American Institute of Mathematical Sciences

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October  2011, 16(3): 835-866. doi: 10.3934/dcdsb.2011.16.835

Robustness of signaling gradient in drosophila wing imaginal disc

Jinzhi Lei 1, , Frederic Y. M. Wan 2, , Arthur D. Lander 3, and  Qing Nie 4,

1. 

Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing 100084
2. 

Department of Mathematics, Center for Complex Biological Systems, University of California, Irvine, California, 92697-3875
3. 

Department of Developmental and Cell Biology, Center for Complex Biological Systems, University of California, Irvine, California, 92697-2300, United States
4. 

Department of Mathematics, Center for Complex Biological Systems & Center for Mathematical and Computational Biology, University of California, Irvine, California, 92697-3875, United States

Received  July 2010 Revised  October 2010 Published  June 2011

Quasi-stable gradients of signaling protein molecules (known as morphogens or ligands) bound to cell receptors are known to be responsible for differential cell signaling and gene expressions. From these follow different stable cell fates and visually patterned tissues in biological development. Recent studies have shown that the relevant basic biological processes yield gradients that are sensitive to small changes in system characteristics (such as expression level of morphogens or receptors) or environmental conditions (such as temperature changes). Additional biological activities must play an important role in the high level of robustness observed in embryonic patterning for example. It is natural to attribute observed robustness to various type of feedback control mechanisms. However, our own simulation studies have shown that feedback control is neither necessary nor sufficient for robustness of the morphogen decapentaplegic (Dpp) gradient in wing imaginal disc of Drosophilas. Furthermore, robustness can be achieved by substantial binding of the signaling morphogen Dpp with nonsignaling cell surface bound molecules (such as heparan sulfate proteoglygans) and degrading the resulting complexes at a sufficiently rapid rate. The present work provides a theoretical basis for the results of our numerical simulation studies.
Keywords: robustness, nonlinear boundary value problem, mathematical modeling., Morphogen gradient.
Mathematics Subject Classification: Primary: 34B15, 92C15; Secondary: 34B60, 35K5.
Citation: Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander, Qing Nie. Robustness of signaling gradient in drosophila wing imaginal disc. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 835-866. doi: 10.3934/dcdsb.2011.16.835
References:
[1]

G. H. Baeg and N. Perrimon, Functional binding of secreted molecules to heparan sulfate proteoglycans in Drosophila, Curr. Opin. Cell. Biol., 12 (2000), 575-580. doi: 10.1016/S0955-0674(00)00134-4.

[2]

G. H. 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), 89-100. doi: 10.1016/j.ydbio.2004.08.023.

[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), 231-244. doi: 10.1016/j.cell.2004.09.031.

[4]

D. J. Bornemann, J. E. Duncan, W. Staatz, S. Seleck and R. Warrior, Abrogation of heparan sulfate synthesis in Drosophila disrupts the Wingless, hedgehog and Decapentaplegic signaling pathways, Development, 131 (2004), 1927-1938. doi: 10.1242/dev.01061.

[5]

K. M. Cadigan, M. P. Fish, E. J. 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.

[6]

A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B. Z. Shilo and N. Barkai, Robustness of MmP morphogen gradient in Drosophila embryonic patterning, Nature, 419 (2002), 304-308. doi: 10.1038/nature01061.

[7]

A. Eldar, D. Rosin, B. Z. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen graients, Dev. Cell, 5 (2003), 635-646. doi: 10.1016/S1534-5807(03)00292-2.

[8]

A. Eldar, B. Z. Shilo and N. Barkai, Elucidating mechanisms underlying robustness of morphogen gradients, Curr. Opin. Genet. Dev., 14 (2004), 435-439.

[9]

E. V. Entchev, A. Schwabedissen and M. Gonzá lez-Gaitán, Grandient formation of the TGF-$\beta$ homolog dpp, Cell, 103 (2000), 981-991. doi: 10.1016/S0092-8674(00)00200-2.

[10]

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), 1515-1522. doi: 10.1242/dev.00379.

[11]

Y. Funakoshi, M. Minami and T. Tabata, Mtv shapes the activity gradient of the Dpp morphogen through regulation of thickveins, Development, 128 (2001), 67-74.

[12]

C. Han, T. Y. Belenkaya, M. Khodoun, M. Tauchi, X. D. Lin and X. H. Lin, Distinct and collaborative roles of Drosophila EXT family proteins in morphogen signalling and gradient formation, Development, 131 (2004), 1563-1575. doi: 10.1242/dev.01051.

