2005, 2(2): 239-262. doi: 10.3934/mbe.2005.2.239

Spatially Distributed Morphogen Production and Morphogen Gradient Formation

1. 

Department of Developmental and Cell Biology, University of California, Irvine, CA 92697-3875, United States

2. 

Department of Mathematics and Department of Biomedical Engineering, University of California, Irvine, CA 92697-3875, United States

3. 

Department of Mathematics, Center for Complex Biological Systems, University of California, Irvine, California, 92697-3875, United States

Received  December 2004 Revised  March 2005 Published  March 2005

Partial differential equations and auxiliary conditions governing the activities of the morphogen Dpp in Drosophila wing imaginal discs were formulated and analyzed as Systems B, R, and C in [7][9][10]. All had morphogens produced at the border of anterior and posterior chamber of the wing disc idealized as a point, line, or plane in a one-, two-, or three-dimensional model. In reality, morphogens are synthesized in a narrow region of finite width (possibly of only a few cells) between the two chambers in which diffusion and reversible binding with degradable receptors may also take place. The present investigation revisits the extracellular System R, now allowing for a finite production region of Dpp between the two chambers. It will be shown that this more refined model of the wing disc, designated as System F, leads to some qualitatively different morphogen gradient features. One significant difference between the two models is that System F impose no restriction on the morphogen production rate for the existence of a unique stable steady state concentration of the Dpp-receptor complexes. Analytical and numerical solutions will be obtained for special cases of System F. Some applications of the results for explaining available experimental data (to appear elsewhere) are briefly indicated. It will also be shown how the effects of the distributed source of System F may be aggregated to give an approximating point source model (designated as the aggregated source model or System A for short) that includes System R as a special case. System A will be analyzed in considerable detail in [6], and the limitation of System R as an approximation of System F will also be delineated there.
Citation: 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
[1]

David Iron, Adeela Syed, Heidi Theisen, Tamas Lukacsovich, Mehrangiz Naghibi, Lawrence J. Marsh, Frederic Y. M. Wan, Qing Nie. The role of feedback in the formation of morphogen territories. Mathematical Biosciences & Engineering, 2008, 5 (2) : 277-298. doi: 10.3934/mbe.2008.5.277

[2]

Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i

[3]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081

[4]

Adélia Sequeira, Rafael F. Santos, Tomáš Bodnár. Blood coagulation dynamics: mathematical modeling and stability results. Mathematical Biosciences & Engineering, 2011, 8 (2) : 425-443. doi: 10.3934/mbe.2011.8.425

[5]

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

[6]

Avner Friedman. PDE problems arising in mathematical biology. Networks & Heterogeneous Media, 2012, 7 (4) : 691-703. doi: 10.3934/nhm.2012.7.691

[7]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

[8]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[9]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

[10]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217

[11]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809

[12]

Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609

[13]

Monique Chyba, Benedetto Piccoli. Special issue on mathematical methods in systems biology. Networks & Heterogeneous Media, 2019, 14 (1) : ⅰ-ⅱ. doi: 10.3934/nhm.20191i

[14]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111

[15]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

[16]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[17]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339

[18]

Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056

[19]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867

[20]

H. Malchow, F.M. Hilker, S.V. Petrovskii. Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 705-711. doi: 10.3934/dcdsb.2004.4.705

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]