Mathematical Modeling of Planar Cell Polarity to Understand Domineering Nonautonomy

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Science  21 Jan 2005:
Vol. 307, Issue 5708, pp. 423-426
DOI: 10.1126/science.1105471


Planar cell polarity (PCP) signaling generates subcellular asymmetry along an axis orthogonal to the epithelial apical-basal axis. Through a poorly understood mechanism, cell clones that have mutations in some PCP signaling components, including some, but not all, alleles of the receptor frizzled, cause polarity disruptions of neighboring wild-type cells, a phenomenon referred to as domineering nonautonomy. Here, a contact-dependent signaling hypothesis, derived from experimental results, is shown by reaction-diffusion, partial differential equation modeling and simulation to fully reproduce PCP phenotypes, including domineering nonautonomy, in the Drosophila wing. The sufficiency of this model and the experimental validation of model predictions reveal how specific protein-protein interactions produce autonomy or domineering nonautonomy.

As the understanding of cellular regulatory networks grows, system behaviors resulting from feedback effects have proven sufficiently complex so as to preclude intuitive understanding. The challenge now is to show that enough of a network is understood to explain such behaviors. Using mathematical modeling, we show the sufficiency of a proposed biological model and study its properties, to demonstrate that it can explain complex PCP phenotypes and to provide insight into the system dynamics that govern them.

Many epithelia are polarized along an axis orthogonal to the apical-basal axis. On the Drosophila adult cuticle, each hexagonally packed cell elaborates an actin-rich hair that develops from the distal vertex and points distally (Fig. 1A). Genetic analyses have identified a group of PCP proteins whose activities are required to correctly polarize these arrays (1, 2). The domineering nonautonomy adjacent to cell clones mutant for some, but not other, PCP genes has not yet been adequately explained (1). For example, in the Drosophila wing, Van Gogh/strabismus (Vang; encoding a four-pass transmembrane protein) (3, 4) clones disrupt polarity proximal to the mutant tissue (3), whereas null frizzled (fz; encoding a seven-pass transmembrane protein) alleles disrupt polarity distal to the clone (5, 6). Models to explain this phenomenon have often invoked diffusible factors, referred to as factor X or Z, because they have not yet been identified (1, 715). We propose instead that the observed behaviors of known PCP proteins are sufficient to explain domineering nonautonomy.

Fig. 1.

Illustrations of PCP. (A) Hexagonally packed wing cells accumulate Fz and Dsh distally (blue/green) and Pk and Vang proximally (pink/orange). A hair is elaborated from the distal vertex of each cell. (B) A local feedback loop model for a PCP signaling mechanism. Black lettering indicates protein accumulation. Gray lettering indicates proteins whose concentrations are decreased. (C) Representation of proposed binding interactions. (D) Inhibition model representing disruption of the Dsh-Fz interaction by Pk and Vang.

Fz and other PCP signaling components accumulate selectively on the distal or proximal side of wing cells (Fig. 1A). Evidence has been provided that these proteins function in a feedback loop that amplifies an asymmetry cue, which converts uniform distributions of PCP proteins into highly polarized distributions (11, 1618) (Fig. 1B). The proposed feedback mechanism depends on several functional relationships (18). Fz recruits Dishevelled (Dsh; a cytoplasmic protein) to the cell membrane (16, 19). In addition, Fz promotes the recruitment of Prickle-spiny-legs (Pk; a LIM domain protein) (5, 18, 20) and Vang (21) to the cell membrane of the adjacent cell. Feedback is provided by the ability of Pk [and Vang; supporting online material (SOM)] to cell-autonomously block Fz-dependent recruitment of Dsh (18). This feedback loop functions strictly locally, between adjacent cells. Global directionality is imposed through the agency of the novel transmembrane protein Four-jointed and the cadherins Dachsous and Fat (Ft) (2224). Widerborst, a regulatory subunit of protein phosphatase 2A, accumulates asymmetrically within each cell and is required to bias the feedback loop (25). Although the mechanism by which Ft biases the direction of the feedback loop is unknown, one possibility is that Ft may direct Widerborst distribution.

However, it is not readily apparent that this biological model does not readily explain the complex patterns observed in fields of cells containing mutant clones, and others have argued that it cannot account for some of the observed phenotypes (1, 26). Indeed, progress in understanding PCP signaling has been hampered by an inability to deduce, given a particular signaling network hypothesis, definitive links between molecular genetic interventions and tissue patterning effects. For example, although it is apparent that removing Dsh or Fz would disrupt the feedback loop, it is not obvious how the feedback loop in adjacent wild-type cells responds, such that dsh mutant clones behave autonomously, whereas for most fz alleles, mutant clones behave nonautonomously. Interestingly, though, for some fz alleles, mutant clones produce an almost cell-autonomous phenotype (5, 6, 27). As another example, Pk overexpression promotes the asymmetric accumulation of Dsh and Fz (18), despite the role of Pk in the feedback loop as an inhibitor of Dsh membrane recruitment.

