Spatially Distributed Morphogen Production and Morphogen Gradient Formation
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1.
Department of Developmental and Cell Biology, University of California, Irvine, CA 92697-3875
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2.
Department of Mathematics and Department of Biomedical Engineering, University of California, Irvine, CA 92697-3875
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3.
Department of Mathematics, Center for Complex Biological Systems, University of California, Irvine, California, 92697-3875
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Received:
01 December 2004
Accepted:
29 June 2018
Published:
01 March 2005
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MSC :
92C15.
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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[J]. Mathematical Biosciences and Engineering, 2005, 2(2): 239-262. doi: 10.3934/mbe.2005.2.239
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Abstract
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.
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