2018 Nov 27
Theoretical Tool Bridging Cell Polarities With Development of Robust Morphologies
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Theoretical Tool Bridging Cell Polarities With Development of Robust Morphologies
Despite continual renewal and damages, a multicellular organism is able to maintain its complex morphology. How is this stability compatible with the complexity and diversity of living forms? Looking for answers at protein level may be limiting as diverging protein sequences can result in similar morphologies. Inspired by the progressive role of apical-basal and planar cell polarity in development, we propose that stability, complexity, and diversity are emergent properties in populations of proliferating polarized cells. We support our hypothesis by a theoretical approach, developed to effectively capture both types of polar cell adhesions. When applied to specific cases of development - gastrulation and the origins of folds and tubes - our theoretical tool suggests experimentally testable predictions pointing to the strength of polar adhesion, restricted directions of cell polarities, and the rate of cell proliferation to be major determinants of morphological diversity and stability.
apical-basal; convergent extension; developmental biology; gastrulation; metastable topologies; none; organogenesis; pcp; physics of living systems.
© 2018, Nissen et al.
Conflict of interest statement
SN, SR, AT, KS No competing interests declared
Figure 1.. 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 tube formation (axial symmetry). Complex morphologies can be formed by combining cells with none, one, or two polarities. In Figure 1—figure supplement 1, we schematically illustrate how existing models capture different elements of development.
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.. 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 (orange/bottom part). ( B–C) Two cells interact only if no other cells block the line of sight between them. ( B) Cell i and j do not interact if ij’s midpoint (black dot) is inside of the Voronoi diagram for cell k (shaded in grey). ( C) Cell i and j interact because cell k is further away than the distance ij/2 and ij’s midpoint therefore lie outside of cell k’s Voronoi diagram. In the related Figure 2—figure supplement 1A–D, we test the sensitivity of our model to the details of the potential and neighborhood assignments. In Figure 2—figure supplement 2, we relate changes in cell shapes to the model components, and in Figure 2—figure supplement 3 (Figure 2—video 1), we illustrate how altering polarity affects the dynamics of the systems with two and six cells.
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 = 10 −4). ( B) Full Voronoi interactions with a cut-off does also lead to a stable state, although a few cells might lose interaction with the majority of cells (cut-off at three shown). ( C) However, a simple cut-off (and no Voronoi) does not result in stable morphologies but broken sheets on top of each other (cut-off at 2.5 shown). ( D) The same happens, when all cells always interact with their six nearest neighbors. ( E) With changed initial conditions compared to Figure 3, the system reaches a different stable state with low noise ( η = 10 −4). ( F) However, this is comparable to the state obtained under high noise ( η = 10 −1). ( G) Increasing the noise in ( E) at time log(t) = 4 from η = 10 −4 to η = 10 −1, it results in fewer permutations than having high noise during the entire simulation which is shown in ( F). The lower-case letters in (E –G) point at macroscopic features (a –h on one side in blue and r –u on the other side in red). The initial positions and polarities are identical for all seven simulations.
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 constriction (narrowing of the apical side) giving rise to a central bottle cell. ( C) If the AB polarity is shortened, the neighboring cells will reorient in a similar way. These results are obtained under the assumption that the AB polarity tends to orient perpendicular to the distance vectors (black lines) connecting cells’ centers of masses (black dots) which are central to the model presented.
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 away from another in a two-cell system, it tilts the axis of position. ( D) Similar to ( C), if one cell points towards another, it tilts in the other direction. ( E) Similar to ( B), but with six cells instead of two. In this case, the final axis of position is only tilted by five degrees compared to the initial axis. ( F) Similar to ( C) and ( E), with three cells pointing up and three cells pointing to the right. This also gives a final axis of position that is tilted by five degrees compared to the initial axis.
Figure 3.. 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 and blue marking two opposite sides of the polar cells. Cells closest to the viewer are marked red/blue, whereas cells furthest away are yellow/white. ( E) Full system at the time point shown in ( D). ( F) Development of the number of neighbors per cell (red) and the energy per cell (blue), as defined by the potential between neighbor cells in Figure 2. Dark colors show the mean over all cells while light-shaded regions show the cell–cell variations. The yellow dot marks the energy for a hollow sphere with the same number of cells. See Figure 3—video 1 for full time series. In Figure 2—figure supplement 1E–G and Figure 3—figure supplement 1, we study how the final morphology depends on noise. In Figure 3—figure supplement 2, we show how the outer surface self-seals, and that the shape is maintained when cells divide.
