Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies

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May 2014, 19(3): 775-787. doi: 10.3934/dcdsb.2014.19.775

Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies

Wing-Cheong Lo ^1,

1.	Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, United States

Received September 2013 Revised November 2013 Published February 2014

Abstract
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Robust morphogen gradient formation is important for embryo development. Patterns of developmental tissue are encoded by the morphogen gradient that drives the process of cell differentiation in response to different morphogen levels. Experiments have shown that tissue patterning is robust with respect to morphogen overexpression. However, the mechanisms for this robust patterning remain unclear. The expansion-repression mechanism, which was proposed for achieving scaling of patterning with organ size, is a type of self-enhanced clearance through a non-local feedback regulation and may contribute to the robustness with respect to morphogen overexpression. In this paper, we study the role of the expansion-repression mechanism in morphogen gradient robustness through a two-equation model with general forms of feedback functions. We prove the existence of steady-state solutions, and, through model reduction and simplification, show that the expansion-repression mechanism is able to improve the robustness against changes in the morphogen production rate. However, this improvement is restricted by the biological requirement of multi-fate long-range morphogen gradient.

Keywords: matheamatical modeling, expansion-repression mechanism., Morphogen gradient, robustness.

Mathematics Subject Classification: Primary: 34B15, 92C15; Secondary: 34B60, 35K5.

Citation: Wing-Cheong Lo. Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 775-787. doi: 10.3934/dcdsb.2014.19.775

References:

[1]	M. Affolter and K. Basler, The Decapentaplegic morphogen gradient: From pattern formation to growth regulation, Nature Rev. Genet., 8 (2007), 663-674. doi: 10.1038/nrg2166.
[2]	D. Ben-Zvi and N. Barkai, Scaling of morphogen gradients by an expansion-repression integral feedback control, Proc. Natl. Acad. Sci. USA, 107 (2010), 6924-6929. doi: 10.1073/pnas.0912734107.
[3]	D. Ben-Zvi, G. Pyrowolakis, N. Barkai and B. Z. Shilo, Expansion-repression mechanism for scaling the dpp activation gradient in drosophila wing imaginal discs, Curr Biol, 21 (2011), 1391-1396. doi: 10.1016/j.cub.2011.07.015.
[4]	D. Ben-Zvi, B. Z. Shilo and N. Barkai, Scaling of morphogen gradients, Curr Opin Genet Dev, 21 (2011), 704-710. doi: 10.1016/j.gde.2011.07.011.
[5]	S. Bergmann, O. Sandler, H. Sberro, S. Shnider, E. Schejter, B.-Z. Shilo and N. Barkai, Pre-steady-state decoding of the bicoid morphogen gradient, PLoS Biol, 5 (2007), e46. doi: 10.1371/journal.pbio.0050046.
[6]	T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Robust formation of morphogen gradients, Phys Rev Lett., 94 (2005), 018103. doi: 10.1103/PhysRevLett.94.018103.
[7]	K. Cadigan, M. Fish, E. Rulifson and R. Nusse, Wingless repression of Drosophila frizzled 2 expression shapes the wingless morphogen gradient in the wing, Cell, 93 (1998), 767-777. doi: 10.1016/S0092-8674(00)81438-5.
[8]	A. Eldar, D. Rosin, B. Z. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients, Developmental Cell, 5 (2003), 635-646. doi: 10.1016/S1534-5807(03)00292-2.
[9]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998. doi: 10.1007/978-3-642-61798-0.
