May  2013, 18(3): 721-739. doi: 10.3934/dcdsb.2013.18.721

Robustness of Morphogen gradients with "bucket brigade" transport through membrane-associated non-receptors

1. 

Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing 100084

2. 

Department of mathematics, University of California, CA, 92697-3875, United States, United States, United States

3. 

College of software, Beihang University, Beijing, 100083, China

Received  July 2012 Revised  October 2012 Published  December 2012

Robust multiple-fate morphogen gradients are essential for embryo development. Here, we analyze mathematically a model of morphogen gradient (such as Dpp in Drosophila wing imaginal disc) formation in the presence of non-receptors with both diffusion of free morphogens and the movement of morphogens bound to non-receptors. Under the assumption of rapid degradation of unbound morphogen, we introduce a method of functional boundary value problem and prove the existence, uniqueness and linear stability of a biologically acceptable steady-state solution. Next, we investigate the robustness of this steady-state solution with respect to significant changes in the morphogen synthesis rate. We prove that the model is able to produce robust biological morphogen gradients when production and degradation rates of morphogens are large enough and non-receptors are abundant. Our results provide mathematical and biological insight to a mechanism of achieving stable robust long distance morphogen gradients. Key elements of this mechanism are rapid turnover of morphogen to non-receptors of neighoring cells resulting in significant degradation and transport of non-receptor-morphogen complexes, the latter moving downstream through a "bucket brigade" process.
Citation: Jinzhi Lei, Dongyong Wang, You Song, Qing Nie, Frederic Y. M. Wan. Robustness of Morphogen gradients with "bucket brigade" transport through membrane-associated non-receptors. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 721-739. doi: 10.3934/dcdsb.2013.18.721
References:
[1]

T. Akiyama, K. Kamimura, C. Firkus, S. Takeo, O. Shimmi and H. Nakato, Dally regulates Dpp morphogen gradient formation by stabilizing Dpp on the cell surface,, Dev. Biol., 313 (2008), 408.   Google Scholar

[2]

G. Baeg, E. M. Selva, R. M. Goodman, R. Dasgupta and N. Perrimon, The Wingless morphogen gradient is established by the cooperative action of Frizzled and heparan sulfate proteoglycan receptors,, Dev. Biol., 276 (2004), 89.   Google Scholar

[3]

T. Y. Belenkaya, C. Han, D. Yan, R. J. Opoka, M. Khodoun, H. Liu and X. Lin, Drosophila Dpp morphogen movement is independent of dynamin-mediated endocytosis but regulated by the glypican members of heparan sulfate proteoglycans,, Cell, 119 (2004), 231.   Google Scholar

[4]

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

[5]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in epithelia,, Phys. Rev. E., 75 (2007).   Google Scholar

[6]

D. J. Bornemann, J. E. Duncan, W. Staatz, S. Selleck and R. Warrior, Abrogation of heparan sulfate synthesis in Drosophila disrupts the Wingless, Hedgehog and Decapentaplegic signaling pathways,, Development, 131 (2004), 1927.   Google Scholar

[7]

A. Eldar, D. Rosin, B. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients,, Dev. Cell, 5 (2003), 635.   Google Scholar

[8]

E. V. Entchev, A. Schwabedissen and M. Gonzalez-Gaitan, Gradient formation of the TGF-$\beta$ homolog Dpp,, Cell, 103 (2000), 981.   Google Scholar

[9]

M. Fujise, S. Takeo, K. Kamimura, T. Matsuo, T. Aigaki, S. Izumi and H. Nakato, Dally regulates Dpp morphogen gradient formation in the Drosophila wing,, Development, 130 (2003), 1515.   Google Scholar

[10]

L. Hufnagel, J. Kreuger, S. M. Cohen and B. I. Shraiman, On the role of glypicans in the process of morphogen gradient formation,, Dev. Biol., 300 (2006), 512.   Google Scholar

[11]

M. Kerszberg, Accurate reading of morphogen concentrations by nuclear receptors: A formal model of complex transduction pathways,, J. Theor. Biol., 183 (1996), 95.   Google Scholar

[12]

M. Kerszberg and L. Wolpert, Mechanisms for positional signalling by morphogen transport: A theoretical study,, J. Theor. Biol., 191 (1998), 103.   Google Scholar

[13]

A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Jülicher and M. González-Gaitán, Kinetics of morphogen gradient formation,, Science, 315 (2007), 521.   Google Scholar

[14]

