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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 721 - 739, Volume 18, Issue 3, May 2013      doi:10.3934/dcdsb.2013.18.721

 
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Jinzhi Lei - Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing 100084, China (email)
Dongyong Wang - Department of mathematics, University of California, CA, 92697-3875, United States (email)
You Song - College of software, Beihang University, Beijing, 100083, China (email)
Qing Nie - Department of mathematics, University of California, CA, 92697-3875, United States (email)
Frederic Y. M. Wan - Department of mathematics, University of California, CA, 92697-3875, United States (email)

Abstract: 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.

Keywords:  Morphogen gradient, reaction diffusion equation, existence and uniqueness, stability, robustness, functional boundary value problem.
Mathematics Subject Classification:  Primary: 34B15, 92C15; Secondary: 34B60, 35K57.

Received: July 2012;      Revised: October 2012;      Available Online: December 2012.

 References