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 and 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), 408-419.

[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), 89-100.

[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.

[4]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Robust formation of morphogen gradients, Phys. Rev. Lett., 94 (2005), 018103.

[5]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in epithelia, Phys. Rev. E., 75 (2007), 011901.

[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), 1927-1938.

[7]

A. Eldar, D. Rosin, B. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients, Dev. Cell, 5 (2003), 635-646.

[8]

E. V. Entchev, A. Schwabedissen and M. Gonzalez-Gaitan, Gradient formation of the TGF-$\beta$ homolog Dpp, Cell, 103 (2000), 981-991.

[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), 1515-1522.

[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), 512-522.

[11]

M. Kerszberg, Accurate reading of morphogen concentrations by nuclear receptors: A formal model of complex transduction pathways, J. Theor. Biol., 183 (1996), 95-104.

[12]

M. Kerszberg and L. Wolpert, Mechanisms for positional signalling by morphogen transport: A theoretical study, J. Theor. Biol., 191 (1998), 103-114.

[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), 521-525.

[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), 513-523.

[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), 4843-4856.

[16]

A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion?, Dev. Cell, 2 (2002), 785-796.

[17]

A. D. Lander, Q. Nie and F. Y. M. Wan, Spatially distributed morphogen production and morphogen gradient formation, Math. Biosci. Eng., 2 (2005), 239-262. doi: 10.3934/mbe.2005.2.239.

[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), 232-251. doi: 10.1016/j.cam.2004.11.054.

[19]

A. D. Lander, Morpheus unbound: Reimagining the morphogen gradient, Cell, 128 (2007), 245-256.

[20]

A. D. Lander, Q. Nie and F. Y. M. Wan, Membrane-associated non-receptors and morphogen gradients, Bull. Math. Biol., 69 (2007), 33-54. doi: 10.1007/s11538-006-9152-2.

[21]

A. D. Lander, F. Y. M. Wan and Q. Nie, Multiple paths to morphogen gradient robustness, preprint, Univ. California, Irvine, http://www.ics.uci.edu/~xhx/courses/SDE/Draft5.pdf.

[22]

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

[23]

J. Lei and Y. Song, Mathematical model of the formation of morphogen gradients through membrane-associated non-receptors, Bull. Math. Biol., 72 (2010), 805-829. doi: 10.1007/s11538-009-9470-2.

[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), 835-866. doi: 10.3934/dcdsb.2011.16.835.

[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), 183-215.

[26]

Y. Lou, Q. Nie and F. Y. M. Wan, Effects of Sog on Dpp-receptor binding, SIAM J. Appl. Math., 65 (2005), 1748-1771. doi: 10.1137/S0036139903433219.

[27]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana University Math. J., 21 (1971/72), 979-1000.

[28]

A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc, Cell, 103 (2000), 971-980.

[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), 633-639.

[30]

J. Vincent and L. Dubois, Morphogen transport along epithelia and integrated trafficking problem, Dev. Cell., 3 (2002), 615-623.

[31]

L. Wolpert, Position information and patterning revisited, J. Theor. Biol., 269 (2011), 359-365.

[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), 579-589. doi: 10.1016/j.jtbi.2007.05.026.

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), 408-419.

[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), 89-100.

[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.

[4]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Robust formation of morphogen gradients, Phys. Rev. Lett., 94 (2005), 018103.

[5]

T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in epithelia, Phys. Rev. E., 75 (2007), 011901.

[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), 1927-1938.

[7]

A. Eldar, D. Rosin, B. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients, Dev. Cell, 5 (2003), 635-646.

[8]

E. V. Entchev, A. Schwabedissen and M. Gonzalez-Gaitan, Gradient formation of the TGF-$\beta$ homolog Dpp, Cell, 103 (2000), 981-991.

[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), 1515-1522.

[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), 512-522.

