Establishment and maintenance of planar epithelial cell polarity by asymmetric cadherin bridges: A computer model

Authors

  • Jean-François Le Garrec,

    1. Modélisation dynamique des systèmes biologiques intégrés, CNRS UMR 7138 Systématique, Adaptation, Évolution, Université Pierre et Marie Curie, Paris, France
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  • Philippe Lopez,

    1. Modélisation dynamique des systèmes biologiques intégrés, CNRS UMR 7138 Systématique, Adaptation, Évolution, Université Pierre et Marie Curie, Paris, France
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  • Michel Kerszberg

    Corresponding author
    1. Modélisation dynamique des systèmes biologiques intégrés, CNRS UMR 7138 Systématique, Adaptation, Évolution, Université Pierre et Marie Curie, Paris, France
    • Modélisation dynamique des systèmes biologiques intégrés, CNRS UMR 7138 Systématique, Adaptation, Évolution, Université Pierre et Marie Curie, 7 quai St Bernard, Bât. A 4ème ét. Case 05, 75252 Paris, France
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Abstract

Animal scales, hairs, feathers, and cilia are oriented due to cell polarization in the epithelial plane. Genes involved have been identified, but the signal and mechanism remain unknown. In Drosophila wing polarization, the action of a gradient of Frizzled activity is widely assumed; and cell–cell signalling by cadherins such as Flamingo surely plays a major role. We present a computer model where reading the Frizzled gradient occurs through biased, feedback-reinforced formation of Flamingo-based asymmetric intercellular complexes. Through these complexes neighboring cells are able to compare their Frizzled activity levels. Our computations are highly noise-resistant and reproduce both wild-type and all known mutant wing phenotypes; other phenotypes are predicted. The model puts stringent limits on a Frizzled activation signal, which should exhibit unusual properties: (1) the extracellular Frizzled signalling gradient should be counterdirectional—decreasing from proximal (P) to distal (D), whereas during polarization, the intracellular Frizzled gradient builds up from P to D; (2) the external gradient should be relatively weak and short-lived, lest it prevent inversion of intracellular Frizzled. These features, largely independent of model details, may provide useful clues for future experimental efforts. Developmental Dynamics 235:235–246, 2006. © 2005 Wiley-Liss, Inc.

INTRODUCTION

The planar cell polarity (PCP) signalling pathway regulates tissue polarity in bilaterians (reviewed in Strutt,2003). Such polarity is obvious on the epidermis of most animals bearing scales, feathers, or hair. Proper functioning of many internal epithelia depends on polarized cilia, for instance in the mammalian uterus, trachea, or cochlea. The same pathway is involved in orienting clusters of differentiated cells, as happens in the insect compound eye (Wehrli and Tomlinson,1998). PCP signalling was also shown to regulate convergent extension during vertebrate gastrulation and neurulation (reviewed in Wallingford et al.,2002). The Drosophila wing, covered by distally pointing hairs, is the most thoroughly studied genetic model of tissue polarization (reviewed in Eaton,2003); therefore, we have focussed on understanding the mechanisms at work in this system.

The PCP pathway consists of three tiers: upstream molecules deliver directional information, core polarity proteins read this signal, and downstream molecules mediate interaction with the cytoskeleton (Tree et al.,2002a). Early in pupal life, the six core PCP proteins frizzled (fz, Vinson et al.,1989), dishevelled (dsh, Klingensmith et al.,1994), van gogh (vang also known as strabismus, Taylor et al.,1998), prickle (pk, Gubb et al.,1999), flamingo (fmi, also known as starry night, Usui et al.,1999), and diego (dgo, Feiguin et al.,2001) accumulate at the apicolateral adhesion junctions of wing cells. These molecules then progressively segregate to the distal and proximal cell boundaries until approximately 30 hr after pupa formation (reviewed in Strutt,2002). The actin pre-hair finally appears at the distal vertex of each cell and grows to become the distally pointing hair of the adult wing cell.

All the core molecules are required for proximodistal (P/D) segregation in the wing, but despite extensive work, the process is poorly understood. Investigators have proposed various mechanisms, including new putative molecules, some generally considered as plausible, others less so; these proposals may be summarized as follows:

  • 1It is usually admitted that a wing–scale gradient of molecular activity must be involved; it is possible that an extracellular ligand, maybe a Wnt molecule secreted from a localized source, is propagated over the system (Vinson and Adler,1987). Such a morphogenetic ligand, however, remains to be identified.
  • 2Assuming an Fz ligand is involved, a more specialized hypothesis is that, on each cell membrane, the Fz receptors of Wnt are inactivated on the proximal side. On the distal side, unbound Fz activates pre-hair initiation, secretion of Wnt, and desensitization of Fz to ligand (Park et al.,1994a).
  • 3While there is a morphogen gradient established along the P/D axis (see 1), each cell would secrete another, secondary signalling molecule in proportion to the local morphogen concentration (Zheng et al.,1995; Adler et al.,2000; Adler,2002). This cell–cell communication has not been demonstrated but would help explain non–cell-autonomous phenotypes around fz or vang clones (see below).
  • 4The initial, morphogen-dependent cellular asymmetry could be amplified by a feedback loop involving the known core PCP proteins, dispensing with any secondary signal. This amplification process might lead to the asymmetric P/D accumulation of the molecules, before reorganization of the cytoskeleton (Tree et al.,2002b; Ma et al.,2003). A computationally unified view of 12 phenotypes has been presented recently, based on this assumption (Amonlirdviman et al.,2005). This computer model succeeded remarkably, but its intention admittedly was not to explain how each cell initially becomes polarized.
  • 5Neighboring cells have been proposed to be able to compare their readout of a morphogen gradient, communicating maybe by means of the extracellular tails of Fmi cadherins. Hairs would point toward the neighbor cell with the lowest level of Fz activity (Lawrence et al.,2004). How Fmi may carry out this activity remains unknown, however.

