Pattern-generating travelling waves in a discrete multicellular system with lateral inhibition

Physica D: Nonlinear Phenomena

Volumn 226, Issue 2, 2007, Pages 117-128

  Plahte, Erik ^a   Øyehaug, Leiv ^a  

^a NORWEGIAN UNIVERSITY OF LIFE SCIENCES   (Norway)

Author keywords

Delta; Lateral inhibition; Notch; Pattern; Signal propagation; Travelling wave; Wave speed

Indexed keywords

COMPUTER SIMULATION; MATHEMATICAL MODELS; RANDOM PROCESSES; ROBUSTNESS (CONTROL SYSTEMS); SIGNAL PROCESSING; WAKES;

DELTA; LATERAL INHIBITION; TRAVELLING WAVE; WAVE SPEED;

ELECTROMAGNETIC WAVES;

EID: 33846564324     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2006.11.011     Document Type: Article

References (37)

1. Artavanis-Tsakonas S., Rand M.D., and Lake R.J. Notch signaling: Cell fate control and signal integration in development. Science 284 5415 (1999) 770-776
2. Baker R.E., Schnell S., and Maini P.K. A mathematical investigation of a clock and wave front model for somitogenesis. J. Math. Biol. 52 4 (2006) 458-482
3. Benguria R.D., and Depassier M.C. On the transition from pulled to pushed monotonic fronts of the extended Fisher-Kolmogorov equation. Physica A 356 1 (2005) 61-65
4. Collier J.R., Monk N.A.M., Maini P.K., and Lewis J.H. Pattern formation by lateral inhibition with feedback: A mathematical model of Delta-Notch intercellular interaction. J. Theoret. Biol. 183 (1996) 429-446
5. Ebert U., and van Saarloos W. Universal algebraic relaxation of fronts propagating into an unstable state and implications for moving boundary approximations. Phys. Rev. Lett. 80 8 (1998) 1650-1653
6. Ebert U., and van Saarloos W. Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts. Physica D 146 1-4 (2000) 1-99
7. Ebert U., and van Saarloos W. Breakdown of the standard perturbation theory and moving boundary approximation for pulled fronts. Phys. Rep. 337 1-2 (2000) 139-156
8. Ebert U., van Saarloos W., and Peletier B. Universal algebraic convergence in time of pulled fronts: The common mechanism for difference-differential and partial differential equations. European J. Appl. Math. 13 (2002) 53-66
9. Ebert U., Spruijt W., and van Saarloos W. Pattern forming pulled fronts: Bounds and universal convergence. Physica D 199 1-2 (2004) 13-32
10. Eddison M., Le Roux I., and Lewis J. Notch signaling in the development of the inner ear: Lessons from Drosophila. Proc. Natl. Acad. Sci. USA 97 22 (2000) 11692-11699
11. Fisher R.A. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937) 353-369
12. Gierer A., and Meinhard H. Theory of biological pattern formation. Kybernetik 12 1 (1972) 30-39
13. Haddon C., Jiang Y.J., Smithers L., and Lewis J. Delta-Notch signalling and the patterning of sensory cell differentiation in the zebrafish ear: Evidence from the mind bomb mutant. Development 125 23 (1998) 4637-4644
14. Heberlein U., Borod E., and Chanut F. Dorsoventral patterning in the Drosophila retina by wingless. Development 125 4 (1998) 567-577
15. Iwaki D.D., and Lengyel J.A. A Delta-Notch signaling border regulated by engrailed/invected repression specifies boundary cells in the Drosophila hindgut. Mech. Dev. 114 1-2 (2002) 71-84
16. Justice N.J., and Jan Y.N. Variations on the Notch pathway in neural development. Curr. Opin. Neurobiol. 12 1 (2002) 64-70
17. Kolmogorov A., Petrovsky I., and Piscounov N. Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem. Moscow Univ. Bull. Math. 1 (1937) 1-15
18. Kumar J.P., and Moses K. The EGF receptor and Notch signaling pathways control the initiation of the morphogenetic furrow during Drosophila eye development. Development 128 (2001) 2689-2697
19. Lewis J. Notch signalling and the control of cell fate choices in vertebrates. Cell Dev. Biol. 9 (1998) 583-589
20. Lucia M., Muratov C.B., and Novaga M. Linear vs. nonlinear selection for the propagation of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium. Comm. Pure Appl. Math. LVII (2004) 0616-0636
21. Ma C.Y., and Moses K. Wingless and patched are negative regulators of the morphogenetic furrow and can affect tissue polarity in the developing Drosophila compound eye. Development 121 (1995) 2279-2289
22. Meir E., von Dassow G., Munro E., and Odell G.M. Robustness, flexibility, and the role of lateral inhibition in the neurogenic network. Curr. Biol. 12 10 (2002) 778-786
23. Monk N.A.M. Restricted range gradients and travelling fronts in a model of juxtacrine cell relay. Bull. Math. Biol. 60 (1998) 901-918
24. Nagel A.C., Wech I., and Preiss A. Scalloped and strawberry notch are target genes of Notch signaling in the context of wing margin formation in Drosophila. Mech. Dev. 109 2 (2001) 241-251
25. Othmer H.G., and Scriven L.E. Instability and dynamic patterns in cellular networks. J. Theoret. Biol. 32 (1971) 507-537
26. Owen M.R., and Sherratt J.A. Mathematical modelling of juxtacrine cell signalling. Math. Biosci. 153 (1998) 125-150
27. Owen M.R., Sherratt J.A., and Wearing H.J. Lateral induction by juxtacrine signaling is a new mechanism for pattern formation. Dev. Biology 217 1 (2000) 54-61
28. Owen M.R. Waves and propagation failure in discrete space models with nonlinear coupling and feedback. Physica D 173 1-2 (2002) 59-76
29. Panja D., and van Saarloos W. Fronts with a growth cutoff but with speed higher than the linear spreading speed. Phys. Rev. E 66 1 (2002)
30. Papayannopoulos V., Tomlinson A., Panin V.M., Rauskolb C., and Irvine K.D. Dorsal-ventral signaling in the Drosophila eye. Science 281 5385 (1998) 2031-2034
31. Plahte E. Pattern formation in discrete cell lattices. J. Math. Biol. 43 (2001) 411-445
32. Pourquie O. Vertebrate somitogenesis: A novel paradigm for animal segmentation?. Int. J. Dev. Biol. 47 7-8 (2003) 597-603
33. Rones M.S., McLaughlin K.A., Raffin M., and Mercola M. Serrate and Notch specify cell fates in the heart field by suppressing cardiomyogenesis. Development 127 17 (2000) 3865-3876
34. Storm C., Spruijt W., Ebert U., and van Saarloos W. Universal algebraic relaxation of velocity and phase in pulled fronts generating periodic or chaotic states. Phys. Rev. E 61 6 (2000) R6063-R6066
35. Turing A.M. The chemical basis of morphogenesis. Philos. Trans. R. Soc. (part B) 237 (1952) 37-72 Reprinted in
36. Turing A.M. The chemical basis of morphogenesis. Bull. Math. Biol. 52 (1990) 153-197
37. van Saarloos W. Front propagation into unstable states. Phys. Rep. 386 (2003) 29-222