Fig 1.
Sub-cellular potts model for planar cell polarity:
(A) Schematic of an epithelial cell arrangement with PCP protein distributions. Initially, the PCP proteins are uniformly distributed within each cell. Over time, Fz, Dsh, and Dgo (magenta) localize to the proximal/anterior edge, while Vang and Pk (green) segregate to the distal/posterior side. (B) Schematic of two epithelial cells with cluster ids 1 (orange) and 2 (blue) respectively, along with the respective domain ids. The cluster id indicates the association of each domain with a specific biological cell. (C) Model Overview: Each epithelial cell is subdivided into four compartments—proximal, distal, lateral, and cytoplasm. Seed cells are placed randomly within the lattice, which grow and rearrange their shapes to efficiently pack in the space. PCP domains are introduced with random initial distributions. The proximal and distal domains spatially segregate into opposing ends of each cell over time. Blue box highlights the evolution of a single cell during this process. (D) The vector field representation of PCP orientations shows the transition from a homogeneous distribution () of global polarization to complete alignment (
). (Panel A adapted from [25]).
Table 1.
Internal contact energies (j) between PCP domains.
Table 2.
External contact energies (J) between PCP domains and medium.
Fig 2.
Time evolution of global polarization under different boundary conditions and domineering non-autonomy:
(A) Polarization outcomes under periodic boundary conditions applied along both the x and y directions. The system exhibits four distinct final states, with the mean tissue polarization vector aligning along one of the four directions: right, left, up, or down. (B) Polarization under a directional cue applied at the left boundary, with periodic boundary conditions maintained along the y axis. This configuration introduces a bias to the right and sets a definitive proximal-distal axis to the right. (C) Distribution of individual cell polarization angles in a system comprising cells. The first panel (green) shows the initial distribution of cell polarization vectors. The second panel (red) depicts the final distribution under periodic boundary conditions for a system that polarizes towards the left direction, similar to the output highlighted in red in panel (A). The third panel (blue) shows the final distribution under the left-boundary orienting signal, with polarization vectors highly aligned along the proximal-distal axis. (D) Time evolution of global polarization ϕ under periodic (red) and left-boundary (blue dashed) conditions in an
system over 107 MCS; shaded regions indicate the standard error (SE). Global polarization increases with increasing time. (E) Domineering non-autonomous behaviour of cells in loss-of-function mutants for Fz. In Fz loss-of-function mutants (magenta), polarity vectors distal to the mutant patch point towards the mutant clone. (E’) Domineering non-autonomous behaviour of cells in loss-of-function mutants for Vang. In Vang loss-of-function mutants (green), polarity vectors proximal to the mutant patch point away from the mutant clone.
Fig 3.
Effect of system size on polarization for periodic boundary and left boundary signal configurations.
(A) Final average global polarization (ϕ) for different system sizes under periodic and left boundary signal configurations (average over 100 simulations). As system size increases, global polarization decreases for both boundary conditions. (B) Final tissue angle distribution of polarization for the system under periodic (red) and left boundary (blue) conditions. For periodic boundaries, the polarization angles become nearly uniform, while for the left boundary signal, the alignment direction is more dispersed around the proximal-distal axis. (C) Final cell polarity vectors for the
(top panel) and
(bottom panel) system with periodic boundary conditions. In the smaller
system, all cells align in a single direction (upward in this case). For the
system, local alignment occurs within small domains, and swirling patterns emerge. (D) Final cell polarity vectors for the
(top panel) and
(bottom panel) systems with left boundary signal configuration. For the smaller
system, all cells align along proximal-distal direction. For the
system, alignment is maintained in the first few columns but dissipates as cells move farther from the boundary. (E) Local polarization as a function of radial distance for periodic boundary configuration (top panel) and as a function of distance from the boundary for left boundary signal configuration (bottom). Local polarization decreases sharply with distance in the periodic boundary case, whereas the decrease in polarization for the left boundary signal configuration is less steep.
Fig 4.
Cell proliferation-based mechanism for maintaining polarity.
(A) Schematic of the system used to study the effect of cell proliferation. 10 rows of cells are arranged in a box with periodic conditions along the vertical direction, a boundary signal on the left, and an open space on the right. (B) Cells grow to double their volume and then divide. After each division, daughter cells inherit the PCP domains equally, lose polarity, and redistribute these domains based on neighbouring cells’ polarity. (C) Simulation showing cell growth and division at the distal/front boundary of a system. (D) Simulation showing uniform cell proliferation. Cells about to divide are indicated by an ’X’ mark. (E) Comparison of global polarization ϕ for systems with and without cell proliferation as a function of the number of cells. For fixed tissue sizes, global polarization decreases with system size (red). Systems with cell proliferation on front/distal boundary stabilize at a higher ϕ, approximately 0.89 (blue). Uniform cell proliferation further enhances global polarization at a higher value 0.92 (green). (F) Distribution of mean angle of tissue polarization for systems with proliferating cells, showing a shift towards for front/distal boundary proliferation and proximal-distal alignment for uniform cell proliferation. (G) Comparison of final global polarization when number of cells in the system is 900 for varying growth rates for front/distal proliferation (blue) and uniform cell proliferation (green). For uniform cell proliferation, the refractory time between cell divisions is set to 104 MCS. (H) Simulation output showing polarity vector alignment for the configuration without cell proliferation. Domains of local alignment can be observed. (I) Simulation output showing polarity vector alignment for the proliferation on front configuration. (J) Simulation output showing preference for alignment along the proximal-distal axis for the uniform cell proliferation configuration.
Fig 5.
Cell non-autonomous polarization.
(A) Simulation snapshots showing cell-autonomous (as observed in Drosophila wing) and cell non-autonomous polarization (as observed in mouse epidermis), where the PCP proteins can either self-polarize or not in single cells. (B) Comparison of the distance (left panel) and interface (right panel) between proximal and distal compartments over time for both model versions. In the cell-autonomous case, the compartments have no common interface and remain spatially separated, while in the cell non-autonomous case the interface area is high and their distance fluctuates at values lower than 0.8, indicating mixing. (C) Global polarization of cell non-autonomous PCP model as a function of system size for periodic and left boundary signal configurations. As system size increases, global polarization decreases under both boundary conditions. (D) Final mean angle of tissue polarization for a cell system: under periodic boundary conditions (left, red), the angles are uniformly distributed between
and
; under the left boundary condition (right, blue), the angles are more dispersed along the proximal–distal axis. (E) Final cell polarity vectors for a
system with periodic boundary conditions (left): local alignment occurs within small domains. For the left boundary signal configuration (right), alignment is maintained in the first few columns but dissipates with distance from the boundary. (F) Comparison of global polarization for systems with and without cell proliferation as a function of the number of cells. For fixed tissue sizes, global polarization decreases with system size (red). Uniform cell proliferation enhances global polarization at a higher value 0.93 (green). (G) Simulation output showing preference for alignment along the proximal-distal axis for the uniform cell proliferation configuration.