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 same set of aggregates, the angle between the pairwise polarities is calculated. The initial positions are the same for all systems. Each system has 8000 cells. Cells are pairs if they were initiated with identical position. Here, the noise level is η = 10 −4. Two-sample Kolmogorov-Smirnov tests showed p < 0.001 statistical significance (marked by *). Comparing noise levels give similar results as comparing noise seed. The initial polarities are random like in Figure 3.
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 3—video 1), some of the outer surfaces subsequently reorganize to form a less disrupted torus-like structure with multiple handles. ( D–F) The final structure in ( C) (and Figure 3D–E together with the end of Figure 3—video 1) is robust to cell divisions. For every 10th time step, we select a cell by random, and let it divide in an arbitrary direction. We see that the overall shape of the structure is maintained, and that it expands equally in all directions.
Figure 4.. 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) Two flat planes pointing in opposite directions are obtained if polarities point away from a central plane. ( D) For all three initial conditions ( A–C), if the polarities are allowed to change dynamically and the noise is high ( η = 10 0 compared to η = 10 −1 in A–C), the resulting shape consists of three nested ‘Russian doll’-like hollow spheres that will never merge due to opposing polarities. In contrast to the random initial condition in Figure 3, the initial conditions in ( D) are symmetric.
Figure 5.. 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), t G −1. In silico organoids grow from 200 cells up to 8000, 12,000, or 16,000 cells with different generation times and no outer pressure. ( B) Number of local minima as a function of pressure, P. In silico organoids grow to the same size with the same 1/(generation time), t G −1 = 1.4⋅10 −4 but different outer pressure. The images illustrate the 16,000 cells stage. Blue dots mark the average, while light shaded regions show the SEM based on triplicates. See also Figure 5—figure supplement 1 for additional measurements on the differences between rapid growth and pressure.
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 these ‘lakes’. (
A–C) Deepest point of the ‘lakes’ (folds) relative to the water level. The probability of having a lake at a given depth is normalized to the number of ‘lakes’. ( D–F) Length of the ‘lakes’ relative to the entire circumference at this same level. Length of a lake is defined from the angle between the two cells at lake shore that are the furthest away from each other. Pressure and 1/(generation time) increase from upper to lower panels. Two-sample Kolmogorov-Smirnov tests showed p < 0.001 statistical significance (marked by *). The shown histograms are for the 16,000 cell stage, which compares to the dark blue line in Figure 5.
Figure 6.. The length and width of tubes are set by the strength of planar cell polarity (PCP, λ
For each value of λ
3, 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) and semi-minor axis (light blue) are measured at the final stage (Materials and methods). Images show the final state. Throughout the figure, λ 2 = 0.5 and λ 1 = 1 - λ 2 - λ 3. The animated evolution from sphere to tube is shown in Figure 6—video 1. See also Figure 6—figure supplement 1 where we show that tubes also form when we disable the direct influence of PCP on apical-basal polarity, and Figure 6—figure supplement 2 where we vary the degree of PCP along the axis of the tube. In Figure 6—figure supplement 3, we show that cell intercalations result in experimentally reported T1 neighbor exchanges during convergent extension.
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 λ
2 = 0 when updating AB polarity (λ 2 = 0.5 when updating position and PCP as in Figure 6). The strength of PCP (λ 3) is defined as shown along the x-axis. λ 1 = 1 - λ 2 - λ 3 for updating position and PCP, and for updating AB polarity λ 1 = 0.5 - λ 3. This way λ 1 and λ 3 are the same for position, AB polarity, and PCP, and the only change is the value of λ 2. The final tubes are slightly wider and shorter compared to Figure 6 since the tips become more rounded when PCP does not affect AB polarity. Throughout, all the simulations in this figure, dt = 0.2 and the noise parameter η = 5⋅10 −5.
