (2005). https://doi.org/10.1007/s00285-004-0304-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-004-0304-4

Keywords

Search

Navigation