Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Bifurcations of nonlinear reaction-diffusion systems in oblate spheroids

  • 50 Accesses

  • 9 Citations

Abstract

Spontaneous pattern formation may arise in biological systems as primary and secondary bifurcations to nonlinear parabolic partial differential equations describing chemical reaction-diffusion systems. Bipolarity in mitosis and cleavage planes in cytokinesis may be related to this formation of prepatterns. Three dimensional prepatterns are investigated, as they emerge in flattened spheres (i.e. oblate spheroids). Pattern sequences and selection rules are established numerically. The results confirm previously recorded results of the spherical and prolate regions, upon which a prepattern theory of mitosis and cytokinesis is based. Especially, the phenomenon of 90 degree axis tilting and the formation of a highly symmetrical saddle shaped pattern, crucial for the prepattern theory of mitosis and cytokinesis, is examined. Present results show, that these phenomena are stabilized in oblate spheroids. The bipolar “mitosis” prepattern is found as well, although the polar axis may appear with an angle toward the axis of the oblate spheroid. These results are thus further support for the prepattern theory of mitosis and cytokinesis.

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

References

  1. 1.

    Billing, G. D., Hunding, A.: Bifurcation analysis of non-linear reaction-diffusion systems: Dissipative structures in a sphere. J. Chem. Phys. 69, 3603–3610 (1978)

  2. 2.

    Boiteux, A., Hess, B.: Spatial dissipative structures in yeast extracts. Ber. Bunsen-Ges. Phys. Chem. 84, 392–398 (1980)

  3. 3.

    Bunow, B., Kernevez, J.-P., Joly, G., Thomas, D.: Pattern formation by reaction-diffusion instabilities: Application to morphogenesis in Drosophila. J. Theoret. Biology 84, 629–649 (1980)

  4. 4.

    Bus, J. C. P., Dekker, T. J.: Two efficient algorithms with guaranteed convergence for finding a zero of a function. ACM Trans. Math. Software 1, 330–345 (1975)

  5. 5.

    Flammer, C.: Spheroidal wave functions. Stanford, CA: Stanford University Press 1957

  6. 6.

    Goodwin, B. C., Treanor, L. E. H.: A field description of the cleavage process in embryogenesis. J. Theoret. Biology 86, 757–770 (1980)

  7. 7.

    Haken, H.: Synergetics. Bedin-Heidelberg-New York: Springer 1978

  8. 8.

    Hodge, D. B.: Eigenvalue and eigenfunctions of the spheroidal wave equation. J. Math. Phys. 11, 2308–2312 (1970)

  9. 9.

    Hunding, A.: Dissipative structures in reaction-diffusion systems: Numerical determination of bifurcations in the sphere. J. Chem. Phys. 72, 5241–5248 (1980)

  10. 10.

    Hunding, A.: Possible prepatterns governing mitosis: The mechanism of spindle-free chromosome movement in Aulacantha Scolymantha. J. Theoret. Biology 89, 353–385 (1981)

  11. 11.

    Hunding, A.: Bifurcations of nonlinear reaction-diffusion systems in prolate spheroids. J. Math. Biol. 17, 223–239 (1983)

  12. 12.

    Hunding, A., Billing, G. D.; Spontaneous pattern formation in spherical nonlinear reaction-diffusion systems. Selection rules favor the bipolar “mitosis” pattern. J. Chem. Phys. 75, 486–488 (1981)

  13. 13.

    Kauffman, S. A., Shymko, R. M., Trabert, K.: Control of sequential compartment formation in Drosophila. Science 199, 259–270 (1978)

  14. 14.

    Meinhardt, H.: Models of biological pattern formation. New York-San Francisco-London: Academic Press 1982

  15. 15.

    Morse, P. M., Feshbach, H.: Methods of theoretical physics. New York-Toronto-London: McGraw-Hill 1953

  16. 16.

    Nicolis, G., Prigogine, I.: Self-organization in non-equilibrium systems. New York-London-Sydney-Toronto: Wiley

  17. 17.

    Sel'kov, E. E.: Self-oscillations in glycolysis. Eur. J. Biochem. 4, 79–86 (1968)

  18. 18.

    Turing, A. M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London, Ser. B237, 37–72 (1952)

  19. 19.

    Tyson, J. J.: The Belousov-Zhabotinskii reaction. Lecture notes in biomathematics, vol. 10. Berlin-Heidelberg-New York: Springer 1976

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hunding, A. Bifurcations of nonlinear reaction-diffusion systems in oblate spheroids. J. Math. Biology 19, 249–263 (1984). https://doi.org/10.1007/BF00277098

Download citation

Key words

  • Mitosis
  • prepatterns
  • morphogenesis