Although such proposals attempt to outline the mechanisms at work, none has provided a dynamic demonstration that the set of core proteins, with their known biochemical properties, is able to polarize a large group of initially unbiased cells. It seems to us that predictive understanding of the PCP pathway requires the design of computer models of precisely this process, which models then may be supported or falsified as computational behavior is compared with current and future experiments. We introduce here such a predictive, dynamic model of planar polarization at the individual cell and global wing levels: the model centers on the role of the protocadherin Fmi, and posits that distinct Fmi-containing complexes may bind asymmetrically across adjacent cell membranes; it attempts to incorporate an unprecedented amount of the relevant experimental data on biochemical interactions and molecular localization.

We show that our postulated mechanism ensures a robust, uniform response of all cells to a weak and transient gradient of Fz activity. Built to reproduce the wild-type (WT) and the loss-of-function clones phenotypes, the model succeeds in reproducing other phenotypes with no further parameter tuning and predicts new ones as well. In a recent review, Klein and Mlodzik (2005) have proposed a somewhat analogous Fmi-centered mechanism for PCP. Although our simulations were completed before we became aware of this study, we discuss later some of its content.

RESULTS

Proposed Mechanism

The model proposes a mechanism of cell–cell communication mediated by the Fmi protocadherin and amplified by a feedback loop consisting of three tiers (Fig. 1a): (1) there is a mechanism that amplifies differences of composition between juxtaposed cell faces (this effect is mediated by Fmi–Fmi complexes, see below)—it is the only cell–cell interaction in the model; (2) a biochemical cross-inhibitory feedback system further amplifies cell-autonomously the difference between cell face compositions by locally favoring the formation of certain complexes over others at those faces; this is all the more effective that the complexes in question all require the Fmi component and their formation, therefore, is competitive; finally (3) there is cell-wide diffusion of the various components of the complexes over the cell membranes, driven by the gradients in concentration created through mechanisms 1 and 2; this results in a further amplification of their effects. In the following paragraph, we describe the proposed mechanism in molecular detail.

Figure 1.

“Asymmetric Flamingo Bridge” model of Drosophila wing polarization. a: The three tiers of the polarization cascade in the present model. (1) GLOBAL: this is the preexisting graded signal. (2) CELL–CELL: the local intercellular asymmetric complexes. (3) INTRACELL: this comprises the local intracellular inhibition loops (see part b) and the membrane diffusion driven by concentration differences on anterior and posterior cell faces. b: Top: A putative extracellular ligand (see text for details), distributed in a proximodistal (P/D) gradient, binds the Fz receptor, generating a gradient of Fz activity (Fz*, green). Fmi recruits either Fz* or Vang to membrane-bound dimers. The ligand quickly becomes depleted but not before two opposing gradients are established: an Fz*Fmi gradient, decreasing from the P-edge of the wing and, due to competition for Fmi, an FmiVang gradient, decreasing from the D-edge of the wing (see Fig. 9). The model assumes that, through its extracellular cadherin motifs, Fmi may now form asymmetrical, cell-bridging Fz*Fmi-FmiVang tetrameric complexes. The shallow Fz*Fmi and FmiVang gradients clearly favor formation, e.g., of slightly more Fz*-containing complexes on the P-side than on the D-side of a boundary; this asymmetry is amplified by the first feedback loop (purple dashed line), whereby Fz* inhibits Vang binding. Bottom: Dsh molecules bind the Fz*Fmi subunit in a bridging tetramer and become activated (Dsh*), whereas Pk binds the FmiVang subunit; a second feedback loop (purple dashed line) further amplifies the asymmetry, as Dsh* now inhibits Pk binding. In addition, monomers propagate down their own concentration gradients but up the gradient that is being built by the complexes (black arrows) and contribute to forming progressively more intercellular polarized complexes. After 30 hr (simulated biological time), Fz* and Dsh* are localized mainly on the distal membrane of each cell, whereas Vang and Pk have accumulated on the proximal side. Actin polymerization and hair outgrowth are assumed to take place at the highest point of Dsh* intracellular concentration (red).

Figure 9.

Concentration profiles of the intermediate molecular complexes along the proximal–distal (P/D, left–right) axis, as measured at the end of a 30-hr simulation. The concentrations in all the pixels on a vertical row are added for each position on the P/D axis and are plotted after normalization on the vertical axis. Short-range oscillations reflect the cellular pattern. Approximate peak values of the long-range gradient, as measured on cells along a P/D transect are given below for each figure. a: Profile of Fz*: the long-range gradient decreases from approximately 1.2 (proximal) to 0.2 (distal). b: Profile of Fz*Fmi: the gradient decreases from approximately 0.6 (proximal) to 0.07 (distal). c: Profile of VangFmi: the gradient decreases from approximately 0.1(distal) to 0.02 (proximal). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

