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]

Dev. Biol., 313 (2008), 408-419. Google Scholar

[2]

Dev. Biol., 276 (2004), 89-100. Google Scholar

[3]

Cell, 119 (2004), 231-244. Google Scholar

[4]

Phys. Rev. Lett., 94 (2005), 018103. Google Scholar

[5]

Phys. Rev. E., 75 (2007), 011901. Google Scholar

[6]

Development, 131 (2004), 1927-1938. Google Scholar

[7]

Dev. Cell, 5 (2003), 635-646. Google Scholar

[8]

Cell, 103 (2000), 981-991. Google Scholar

[9]

Development, 130 (2003), 1515-1522. Google Scholar

[10]

Dev. Biol., 300 (2006), 512-522. Google Scholar

[11]

J. Theor. Biol., 183 (1996), 95-104. Google Scholar

[12]

J. Theor. Biol., 191 (1998), 103-114. Google Scholar

[13]

Science, 315 (2007), 521-525. Google Scholar

[14]

Dev. Cell, 7 (2004), 513-523. Google Scholar

[15]

Development, 131 (2004), 4843-4856. Google Scholar

[16]

Dev. Cell, 2 (2002), 785-796. Google Scholar

[17]

Math. Biosci. Eng., 2 (2005), 239-262. doi: 10.3934/mbe.2005.2.239.  Google Scholar

[18]

J. Comp. Appl. Math., 190 (2006), 232-251. doi: 10.1016/j.cam.2004.11.054.  Google Scholar

[19]

Cell, 128 (2007), 245-256. Google Scholar

[20]

Bull. Math. Biol., 69 (2007), 33-54. 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]

Chinese Sci. Bull., 55 (2010), 984-991. Google Scholar

[23]

Bull. Math. Biol., 72 (2010), 805-829. doi: 10.1007/s11538-009-9470-2.  Google Scholar

[24]

Disc. Cont. Dyns. Syst. B, 16 (2011), 835-866. doi: 10.3934/dcdsb.2011.16.835.  Google Scholar

[25]

Studies in Applied Mathematics, 113 (2004), 183-215.  Google Scholar

[26]

SIAM J. Appl. Math., 65 (2005), 1748-1771. 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]

Cell, 103 (2000), 971-980. Google Scholar

[29]

Mol. Cell., 4 (1999), 633-639. Google Scholar

[30]

Dev. Cell., 3 (2002), 615-623. Google Scholar

[31]

J. Theor. Biol., 269 (2011), 359-365. Google Scholar

[32]

J. Theor. Biol., 248 (2007), 579-589. doi: 10.1016/j.jtbi.2007.05.026.  Google Scholar

show all references

References:
[1]

Dev. Biol., 313 (2008), 408-419. Google Scholar

[2]

Dev. Biol., 276 (2004), 89-100. Google Scholar

[3]

Cell, 119 (2004), 231-244. Google Scholar

[4]

Phys. Rev. Lett., 94 (2005), 018103. Google Scholar

[5]

Phys. Rev. E., 75 (2007), 011901. Google Scholar

[6]

Development, 131 (2004), 1927-1938. Google Scholar

[7]

Dev. Cell, 5 (2003), 635-646. Google Scholar

[8]

Cell, 103 (2000), 981-991. Google Scholar

[9]

Development, 130 (2003), 1515-1522. Google Scholar

[10]

Dev. Biol., 300 (2006), 512-522. Google Scholar

[11]

J. Theor. Biol., 183 (1996), 95-104. Google Scholar

[12]

J. Theor. Biol., 191 (1998), 103-114. Google Scholar

[13]

Science, 315 (2007), 521-525. Google Scholar

[14]

Dev. Cell, 7 (2004), 513-523. Google Scholar

[15]

Development, 131 (2004), 4843-4856. Google Scholar

[16]

Dev. Cell, 2 (2002), 785-796. Google Scholar

[17]

Math. Biosci. Eng., 2 (2005), 239-262. doi: 10.3934/mbe.2005.2.239.  Google Scholar

[18]

J. Comp. Appl. Math., 190 (2006), 232-251. doi: 10.1016/j.cam.2004.11.054.  Google Scholar

[19]

Cell, 128 (2007), 245-256. Google Scholar

[20]

Bull. Math. Biol., 69 (2007), 33-54. 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]

Chinese Sci. Bull., 55 (2010), 984-991. Google Scholar

[23]

Bull. Math. Biol., 72 (2010), 805-829. doi: 10.1007/s11538-009-9470-2.  Google Scholar

[24]

Disc. Cont. Dyns. Syst. B, 16 (2011), 835-866. doi: 10.3934/dcdsb.2011.16.835.  Google Scholar

[25]

Studies in Applied Mathematics, 113 (2004), 183-215.  Google Scholar

[26]

SIAM J. Appl. Math., 65 (2005), 1748-1771. 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]

Cell, 103 (2000), 971-980. Google Scholar

[29]

Mol. Cell., 4 (1999), 633-639. Google Scholar

[30]

Dev. Cell., 3 (2002), 615-623. Google Scholar

[31]

J. Theor. Biol., 269 (2011), 359-365. Google Scholar

[32]

J. Theor. Biol., 248 (2007), 579-589. doi: 10.1016/j.jtbi.2007.05.026.  Google Scholar

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