[11]

M. Kerszberg, Accurate reading of morphogen concentrations by nuclear receptors: A formal model of complex transduction pathways, J. Theor. Biol., 183 (1996), 95-104.

[12]

M. Kerszberg and L. Wolpert, Mechanisms for positional signalling by morphogen transport: A theoretical study, J. Theor. Biol., 191 (1998), 103-114.

[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), 521-525.

[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), 513-523.

[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), 4843-4856.

[16]

A. D. Lander, Q. Nie and F. Y. M. Wan, Do morphogen gradients arise by diffusion?, Dev. Cell, 2 (2002), 785-796.

[17]

A. D. Lander, Q. Nie and F. Y. M. Wan, Spatially distributed morphogen production and morphogen gradient formation, Math. Biosci. Eng., 2 (2005), 239-262. doi: 10.3934/mbe.2005.2.239.

[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), 232-251. doi: 10.1016/j.cam.2004.11.054.

[19]

A. D. Lander, Morpheus unbound: Reimagining the morphogen gradient, Cell, 128 (2007), 245-256.

[20]

A. D. Lander, Q. Nie and F. Y. M. Wan, Membrane-associated non-receptors and morphogen gradients, Bull. Math. Biol., 69 (2007), 33-54. doi: 10.1007/s11538-006-9152-2.

[21]

A. D. Lander, F. Y. M. Wan and Q. Nie, Multiple paths to morphogen gradient robustness, preprint, Univ. California, Irvine, http://www.ics.uci.edu/~xhx/courses/SDE/Draft5.pdf.

[22]

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

[23]

J. Lei and Y. Song, Mathematical model of the formation of morphogen gradients through membrane-associated non-receptors, Bull. Math. Biol., 72 (2010), 805-829. doi: 10.1007/s11538-009-9470-2.

[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), 835-866. doi: 10.3934/dcdsb.2011.16.835.

[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), 183-215.

[26]

Y. Lou, Q. Nie and F. Y. M. Wan, Effects of Sog on Dpp-receptor binding, SIAM J. Appl. Math., 65 (2005), 1748-1771. doi: 10.1137/S0036139903433219.

[27]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana University Math. J., 21 (1971/72), 979-1000.

[28]

A. A. Teleman and S. M. Cohen, Dpp gradient formation in the Drosophila wing imaginal disc, Cell, 103 (2000), 971-980.

[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), 633-639.

[30]

J. Vincent and L. Dubois, Morphogen transport along epithelia and integrated trafficking problem, Dev. Cell., 3 (2002), 615-623.

[31]

L. Wolpert, Position information and patterning revisited, J. Theor. Biol., 269 (2011), 359-365.

[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), 579-589. doi: 10.1016/j.jtbi.2007.05.026.

[1]

Georg Hetzer. Global existence for a functional reaction-diffusion problem from climate modeling. Conference Publications, 2011, 2011 (Special) : 660-671. doi: 10.3934/proc.2011.2011.660

[2]

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

[3]

Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems and Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709

[4]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[5]

Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371

[6]

Arthur D. Lander, Qing Nie, Frederic Y. M. Wan. Spatially Distributed Morphogen Production and Morphogen Gradient Formation. Mathematical Biosciences & Engineering, 2005, 2 (2) : 239-262. doi: 10.3934/mbe.2005.2.239

[7]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617

[8]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations and Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[9]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020

[10]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431

[11]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[12]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks and Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003

[13]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks and Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011

[14]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[15]

John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291

[16]

Feliz Minhós, T. Gyulov, A. I. Santos. Existence and location result for a fourth order boundary value problem. Conference Publications, 2005, 2005 (Special) : 662-671. doi: 10.3934/proc.2005.2005.662

[17]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436

[18]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

[19]

Kateryna Marynets. Stability analysis of the boundary value problem modelling a two-layer ocean. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2433-2445. doi: 10.3934/cpaa.2022083

[20]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (7)

[Back to Top]