We assume that two distinct molecular complexes, which we may call Prox (for proximal) and Dist (for distal), can form on the cellular membrane at the adhesion junction, each including an Fmi molecule. This introduces a competition between the two complexes for a limited stock of Fmi. Complexes facing each other in adjacent cells bind selectively by homophilic interaction between the two Fmi molecules. Prox–Dist binding is assumed to be far more likely than Prox–Prox or Dist–Dist binding, as the cadherin tails interact more readily when joining distinct complexes. Polarization, i.e., the accumulation of Prox complexes on one face and of Dist complexes on the other face, then results from two processes. First, a polarizing signal produces a small imbalance in favor of either complex Prox or Dist on a given face. Second, one of the two complexes inhibits the formation of the other on the same face in the same cell, thus taking advantage in the competition for Fmi and amplifying the initial imbalance. In our model Prox is the aggregation of Fmi with Vang and Pk, whereas Dist is the aggregation of Fmi with Fz and Dsh (Fig. 1b). Fz inhibits Vang binding to Fmi, whereas Dsh inhibits Pk binding. This creates two feedback loops: the more Fz and Dsh accumulate into Dist complexes, the more they inhibit formation of Prox complexes, thus favoring further their own accumulation. In this way, each cell becomes progressively polarized by the internal segregation of its molecules. The direction of polarization, however, is given by the adjacent cells, which are also in the process of becoming polarized. Thus, the signal does not propagate from one end of the wing to the other but is initially given to all cells as a weak asymmetry and progressively amplified in a coordinated way by all the cells in the tissue.

Model Building by Integration of Experimental Evidence

The model's central molecular tenet (Fig. 1) is that Fmi, a transmembrane protocadherin (Usui et al.,1999; Chae et al.,1999), may play a key role as scaffold for an asymmetric complex accumulating at P/D cell boundaries. Evidence for the progressive build-up of an intercellular complex comes from Fmi being essential for all other core molecules to localize apicolaterally (Feiguin et al.,2001; Axelrod,2001; Strutt,2001; Tree et al.,2002b; Bastock et al.,2003), Fz and Vang also being necessary for localization of Dsh and Pk, respectively (Axelrod,2001; Bastock et al.,2003). For the sake of computation, we assume that Frizzled activation occurs through a ligand gradient initially established over the whole wing tissue. Putative ligand molecules bind and activate the Fz transmembrane receptors. The net effect is to create a finite space-dependent concentration of activated Fz (Fz*). Note that this could be achieved by other mechanisms, as we discuss later.

Fz* competes with Vang for binding Fmi, as some Fz molecules were shown to interact directly with protocadherins (Medina et al.,2004). Two dimers, Fz*Fmi in one cell and VangFmi in a neighboring cell, are further assumed to be able to bind across juxtaposed membranes by homophilic interaction (Usui et al.,1999) between the Fmi extracellular domains (Fig. 1). In our model, asymmetric intercellular complexes are exclusively built, but it is only required that asymmetric complexes be more likely to form than symmetric ones. This finding may result for instance from allostery, Fmi assuming different conformations when bound to either Fz* or Vang. With the aggregation of such asymmetric complexes, each cell in effect compares the Fz activity of its neighbors: Fz*Fmi will bind preferentially with adjacent cells presenting the highest concentration of FmiVang, whereas FmiVang will preferentially bind cells with highest Fz*Fmi. Onto the asymmetric tetramer Dsh and Pk may now aggregate, Dsh binding the FmiFz* half-tetramer (Wong,2003) and Pk binding the FmiVang half (Bastock et al.,2003; Jenny et al.,2003). Dsh is phosphorylated (Dsh*) when binding the Fz*Fmi dimer (Rothbächer et al.,2000), a process known to depend on Fz itself being activated previously (Axelrod et al.,1998) and on the presence of Fmi (Axelrod,2001). Evidence for the existence of a Dsh*Fz*Fmi heterotrimeric complex is strong (Usui et al.,1999; Shimada et al.,2001). Two hypothetical feedback loops amplify the initial reading of the signal: Fz* inhibits binding of Vang to Fmi, whereas Dsh* inhibits binding of Pk to FmiVang. These loops were selected among all other possible combinations to reproduce the WT polarization phenotype (see Experimental Procedures section). This feedback mechanism differs from another model where Pk and Vang inhibit Dsh binding to Fz (Amonlirdviman et al.,2005), as discussed later. Finally, activated Dsh* triggers a signalling cascade ultimately controlling actin accumulation and hair growth (Strutt et al.,1997; Winter et al.,2001). We have ignored Diego (Feiguin et al.,2001), because it has not been demonstrated to play a role in all known PCP instances (Mlodzik,2002); also, it is difficult currently to reconcile that Diego physically interacts with Vang and Pk (Das et al.,2004), with the observation that it binds Dsh and accumulates on the distal side with Fz and Dsh (Jenny et al.,2005).

To mathematically define and solve the model (see Experimental Procedures section), we use ctrl-Dev, an application designed to simulate molecular processes in cells and tissues (Kerszberg and Changeux,1998; see Experimental Procedures section). Cell compartments are represented in a two-dimensional space, where molecular reactions and diffusion proceed stochastically. Each computation run simulates a field of ca. One hundred fifty roughly hexagonal cells for 30 hr of development time (see Experimental Procedures section and Supplementary Materials for parameter values and cell definition; Supplementary Materials can be viewed at http://www.interscience.wiley.com/jpages/1058-8388/suppmat). In Figures 2–8, the vector sum of Dsh* gradient in each cell, representing the predicted direction of hair outgrowth, is shown at the end of each run (see time course video in the Supplementary Materials).

Figure 2.

Simulations of the wild-type (WT) wing. In this and all the following figures, proximal is to the left, distal to the right, anterior is at the top, posterior at the bottom. The shading scale is different (to improve contrast) and is indicated for each panel. a: Dsh* concentration on the simulated wing epithelium of a WT fly. Dsh* exhibits the typical zig-zag pattern of core polarity molecules, as it accumulates on the distal membrane of each cell. b: Magnified view of a cell in a, differing proximal and distal Dsh* membrane concentrations are apparent (the white pixels concentration is near 1.0, whereas dark blue ones are close to 0.1). The Dsh* gradient vector (red) shows the direction of hair outgrowth, pointing distally. c: Pattern of total Vang (whether as a free monomer or bound in a complex) concentration in a typical WT cell (white is approximately 2 and dark grey close to 1). d: Pattern of total Pk concentration in a WT cell (light = 2, dark = 1). e: Pattern of total Fz in a WT cell (light = 3, dark = 4.5). f: Pattern of total Fmi in a WT cell (light = 3.7, dark = 2.8). Here, proximal and distal membranes show similar concentrations, but these concentrations are higher than the anterior and posterior concentrations.