[10]	F. Hamaratoglu, A. M. de Lachapelle, G. Pyrowolakis, S. Bergmann and M. Affolter, Dpp signaling activity requires pentagone to scale with tissue size in the growing drosophila wing imaginal disc, PLoS Biol, 9 (2011), e1001182. doi: 10.1371/journal.pbio.1001182.
[11]	H. Hardway, B. Mukhopadhyay, T. Burke, T. J. Hitchman and R. Forman, Modeling the precision and robustness of hunchback border during drosophila embryonic development, Journal of Theoretical Biology, 254 (2008), 390-399. doi: 10.1016/j.jtbi.2008.05.021.
[12]	A. D. Lander, W. C. Lo, Q. Nie and F. Y. M. Wan, The measure of success: Constraints, objectives, and tradeoffs in morphogen-mediated patterning, Cold Spring Harbor Perspectives in Biology, 1 (2009), a002022.
[13]	A. D. Lander, Q. Nie and F. Y. M. Wan, Membrane-associated non-receptors and morphogen gradients, Bulletin of Mathematical Biology, 69 (2007), 33-54. doi: 10.1007/s11538-006-9152-2.
[14]	A. Lander, Q. Nie, B. Vargas and F. Wan, Size-normalized robustness of dpp gradient in Drosophila wing imaginal disc, Journal of Mechanics of Materials and Structures, 6 (2011), 321-350.
[15]	A. D. Lander, Q. Nie and F. Y. M. Wan, Spatially distributed morphogen production and morphogen gradient formation, Math Biosci Eng, 2 (2005), 239-262. doi: 10.3934/mbe.2005.2.239.
[16]	T. Lecuit, W. J. Brook, M. Ng, M. Calleja, H. Sun and S. M. Cohen, Two distinct mechanisms for long-range patterning by Decapentaplegic in the Drosophila wing, Nature, 381 (1996), 387-393. doi: 10.1038/381387a0.
[17]	J. Lei, F. Wan, A. Lander and Q. Nie, Robustness of signaling gradient in drosophila wing imaginal disc, Discrete and Continuous Dynamical Systems, Series B, 16 (2011), 835-866. doi: 10.3934/dcdsb.2011.16.835.
[18]	J. Lei, D. Wang, Y. Song, Q. Nie and F. Y. M. Wan, Robustness of morphogen gradients with "bucket brigade" transport through membrane-associated non-receptors, Discrete and Continuous Dynamical Systems, Series B, 18 (2013), 721-739. doi: 10.3934/dcdsb.2013.18.721.
[19]	J. Lei and Y. Song, Mathematical model of the formation of morphogen gradients through membrane-associated non-receptors, Bull Math Biol, 72 (2010), 805-829. doi: 10.1007/s11538-009-9470-2.
[20]	S. Morimura, L. Maves, Y. Chen and F. M. Hoffmann, decapentaplegic overexpression affects Drosophila wing and leg imaginal disc development and wingless expression, Dev Biol., 177 (1996), 136-151.
[21]	G. Reeves and S. E. Fraser, Biological systems from an engineer's point of view, PLoS Biology, 7 (2009), e1000021. doi: 10.1371/journal.pbio.1000021.
[22]	W. Rudin, Real and Complex Analysis, McGraw-Hill Education, 1987.
[23]	D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana University Math. J., 21 (1972), 979-1000.
[24]	M. Strigini and S. M. Cohen, A hedgehog activity gradient contributes to AP axial patterning of the Drosophila wing, Development, 124 (1997), 4697-4705.
[25]	A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc, Cell, 103 (2000), 971-980. doi: 10.1016/S0092-8674(00)00199-9.
[26]	L. Wolpert, Positional information and spatial pattern of cellular differentiation, Journal of Theoretical Biology, 25 (1969), 1-47. doi: 10.1016/S0022-5193(69)80016-0.
[27]	L. Wolpert, Positional information and patterning revisited, Journal of Theoretical Biology, 269 (2011), 359-365. doi: 10.1016/j.jtbi.2010.10.034.
[28]	M. Zecca, K. Basler and G. Struhl, Direct and long-range action of a wingless morphogen gradient, Cell, 87 (1996), 833-844. doi: 10.1016/S0092-8674(00)81991-1.