We have developed a mathematical model based on the described feedback loop and an initial asymmetry input representing the global directional cue (18, 22, 23). Although mathematical modeling cannot prove the correctness of the underlying biological model, the ability of the mathematical model to capture the known behaviors of the system proves the feasibility of the biological model, provides testable hypotheses, and yields insight into the factors contributing to autonomy and nonautonomy.

We have represented the features of the biological feedback loop model as a mathematical reaction-diffusion model (28) that describes the concentrations of Dsh, Fz, Vang, and Pk throughout a network of cells. Although the mechanisms that underlie the local feedback loop are not fully understood, the essential logic of this feedback loop is preserved by representing these interactions as binding to form protein complexes (Fig. 1, C and D).

Inhibition of Dsh membrane recruitment by Pk and Vang (Fig. 1, C and D, SOM) is represented in the mathematical model as an increase in the backward reaction rate of reactions in which Dsh binds Fz (or Fz complexes) by a factor dependent on the local concentration of Pk and Vang. The specific mechanism for the introduction of the directional bias into the feedback loop network is not known. Two forms of a global biasing signal were therefore implemented, and the results using either of these models were similar (SOM). The resulting mathematical model consists of a system of 10 nonlinear partial differential equations. With a given set of model parameters, an array of cells could then be simulated, and the resulting hair pattern assigned on the basis of the final distribution of Dsh.

The model parameters, including the initial protein concentrations, reaction rates, and diffusion constants, were not known, and so these parameters were identified by constraining them to result in specific qualitative features of the hair pattern phenotypes. A sensitivity analysis showed that the model results are not highly sensitive to the precise parameter values and suggests that our conclusions regarding the feasibility of the model are valid for considerable ranges of parameters. The full development and discussion of assumptions for the mathematical model are in the SOM text.

In simulated wild-type cells (Fig. 2, A to E), Dsh and Fz localize to the distal membrane, and Vang and Pk localize to the proximal membrane, as is seen in vivo (1618, 21). Simulated clones of cells lacking fz function disrupt polarity in wild-type cells distal to the clones [Fig. 2F, compare with 2H (5, 6)], whereas simulated clones lacking Vang function disrupt polarity on the proximal side of the clones [Fig. 2G, compare with 2, I and J; (3)]. Simulated clones lacking dsh function result in the disruption of polarity within the mutant cells, but only show a mild effect outside of the clones (Fig. 2K). The nearly, though not fully cell autonomous, phenotype is similar to that which we observed experimentally [Fig. 2, L and M; and (29)]. Clones lacking all pk function show only a subtle phenotype [Fig. 2, N and O; (20)]. Examination of protein distributions (Fig. 2; fig. S7) shows that the results are highly concordant with published observations (table S2). Similarly, simulated overexpression clones produce results closely mimicking observed experimental results (fig. S8). In simulations and in wings, relatively small clones lacking a global biasing signal show no phenotype, demonstrating that not all cells need to respond to the global directional signal for the feedback loop to co-operatively align all of the cells (23, 30).

Fig. 2.

Simulation results. 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 to 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 null clones. (I) and (J) VangA3 mutant clones. (L and M) dshV26 mutant clones. (O) pk-sple13 mutant clones. Asterisks mark nonautonomous effects near dsh clones.

Previously, we found that Pk overexpression in the posterior wing domain enhanced the accumulation of Fz and Dsh at cell boundaries, despite the observed ability of Pk and Vang to block Dsh recruitment (18). Consistent with these results, Dsh and Fz are seen in a simulation of this experiment to accumulate to higher levels in the region overexpressing pk than in the wild-type region, and they accumulate perpendicular to the wild-type orientation near the anterior-posterior boundary (Fig. 3).

Fig. 3.

pk overexpression. Pk (A), Fz::GFP (B) and Dsh::GFP (C) near the anterior-posterior boundary of pupal wings overexpressing pk in the engrailed domain. (D to F) Simulation of a periodic horizontal band of random 1x – 11.58x pk overexpression, showing Pk (D), Fz (E) and Dsh (F). Reorientation of Fz accumulation is denoted with arrowheads. (A to C) are different wings.