Figure 3.

Simulations of loss-of-function clones. Initial concentration of the molecule in the clone was set to zero. a: Non–cell-autonomous phenotype of a fz clone. Hairs of wild-type (WT) cells outside the clone point toward it. The disturbance extends to four cell rows on the distal side. b: Non–cell-autonomous phenotype of a vang clone: hairs of WT cells point away from the clone. c: Nearly cell-autonomous phenotype of a dsh clone: WT cells immediately adjacent to the clone have slightly deviated hairs, but this disturbance is much weaker than that caused by the fz clone (compare with a). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 4.

Fmi phenotypes. a: Nearly cell-autonomous phenotype of an fmi loss-of-function clone. Wild-type (WT) cells on the proximal side of the clone are slightly influenced, but this effect is much weaker than with vang clones (compare Fig. 3b). b: Experimental fmi clone marked with the hair marker pwn. On the proximal side of the clone, some WT cells show abnormal polarity (arrow) or multiple hairs (arrowhead; reproduced, with permission, from Chae et al.,1999). c: fmi overexpression clone: hairs of WT cells point toward the clone. d: Computational “experiment”: when a gradient expression of fmi is induced perpendicular to the proximodistal (P/D) axis (shape shown), hairs orient toward the Fmi peak concentration (compare with e). e: Experiment where a ptc-Gal4 promoter drives gradient expression of fmi (peak between arrows; reproduced, with permission, from Usui et al.,1999). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 5.

Simulations of overexpression clones. a: dsh overexpression clone. Hairs of the wild-type (WT) cells point away from the clone, and polarity is slightly disturbed on the proximal side. This finding is in accordance with unpublished observations as reported in Amonlirdviman et al. (2005). b: vang overexpression clone. Hairs of the WT cells are attracted by the clone, and polarity is mostly disturbed on the distal side. This finding is in accordance with unpublished observations as reported in Amonlirdviman et al. (2005). c: pk overexpression clone. Hairs of the WT cells are attracted by the clone, and polarity is mostly disturbed on the distal side. This finding is in accordance with experimental observations in Tree et al. (2002b). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 6.

Ectopic expression simulations. The shape of the ectopic gradient is shown next to each figure. a: An attempt to replicate experiments where hot wax activates a heat shock gene promoter driving fz (Adler, P. et al.,1997). Initial Fz concentration was set at 4 (i.e. 4 times the minimum ligand concentration) on the three columns of distal-most cells. Fz concentration then decreases exponentially and reaches a value of 1.47 in the cells at the proximal edge. As seen experimentally, the polarity of proximal wing hairs is reversed. b: Fz ectopic anteroposterior (A/P) gradient. A double-gradient of Fz is induced to simulate a ptc-Gal4 driver in a wild-type (WT) background. Dark cells at the mid-line express fz at level 2. Other cells express fz in an exponentially decreasing manner (from 2 down to 0.735), as their distance from the mid-line increases. Hairs are deviated, in comparison to the WT, toward the lowest point of the Fz gradient. ce: Dsh (c), Vang (d), and Pk (e) ectopic A/P gradients. A double-gradient is induced to simulate a ptc-Gal4 driver in a WT background. Dark cells at the mid-line express the gene at 4 times the WT level. Other cells express the gene in an exponentially decreasing manner (from 4 down to 1.5), as their distance from the mid-line increases. Hairs are deviated, in comparison to the WT, toward the lowest point of the Dsh gradient (as for Fz, see b.), whereas they are deviated toward the peak of the Vang or Pk gradients (as for Fmi, see Fig. 4d). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 7.

Simulations of various Fz alleles. a: Clone of an Fz allele impaired for its capacity to bind Dsh (the forward rates in Eq. 6 and 7 of the Experimental Procedures section were set at zero). The phenotype of this allele is clearly cell-autonomous. This finding suggests that the first cytoplasmic loop of the transmembrane region in the Fz protein interacts with Dsh, as this loop includes the same missense mutation in all cell-autonomous Fz alleles. b: Clone of an Fz allele impaired for its capacity to bind the ligand (the forward rate in Eq. 1 of the Experimental Procedures section was set at zero). This allele shows a strong nonautonomous phenotype, with hairs pointing toward the clone. c: Clone of an Fz allele impaired for its capacity to bind Fmi (the forward rate in Eq. 2 of the Experimental Procedures section was set at zero). This allele shows a nonautonomous phenotype (albeit somewhat weaker than in b), as hairs of wild-type (WT) cells point toward the clone. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 8.