[13]

B. Houchmandzadeh, E. Wieschaus and S. Leibler, Establishment of developmental precision and proportions in the early Drosophila embryo, Nature, 415 (2002), 798-802.

[14]

N. T. Ingolia, Topology and robustnessin the Drosophila segment polarity network, PLoS Biol., 2(2004), 0805-0815.

[15]

M. Khong, F. Y. M. Wan, Negative feedback in morphogen gradients, Frontiers of Applied Mathematics (Proc. of 2nd International Symposium, June, 2006, Beijing), Ed. D.-Y. Hsieh, M. Zhang and W. Sun, World Scientific, NJ, 2007, 29-51.

[16]

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), 513-523. doi: 10.1016/j.devcel.2004.08.004.

[17]

J. Kreuger, L. Perez, A. J. Giraldez and S. M. Choen, Opposing activities of Dally-like glypican at high and low levels of Wingless morphogen activity, Dev. Cell, 7 (2004), 503-512. doi: 10.1016/j.devcel.2004.08.005.

[18]

A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion? Dev. Cell, 2 (2002), 785-796. doi: 10.1016/S1534-5807(02)00179-X.

[19]

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.

[20]

A. D. Lander, F. Y. M. Wan and Q. Nie, Multiple paths to morphogen gradient robustness, preprint, (2005).

[21]

A. D. Lander, Q. Nie, B. Vargas and F. Y. M. Wan, Size-normalized robustness of Dpp gradient in drosophila wing imaginal disc, J. Mech. Mat. Struct. Accepted, (2010).

[22]

T. Lecuit and S. M. Cohen, Dpp receptor levels contribute to shaping the Dpp morphogen gradient in the Drosophila wing imaginal disc, Development, 125 (1998), 4901-4907.

[23]

J. Lei, Mathematical model of the Dpp gradient formation in drosophila wing imaginal disc, Chinese Sci. Bull., 55 (2010), 984-991. doi: 10.1360/972009-1522.

[24]

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.

[25]

X. Lin, Functions of heparan sulfate proteoglygans in cell signaling during development, Development, 131 (2004), 6009-6021. doi: doi:10.1242/dev.01522.

[26]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equation,'' Springer, Berlin, 2004.

[27]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana University Math. J., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079.

[28]

M. Strigini and S. M. Cohen, Wingless gradient formation in the Drosophila wing, Curr. Biol., 10 (2000), 293-300. doi: 10.1016/S0960-9822(00)00378-X.

[29]

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.

[30]

I. The, Y. Bellaiche and N. Perrimon, Hedgehog movement is regulated through tout velu-dependent synthesis of a heparan sulfate proteglycan, Mol. Cell, 4 (1999), 633-639. doi: 10.1016/S1097-2765(00)80214-2.

[31]

G. von Dassow, E. Meir, E. M. Munro and G. M. Odell, The segment polarity network is a robust developmental module, Nature, 406 (2000), 188-192. doi: 10.1038/35018085.

[32]

G. von Dassow, and G. M. Odell, Design and constraints of the Drosophila segment polarity module: robust spatial patterning emerges from intertwined cell state switches, J. Exp. Zool., 294 (2002), 179-215. doi: 10.1002/jez.10144.

show all references

References:
[1]

G. H. Baeg and N. Perrimon, Functional binding of secreted molecules to heparan sulfate proteoglycans in Drosophila, Curr. Opin. Cell. Biol., 12 (2000), 575-580. doi: 10.1016/S0955-0674(00)00134-4.

[2]

G. H. 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), 89-100. doi: 10.1016/j.ydbio.2004.08.023.

[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), 231-244. doi: 10.1016/j.cell.2004.09.031.

[4]

D. J. Bornemann, J. E. Duncan, W. Staatz, S. Seleck and R. Warrior, Abrogation of heparan sulfate synthesis in Drosophila disrupts the Wingless, hedgehog and Decapentaplegic signaling pathways, Development, 131 (2004), 1927-1938. doi: 10.1242/dev.01061.

[5]

K. M. Cadigan, M. P. Fish, E. J. 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.

[6]

A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B. Z. Shilo and N. Barkai, Robustness of MmP morphogen gradient in Drosophila embryonic patterning, Nature, 419 (2002), 304-308. doi: 10.1038/nature01061.

