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WIREs Syst Biol Med
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Modeling the control of planar cell polarity

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Abstract A growing list of medically important developmental defects and disease mechanisms can be traced to disruption of the planar cell polarity (PCP) pathway. The PCP system polarizes cells in epithelial sheets along an axis orthogonal to their apical–basal axis. Studies in the fruitfly, Drosophila, have suggested that components of the PCP signaling system function in distinct modules, and that these modules and the effector systems with which they interact function together to produce emergent patterns. Experimental methods allow the manipulation of individual PCP signaling molecules in specified groups of cells; these interventions not only perturb the polarization of the targeted cells at a subcellular level, but also perturb patterns of polarity at the multicellular level, often affecting nearby cells in characteristic ways. These kinds of experiments should, in principle, allow one to infer the architecture of the PCP signaling system, but the relationships between molecular interactions and tissue‐level pattern are sufficiently complex that they defy intuitive understanding. Mathematical modeling has been an important tool to address these problems. This article explores the emergence of a local signaling hypothesis, and describes how a local intercellular signal, coupled with a directional cue, can give rise to global pattern. We will discuss the critical role mathematical modeling has played in guiding and interpreting experimental results, and speculate about future roles for mathematical modeling of PCP. Mathematical models at varying levels of inhibition have and are expected to continue contributing in distinct ways to understanding the regulation of PCP signaling. WIREs Syst Biol Med 2011 3 588–605 DOI: 10.1002/wsbm.138 This article is categorized under: Developmental Biology > Developmental Processes in Health and Disease

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Planar cell polarity. (a) Cartoon of cochlear hair cells showing asymmetrically localized and coordinately oriented kinoclila (purple) and stereocilia (blue). (b) Wild‐type and (c) Lp/Lp mutant cochlear hair cells stained with phalloidin to show actin. Orientation of some cells is denoted with arrows. (d) Cartoon of Drosophila pupal wing cells with nascent hairs emerging from the distal vertex and pointing distally. (e) Wild‐type and (f) dishevelled‐1 wings showing hair polarity (distal is right). (g) Chiral and oriented ommatidia in the fly eye. Equipotent R3/R4 cells (green) differentiate to R3 (equatorial; yellow) and R4 (polar; blue) according to their relative position along the equatorial‐polar axis. Ommatidia then rotate in opposite directions in the dorsal (top) and ventral (bottom) eye fields. (h) Wild‐type eye showing ommatidia in the dorsal (top) and ventral (bottom) eye fields. Orientation is noted with blue arrows. Images in (b) and (c) were kindly provided by M. Montcouquiol and M. Kelley.

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Signaling mechanism modeled by Le Garrec et al.125 using reaction–diffusion stochastic difference equations. Note that the feedback loops differ from those in Figure 5.

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Global signaling paradigms tested by the Amonlirdviman et al. model.65 (a) Nonglobally varying, constant slope within each cell, as suggested by the global signaling model depicted in Figure 5. (b) Globally varying, uniform within each cell, similar to that proposed by Lawrence et al.75 (c) Globally varying, constant slope. This signal can be viewed as a combination of the forms shown in (a) and (b). Depending on whether the intercellular or intracellular components predominate, the slope can be in either direction. As shown, the subcellular asymmetry dominates, as in (a).

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Simulation results from the Amonlirdviman et al. model. Wild‐type results showing the final distributions of Dsh (a), Fz (b), Pk (c), and Vang (d). The same color scale is used in all figures, where 1 is scaled to the initial uniform concentration of Dsh and the scale is truncated so that concentrations greater than 3 are shown in red. (e) Simulated distribution of Dsh displayed as an intensity representing total Dsh concentrations, corresponding to the appearance of Dsh::GFP in wild‐type experiments. Simulation results of several PCP phenotypes showing the final distribution of Dsh with predicted hair growth directions derived from the vector sum of Dsh (f, g, k, and n) and corresponding pupal wings (h–j, l, m, and o). Greater Dsh asymmetry is represented by hair placement at increasing distances from the cell center. When Dsh asymmetry does not exceed the threshold value, the hair is depicted at the center of the cell. Mutant cells are designated with yellow. (h) fzR52 clones. (i and j) VangA3 clones. (l and m) dshV26 mutant clones. (o) pk‐sple13 clones. Asterisks mark nonautonomous effects near dsh clones. (Reprinted with permission from Ref 65. Copyright 2005 American Association for the Advancement of Science.)

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Schematics of the Lawrence et al. algebraic model. (a) In wild type, the factor X gradient (gray) initiates Fz activation at scalar levels in each cell. After averaging, the resulting Fz levels are compared with neighbors to produce a vector determining polarity (arrows). (b) A wing with a fz clone cannot produce Fz within the clone, and after averaging, the scalar levels of Fz in neighboring cells are affected, altering the resulting polarity. Cells at the edges of the clone may be repolarized, depending on the mechanism of comparison and vector determination (arrowheads).

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Cartoon of the Fz feedback loop proposed by Tree et al. (without Fmi and Vang) as subsequently modified by Amonlirdviman et al.65,100(with Fmi and Vang; bottom). A cartoon of a possible mechanism for Fj, Ds, and Ft function that biases Fz feedback loop function is also shown (top). Pale symbols represent species with reduced concentration.

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Cellular phenotypes of wing hair cells. (a) Wild‐type, (b) core gene mutants, (c) downstream effector mutants, and (d) global directional mutants.

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Diffusible factor model. An upstream signal (a; blue), such as Wg in the eye, induces a proportional expression of a diffusible factor X (gray). (b) In wild type, factor X distributes in a gradient (green) whose slope determines polarity (arrows). (c) In clones that do not respond to the upstream signal, the factor X expression profile results in a distribution profile (green) with slope reversal, causing polarity reversal distal to the clone (red arrow). (d) Clones with augmented response to the upstream factor result in a distribution profile showing polarity reversal on the proximal side of the clone. (e) Factor X and factor Z signaling scheme.

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Cartoons of clonal phenotypes. (a) Wild‐type polarity with all hairs pointing distally. (b) A cell autonomous mutant clone (orange outline) showing loss of polarity within the clone (triangles) and normal polarity surrounding the clone. (c) Mutant clone with distal domineering nonautonomy (gray shaded area) characteristic of fz mutant clones. (d) Mutant clone showing proximal domineering nonautonomy characteristic of Vang mutant clones. See Figure 7 for phalloidin stained pupal wing clones showing these phenotypes.

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