Theoretical tool bridging cell polarities with development of robust morphologies
- University of Copenhagen, Denmark
Figures
Figure 1 with 1 supplement
Two symmetry-breaking events, gain of apical-basal (AB) polarity and planar cell polarity (PCP), on cellular level coincide with the appearance of a rich set of morphologies.
Starting from an aggregate of non-polarized cells (globular symmetry), individual cells can gain AB polarity and form one or multiple lumens (spherical symmetry). Additional, gain of PCP allows for …
Figure 1—figure supplement 1
Overview of the existing literature on models addressing specific developmental events discussed in our work.
For more references on vertex models see Alt et al. (2017).
Figure 2 with 4 supplements
Cells are modeled as interacting particles with a polarity-dependent potential.
(A) Potential between two interacting cells with apical-basal polarity (see Equation 6). Cells repulse when polarities are antiparallel (top/green part) and attract when they are parallel …
Figure 2—figure supplement 1
Dependence on the shape of the physical potential, the interaction partners, and noise.
(A) Applying the neighborhood function shown in Figure 2, but changing the shape of the potential to the short-range potential written in Equation 2, the system unfolds and reaches a stable state (η …
Figure 2—figure supplement 2
Changes in cell shapes may reorient apical-basal (AB) polarity.
(A) In an epithelial sheet AB polarity (yellow arrows) points perpendicular to the sheet. (B) Regulated changes in planar cell polarity can reorient the AB polarities in neighboring cells by apical …
Figure 2—figure supplement 3
Examples of simple systems consisting of only two or six cells (see also Figure 2—video 1).
(A) Two cells initially aligned do not result in any movement. (B) If both cells’ polarities are 45 degrees to the plane, the axis of position becomes tilted by 30 degrees. (C) If one cell points …
Figure 2—video 1
Dynamics for two and six interacting cells.
(A) No movement occurs if the polarities are perfectly aligned and the distance between the cells is at steady state. (B) If both cells initially have their polarities 45 degrees to the axis of …
Figure 3 with 3 supplements
Development of 8000 cells from a compact aggregate starting at time 0.
(A) Cells are assigned random apical-basal polarity directions and attract each other through polar interactions (see Equation 6). (A–D) Cross-section of the system at different time points with red …
Figure 3—figure supplement 1
The final shapes are more sensitive to initial polarities than to noise.
(A) The pairwise distance between cells for three systems with identical initial polarities but different noise and three systems with identical noise but different initial polarities. (B) For the …
Figure 3—figure supplement 2
The complex morphology in Figure 3 self-seals and is robust to overall system growth.
(A–C) Self-sealing properties of polarized cell surfaces when close to a final stable state in Figure 3. While the internal morphology remains the same from time log(t) = 3.6 (Figure 3C–D and Figure …
Figure 3—video 1
An aggregate of 8000 cells with initial random polarities unfolds into a stable complex morphology.
During the simulation, the polarities and positions are updated dynamically with equal speed and noise (dt = 0.1 and η = 10−3). There is no planar cell polarity (λ1 = 1 and λ2 = λ3 = 0). (A) The …
Figure 4 with 1 supplement
Different morphologies can be obtained by varying boundary conditions (Figure 4—video 1).
(A) A hollow sphere emerges if polarities are fixed and initially point radially out from the center of mass. (B) A hollow tube is obtained if polarities point radially out from a central axis. (C) …
Figure 4—video 1
Dynamics when the polarities have restricted orientations.
In all three simulations, an aggregate consisting of 8000 cells develops into three simple morphologies due to polarities being fixed in different directions. (A) One big lumen forms when the …
Figure 5 with 2 supplements
The number of complex folds in a growing organoid depends on the generation time and the pressure from the surrounding medium (Figure 5—video 1).
(A) Number of local minima as a function of 1/(generation time), tG−1. In silico organoids grow from 200 cells up to 8000, 12,000, or 16,000 cells with different generation times and no outer …
Figure 5—figure supplement 1
Organoids grown under external pressure have deeper and longer folds compared to organoids grown with rapid cell proliferation.
To quantify the folds, we fill the surface of the organoids with 'water' until halfway between the maximum and minimum radius of the system. Then we measure the relative depth and circumference of …
Figure 5—video 1
In silico organoids grown from 200 up to 16,000 cells.
(A) Increasingly many near-surface folds emerge when the organoid is grown with rapid cell proliferation (tG−1 = 7⋅10−3, p = 0, Figure 5A). (B) Number of deep folds saturates when the organoid is …
Figure 6 with 4 supplements
The length and width of tubes are set by the strength of planar cell polarity (PCP, λ3).
For each value of λ3, we initialize 1000 cells on a hollow sphere with PCP whirling around an internal axis (PCP orientation marked by cyan arrows in the top-left inset). Semi-major axis (dark blue) …
Figure 6—figure supplement 1
Removing the influence of planar cell polarity (PCP) on apical-basal (AB) polarity.
This figure is identical to Figure 6 with the only difference that now λ2 = 0 when updating AB polarity (λ2 = 0.5 when updating position and PCP as in Figure 6). The strength of PCP (λ3) is defined …
Figure 6—figure supplement 2
A lumen forms inside a developing tube in areas that lack planar cell polarity (PCP).
(A) Similar to Figure 6, a hollow sphere of cells is initialized. However, in this example only cells inside zone (i) have PCP while cells inside zone (ii) do not have PCP. (B) At the final stage, …
Figure 6—figure supplement 3
T1 exchanges occur during sphere–tube transition.
Two consecutive time frames of the most extreme scenario in Figure 6 (λ1 = 0.41, λ2 = 0.5, and λ3 = 0.09, see also Figure 6—video 1). (A) Snapshot of the entire system at time t = 1259.0. (B) …
Figure 6—video 1
Model of tubulogenesis.
Planar cell polarity (PCP) is tunred on in a spherical lumen consisting of 1000 cells with apical-basal polarity pointing radially out. In equilibrium, PCP will curl around an internal axis. …
Figure 7 with 2 supplements
External constraints on apical-basal (AB) polarity and planar cell polarity (PCP) can initiate invagination and drive gastrulation in sea urchin.
(A) The lower third of the cells in a blastula with AB polarity (apical is blue–white, basal is red–orange) pointing radially out acquire PCP (cyan–green) in apical plane pointing around the …
Figure 7—figure supplement 1
Directed changes in the direction of planar cell polarity (PCP) may drive invagination in gastrulation and neurulation.
(A–D) Gastrulation in sea urchin modeled without the apical constriction in Figure 7. (A) The lower third of the cells in the blastula acquire PCP (cyan–green) pointing opposite to the apical-basal …
Figure 7—video 1
Model of sea urchin gastrulation.
Starting from a hollow sphere of 1000 cells with apical-basal polarity pointing radially out. The bottom flattens and invaginates by applying an external force to mimic apical constriction (see …
Author response image 1
Changing the polarized direction of a plane of cells does not rotate the plane as a whole but breaks it into smaller planes.
Time t = 0 shows a plane consisting of 500 cells with AB polarity pointing to the right. At time t = 0.1, the direction of the polarity is shifted by 45 degrees. Since the polarity is fixed in time, …
Author response image 2
At cell division, the daughter cell equilibrates by half a cell radius in one time unit, which is of order 1/1000 the generation time.
Here, we show how a new cell (in blue) reaches equilibrium (in red). Cell division happens at time 0. At time 5, the next consecutive division happens in the system (not shown). The systems consists …
Additional files
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Source code 1
MATLAB script.
- https://doi.org/10.7554/eLife.38407.027
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Transparent reporting form
- https://doi.org/10.7554/eLife.38407.028