[7]

A. Eldar, D. Rosin, B. Z. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen graients, Dev. Cell, 5 (2003), 635-646. doi: 10.1016/S1534-5807(03)00292-2.

[8]

A. Eldar, B. Z. Shilo and N. Barkai, Elucidating mechanisms underlying robustness of morphogen gradients, Curr. Opin. Genet. Dev., 14 (2004), 435-439.

[9]

E. V. Entchev, A. Schwabedissen and M. Gonzá lez-Gaitán, Grandient formation of the TGF-$\beta$ homolog dpp, Cell, 103 (2000), 981-991. doi: 10.1016/S0092-8674(00)00200-2.

[10]

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), 1515-1522. doi: 10.1242/dev.00379.

[11]

Y. Funakoshi, M. Minami and T. Tabata, Mtv shapes the activity gradient of the Dpp morphogen through regulation of thickveins, Development, 128 (2001), 67-74.

[12]

C. Han, T. Y. Belenkaya, M. Khodoun, M. Tauchi, X. D. Lin and X. H. Lin, Distinct and collaborative roles of Drosophila EXT family proteins in morphogen signalling and gradient formation, Development, 131 (2004), 1563-1575. doi: 10.1242/dev.01051.

[13]

B. Houchmandzadeh, E. Wieschaus and S. Leibler, Establishment of developmental precision and proportions in the early Drosophila embryo, Nature, 415 (2002), 798-802.

[14]

N. T. Ingolia, Topology and robustnessin the Drosophila segment polarity network, PLoS Biol., 2(2004), 0805-0815.

[15]

M. Khong, F. Y. M. Wan, Negative feedback in morphogen gradients, Frontiers of Applied Mathematics (Proc. of 2nd International Symposium, June, 2006, Beijing), Ed. D.-Y. Hsieh, M. Zhang and W. Sun, World Scientific, NJ, 2007, 29-51.

[16]

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), 513-523. doi: 10.1016/j.devcel.2004.08.004.

[17]

J. Kreuger, L. Perez, A. J. Giraldez and S. M. Choen, Opposing activities of Dally-like glypican at high and low levels of Wingless morphogen activity, Dev. Cell, 7 (2004), 503-512. doi: 10.1016/j.devcel.2004.08.005.

[18]

A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion? Dev. Cell, 2 (2002), 785-796. doi: 10.1016/S1534-5807(02)00179-X.

[19]

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.

[20]

A. D. Lander, F. Y. M. Wan and Q. Nie, Multiple paths to morphogen gradient robustness, preprint, (2005).

[21]

A. D. Lander, Q. Nie, B. Vargas and F. Y. M. Wan, Size-normalized robustness of Dpp gradient in drosophila wing imaginal disc, J. Mech. Mat. Struct. Accepted, (2010).

[22]

T. Lecuit and S. M. Cohen, Dpp receptor levels contribute to shaping the Dpp morphogen gradient in the Drosophila wing imaginal disc, Development, 125 (1998), 4901-4907.

[23]

J. Lei, Mathematical model of the Dpp gradient formation in drosophila wing imaginal disc, Chinese Sci. Bull., 55 (2010), 984-991. doi: 10.1360/972009-1522.

[24]

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.

[25]

X. Lin, Functions of heparan sulfate proteoglygans in cell signaling during development, Development, 131 (2004), 6009-6021. doi: doi:10.1242/dev.01522.

[26]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equation,'' Springer, Berlin, 2004.

[27]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana University Math. J., 21 (1972), 979-1000. doi: 10.1512/iumj.1972.21.21079.

[28]

M. Strigini and S. M. Cohen, Wingless gradient formation in the Drosophila wing, Curr. Biol., 10 (2000), 293-300. doi: 10.1016/S0960-9822(00)00378-X.

[29]

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.

[30]

I. The, Y. Bellaiche and N. Perrimon, Hedgehog movement is regulated through tout velu-dependent synthesis of a heparan sulfate proteglycan, Mol. Cell, 4 (1999), 633-639. doi: 10.1016/S1097-2765(00)80214-2.

[31]

G. von Dassow, E. Meir, E. M. Munro and G. M. Odell, The segment polarity network is a robust developmental module, Nature, 406 (2000), 188-192. doi: 10.1038/35018085.

[32]

G. von Dassow, and G. M. Odell, Design and constraints of the Drosophila segment polarity module: robust spatial patterning emerges from intertwined cell state switches, J. Exp. Zool., 294 (2002), 179-215. doi: 10.1002/jez.10144.

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