Wild-type (WT) simulation with a low signal-to-noise ratio, illustrating the robustness of the model. Noise is added to the initial exponential ligand-gradient by setting N(x,t0), such that N reaches maximum values Nmax = 1 in the equation: [Ld]0 = A. exp {(L − X)/B } + N(x,t0) (see the Experimental Procedures section). Thus, the signal-to-noise ratio decreases from 2.7 (proximally) to 1 (distally). Polarization is only slightly deteriorated, with a typical D/P Dsh* ratio reaching 5 after 30 hr, compared with 10 without noise (Nmax = 0). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

The Wild-Type: Emergence of Cell and Tissue Polarization

During our WT simulations, Fz progressively colocalizes with Dsh on the distal membrane, whereas Vang and Pk preferentially localize on the proximal membrane (Fig. 2 and video in the Supplementary Materials), as observed in vivo. From an initial homogeneous distribution of all the molecules, the simulations end up, after 300,000 steps (a time-equivalent of approximately 30 hr; see Experimental Procedures section), with a D/P concentration ratio for Dsh* that is higher than 10. This accumulation of Dsh* on the distal membranes leads to a robust alignment of the gradient vectors, simulating the alignment of all hairs along the P/D axis (Fig. 2a,b). The P/D ratios are close to 2 for total Vang (Fig. 2c) or Pk (Fig. 2d), as is the D/P ratio for total Dsh (activated or not; data not shown). The D/P ratio for Fz* is higher than 2, compared with 1.5 for total Fz (i.e., Fz whether activated or not; Fig. 2e). The P/D Fmi concentration is larger than the anteroposterior (A/P) Fmi concentration by a factor ca. 1.3 (Fig. 2f). As it is difficult to relate immunofluorescence luminosity to protein concentrations, most studies reporting asymmetric membrane accumulation do not include luminosity measurements. Mean pixel luminosity contrasts were published for Dsh, with a D/P ratio of approximately 1.2 (Tree et al.,2002b), and Fmi with a P/D to A/P ratio higher than 2 (Feiguin et al.,2001).

Phenotypes Around Clones

The model replicates observed phenotypes around clones of mutant cells. fz clones attract hairs of adjacent cells and disrupt the polarity of WT cells on the distal side (Vinson and Adler,1987; Fig. 3a). Inversely, hairs point away from vang clones, and polarity is disrupted on the proximal side (Taylor et al.,1998) (Fig. 3b). Around dsh clones, a small one-cell nonautonomy effect is seen on cells immediately adjacent to the clone (Fig. 3c), as observed experimentally (Shimada et al.,2001). fmi clones are also almost cell-autonomous with a slight proximal disturbance (Chae et al.,1999; Usui et al.,1999; Fig. 4a,b). Our model allows replication of the known overexpression phenotypes for fz (Adler et al.,1997), fmi (Usui et al.,1999; Fig. 4c), and pk (Tree et al.,2002; Fig. 5c). Consistent with unpublished observations (Amonlirdviman et al.,2005), we predict that dsh and fz overexpression phenotypes are similar, with WT adjacent hairs pointing away from the clone (Fig. 5a), whereas vang, fmi, or pk overexpression produce the opposite pattern, with WT adjacent hairs being attracted (Figs. 4c, 5b,c). The model provides a mechanistic explanation for all these phenotypes: for example, WT cells around a fz clone will only be able to build one type of intercellular bridge with clone-cells (Fz*Fmi being on the WT side and VangFmi being on the clone side; see Fig. 1); in WT cells distal to the clone, this is contrary to the normal polarization, and this disturbance is propagated to neighboring cells.

Ectopic Expression

Ectopic expression experiments with heat shock gene or ptc-Gal4 promoters (Adler et al.,1997; Usui et al.,1999) have shown how polarity is influenced by an imposed Fz or Fmi gradient. Our simulations replicate these effects without additional parameter adjustment. The polarity of proximal wing hair is reversed in an ectopic Fz distal gradient that replicates the activation by hot wax of a heat shock gene promoter driving fz (Fig. 6a). Hairs orient themselves toward the peak of an Fmi gradient perpendicular to the P/D axis (Fig. 4d,e), whereas they orient down an Fz gradient (Fig. 6b). We make novel, unequivocal predictions on the influence of ectopic Dsh, Vang, or Pk gradients, which are experimentally testable: hairs orient down a Dsh gradient (Fig. 6c), whereas they orient up a Vang or Pk gradient (Fig. 6d,e). Again, the model provides a mechanistic explanation for the effects of ectopic gradients. Why hairs orient up an Fmi gradient was difficult to explain by experimental results, as Fmi is symmetrically aggregated on both proximal and distal membranes of each cell. In our model, ligand concentration limits the formation of Fz*Fmi dimers, whereas that of VangFmi is ligand-independent: overexpression of Fmi at the peak of a gradient will accordingly favor the formation of VangFmi dimers, giving a phenotype similar to Vang overexpression.

Frizzled Alleles

Mutagenesis has produced both autonomous and nonautonomous fz alleles (Vinson and Adler,1987; Jones et al.,1996). As Fz interacts with three molecules in our model, we could simulate various deficiencies of the Fz protein. A clone of cells impaired for Fz capacity to bind Dsh shows a nearly cell-autonomous phenotype (Fig. 7a), in accordance with the results of others (Amonlirdviman et al.,2005). Because all cell-autonomous fz alleles share missense mutations at proline residue 278, in the first cytoplasmic loop of the transmembrane region, our model suggests that this domain interacts with Dsh. Simulated clones of fz alleles unable to bind the ligand or Fmi are non–cell-autonomous (Fig. 7b,c). Some reported nonautonomous fz alleles are mutated in the transmembrane domains, which may form a ligand-binding pocket in this putative G-protein–coupled receptor (Katanaev et al.,2005). It is, therefore, tempting to speculate that these alleles are ligand-binding deficient. Inference about interaction sites between Fz and Fmi is less straightforward, because nonautonomous mutations may affect extracellular, transmembrane, or cytoplasmic domains (Jones et al.,1996).

Robustness of the Model

Despite the weak, transient signal, the model is remarkably resilient against signal fluctuations, thermodynamic noise, cell shape, and contact fluctuations (see Experimental Procedures section), as well as against boundary effects as may happen, say, near the wing veins. We simulated a WT wing where random noise, with maximum amplitude reaching one third of total signal strength, is added to the initial ligand gradient. Polarization is only slightly deteriorated, with Dsh* accumulation on the D-side still fivefold higher than on the P-side (Fig. 8).

