Waves of Autocrine Signaling in Patterned Epithelia

C. B. Muratov; S. Y. Shvartsman

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 5, page 46-63
  • ISSN: 0973-5348

Abstract

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A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic pre-pattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is possible, using a variational approach. It is shown that under a number of assumptions which can be easily verified for particular sets of model parameters, the equation admits a unique (up to translations) variational traveling wave solution. A global bifurcation structure of these solutions is investigated in a number of particular cases. It is demonstrated that the considered setting may provide a robust developmental regulatory mechanism for delivering chemical signals across large distances in developing epithelia.

How to cite

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Muratov, C. B., and Shvartsman, S. Y.. "Waves of Autocrine Signaling in Patterned Epithelia." Mathematical Modelling of Natural Phenomena 5.5 (2010): 46-63. <http://eudml.org/doc/197683>.

@article{Muratov2010,
abstract = {A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic pre-pattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is possible, using a variational approach. It is shown that under a number of assumptions which can be easily verified for particular sets of model parameters, the equation admits a unique (up to translations) variational traveling wave solution. A global bifurcation structure of these solutions is investigated in a number of particular cases. It is demonstrated that the considered setting may provide a robust developmental regulatory mechanism for delivering chemical signals across large distances in developing epithelia.},
author = {Muratov, C. B., Shvartsman, S. Y.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {cell communication; front propagation; traveling waves; strong heterogeneity; global bifurcation structure},
language = {eng},
month = {7},
number = {5},
pages = {46-63},
publisher = {EDP Sciences},
title = {Waves of Autocrine Signaling in Patterned Epithelia},
url = {http://eudml.org/doc/197683},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Muratov, C. B.
AU - Shvartsman, S. Y.
TI - Waves of Autocrine Signaling in Patterned Epithelia
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/7//
PB - EDP Sciences
VL - 5
IS - 5
SP - 46
EP - 63
AB - A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic pre-pattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is possible, using a variational approach. It is shown that under a number of assumptions which can be easily verified for particular sets of model parameters, the equation admits a unique (up to translations) variational traveling wave solution. A global bifurcation structure of these solutions is investigated in a number of particular cases. It is demonstrated that the considered setting may provide a robust developmental regulatory mechanism for delivering chemical signals across large distances in developing epithelia.
LA - eng
KW - cell communication; front propagation; traveling waves; strong heterogeneity; global bifurcation structure
UR - http://eudml.org/doc/197683
ER -

References

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  1. D. G. Aronson, H. F. Weinberger. Multidimensional diffusion arising in population genetics. Adv. Math., 30 (1978), 33–58. 
  2. H. L. Ashe, J. Briscoe. The interpretation of morphogen gradients. Development, 133 (2006), 385–394. 
  3. H. Berestycki, F. Hamel. Generalized travelling waves for reaction-diffusion equations. Perspectives in nonlinear partial differential equations, volume 446 of Contemp. Math., pages 101–123. Amer. Math. Soc., Providence, RI, 2007.  
  4. J. D. Buckmaster, G. S. S. Ludford. Theory of laminar flames. Cambridge University Press, Cambridge, 1982.  
  5. G. Chapuisat. Existence and nonexistence of curved front solution of a biological equation. J. Differential Equations236 (2007), 237–279. 
  6. G. Chapuisat and R. Joly, Asymptotic profiles for a travelling front solution of a biological equation. Preprint.  
  7. G. Dal Maso. An Introduction to Γ-Convergence. Birkhäuser, Boston, 1993.  
  8. P. C. Fife.Mathematical Aspects of Reacting and Diffusing Systems. Springer-Verlag, Berlin, 1979.  
  9. M. Freeman. Feedback control of intercellular signalling in development. Nature, 408 (2000), 313–319. 
  10. D. Gilbarg, N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1983.  
  11. U. Heberlin, K. Moses. Mechanisms of Drosophila retinal morphogenesis: the virtues of being progressive. Cell, 81 (1995), 987–990. 
  12. B. Kazmierczak, V. Volpert. Travelling calcium waves in systems with non-diffusing buffers. Math. Models Methods Appl. Sci., 18 (2008), 883–912. 
  13. J. Lembong, N. Yakoby, S. Y. Shvartsman. Pattern formation by dynamically interacting network motifs. Proc. Natl. Acad. Sci. USA, 106 (2009), 3213-3218. 
  14. M. Lucia, C. B. Muratov, M. Novaga. Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders. Arch. Rational Mech. Anal., 188 (2008), 475–508. 
  15. A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems, volume 16 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Basel, 1995.  
  16. A. Martinez-Arias, A. Stewart. Molecular principles of animal development. Oxford University Press, New York, 2002.  
  17. C. B. Muratov. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete Cont. Dyn. S., Ser. B, 4 (2004), 867–892.  
  18. C. B. Muratov, M. Novaga. Front propagation in infinite cylinders. I. A variational approach. Comm. Math. Sci., 6 (2008), 799–826. 
  19. C. B. Muratov, F. Posta, S. Y. Shvartsman. Autocrine signal transmission with extracellular ligand degradation. Phys. Biol., 6 (2009), 016006. 
  20. C. B. Muratov, S. Y. Shvartsman. Signal propagation and failure in discrete autocrine relays. Phys. Rev. Lett., 93 (2004), 118101. 
  21. M. Přibyl, C. B. Muratov, S. Y. Shvartsman. Discrete models of autocrine cell communication in epithelial layers. Biophys. J., 84 (2003), 3624–3635. 
  22. M. Přibyl, C. B. Muratov, S. Y. Shvartsman. Long-range signal transmission in autocrine relays. Biophys. J., 84 (2003), 883–896. 
  23. N. Shigesada, K. Kawasaki. Biological invasions: theory and practice. Oxford Series in Ecology and Evolution. Oxford Univ. Press, Oxford, 1997.  
  24. T. Tabata, Y. Takei. Morphogens, their identification and regulation. Development, 131 (2004), 703–712. 
  25. J. J. Tyson, K. Chen, B. Novak. Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol., 2 (2001), 908–916. 
  26. A. I. Volpert, V. A. Volpert, V. A. Volpert. Traveling wave solutions of parabolic systems. AMS, Providence, 1994.  
  27. V. Volpert, A. Volpert. Existence of multidimensional travelling waves in the bistable case. C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 245–250. 
  28. H. S. Wiley, S. Y. Shvartsman, D. A. Lauffenburger. Computational modeling of the EGF-receptor system: a paradigm for systems biology. Trends Cell Biol., 13 (2003), 43–50. 
  29. L. Wolpert, R. Beddington, T. Jessel, P. Lawrence, E. Meyerowitz. Principles of Development. Oxford University Press, Oxford, 1998.  
  30. J. Xin. Front propagation in heterogeneous media. SIAM Review, 42 (2000), 161–230. 
  31. N. Yakoby, J. Lembong, T. Schüpbach, S. Y. ShvartsmanDrosophila eggshell is patterned by sequential action of feedforward and feedback loops. Development, 135 (2008), 343–351. 

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