Lateral Inhibition through Delta-Notch Signaling: A Piecewise Affine Hybrid Model

  • Ronojoy Ghosh
  • Claire J. Tomlin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2034)

Abstract

Biological cell networks exhibit complex combinations of both discrete and continuous behaviors: indeed, the dynamics that govern the spatial and temporal increase or decrease of protein concentration inside a single cell are continuous differential equations, while the activation or deactivation of these continuous dynamics are triggered by discrete switches which encode protein concentrations reaching given thresholds. In this paper, we model as a hybrid system a striking example of this behavior in a biological mechanism called Delta-Notch signaling, which is thought to be the primary mechanism of cell differentiation in a variety of cell networks. We present results in both simulation and reachability analysis of this hybrid system.We emphasize how the hybrid system model is computationally superior (for both simulation and analysis) to other nonlinear models in the literature, without compromising faithful modeling of the biological phenomena.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lewis Wolpert. Principles of Development. Current Biology Ltd and Oxford University Press, 1998.Google Scholar
  2. 2.
    Julian Lewis. Notch signalling and the control of cell fate choices in vertebrates. Seminars in Cell & Developmental Biology, 9:583–589, 1998.CrossRefGoogle Scholar
  3. 3.
    G. Marnellos, G. A. Deblandre, E. Mjolsness, and C. Kintner. Delta-notch lateral inhibitory patterning in the emergence of ciliated cells in Xenopus: experimental observations and a gene network model. In Pacific Symposium on Biocomputing, pages 5:326–337, 2000.Google Scholar
  4. 4.
    Pascal O. Luthi, Bastien Chopard, Anette Preiss, and Jeremy J. Ramsden. A cellular automaton model for neurogenesis in Drosophila. DrosophilaPhysica D, (118):151–160, 1998.MATHCrossRefGoogle Scholar
  5. 5.
    G. Marnellos and E. Mjolsness. A gene network approach to modeling early neurogenesis in Drosophila. In Pacific Symposium on Biocomputing, pages 3:30–41, 1998.Google Scholar
  6. 6.
    Catherine Haddon, Yun-Jin Jiang, Lucy Smithers, and Julian Lewis. Delta-notch signalling and the patterning of sensory cell differentiation in the zebrafish ear: evidence from the mind bomb mutant. Development, 125:4637–4644, 1998.Google Scholar
  7. 7.
    Rebecca Crowe, Domingos Henrique, David Ish-Horowicz, and Lee Niswander. A new role for notch and delta in cell fate decisions: pattering the feather array. Development, 125:767–775, 1998.Google Scholar
  8. 8.
    Stacey S. Huppert, Thomas L. Jacobsen, and Marc A. T. Muskavitch. Feedback regulation is central to delta-notch signalling required for Drosophila wing vein morphogenesis. Development, 124:3283–3291, 1997.Google Scholar
  9. 9.
    Daniel J. Moloney, Vladislav M. Panin, Stuart H. Johnston, Jihua Chen, Li Shao, Richa Wilson, Yang Wang, Pamela Stanley, Kenneth D. Irvine, Robert S. Haltiwanger, and Thomas F. Vogt. Fringe is a glycosyltransferase that modifies notch. Nature, 406(27):369–375, July 2000.CrossRefGoogle Scholar
  10. 10.
    Sarah Bray. Notch signalling in Drosophila: three ways to use a pathway. Seminars in Cell & Developmental Biology, 9:591–597, 1998.CrossRefGoogle Scholar
  11. 11.
    A. M. Turing. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, 237(641):37–72, August 1952.CrossRefGoogle Scholar
  12. 12.
    George F. Oster. Lateral inhibition models of developmental processes. Mathematical Biosciences, 90:265–286, 1988.CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Joanne R. Collier, Nicholas A. M. Monk, Philip K. Maini, and Julian H. Lewis. Pattern formation by lateral inhibition with feedback: a mathematical model of delta-notch intercellular signalling. J. theor. Biol., (183):429–446, 1996.CrossRefGoogle Scholar
  14. 14.
    Eric Mjolsness, David H. Sharp, and John Reinitz. A connectionist model of development. J. theor. Biol., 152:429–453, 1991.CrossRefGoogle Scholar
  15. 15.
    Harley H. McAdams and Adam Arkin. Simulation of prokaryotic genetic circuits. Annu. Rev. Biophys. Biomol. Struct., 27:199–224, 1998.CrossRefGoogle Scholar
  16. 16.
    John J. Tyson, Bela Novak, Garrett M. Odell, Kathy Chen, and C. Dennis Thron. Chemical kinetic theory: understanding cell-cycle regulation. TIBS, 21:89–96, March 1996.Google Scholar
  17. 17.
    C. Tomlin, J. Lygeros, and S. Sastry. Controller design for hybrid systems. Proceedings of the IEEE, 88(7), July 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Ronojoy Ghosh
    • 1
  • Claire J. Tomlin
    • 1
  1. 1.Stanford UniversityStanfordUSA

Personalised recommendations