C. A. Kirkpatrick, B. D. Dimitroff, J. M. Rawson and S. B. Selleck, Spatial regulation of Wingless Morphogen distribution and signaling by Dally-like protein,, Dev. Cell, 7 (2004), 513.   Google Scholar

[15]

K. Kruse, P. Pantazis, T. Bollenbach, F. Jülicher and M. González-Gaitán, Dpp gradient formation by dynamin-dependent endocytosis: Receptor trafficking and the diffusion model,, Development, 131 (2004), 4843.   Google Scholar

[16]

A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion?,, Dev. Cell, 2 (2002), 785.   Google Scholar

[17]

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

[18]

A. D. Lander, Q. Nie and F. Y. M. Wan, Internalization and end flux in morphogen gradient formation,, J. Comp. Appl. Math., 190 (2006), 232.  doi: 10.1016/j.cam.2004.11.054.  Google Scholar

[19]

A. D. Lander, Morpheus unbound: Reimagining the morphogen gradient,, Cell, 128 (2007), 245.   Google Scholar

[20]

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

[21]

A. D. Lander, F. Y. M. Wan and Q. Nie, Multiple paths to morphogen gradient robustness,, preprint, ().   Google Scholar

[22]

J. Lei, Mathematical model of the Dpp gradient formation in drosophila wing imaginal disc,, Chinese Sci. Bull., 55 (2010), 984.   Google Scholar

[23]

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

[24]

J. Lei, F. Y. M. Wan, A. D. Lander and Q. Nie, Robustness of signaling gradient in Drosophila wing imaginal disc,, Disc. Cont. Dyns. Syst. B, 16 (2011), 835.  doi: 10.3934/dcdsb.2011.16.835.  Google Scholar

[25]

Y. Lou, Q. Nie and F. Y. M. Wan, Nonlinear eigenvalue problems in the stability analysis of morphogen gradients,, Studies in Applied Mathematics, 113 (2004), 183.   Google Scholar

[26]

Y. Lou, Q. Nie and F. Y. M. Wan, Effects of Sog on Dpp-receptor binding,, SIAM J. Appl. Math., 65 (2005), 1748.  doi: 10.1137/S0036139903433219.  Google Scholar

[27]

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

[28]

A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc,, Cell, 103 (2000), 971.   Google Scholar

[29]

I. The, Y. Bellaiche and N. Perrimon, Hedghog movement is regulated through tout velu-dependent synthesis of a haparan sulfate proteoglycan,, Mol. Cell., 4 (1999), 633.   Google Scholar

[30]

J. Vincent and L. Dubois, Morphogen transport along epithelia and integrated trafficking problem,, Dev. Cell., 3 (2002), 615.   Google Scholar

[31]

L. Wolpert, Position information and patterning revisited,, J. Theor. Biol., 269 (2011), 359.   Google Scholar

[32]

Y.-T. Zhang, A. D. Lander and Q. Nie, Computational analysis of BMP gradients in dorsal-ventral patterning of the zebrafish embryo,, J. Theor. Biol., 248 (2007), 579.  doi: 10.1016/j.jtbi.2007.05.026.  Google Scholar

show all references

References:
[1]

T. Akiyama, K. Kamimura, C. Firkus, S. Takeo, O. Shimmi and H. Nakato, Dally regulates Dpp morphogen gradient formation by stabilizing Dpp on the cell surface,, Dev. Biol., 313 (2008), 408.   Google Scholar

[2]

G. Baeg, E. M. Selva, R. M. Goodman, R. Dasgupta and N. Perrimon, The Wingless morphogen gradient is established by the cooperative action of Frizzled and heparan sulfate proteoglycan receptors,, Dev. Biol., 276 (2004), 89.   Google Scholar

[3]

T. Y. Belenkaya, C. Han, D. Yan, R. J. Opoka, M. Khodoun, H. Liu and X. Lin, Drosophila Dpp morphogen movement is independent of dynamin-mediated endocytosis but regulated by the glypican members of heparan sulfate proteoglycans,, Cell, 119 (2004), 231.   Google Scholar

[4]

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

[5]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in epithelia,, Phys. Rev. E., 75 (2007).   Google Scholar

[6]

D. J. Bornemann, J. E. Duncan, W. Staatz, S. Selleck and R. Warrior, Abrogation of heparan sulfate synthesis in Drosophila disrupts the Wingless, Hedgehog and Decapentaplegic signaling pathways,, Development, 131 (2004), 1927.   Google Scholar

[7]