DISCUSSION

In the present model, the known molecules and their properties are sufficient to explain how planar cell polarity is built up in the wing epithelium, downstream of a hypothetical polarizing gradient. There is no need for a secondary molecule, secreted by each cell to explain the nonautonomous phenotypes, as has been postulated by some (Zheng et al.,1995; Wehrli and Tomlinson,1998; Adler,2002). By incorporating Fmi, this model is uniquely able to explain fmi clones, fmi overexpression clones, and fmi ectopic expression experiments. On the other hand, our model shares some components with existing ones: the gradient of Fz activation is a case in point. Some of the similarities and differences are discussed below.

Upstream Signal

The model puts strict limits on the direction, strength, and duration of the assumed Fz activity gradient. Most authors have proposed that Fz activity should be positively biased on the distal side (Strutt,2002; Uemura and Shimada,2003; Amonlirdviman et al.,2005). Quite unexpectedly, although the concentration of Fz* ends up being higher at the distal side of each cell, our model predicts that Fz activity should be lower at the distal edge of the wing. This is a consequence of the built-in signal readout mechanism (Fig. 1), as underlined in a recent review proposing a somewhat similar mechanism (Klein and Mlodzik,2005). A global P/D gradient of Fz*Fmi is established and maintained well after depletion of the free ligand activating Fz (Fig. 9a,b). Because of the competition for Fmi, an opposite global VangFmi gradient also arises (Fig. 9c). As a result, a cell's proximal neighbor has a higher Fz*Fmi concentration than its distal neighbor (and vice versa for the VangFmi concentration). Asymmetric complex formation, thus, will enrich the proximal side of the cell with VangFmi and the distal side with Fz*Fmi: the intracellular gradient is inverse of the global one (and this will be stabilized by aggregation of Dsh to Fz*Fmi and of Pk to VangFmi). Because of this opposition, we predict that, if Frizzled activity is set up by a ligand, too strong a ligand concentration will actually hinder polarization.

The upstream signal has indeed long been thought to be subtle, with long-range gradients of the Four-jointed (Fj) and Dachsous (Ds) proteins possibly acting through Fat(Ft) to activate Fz, if no Wnt is involved. The exact mechanism may in fact differ between the Drosophila wing and eye: Fj and Ds uniform expression is sufficient for polarization to occur in the wing (Ma et al.,2003; Matakatsu and Blair,2004), whereas in the eye, their gradients were shown to specify the polarization axis, and it has been suggested that the ensuing weak Ft gradient could act as signal (Simon,2004).

The simulations suggest that the signal's lifetime cannot overlap much with the period during which polarity is established: the putative ligand molecules are almost all bound to Fz after a time-equivalent of 7 min, whereas polarization takes approximately 30 hr. Increasing signal strength or maintaining it much beyond 7 min is actually detrimental. This finding is consistent with ablation experiments showing that any long-range signal must propagate at least 22 hr before hairs emerge (Adler et al.,2000). If we assume the upstream signal to be a by-product of earlier patterning, the polarization process might be triggered either by the recruitment of the core molecules at the proper membrane location or by Fz becoming sensitive to activation. As to the first possibility, there is evidence that Fz and Fmi undergo a burst of transcription early in pupal life (Park et al.,1994b; Chae et al.,1999), and this is coincident with progressive apicolateral aggregation of all the core proteins. As to the second possibility, it was shown recently that apical Fz is permanently inactivated by aPKC (atypical protein kinase C) and the apical determinant dPatj in most cells. The PDZ domain protein Bazooka counteracts this inhibition in cells where and when a response to PCP signalling is needed (Djiane at al.,2005). In this context, note also that we have not introduced any production or degradation of the molecules in the model, as there are no data indicating that such processes might be significant over the 30 hr required for polarization.

Our computations show that polarity does not require a gradient steeper than 4% per cell, much less than the 40% estimate for typical morphogen gradients (Gurdon and Bourillot,2001). Over a wing length of ca. 200 cells, this rate suggests a P/D concentration ratio of 2.5 × 103. Restricted diffusion mechanisms, involving glypicans, extend the range of Wg signalling in the wing and may help build such shallow gradients (Franch-Marro et al.,2005). However, because no Wnt has been identified in the PCP pathway, the signal molecules could act in a very different way. Whereas steep morphogen gradients have been proposed to convey precise positional value information, the PCP signal delivers a global, longer range, directional information. This signal could be weak and inaccurate, provided that cell–cell communication and an amplification mechanism ensure proper reading in the whole organ.

Amplification Process

Under the hypotheses of our model, simulations show that, if one keeps the parameters within physiological limits, only the scheme proposed here will produce the observed phenotypes, i.e., the most efficient loop involves Fz* and Dsh*, respectively, inhibiting Vang and Pk binding to the intercellular complexes. This scheme is at odds with a privileged role for Pk, as was proposed in a report investigating Pk localization and interactions with other molecules (Tree et al.,2002b), and further refined and computationally tested in a model where the feedback consists of Pk and Vang inhibiting Dsh binding to Fz (Amonlirdviman et al.,2005). This inhibition was proposed as Pk and Vang bind Dsh in vitro, thus possibly antagonizing binding of Dsh to Fz. However, some experimental results do not support such a mechanism (Bastock et al.,2003). Most strikingly, pk overexpression in vivo seems to promote apicolateral membrane recruitment of other core proteins, including Dsh (Tree et al.,2002b; Bastock et al.,2003). Thus, there seems to exist, to the best of our knowledge, no unambiguous experimental data that would rule out our inhibition scheme.

