Skip to main content

Advertisement

SpringerLink
Log in

Oscillations and patterns in spatially discrete models for developmental intercellular signalling

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

Abstract.

We extend previous models for nearest neighbour ligand-receptor binding to include both lateral induction and inhibition of ligand and receptor production, and different geometries (strings of cells and hexagonal arrays, in addition to square arrays). We demonstrate the possibility of lateral inhibition giving patterns with a characteristic length scale of many cell diameters, when receptor production is included. In contrast, lateral induction combined with inhibition of receptor synthesis cannot give rise to a patterning instability under any circumstances. Interesting new dynamics include the analytical prediction and consequent numerical observation of spatiotemporal oscillations, this depends crucially on the production terms and on the relationship between the decay rates of ligand and free receptor. Our approach allows for a detailed comparison with the model for Delta-Notch interactions of Collier et al. [4], and we find that a formal reduction may be made only when the ligand receptor binding kinetics are very slow. Without such very slow receptor kinetics, spatial pattern formation via lateral inhibition in hexagonal cellular arrays requires significant activation of receptor production, a feature that is not apparent from previous analyses.

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. Brandt, R.E., Vickery, L.E.: Cooperativity and dimerization of recombinant human estrogen receptor hormone-binding domain. J. Biol. Chem. 272, 4843–4849 (1997)

    Article  Google Scholar 

  2. Clark, A.J.L., Ishii, S., Richert, N., Merlino, G.T., Pastan, I.: Epidermal growth factor regulates the expression of its own receptor. Proc. Nat. Acad. Sci. USA 82, 8374–8378 (1985)

    Google Scholar 

  3. Coffey, R.J., Derynck, R., Wilcox, J.N., Bringman, T.S., Goustin, A.S., Moses, H.L., Pittelkow, M.R.: Production and auto-induction of transforming growth Factor-α in human keratinocytes. Nature 328, 817–820 (1987)

    Article  Google Scholar 

  4. Collier, J.R., Monk, N.A.M., Maini, P.K., Lewis, J.H.: Pattern formation by lateral inhibition with feedback: A mathematical model of delta–notch intercellular signalling. J. Theor. Biol. 183, 429–446 (1996)

    Article  Google Scholar 

  5. de Celis, J.F., Bray, S.: Feedback mechanisms affecting notch activation at the dorsoventral boundary in the Drosophila wing. Development 124, 3241–3251 (1997)

    Google Scholar 

  6. Doedel, E.J., Champneys, A.R., Fairgrieve, T.R., Kuznetsov, Y.A., Sandstede, B., Wang, X.J., AUTO 97: Continuation and bifurcation software for ordinary differential equations, 1997 Available from http://indy.cs.concordia.ca/auto/main.html

  7. Fagotto, F., Gumbiner, B.M.: Cell contact dependent signalling. Dev. Biol. 180, 445–454 (1996)

    Article  Google Scholar 

  8. Fanger, B., Stephens, J.E., Staros, J.V.: High yield trapping of EGF-induced receptor dimers by cross-linking. FASEB 3, 71–75 (1989)

    Google Scholar 

  9. Fretto, L.J., Snape, A.J., Tomlinson, J.E., Seroogy, J.J., Wolf, D.L., LaRochelle, W.J., Giese, N.A.: Mechanism of platelet-derived growth-factor (PDGF)-AA, (PDGF)-AB, and (PDGF)-BB binding to alpha-PDGF and beta -PDGF receptor. J. Biol. Chem. 268, 3625–3631 (1993)

    Google Scholar 

  10. Hirata, H., Yoshiura, S., Ohtsuka, T., Bessho, Y., Harada, T., Yoshikawa, K., Kageyama, R.: Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science 298, 840–843 (2002)

    Article  Google Scholar 

  11. Hughes, A.D., Clunn, G.F., Refson, J., Demoliou-Mason, C.: Platelet-derived growth factor (PDGF): Actions and mechanisms in vascular smooth muscle. Gen. Pharmac. 27, 1079–1089 (1996)

    Article  Google Scholar 

  12. Huppert, S.S., Jacobson, T.L., Muskavitch, M.A.T.: Feedback regulation is central to Delta-Notch signalling required for Drosophila wing vein morphogenesis. Development 124, 3283–3291 (1997)

    CAS  PubMed  Google Scholar 

  13. Jiang, Y., Aerne, B.L., Smithers, L., Haddon, C., Ish-Horowicz, D., Lewis, J., Notch signalling and the synchronization of the somite segmentation clock. Nature 408, 475–479 (2000)

  14. Jouve, C., Palmeirim, I., Henrique, D., Beckers, J., Gossler, A., Ish-Horowicz, D., Pourquié, O.: Notch signalling is required for cyclic expression of the hairy-like gene HES1 in the presomitic mesoderm. Development 127, 1421–1429 (2000)

    Google Scholar 

  15. Kerszberg, M., Wolpert, L.: Mechanisms for positional signalling by morphogen transport: a theoretical study. J. Theor. Biol. 191, 103–114 (1998)

    Article  Google Scholar 

  16. Kimble, J., Simpson, P.: The LIN-12/Notch signalling pathway and its regulation. Ann. Rev. Cell Dev. Biol. 13, 333–361 (1997)

    Article  Google Scholar 

  17. Klueg, K.M., Parody, T.R., Muskavitch, M.A.: Complex proteolytic processing acts on Delta, a transmembrane ligand for Notch, during Drosophila development. Mol. Biol. Cell. 9, 1709–1723 (1998)

