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WIREs Dev Biol
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Mathematical models of morphogen gradients and their effects on gene expression

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Abstract An introduction to mathematical models of pattern formation by morphogen gradients is presented, using the early embryo of the fruit fly Drosophila as the main experimental example. Analysis of morphogen gradient formation is based on the source–diffusion–degradation models and a formalism of local accumulation times. Transcriptional control by morphogens is discussed within the framework of thermodynamic site occupancy models of gene regulatory regions. WIREs Dev Biol 2012 doi: 10.1002/wdev.55 For further resources related to this article, please visit the WIREs website.

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Graded distributions of transcription factors in the early Drosophila embryo. (a) Spatially uniform arrangement of nuclei in a fixed embryo, ∼2 h after fertilization. No morphological asymmetries are apparent at this point of development. Nuclei are visualized by DAPI staining. (b) Adult Drosophila, about to take off. Schematic on the right provides the orientation of the anteroposterior (AP) and dorsoventral (DV) body axes. (c) Concentration gradient of Bicoid (Bcd), a transcription factor that organizes anterior development of the embryo. The Bcd distribution is visualized by antibody staining in a fixed embryo. (d) Fluorescence intensity profile of the signal from an embryo stained with a‐Bcd antibody. (e) Nuclear localization gradient of Dorsal (Dl), a transcription factor that organizes DV patterning of the embryo. In this image, the nuclear distribution of Dl is visualized in a vertically oriented embryo. (f) Quantified fluorescence intensity of nuclear Dl, based on a‐Dl antibody staining. (g) Fixed embryo stained with an antibody that recognizes Capicua (Cic), a transcriptional repressor downregulated at the terminal regions of the embryo. (h) The graded pattern of Cic concentration has minima at both anterior and posterior poles.

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Spatial signal processing by the positive feedback loop. (a) Possible configurations of a gene regulatory region controlled by two activators, one of which has a cluster of homotypic binding sites. (b) Structure of the positive feedback circuit. (c) Geometrical analysis of steady states and their stability at zero level of morphogen concentration. A case with a single stable steady state. Model parameters are as follows (see text for details): \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$C_{B}^{\max} =3$ \end{document}, KA = 10, KB = 2. (d) Geometrical analysis of steady states and their stability at zero level of morphogen concentration. A case with steady state multiplicity. Parameters are as in (c), but \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$C_{B}^{\max} =5$ \end{document}. (e) Parametric dependence of steady states on morphogen concentration (CA). The ‘off’ steady state (blue) disappears beyond critical concentration of the morphogen. Stable and unstable steady states are shown by solid and dashed curves, respectively. (f) Discontinuous spatial profile of the steady state generated by a positive feedback responding to a morphogen gradient.

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Spatial signal processing by the incoherent feedforward loop. (a) Possible configurations of a regulatory region controlled by an activator and repressor. (b) Structure of an incoherent feedforward loop. (c) Normalized activity of the regulatory region as a function of morphogen concentration. Model parameters are as follows (see text for details): \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$K_{A}^{C} =0.05$ \end{document},\documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$K_{B}^{C} =0.01$ \end{document}, \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$K_{A}^{B} =50$ \end{document}, \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$C_{B}^{\max} =35$ \end{document}. (d) Normalized spatial distribution of the morphogen (black) and normalized spatial distribution of the activity of the regulatory region of gene C (blue).

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Gene regulation by combination of graded and uniform signals. (a) Possible configurations of a regulatory region with two binding sites for activator A (oval) and B (pentagon). (b) Input/output map of the regulatory region as a function of concentration of factor A, at different concentrations of factor B, or at different values of the cooperativity parameter (see text for details). Model parameters are as follows (see text for details):KA = 1, KB = 8,4,200, and ω = 200. (c) Spatial distributions of two transcription factors. (d) Normalized spatial pattern of the activity of the regulatory region.

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Dynamics in the model of Bicoid (Bcd) gradient formation. (a) Local accumulation time as a function of position, calculated for D = 4 µm2/s, 1/k = 50 min, λs = 0 µm (black) and λs = 50 µm (red). (b) Morphogen concentration profiles, evaluated at 30 min, 60 min, and at steady state, based on exact and approximate solutions (black and red curves, respectively).

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Local accumulation time. (a) Schematic of morphogen gradient formation by synthesis, diffusion, and spatially uniform degradation. The source of morphogen production is distributed exponentially. (b) Concentration profile at four different times: t = 1/2k,1/k,2/k,∞. The concentration profile has been plotted for λs = 0, and scaled by the maximal concentration at x = 0, \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$C_{s} (0)=Q/\sqrt{Dk}$ \end{document}. The spatial coordinate has been rescaled by the dynamic length scale, \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$\lambda =\sqrt{D/k}$ \end{document}. (c) Concentration dynamics at x = λ/2 and x = 2λ. (d) Local kinetics of the fractional deviation from the steady state concentration, plotted at two different positions, x = λ/2 and x = 2λ.

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Fragments of regulatory networks that convert maternal morphogen gradients into patterns of zygotic gene expression. (a) Anterior expression pattern of tailless (tll) depends on its direct activation by Bicoid (Bcd) and repression by Capicua (Cic). (b) Dorsal (Dl) expression of zerknüllt (zen) is generated by a combination of graded Dl, which acts a direct repressor, and spatially uniform Zelda (Zld), an early activator of zygotic transcription. Dl‐dependent repression of zen also depends on Cic (not shown). (c) Lateral stripes of rhomboid (rho) expression along the dorsoventral axis of the embryo are generated by a network that combines a coherent feedforward loop, an incoherent feedforward loop, and a positive feedback autoregulation motif. Dl regulates rho both directly and through Twist (twi, an activator) and Snail (sna, a repressor). In addition to sna and rho, Twi also regulates its own expression.

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Fluorescent in situ hybridization images showing the expression patterns of some of the genes regulated by Bicoid (Bcd), Capicua (Cic) and Dorsal (Dl). (a) Expression domains of Bcd and Cic targets along the anteroposterior (AP) axis of the embryo. Note that hunchback (hb; yellow), Krüppel (Kr; green), knirps (kni; red), giant (gt; blue), tailless (tll; magenta) and huckebein (hkb; orange) are expressed in distinct domains along the AP axis. In anterior region of the embryo, the expression domains of these genes dependend on the activating functions of the Bcd gradient and the repressive functions of Cic. Anterior: left. (b) Expression domains of Dl targets. snail (sna; blue) is only detected in the ventral regions of the embryo, while rhomboid (rho; green) and short gastrulation (sog; yellow) are expressed in more lateral domains and zerknüllt (zen; red) is confined to the dorsal region. Shown are sections through the embryo, ventral: bottom, dorsal: top.

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Gene Expression and Transcriptional Hierarchies > Gene Networks and Genomics
Establishment of Spatial and Temporal Patterns > Gradients
Gene Expression and Transcriptional Hierarchies > Quantitative Methods and Models