Open Issues

Whereas we have been able to integrate most of the available data into a working model, some experimental results clearly show that planar polarization must involve additional mechanisms and molecules. Ever since mutants of core PCP genes were discovered, investigators have noticed that individual hairs on mutant wings tend to align with their neighbors. There must exist, therefore, an additional local mechanism, genetically independent from the main PCP pathway (Wong and Adler,1993). Our model is limited to interactions between the five core molecules and is conditional upon each of those being present. It obviously fails to account for such local alignment in the absence of one or more molecules. To the best of our knowledge, very little experimental data are available regarding this putative additional mechanism.

Immunofluorescence experiments clearly show that Fmi accumulates at the P/D cell boundaries and is comparatively depleted at the A/P boundaries. No model is presently able to reproduce accumulation at such a high level. The cytoskeleton may not only be a target of PCP signalling but participate as well in the amplification of the polarity signal by polarized transport of the core molecules or by strengthening of the cadherin homophilic bond (Baumgartner et al.,2003), which again would further polarize Fmi. Proteins bound into intercellular complexes may also be stabilized and decay less rapidly than monomers, which would provide an additional mechanism of amplification.

Fz overexpression consistently produces strong phenotypes; this is not easily accounted for by existing models, as none assumes Fz to be constitutionally active. A mechanism controlling Fz activity has been suggested recently, ensuring that Fz becomes receptive only at the right time and place (Djiane et al.,2005); such a mechanism should probably be included in a future version of the model.

During the first 24 hr of pupal life, the wing cells continue to grow and divide, with an average doubling time of approximately 8.5 hr. Groups of approximately five cells, spread over the whole wing, divide simultaneously (Milán et al.,1996). Although our model provides for cell shape and contact fluctuations, the simulations do not include cell divisions. After the initial signal has vanished, the intermediate gradients of Fz*Fmi and VangFmi remain and newly divided cells intercalate among already polarized cells. We do not expect divisions to disturb significantly the polarization process but further simulations would be needed to show that there is ample time between two divisions for a cell to become polarized again.

Conclusion

Our simulations show that the observed phenotypes affecting tissue polarity in the Drosophila wing are consistent with a model where polarization is conditional upon three molecular processes. First, a transient long-range signal must be delivered over the whole wing tissue so as to create a weak imbalance of Fz activity. Second, adjacent cells should be able to communicate by comparing their concentrations of intermediate complexes (Fz*Fmi and VangFmi) and favoring the build-up of asymmetric complexes. Third, an intracellular feedback mechanism must be available for amplification of the initial asymmetry. The model we propose incorporates such mechanisms and appears to be robust against noise and changes of parameter values. What may be the origin of such robustness? Is it related to the intricacy of the molecular interactions required for polarization to take place correctly? Fz and Dsh, together with other molecules of the canonical Wnt/Fz pathway, are present in cnidarians (Steele, 2005) and control polarization of isolated cells in bilaterians, such as the nematode oocyte. It is tempting to speculate that both proteins were recruited into a new pathway controlling tissue polarity in the ancestor of bilaterians. Until now Fmi,Vang, Pk, and Dgo have only been found in bilaterian organisms. Although cadherins are likely very ancient proteins (King et al.,2003), the complex structure of Fmi with its EGF, LAG, and GPCR domains may be a major innovation of bilaterians, allowing it to become a scaffold for the other PCP molecules, and thus enabling, in part at least, high fidelity in planar epithelial polarization.

EXPERIMENTAL PROCEDURES

The ctrl-Dev application

Simulations use ctrl-Dev, a modular informatic tool written in C++, embodying an agent-based modeling approach to genetics and developmental cell biology. It uses freely available software such as C++ compilers and the Qt graphic toolkit from TrollTech (Norway). The code for ctrl-Dev is available upon request to MK.

In ctrl-Dev, tissues are represented on a two-dimensional screen (a set of hexagonal color pixels, here 175 × 175 of them) over which biological objects are moving. In the PCP simulations, these objects are cells (connected collections of pixels), membranes (closed loops of pixels enclosing cells), nuclei, cytoplasm, and extracellular environment (pixels not belonging to any cell). In a simulation, cells are generated initially by adding pixels around cell centers disposed in an approximate hexagonal pattern; hence, the cells themselves start out approximately hexagonal (and remain so).

Pixels contain molecules and/or molecular complexes, depending on the object(s) present there. A membrane pixel may harbor several membrane receptors or ligands. Membrane pixel pairs join two neighboring membranes, and may contain oriented molecules bridging two cells.

The processes in the system are molecular diffusion and in-object and between-object reactions (e.g., reaction between two adhesion molecules on adjacent membranes of neighboring cells). Cell motion is implemented by exchange of pixels between cells or between cells and extracellular matrix (according to a statistical, internal tendency of cells to maintain an optimal size). The dynamics of all processes is noisy and described by stochastic difference equations, i.e., random fluctuations in reaction and diffusion rates, as well as cell movements, are allowed at all steps as described above. It can be expected, thus, that the results automatically encompass a significant measure of robustness.

Formal Model of Drosophila Wing Polarization

Initial conditions.

The model features a set of six proteins (Fz, Dsh, Vang, Pk, Fmi, and Ld, the ligand), and the simulation begins with five of these molecules homogeneously distributed on the membranes. The initial ligand concentration varies along the P/D axis (coordinate X, with 0 < X < L) as: [Ld]0 = A. exp {(L − X)/B } + N(x,t0), where parameter A defines the absolute dimensionless ligand level, B its slope, and N(x,t0) is the space-dependent noise intensity (delivered at each pixel by a random-number generator). Having arbitrarily chosen the minimum (distal) ligand concentration as unity (A = 1), all other initial concentrations are expressed as a multiple of this unit. These were set at [Fz]0 = [Fmi]0 = 4, [Dsh]0 = [Vang]0 = [Pk]0 = 2, so that the ligand is the limiting factor and no protein is exhausted at the end of a simulation (for the way in which these dimensionless numbers are connected to actual concentrations, see an example below, Eq. 9).