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, an elongated tube with a central lumen has formed. Images to the left show the entire system while images to the right show a cross-section. Cells inside zone (i) develop with λ 1 = 0.41, λ 2 = 0.5, and λ 3 = 0.09 while cells inside zone (ii) develop with λ 1 = 1, and λ 2 = λ 3 = 0. The central third of the 1000 cells in the system belong to zone (ii). Throughout, the simulation dt = 0.1 and η = 10 −4.
Figure 6—figure supplement 3.. T1 exchanges occur during sphere–tube transition.
Two consecutive time frames of the most extreme scenario in Figure 6 (λ
1 = 0.41, λ 2 = 0.5, and λ 3 = 0.09, see also Figure 6—video 1). ( A) Snapshot of the entire system at time t = 1259.0. ( B) Snapshot slightly later at time t = 1288.3. In both panels, the cell centers (light grey vertices) are triangulated (light grey edges). Triangle centers (red vertices) are calculated in order to get an approximate location of the cell borders (red edges). Inside the black box, two T1 exchanges (bold red lines) are highlighted.
Figure 7.. 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 anterior-posterior (top-bottom) axis (as in the inset to Figure 6). ( B) Flattening of the blastula and ( C) invagination occur due to external force reorienting AB polarity (Materials and methods). ( D–E) Tube elongation is due to PCP-driven convergent extension and ( F) merging with the top of the blastula happens when the tube approaches the top. Throughout the simulation, λ 1 = 0.5, λ 2 = 0.4, and λ 3 = 0.1 for the lower cells while the top cells have λ 1 = 1 and λ 2 = λ 3 = 0. For full time dynamics see Figure 7—video 1. In Figure 7—figure supplement 1, we consider alternative scenarios of sea urchin gastrulation and and neurulation.
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 (AB) polarity (red–yellow). ( B) Flattening of the blastula and invagination occur if direction of PCP is maintained for some time. During this initial phase λ 2 = 0.1, and there is no convergent extension (λ 3 = 0). ( C) In the final phase, we increase λ 2 to 0.4 and turn on λ 3 = 0.1. With this, we let PCP relax so it curls around the bottom, and allow it to change dynamically in time. ( D) As a result, tube narrows and elongates, until it finally connects and merges with the top. ( E–H) Initial conditions on PCP enable neural plate bending and neural tube closure. ( E) Starting with 1000 cells on a plane with AB polarity, we induce PCP along the plane together with two rows where the PCP points parallel and antiparallel to AB polarity (shown with cyan arrows). Here, we simulate the neural plate (cells in the middle, between the two rows with constrained PCP) surrounded by the epidermis (the rest of the cells). The two rows of cells with PCP pointing out of epithelial plane correspond to the cells at the dorsolateral hinge points next to the neural plate (epidermis boundaries). In chick, spinal neural tube can close with only these two hinge points (Nikolopoulou et al., 2017). The bending is driven by apical constriction and PCP is essential for bending, convergent extension and closure. ( F) This enables neural plate bending and formation of the neural groove. ( G) Continuing the simulation leads to contact of the two sides of the neural plate and hereby neural tube closure. ( H) Finally, the system stabilizes with the neural plate on top of the neural tube. Comparing the initial stage to the final stage, the overall direction of PCP in the plate is conserved while in the tube PCP goes around an internal axis. For this simulation, we set λ 2 = 0.5 and λ 3 = 0. Turning on convergent extension (λ 3) at the final stage will allow for elongating the system along the axis going through the tube and narrowing it in another direction. The concept is similar to gastrulation in Drosophila. In both simulations, sea urchin and neurulation, dt = 0.3. In sea urchin (A–D), the noise parameter η = 3.3⋅10 −5, and in neurulation (E–H), η = 3.3⋅10 −2.
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.
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, the planes break into smaller pieces, and at time 10 4 they have merged into two planes that are separated by several cell diameters.
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 of 200 cells placed on a hollow sphere which is the initial condition in our organoid simulations (Figure 5). A new cell is introduced half a cell radius in a random direction away from the mother cell. The red line is the mean distance from the mother cell to all it’s neighbor cells at
t = 1000. The generation time is intermediate (Figure 5A).
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Research Support, Non-U.S. Gov't
Eukaryotic Cells / cytology
Eukaryotic Cells / physiology
Gastrulation / physiology
Proteoglycans / chemistry
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.