Our results suggested a mechanistic explanation for the difference between autonomous and nonautonomous fz alleles. Because the nearly autonomous fz alleles [fzJ22 and fzF31; (27)] have phenotypes similar to dsh clones, we hypothesized that these alleles may be selectively deficient in complexing with Dsh, but normal in their ability to complex with Vang (Fig. 4A). Simulations of clones in which we disrupted the interaction between Dsh and Fz by reducing the corresponding forward reaction rates produced nearly cell autonomous polarity phenotypes (Fig. 4B and fig. S9).

Fig. 4.

Cell autonomy. (A) Interactions for wild-type, Dsh1, Fz autonomous, and Fz nonautonomous alleles. Red dots indicate point mutations, and the faded Fz indicates absent protein in some null alleles. Spaces between proteins indicate loss of interaction. (B) Simulation of a Fz autonomous clone in which Fz has 0.01% of wild-type interaction with Dsh. (C) fzR52 (nonautonomous) and fzF31 (autonomous) clones with Vang::YFP in nonmutant cells. Vang accumulates at all boundaries of the fzF31 but not the fzR52 clones (arrowheads). (D) Dsh::GFP and Flamingo (Fmi) localization in wild-type, fzR52 (null), as well as fzF31 and fzJ22 (autonomous) wings, viewed on the apical surface (top) or in edge cells (bottom three rows). Fmi staining identifies the correct focal plane.

This hypothesis makes two easily testable predictions. First, Fz autonomous proteins should be present in the membrane and should recruit Vang to the adjacent membrane, whereas Fz nonautonomous protein should not recruit Vang. It has previously been shown that GFP-tagged FzJ22, expressed in a wild-type background, is present at the apical cell cortex, but remains symmetrically distributed (17), a distribution in accordance with our simulation of this condition (fig. S11). Examining this further, we found that in cells adjoining clones of the autonomous fzF31 allele, Vang is recruited to the boundary between wild-type and mutant cells, whereas substantially less Vang is recruited to those boundaries in cells adjoining clones of the nonautonomous fzR52 allele (Fig. 4C, arrow-heads). Thus, Fzautonomous proteins recruit Vang to the opposing cell surface, whereas nonautonomous alleles do not. The second prediction is that autonomous Fz proteins should fail to recruit Dsh. Indeed, we find that both are substantially impaired in Dsh recruitment, though somewhat less impaired than the very strong, nonautonomous fzR52 allele (Fig. 4D). Thus, strong fz alleles, many of which fail to accumulate Fz protein (27), display no or severely impaired interaction with Dsh and Vang, whereas autonomous alleles have impaired interaction with Dsh, but retain substantial ability to recruit Vang to the adjacent membrane. Notably, simulated overexpression of Fz with impaired Dsh interaction also produced the correct polarity disruption in cells proximal to the clones [fig. 9; (17)].

The Dsh1 protein produces nearly autonomous clones, and it carries a mutation in its DEP domain, which is required for membrane localization (16, 19); autonomous fz alleles bear point mutations in the first cytoplasmic loop (27), suggesting these mutations may affect the same interaction. A low affinity interaction between the Dsh PDZ domain and a sequence in the cytoplasmic tail of Fz has been demonstrated (31). Our data suggest that sequences in the Dsh DEP domain, and in the Fz first intracellular loop, are also important for Dsh membrane association. Thus, a regulated, bipartite, high affinity association of Dsh with Fz may be selectively disrupted in fzautonomous alleles.

The ability of our mathematical model to simultaneously reproduce all of the most characteristic PCP phenotypes (table S2) demonstrates the feasibility of the underlying biological model as a PCP signaling mechanism. Further, the mathematical model demonstrates how the overall scheme of the model—a local feedback loop between adjacent cells amplifying an initial asymmetry—can explain the autonomous and nonautonomous behavior of PCP mutant clones. Alternative models invoking diffusible factors have not been supported by the identification of such factors (12), and the contact-dependent intercellular signaling model more readily accounts for the slight nonautonomy of dsh and fzautonomous clones than do the diffusible factor models.

The ability of the mathematical model to make predictions and provide a detailed picture of PCP signaling is limited by the lack of complete biological understanding. Although the validity of quantitative model predictions is subject to its assumptions and the set of features used in parameter identification (SOM text), the model has allowed us to directly connect a biological model to the complex behaviors it was hypothesized to explain and to explore the implications of variations in the model.

Supporting Online Material

Materials and Methods

SOM Text

Figs. S1 to S16

Tables S1 and S2

References and Notes

References and Notes

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