Nonlinear analysis of juxtacrine patterns

SIAM Journal on Applied Mathematics

Volumn 62, Issue 1, 2001, Pages 283-309

  Wearing, Helen J ^a   Sherratt, Jonathan A ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Bifurcation analysis; Juxtacrine signaling; Pattern formation; Positive feedback

Indexed keywords

ALGEBRA; BIFURCATION (MATHEMATICS); CELLS; COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; NUMERICAL ANALYSIS;

BIFURCATION ANALYSIS; JUXTACRINE PATTERNS; LINEAR ANALYSIS; PATTERNING DYNAMICS;

NONLINEAR EQUATIONS;

EID: 0036226924     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003613990037220X     Document Type: Article

References (22)

1. J. F. DE CELIS AND S. BRAY, Feedback mechanisms affecting notch activation at the dorsoventral boundary in the Drosophila wing, Development, 124 (1997), pp. 3241-3251.
2. J. F. DE CELIS, S. BRAY, AND A. GARCIA-BELLIDO, Notch signalling regulates veinlet expression and establishes boundaries between veins and interveins in the Drosophila wing, Development, 124 (1997), pp. 1919-1928.
3. S.-N. CHOW, J. MALLET-PARET, AND E. S. VAN VLECK, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4 (1996), pp. 109-178.
4. A. J. L. CLARK, S. ISHII, N. RICHERT, G. T. MERLINO, AND I. PASTAN, Epidermal growth factor regulates the expression of its own receptor, Proc. Natl. Acad. Sci. USA, 82 (1985), pp. 8374-8378.
5. R. J. COFFEY, R. DERYNCK, J. N. WILCOX, T. S. BRINGMAN, A. S. GOUSTIN, H. L. MOSES, AND M. R. PITTELKOW, Production and auto-induction of transforming growth factor-α in human keratinocytes, Nature, 328 (1987), pp. 817-820.
6. J. R. COLLIER, N. A. M. MONK, P. K. MAINI, AND J. H. LEWIS, Pattern formation by lateral inhibition with feedback: A mathematical model of Delta-Notch intercellular signalling, J. Theoret. Biol., 183 (1996), pp. 429-446.
7. E. J. DOEDEL, H. B. KELLER, AND J. P. KERNÉVEZ, Numerical analysis and control of bifurcation problems: (I) Bifurcation in finite dimensions, Internat . J. Bifur. Chaos Appl. Sci. Engrg., 1 (1991), pp. 493-520.
8. S. S. HUPPERT, T. L. JACOBSON, AND M. A. T. MUSKAVITCH, Feedback regulation is central to Delta-Notch signalling required for Drosophila wing vein morphogenesis, Development, 124 (1997), pp. 3283-3291.
9. J. LEWIS, Neurogenic genes and vertebrate neurogenesis, Curr. Op. Neurobiol., 6 (1996), pp. 3-10.
10. J. LEWIS, Notch signalling and the control of cell fate choices in vertebrates, Sem. Cell Dev. Biol., 9 (1998), pp. 583-589.
11. J. MASSAGUÉ, Transforming growth factor-α: A model for membrane-anchored growth factors, J. Biol. Chem., 265 (1990), pp. 21393-21396.
12. J. D. MURRAY, Mathematical Biology, 2nd ed., Springer-Verlag, Berlin, 1993.
13. M. A. T. MUSKAVITCH, Delta-Notch signalling and Drosophila cell fate choice, Dev. Biol., 166 (1994), pp. 415-430.
14. M. R. OWEN AND J. A. SHERRATT, Mathematical modelling of juxtacrine cell signalling, Math. Biosci., 152 (1998), pp. 125-150.
15. M. R. OWEN, J. A. SHERRATT, AND H. J. WEARING, Lateral induction by juxtacrine signalling is a new mechanism for pattern formation, Dev. Biol., 217 (2000), pp. 54-61.
16. V. M. PANIN, V. PAPAYANNOPOULOS, R. WILSON, AND K. D. IRVINE, Fringe modulates Notch-ligand interactions, Nature, 387 (1997), pp. 908-912.
17. E. PLAHTE, Pattern Formation in Discrete Cellular Lattices with Internal Cell Dynamics and Regulated Cell-Cell Interaction, preprint, Department of Mathematical Sciences, Agricultural University of Norway, Aas, Norway, 1999.
18. K. M. REILLY AND D. A. MELTON, Short-range signaling by candidate morphogens of the TGF beta family and evidence for a relay mechanism of induction, Cell, 86 (1996), pp. 743-754.
19. A. M. TURING, The chemical basis of morphogenesis, Phil. Trans. R. Soc., B237 (1952), pp. 37-72.
20. C. M. WATERS, K. C. OBERG, G. CARPENTER, AND K. A. OVERHOLSER, Rate constants for binding, dissociation, and internalization of EGF: Effect of receptor occupancy and ligand concentration, Biochem., 29 (1990), pp. 3563-3569.
21. H. J. WEARING, M. R. OWEN, AND J. A. SHERRATT, Mathematical modelling of juxtacrine patterning, Bull. Math. Biol., 62 (2000), pp. 293-320.
22. B. ZINNER, G. HARRIS, AND W. HUDSON, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), pp. 46-62.