American Institute of Mathematical Sciences

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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

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 & 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.  doi: 10.1038/nrg2166.  Google Scholar

[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.  doi: 10.1073/pnas.0912734107.  Google Scholar

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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.  doi: 10.1016/j.cub.2011.07.015.  Google Scholar

[4]

D. Ben-Zvi, B. Z. Shilo and N. Barkai, Scaling of morphogen gradients,, Curr Opin Genet Dev, 21 (2011), 704.  doi: 10.1016/j.gde.2011.07.011.  Google Scholar

[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).  doi: 10.1371/journal.pbio.0050046.  Google Scholar

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T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Robust formation of morphogen gradients,, Phys Rev Lett., 94 (2005).  doi: 10.1103/PhysRevLett.94.018103.  Google Scholar

[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.  doi: 10.1016/S0092-8674(00)81438-5.  Google Scholar

[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.  doi: 10.1016/S1534-5807(03)00292-2.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1998).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

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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).  doi: 10.1371/journal.pbio.1001182.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2008.05.021.  Google Scholar

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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).   Google Scholar

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A. D. Lander, Q. Nie and F. Y. M. Wan, Membrane-associated non-receptors and morphogen gradients,, Bulletin of Mathematical Biology, 69 (2007), 33.  doi: 10.1007/s11538-006-9152-2.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/mbe.2005.2.239.  Google Scholar

[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.  doi: 10.1038/381387a0.  Google Scholar

[17]

J. Lei, F. Wan, A. Lander and Q. Nie, Robustness of signaling gradient in drosophila wing imaginal disc,, Discrete and Continuous Dynamical Systems, 16 (2011), 835.  doi: 10.3934/dcdsb.2011.16.835.  Google Scholar

[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, 18 (2013), 721.  doi: 10.3934/dcdsb.2013.18.721.  Google Scholar

[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.  doi: 10.1007/s11538-009-9470-2.  Google Scholar

[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.   Google Scholar

[21]

G. Reeves and S. E. Fraser, Biological systems from an engineer's point of view,, PLoS Biology, 7 (2009).  doi: 10.1371/journal.pbio.1000021.  Google Scholar

[22]

W. Rudin, Real and Complex Analysis,, McGraw-Hill Education, (1987).   Google Scholar

[23]

D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana University Math. J., 21 (1972), 979.   Google Scholar

[24]

M. Strigini and S. M. Cohen, A hedgehog activity gradient contributes to AP axial patterning of the Drosophila wing,, Development, 124 (1997), 4697.   Google Scholar

[25]

A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc,, Cell, 103 (2000), 971.  doi: 10.1016/S0092-8674(00)00199-9.  Google Scholar

[26]

L. Wolpert, Positional information and spatial pattern of cellular differentiation,, Journal of Theoretical Biology, 25 (1969), 1.  doi: 10.1016/S0022-5193(69)80016-0.  Google Scholar

[27]

L. Wolpert, Positional information and patterning revisited,, Journal of Theoretical Biology, 269 (2011), 359.  doi: 10.1016/j.jtbi.2010.10.034.  Google Scholar

[28]

M. Zecca, K. Basler and G. Struhl, Direct and long-range action of a wingless morphogen gradient,, Cell, 87 (1996), 833.  doi: 10.1016/S0092-8674(00)81991-1.  Google Scholar

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.  doi: 10.1038/nrg2166.  Google Scholar

[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.  doi: 10.1073/pnas.0912734107.  Google Scholar

[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.  doi: 10.1016/j.cub.2011.07.015.  Google Scholar

[4]

D. Ben-Zvi, B. Z. Shilo and N. Barkai, Scaling of morphogen gradients,, Curr Opin Genet Dev, 21 (2011), 704.  doi: 10.1016/j.gde.2011.07.011.  Google Scholar

[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).  doi: 10.1371/journal.pbio.0050046.  Google Scholar

[6]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Robust formation of morphogen gradients,, Phys Rev Lett., 94 (2005).  doi: 10.1103/PhysRevLett.94.018103.  Google Scholar

[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.  doi: 10.1016/S0092-8674(00)81438-5.  Google Scholar

[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.  doi: 10.1016/S1534-5807(03)00292-2.  Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1998).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[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).  doi: 10.1371/journal.pbio.1001182.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2008.05.021.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s11538-006-9152-2.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/mbe.2005.2.239.  Google Scholar

[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.  doi: 10.1038/381387a0.  Google Scholar

[17]

J. Lei, F. Wan, A. Lander and Q. Nie, Robustness of signaling gradient in drosophila wing imaginal disc,, Discrete and Continuous Dynamical Systems, 16 (2011), 835.  doi: 10.3934/dcdsb.2011.16.835.  Google Scholar

[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, 18 (2013), 721.  doi: 10.3934/dcdsb.2013.18.721.  Google Scholar

[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.  doi: 10.1007/s11538-009-9470-2.  Google Scholar

[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.   Google Scholar

[21]

G. Reeves and S. E. Fraser, Biological systems from an engineer's point of view,, PLoS Biology, 7 (2009).  doi: 10.1371/journal.pbio.1000021.  Google Scholar

[22]

W. Rudin, Real and Complex Analysis,, McGraw-Hill Education, (1987).   Google Scholar

[23]

D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana University Math. J., 21 (1972), 979.   Google Scholar

[24]

M. Strigini and S. M. Cohen, A hedgehog activity gradient contributes to AP axial patterning of the Drosophila wing,, Development, 124 (1997), 4697.   Google Scholar

[25]

A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc,, Cell, 103 (2000), 971.  doi: 10.1016/S0092-8674(00)00199-9.  Google Scholar

[26]

L. Wolpert, Positional information and spatial pattern of cellular differentiation,, Journal of Theoretical Biology, 25 (1969), 1.  doi: 10.1016/S0022-5193(69)80016-0.  Google Scholar

[27]

L. Wolpert, Positional information and patterning revisited,, Journal of Theoretical Biology, 269 (2011), 359.  doi: 10.1016/j.jtbi.2010.10.034.  Google Scholar

[28]

M. Zecca, K. Basler and G. Struhl, Direct and long-range action of a wingless morphogen gradient,, Cell, 87 (1996), 833.  doi: 10.1016/S0092-8674(00)81991-1.  Google Scholar

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