Mathematical modeling of sub-cellular asymmetry of fat-dachsous heterodimer for generation of planar cell polarity

Abstract

Planar Cell Polarity (PCP) is an evolutionarily conserved characteristic of animal tissues marked by coordinated polarization of cells or structures in the plane of a tissue. In insect wing epithelium, for instance, PCP is characterized by en masse orientation of hairs orthogonal to its apical-basal axis and pointing along the proximal-distal axis of the organ. Directional cue for PCP has been proposed to be generated by complex sets of interactions amongst three proteins - Fat (Ft), Dachsous (Ds) and Four-jointed (Fj). Ft and Ds are two atypical cadherins, which are phosphorylated by Fj, a Golgi kinase. Ft and Ds from adjacent cells bind heterophilically via their tandem cadherin repeats, and their binding affinities are regulated by Fj. Further, in the wing epithelium, sub-cellular levels of Ft-Ds heterodimers are seen to be elevated at the distal edges of individual cells, prefiguring their PCP. Mechanisms generating this sub-cellular asymmetry of Ft-Ds heterodimer in proximal and distal edges of cells, however, have not been resolved yet. Using a mathematical modeling approach, here we provide a framework for generation of this sub-cellular asymmetry of Ft-Ds heterodimer. First, we explain how the known interactions within Ft-Ds-Fj system translate into sub-cellular asymmetry of Ft-Ds heterodimer. Second, we show that this asymmetric localization of Ft-Ds heterodimer is lost when tissue-level gradient of Fj is flattened, or when phosphorylation of Ft by Fj is abolished, but not when tissue-level gradient of Ds is flattened or when phosphorylation of Ds is abrogated. Finally, we show that distal enrichment of Ds also amplifies Ft-Ds asymmetry. These observations reveal that gradient of Fj expression, phosphorylation of Ft by Fj and sub-cellular distal accumulation of Ds are three critical elements required for generating sub-cellular asymmetry of Ft-Ds heterodimer. Our model integrates the known experimental data and presents testable predictions for future studies.

Figure 1. Fat (Ft)-Dachsous (Ds)-Four-jointed (Fj) system regulates PCP in Drosophila wing.

(A) Schematic of Drosophila wing showing opposing expression gradients of Ds (green) and Fj (grey) while expression of Ft (not shown) remains uniform. Arrow marks the proximal-distal (P/D) axis of the wing. (B) Arrangement of cells (boxes) seen in the contexts of proximo distally increasing gradient of Fj expression (grey) as well its activity; the latter represented by arrows and hammers symbolizing its activating and repressive roles, respectively, while thicknesses of these lines indicate relative strengths of these activities. (C) Different concentrations of the Ft-Ds heterodimer (represented by thickness of red line) at the proximal and distal edges of the cells provide distinct intensity plot. (C′) A line joining the peaks of these two levels of Ft-Ds heterodimer at the proximal and distal edges would be proximally or distally inclined depending on the concentrations of the Ft-Ds heterodimer at these edges. This line would, however, be horizontally placed when concentrations of the Ft-Ds heterodimer are equal at the proximal and distal edges. (C′′) Simplified visual representations of the concentrations of Ft-Ds heterodimer at the proximal and distal edges of the epithelial cells seen in (C′). (D) Orientation of wing hairs in wild type and in dachsous (ds) mutant adult wings.

Figure 2. Evolution of phosphorylated Ft (Ft^p) and phosphorylated Ds (Ds^p).

(A–D) Evolution of sub-cellular concentrations of phosphorylated Ft () and that of (E–H) phosphorylated Ds () in cells over time. (A) and (E) show the randomized initial concentrations of and , respectively.

Figure 3. Evolution of asymmetric localization of the Ft-Ds heterodimer in cells over time.

(A–H) Evolution of the asymmetric levels of Ft-Ds heterodimer in cells. (A) shows the initial uncoordinated levels of Ft-Ds heterodimer while that shown in (H) represent the final steady state.

Figure 4. Fj gradient, but not that of Ds, is essential for generating the asymmetric localization of the Ft-Ds heterodimer.

(A) Asymmetric localization of Ft-Ds heterodimer (distally heightened top of trapezoid) is seen when both- Fj and Ds are expressed in gradient. Data shown here is the same as that shown in Figure 3H. (B) Uniform over-expression of Fj results in a loss of asymmetry of Ft-Ds heterodimer (flattened top) in all cells, although Ds expression is maintained in a gradient. (C) Conversely, uniform overexpression of Ds does not affect Ft-Ds heterodimer asymmetry when Fj expression remains in a gradient. (D) When both Fj and Ds are uniformly overexpressed, asymmetry of Ft-Ds heterodimer is lost.

Figure 5. Phosphorylation of Ft by Fj kinase is necessary for Ft-Ds asymmetry.

(A) Wild type Fj activity displays characteristic distal enrichment of Ft-Ds heterodimers. Data shown here is same as that shown in Figure 3H. Inset displays the ordered projection of Drosophila wing hair in proximal-distal axis. Arrow and hammer indicate the activating and repressing roles of kinase Fj against Ft and Ds, respectively. (B) Asymmetry of Ft-Ds heterodimer is disrupted when phosphorylation of Ft by Fj kinase is lost in Fj^(Ft) mutant. This would result in swirling of wing hairs (inset). (C) Loss of kinase activity of Fj against Ds in Fj^(Ds) mutant, however, does not influence asymmetric localization of Ft-Ds heterodimer. Orientations of wing hairs now remain intact (inset) like that of wild type wing (A).

Figure 6. Sub-cellular asymmetry of enriches asymmetry of Ft-Ds heterodimer.

(A) Equal contributions of sub-cellular asymmetries of and result in asymmetric localization of Ft-Ds heterodimer (µ = 1). Inset shows the enlarged view with inclined tops of the trapezoids. Data shown here is same as that shown in Figure 3H. (B) Higher contribution of sub-cellular asymmetry of augments the asymmetric localization of Ft-Ds heterodimer (steeper top of the trapezoids) in cells (µ = 10). (C) Higher contribution of sub-cellular asymmetry, however, results in diminished or even reversed asymmetry of Ft-Ds heterodimer in cells (µ = 0.1). Inset shows the enlarged view with flatter tops of the trapezoids indicating weak asymmetry of Ft-Ds heterodimer.