    Google Scholar 

  18. Lander, A.D., Nie, Q., Wan, F.W.M.: Do morphogen gradients arise by diffusion. Dev. Cell. 2, 785–796 (2002)

    Google Scholar 

  19. Lemmon, M.A., Pinchasi, D., Zhou, M., lax, I., Schlessinger, J.: Kit receptor dimerization is driven by bivalent binding of stem cell factor. J. Biol. Chem. 272, 6311–6317 (1997)

    Article  Google Scholar 

  20. Lewis, J.: Neurogenic genes and vertebrate neurogenesis. Curr. Op. Neurobiol. 6, 3–10 (1996)

    Article  Google Scholar 

  21. Lewis, J.: Notch signalling and the control of cell fate choices in vertebrates. Semin. Cell Dev. Biol. 9, 583–589 (1998)

    Article  Google Scholar 

  22. Massagué, J., Pandiella, A.: Membrane-anchored growth factors. Annu. Rev. Biochem. 62, 515–541 (1993)

    Article  Google Scholar 

  23. Meinhardt, H.: Models of biological pattern formation. Academic Press, London, 1982

  24. Meir, E., von Dassow, G., Munro, E., Odell, G.M.: Robustness, flexibility and the role of lateral inhibition in the neurogenic network. Curr. Biol. 12, 778–786 (2002)

    Article  Google Scholar 

  25. Monk, N.A.M.: Restricted-range gradients and travelling fronts in a model of juxtacrine cell relay. Bull. Math. Biol. 60, 901–918 (1998)

    Article  MATH  Google Scholar 

  26. Murray, J.D.: Mathematical Biology. Springer Verlag, Berlin, 1989

  27. Owen, M.R.: Waves and propagation failure in discrete space models with nonlinear coupling and feedback. Physica D 173, 59–76 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Owen, M.R., Sherratt, J.A.: Mathematical modelling of juxtacrine cell signalling. Math. Biosci. 152, 125–150 (1998)

    Article  Google Scholar 

  29. Owen, M.R., Sherratt, J.A., Wearing, H.J.: Lateral induction by juxtacrine signalling is a new mechanism for pattern formation. Dev. Biol. 217, 54–61 (2000)

    Article  Google Scholar 

  30. Painter, K.J., Maini, P.K., Othmer, H.G.: Stripe formation in juvenile Pomecanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Nat. Acad. Sci. USA 96, 5549–5554 (1999)

    Article  Google Scholar 

  31. Panin, V.M., Papayannopoulos, V., Wilson, R., Irvine, K.D.: Fringe modulates Notch-ligand interactions. Nature 387, 908–912 (1997)

    Article  Google Scholar 

  32. Plahte, E.: Pattern formation in discrete cell lattices. J. Math. Biol. 43, 411–445 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Qi, H., Rand, M.D., Wu, X., Sestan, N., Wang, W., Rakic, P., Xu, T., Artavanis-Tsakonas, S.: Processing of the Notch ligand Delta by the metalloprotease Kuzbanian. Science 283, 91–94 (1999)

    CAS  PubMed  Google Scholar 

  34. Reilly, K.M., Melton, D.A.: Short-range signalling by candidate morphogens of the TGF beta family and evidence for a relay mechanism of induction. Cell 86, 743–754 (1996)

    Google Scholar 

  35. Schnell, S., Maini, P.K., Clock and induction model for somitogenesis. Dev. Dyn. 217, 415–420 (2000)

  36. Skeath, J.B., Carroll, S.B.: Regulation of proneural gene expression and cell fate during neuroblast segregation in the drosophila embryo. Development 114, 939–946 (1992)

    Google Scholar 

  37. Spivak-Kroizman, T., Lemmon, M.A., Dikic, I., Ladbury, J.E., Pinchasi, D., Huang, J., Jaye, M., Crumley, G., Schlessinger, G., Lax, I.: Heparin-induced oligomerization of FGF molecules is responsible for FGF receptor dimerization, activation, and cell proliferation. Cell 79, 1015–1024 (1994)

    CAS  PubMed  Google Scholar 

  38. Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)

    Google Scholar 

  39. Waters, C.M., Oberg, K.C., Carpenter, G., and Overholser, K.A.: Rate constants for binding, dissociation, and internalization of EGF: Effect of receptor occupancy and ligand concentration. Biochemistry 29, 3563–3569 (1990)

    Google Scholar 

  40. Wearing, H.J., Owen, M.R., Sherratt, J.A.: Mathematical modelling of juxtacrine patterning. Bull. Math. Biol. 62, 293–320 (2000)

    Article  Google Scholar 

  41. Wearing, H.J., Sherratt, J.A.: Nonlinear analysis of juxtacrine patterns. SIAM J. Appl. Math. 62, 283–309 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Whitfield, T., Haddon, C., Lewis, J.: Intercellular signals and cell-fate choices in the developing inner ear: origins of global and of fine-grained pattern. Sem. Cell Dev. Biol. 8, 239–247 (1997)

    Article  Google Scholar 

  43. Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Science, Loughborough University, Loughborough, LE11 3TU, UK

    Steven D. Webb

  2. Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK

    Markus R. Owen

Authors
  1. Steven D. Webb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus R. Owen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus R. Owen.

Additional information

Send offprint requests to:Markus R. Owen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Webb, S., Owen, M. Oscillations and patterns in spatially discrete models for developmental intercellular signalling. J. Math. Biol. 48, 444–476 (2004). https://doi.org/10.1007/s00285-003-0247-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-003-0247-1

Key words or phrases

Search

Navigation