show all references

References:

[1]	M. Affolter and K. Basler, The Decapentaplegic morphogen gradient: From pattern formation to growth regulation, Nature Rev. Genet., 8 (2007), 663-674. doi: 10.1038/nrg2166.
[2]	D. Ben-Zvi and N. Barkai, Scaling of morphogen gradients by an expansion-repression integral feedback control, Proc. Natl. Acad. Sci. USA, 107 (2010), 6924-6929. doi: 10.1073/pnas.0912734107.
[3]	D. Ben-Zvi, G. Pyrowolakis, N. Barkai and B. Z. Shilo, Expansion-repression mechanism for scaling the dpp activation gradient in drosophila wing imaginal discs, Curr Biol, 21 (2011), 1391-1396. doi: 10.1016/j.cub.2011.07.015.
[4]	D. Ben-Zvi, B. Z. Shilo and N. Barkai, Scaling of morphogen gradients, Curr Opin Genet Dev, 21 (2011), 704-710. doi: 10.1016/j.gde.2011.07.011.
[5]	S. Bergmann, O. Sandler, H. Sberro, S. Shnider, E. Schejter, B.-Z. Shilo and N. Barkai, Pre-steady-state decoding of the bicoid morphogen gradient, PLoS Biol, 5 (2007), e46. doi: 10.1371/journal.pbio.0050046.
[6]	T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Robust formation of morphogen gradients, Phys Rev Lett., 94 (2005), 018103. doi: 10.1103/PhysRevLett.94.018103.
[7]	K. Cadigan, M. Fish, E. Rulifson and R. Nusse, Wingless repression of Drosophila frizzled 2 expression shapes the wingless morphogen gradient in the wing, Cell, 93 (1998), 767-777. doi: 10.1016/S0092-8674(00)81438-5.
[8]	A. Eldar, D. Rosin, B. Z. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients, Developmental Cell, 5 (2003), 635-646. doi: 10.1016/S1534-5807(03)00292-2.
[9]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998. doi: 10.1007/978-3-642-61798-0.
[10]	F. Hamaratoglu, A. M. de Lachapelle, G. Pyrowolakis, S. Bergmann and M. Affolter, Dpp signaling activity requires pentagone to scale with tissue size in the growing drosophila wing imaginal disc, PLoS Biol, 9 (2011), e1001182. doi: 10.1371/journal.pbio.1001182.
[11]	H. Hardway, B. Mukhopadhyay, T. Burke, T. J. Hitchman and R. Forman, Modeling the precision and robustness of hunchback border during drosophila embryonic development, Journal of Theoretical Biology, 254 (2008), 390-399. doi: 10.1016/j.jtbi.2008.05.021.
[12]	A. D. Lander, W. C. Lo, Q. Nie and F. Y. M. Wan, The measure of success: Constraints, objectives, and tradeoffs in morphogen-mediated patterning, Cold Spring Harbor Perspectives in Biology, 1 (2009), a002022.
[13]	A. D. Lander, Q. Nie and F. Y. M. Wan, Membrane-associated non-receptors and morphogen gradients, Bulletin of Mathematical Biology, 69 (2007), 33-54. doi: 10.1007/s11538-006-9152-2.
[14]	A. Lander, Q. Nie, B. Vargas and F. Wan, Size-normalized robustness of dpp gradient in Drosophila wing imaginal disc, Journal of Mechanics of Materials and Structures, 6 (2011), 321-350.
[15]	A. D. Lander, Q. Nie and F. Y. M. Wan, Spatially distributed morphogen production and morphogen gradient formation, Math Biosci Eng, 2 (2005), 239-262. doi: 10.3934/mbe.2005.2.239.
[16]	T. Lecuit, W. J. Brook, M. Ng, M. Calleja, H. Sun and S. M. Cohen, Two distinct mechanisms for long-range patterning by Decapentaplegic in the Drosophila wing, Nature, 381 (1996), 387-393. doi: 10.1038/381387a0.
[17]	J. Lei, F. Wan, A. Lander and Q. Nie, Robustness of signaling gradient in drosophila wing imaginal disc, Discrete and Continuous Dynamical Systems, Series B, 16 (2011), 835-866. doi: 10.3934/dcdsb.2011.16.835.
[18]	J. Lei, D. Wang, Y. Song, Q. Nie and F. Y. M. Wan, Robustness of morphogen gradients with "bucket brigade" transport through membrane-associated non-receptors, Discrete and Continuous Dynamical Systems, Series B, 18 (2013), 721-739. doi: 10.3934/dcdsb.2013.18.721.
[19]	J. Lei and Y. Song, Mathematical model of the formation of morphogen gradients through membrane-associated non-receptors, Bull Math Biol, 72 (2010), 805-829. doi: 10.1007/s11538-009-9470-2.
[20]	S. Morimura, L. Maves, Y. Chen and F. M. Hoffmann, decapentaplegic overexpression affects Drosophila wing and leg imaginal disc development and wingless expression, Dev Biol., 177 (1996), 136-151.
[21]	G. Reeves and S. E. Fraser, Biological systems from an engineer's point of view, PLoS Biology, 7 (2009), e1000021. doi: 10.1371/journal.pbio.1000021.
[22]	W. Rudin, Real and Complex Analysis, McGraw-Hill Education, 1987.
[23]	D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana University Math. J., 21 (1972), 979-1000.
[24]	M. Strigini and S. M. Cohen, A hedgehog activity gradient contributes to AP axial patterning of the Drosophila wing, Development, 124 (1997), 4697-4705.
[25]	A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc, Cell, 103 (2000), 971-980. doi: 10.1016/S0092-8674(00)00199-9.
[26]	L. Wolpert, Positional information and spatial pattern of cellular differentiation, Journal of Theoretical Biology, 25 (1969), 1-47. doi: 10.1016/S0022-5193(69)80016-0.
[27]	L. Wolpert, Positional information and patterning revisited, Journal of Theoretical Biology, 269 (2011), 359-365. doi: 10.1016/j.jtbi.2010.10.034.
[28]	M. Zecca, K. Basler and G. Struhl, Direct and long-range action of a wingless morphogen gradient, Cell, 87 (1996), 833-844. doi: 10.1016/S0092-8674(00)81991-1.

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