Reactions.

During the simulation, reactions take place among molecules situated in the same pixel of a membrane, or molecules situated in adjacent pixels of two neighboring cells. These reactions build seven molecular complexes (Fz*, Fz*Fmi, VangFmi, Fz*Fmi = FmiVang, Dsh*FzFmi = FmiVang, Fz*Fmi = FmiVangPk, and Dsh*FzFmi = FmiVangPk), the last four bridging the membranes of two neighboring cells. The eight reactions are:

equation image(1)
equation image(2)
equation image(3)
equation image(4)
equation image(5)
equation image(6)
equation image(7)
equation image(8)

All reactions, except (1), are reversible. For each reaction between two molecules M1 and M2, the model computes at each time-step the average production (equal to KfM1 M2.[M1].[M2]) and the dissociation (equal to KdM1 M2.[M1M2]) of new complexes M1 M2, where Kf and Kd are the forward and backward reaction rates, respectively. Fluctuations are introduced in reaction rates by defining a probability that each reaction will, at each time step and in a given pixel, take place at a slightly different rate than the average (difference being typically 10%). The 15 reaction rates, 2 for each reaction except reaction (1) which is not reversible, were fine-tuned manually to optimize the WT and mutant clones phenotypes, while maintaining parameters within biochemically reasonable ranges. Polarization is conditional upon the dissociation rates of the last four reactions (5–8) being significantly lower than other dissociation or production rates (see Supplementary Materials). Simulations, thus, suggest that the fully built intercellular complexes should be more stable than the intermediate molecules.

Feedback loop.

In the model, the binding of both Vang and Pk to the intercellular complex is regulated by intracellular inhibitory loops. Reactions (5) and (8) are inhibited by Dsh*-containing complexes, whereas reaction (3) is inhibited by Fz* and Fz*-containing complexes. Inhibition is described as a decrease of the forward reaction rate (Kf is divided by 1+kfinh.[inhibitor]) and an increase of the backward reaction rate (Kd is multiplied by 1+kdinh.[inhibitor]), where kfinh and kdinh are the inhibition coefficients (see Supplementary Materials).

The choice of these loops is the result of a selection process whereby we investigated all possible regulations: binding of any of the four proteins (Fz, Vang, Dsh, Pk) could be either activated or inhibited by an intra- or intercellular loop. We retained the only combination reproducing the WT polarization phenotypes with weak inhibition coefficients (kfinh set at 1.5 and kdinh set at 3.0). Proper amplification could be reached with other loops, but it would necessitate higher inhibition coefficients or a nonlinear influence of the inhibitor concentration (introducing terms in kfinh · [inhibitor]2 for instance).

Diffusion.

All molecules and complexes, apart from the cell-bridging complexes, may diffuse on the membrane. Ligand diffusion through the extracellular matrix (ECM) was not implemented, having checked that its effect is minimal because the ligand is rapidly depleted in our model. Morphogen transport along epithelia remains controversial (Vincent and Dubois,2002) and very likely involves many additional molecules (Franch-Marro et al.,2005). Although Dsh and Pk are cytoplasmic molecules, we tested that cytoplasmic diffusion would not significantly alter polarization, provided that initial concentrations are similar throughout the cell. Diffusion is computed at each time-step by comparing the concentrations of each molecule in adjacent pixels of the same cell. The probability per time step that a molecule will diffuse from one pixel to a neighboring one is Kt, with mass being conserved between the two pixels. Thermodynamic fluctuations in diffusion processes are introduced as for reactions. The diffusion coefficient of membrane proteins, which is of the order of 10−9 cm2 s−1 (Kusumi et al.,1993), may be related to typical dimensions of our model by:

equation image(9)

where Kmath image is the probability per time step that a molecule diffuses from one membrane compartment to the next, L is the size of a pixel (approximately 1 μm), and t the time step. The model reaches stationarity after approximately 300,000 time steps, which, when compared with the 30-hr polarization process in the Drosophila wing, suggests that t ∼ 0.4 sec. Thus, Kt values should be of a magnitude comprised between 0.01 and 0.1 per time step. No data being available on the diffusion of these particular molecules, we retained a uniform value Kmath image = 0.02 for all molecules (see Supplementary Materials).

Numerical solution.

The model is solved numerically by a first-order difference method, iteratively computing the products of each reaction and the outcome of diffusion in each and every pixel of the lattice. This is done by stochastic processes as evident in the code itself.

Wing hair orientation.

The predicted direction of hair growth, assumed to be controlled by Dsh* activity, is obtained by computing the weighted mean gradient vector of Dsh* concentration over all the membrane pixels of a given cell. The origin of this vector is the “center of gravity” of the membrane pixels, each taken with equal weight, while its extremity is the “center of gravity” of the same pixels weighted with their Dsh* concentration.

Generation of mutant clones.

Mutant clones were generated by selecting a small group of cells at the center of the spatial lattice, so as to minimize border effects. All cells with center coordinates belonging to the intervals [X1;X2] and [Y1;Y2] start with initial conditions different from the WT. For instance, loss-of-function fz clone cells start with [Fz]0 = 0. For overexpression clones, the initial concentration of the molecule in the clone was set to 10 times the WT value in each of the simulations.

Ancillary