A. Eldar, D. Rosin, B. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients,, Dev. Cell, 5 (2003), 635.   Google Scholar

[8]

E. V. Entchev, A. Schwabedissen and M. Gonzalez-Gaitan, Gradient formation of the TGF-$\beta$ homolog Dpp,, Cell, 103 (2000), 981.   Google Scholar

[9]

M. Fujise, S. Takeo, K. Kamimura, T. Matsuo, T. Aigaki, S. Izumi and H. Nakato, Dally regulates Dpp morphogen gradient formation in the Drosophila wing,, Development, 130 (2003), 1515.   Google Scholar

[10]

L. Hufnagel, J. Kreuger, S. M. Cohen and B. I. Shraiman, On the role of glypicans in the process of morphogen gradient formation,, Dev. Biol., 300 (2006), 512.   Google Scholar

[11]

M. Kerszberg, Accurate reading of morphogen concentrations by nuclear receptors: A formal model of complex transduction pathways,, J. Theor. Biol., 183 (1996), 95.   Google Scholar

[12]

M. Kerszberg and L. Wolpert, Mechanisms for positional signalling by morphogen transport: A theoretical study,, J. Theor. Biol., 191 (1998), 103.   Google Scholar

[13]

A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Jülicher and M. González-Gaitán, Kinetics of morphogen gradient formation,, Science, 315 (2007), 521.   Google Scholar

[14]

C. A. Kirkpatrick, B. D. Dimitroff, J. M. Rawson and S. B. Selleck, Spatial regulation of Wingless Morphogen distribution and signaling by Dally-like protein,, Dev. Cell, 7 (2004), 513.   Google Scholar

[15]

K. Kruse, P. Pantazis, T. Bollenbach, F. Jülicher and M. González-Gaitán, Dpp gradient formation by dynamin-dependent endocytosis: Receptor trafficking and the diffusion model,, Development, 131 (2004), 4843.   Google Scholar

[16]

A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion?,, Dev. Cell, 2 (2002), 785.   Google Scholar

[17]

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

[18]

A. D. Lander, Q. Nie and F. Y. M. Wan, Internalization and end flux in morphogen gradient formation,, J. Comp. Appl. Math., 190 (2006), 232.  doi: 10.1016/j.cam.2004.11.054.  Google Scholar

[19]

A. D. Lander, Morpheus unbound: Reimagining the morphogen gradient,, Cell, 128 (2007), 245.   Google Scholar

[20]

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

[21]

A. D. Lander, F. Y. M. Wan and Q. Nie, Multiple paths to morphogen gradient robustness,, preprint, ().   Google Scholar

[22]

J. Lei, Mathematical model of the Dpp gradient formation in drosophila wing imaginal disc,, Chinese Sci. Bull., 55 (2010), 984.   Google Scholar

[23]

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

[24]

J. Lei, F. Y. M. Wan, A. D. Lander and Q. Nie, Robustness of signaling gradient in Drosophila wing imaginal disc,, Disc. Cont. Dyns. Syst. B, 16 (2011), 835.  doi: 10.3934/dcdsb.2011.16.835.  Google Scholar

[25]

Y. Lou, Q. Nie and F. Y. M. Wan, Nonlinear eigenvalue problems in the stability analysis of morphogen gradients,, Studies in Applied Mathematics, 113 (2004), 183.   Google Scholar

[26]

Y. Lou, Q. Nie and F. Y. M. Wan, Effects of Sog on Dpp-receptor binding,, SIAM J. Appl. Math., 65 (2005), 1748.  doi: 10.1137/S0036139903433219.  Google Scholar

[27]

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

[28]

A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc,, Cell, 103 (2000), 971.   Google Scholar

[29]

I. The, Y. Bellaiche and N. Perrimon, Hedghog movement is regulated through tout velu-dependent synthesis of a haparan sulfate proteoglycan,, Mol. Cell., 4 (1999), 633.   Google Scholar

[30]

J. Vincent and L. Dubois, Morphogen transport along epithelia and integrated trafficking problem,, Dev. Cell., 3 (2002), 615.   Google Scholar

[31]

L. Wolpert, Position information and patterning revisited,, J. Theor. Biol., 269 (2011), 359.   Google Scholar

[32]

Y.-T. Zhang, A. D. Lander and Q. Nie, Computational analysis of BMP gradients in dorsal-ventral patterning of the zebrafish embryo,, J. Theor. Biol., 248 (2007), 579.  doi: 10.1016/j.jtbi.2007.05.026.  Google Scholar

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