Cells have the power to organise themselves to form complex and stable structures, whether it is to create a fully shaped baby from a single egg, or to allow adult salamanders to grow a new limb after losing a leg. This ability has been scrutinised at many different levels. For example, researchers have looked at the chemical messages exchanged by cells, or they have recorded the different shapes an embryo goes through during development. However, it is still difficult to reconcile the information from these approaches into a description that makes sense at multiple scales.

When an embryo develops, sheets of cells fold and unfold to create complex 3D shapes, like the tubes that make our lungs. Moulding sheets into tubes relies on interactions between cells that are not the same in all directions. In fact, two types of asymmetry (or polarity) guide these interactions. Apical-basal polarity runs across a sheet of cells, which means that the top surface of the sheet differs from the bottom. Planar cell polarity runs along the sheet and distinguishes one end from the other. For instance, apical-basal polarity marks the inner and outer surfaces of our skin, while planar cell polarity controls the direction in which our hair grows.

Nissen et al. set out to investigate how these polarities help cells in an embryo organise themselves to form complicated folds and tubes. To do this, simple mathematical representations of both apical-basal and planar cell polarities were designed. The representations were then combined to create computer simulations of groups of cells as these divide and interact with each other.

Simulations of ‘cells’ with only apical-basal polarity were able to generate different shapes in the ‘tissues’ produced, including many found in living organisms. External conditions, such as how cells were arranged to start with, determined the resulting shape. With both apical-basal and planar cell polarities, the simulations reproduced an important change that occurs during early development. They also replicated how the tubes that transport nutrients and oxygen form.

These results show that simple properties of individual cells, such as polarities, can produce different shapes in developing tissues and organs, without the need for a complicated overarching program. Abnormal changes in cell polarity are also associated with diseases such as cancer. The mathematical model developed by Nissen et al. could therefore be a useful tool to study these events.

Multicellular organisms are amazing in their ability to maintain complex morphology in face of continuous cell renewal and damages. Adult salamander can regenerate entire limbs (Eguchi et al., 2011), and, during development, some regions can maintain patterning when moved to different parts of an embryo or if the size is varied (Lyons et al., 2012). Given the vast complexity and diversity of living shapes, how can we reconcile the robustness to perturbations with flexibility to diversify? While undoubtedly the end result is encoded in the DNA and protein networks, looking for an answer at this level is challenging. Examples of phenotypic plasticity (Libby and Rainey, 2011), convergent evolution, and contrasting rates of morphological and protein evolution (Cherry et al., 1979) show that morphological similarity may not couple to the protein sequence similarity (Gould, 1970). Inspired by the unfolding of morphological complexity in development, we propose that cellular polarity may be the key for reconciling complexity, robustness, and diversity of organismal morphologies.

During early development, the increase in morphological complexity coincides with the progressive polarization of cells – first apical-basal (AB) polarity and then planar cell polarity (PCP) (Müller and Bossinger, 2003; Roignot et al., 2013; Andrew and Ewald, 2010; Li and Bowerman, 2010). This theme is ubiquitous across vertebrates and invertebrates (Figure 1): starting from the single fertilized egg cell, first the morula formed by non-polarized cells turns into the blastula – a hollow sphere of cells with AB polarity. Then, as cells acquire additional PCP, primary head-tail axis forms and elongates during gastrulation and neurulation (Loh et al., 2016). Because of the optical transparency, these stages are particularly prominent in sea urchin. At the morula stage, a lumen in the center is formed and is gradually expanding as cells proliferate and rearrange into the hollow sphere. Next, during gastrulation, a group of cells invaginate and rearrange into a tube that narrows and elongates primarily by cell rearrangement and convergent extension movements (Martik and McClay, 2017). The tube then merges with the sphere at the side opposite to invagination, and as a result, the sphere transforms into a torus. Emerging data suggest that PCP drives both invagination and tube elongation (Nishimura et al., 2012; Croce et al., 2006; Long et al., 2015) – a recurring theme across species.

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Mutations in PCP pathways produce shorter and wider tubes (Ochoa-Espinosa et al., 2012; Saburi et al., 2008; Kunimoto et al., 2017), somites (Song et al., 2010), and embryos (Gong et al., 2004). Formation and elongation of the tubes can proceed without cell division and cell death by cells rearranging along the tube’s axis, termed convergent extension (CE) (Andrew and Ewald, 2010; Tanimizu et al., 2009; Martik and McClay, 2017). While the importance of PCP in gastrulation and tubulogenesis is well established (Andrew and Ewald, 2010; Tanimizu et al., 2009; Martik and McClay, 2017; Kunimoto et al., 2017), it is unclear how polarity may control tube morphology.

The bulk-lumens-folds/tubes transition seen across animal species in early embryogenesis, is also a key feature of the later organ formation. The early stages of organogenesis in the liver, kidney, brain, gut, and pancreas are apparently so robust, that they can be recapitulated in vitro, allowing for advanced quantification and manipulation (Little, 2017). The case of pancreatic organoids is interesting as it illustrates an increase of morphological complexity from spheres to folds. Cells in in vitro pancreatic organoids first grow as a bulk and later acquire AB polarity and develop lumens. Depending on the growth conditions, organoids develop into a hollow sphere or acquire a complex folded shape (Greggio et al., 2013). It is currently unknown what drives the transition from sphere to folded state; the two possible hypotheses are rapid proliferation or physical pressure from growing into a stiff matrigel.

Is the apparent link between cellular polarity and morphological complexity accidental? Or, could it be that morphological transitions, stability, and diversity are emergent features in a population of proliferating polarized cells? If true, can we identify what drives the transition from lumens to folds and tubes? Why are these stable? Can we predict what controls fold depth, and tube length and width? To answer these questions, we lack a unified approach that could bridge polar interactions between single cells to the global features emerging on the scale of thousands of cells in 3D.

Starting with D’Arcy Thompson’s seminal contribution (Thompson, 1942), quantitative models aided in understanding specific morphogenetic events (Figure 1—figure supplement 1). Among these are invagination (Odell et al., 1981; Rauzi et al., 2015; Polyakov et al., 2014; Hočevar Brezavšček et al., 2012), primitive streak formation (Newman, 2008), convergent extension (Collinet et al., 2015; Belmonte et al., 2016), epithelial folding (Buske et al., 2012; Osterfield et al., 2013; Monier et al., 2015; Murisic et al., 2015), emergence of global PCP alignment from local cell–cell coupling (Amonlirdviman et al., 2005; Le Garrec et al., 2006; Burak and Shraiman, 2009), origins of tubulogenesis (Engelberg et al., 2008), and recently statistical properties of branching morphogenesis (Hannezo et al., 2017). However, they are often on either of the two ends of the spectra: those modeling single cells explicitly, often rely on vertex-based approaches and are limited to dozens of cells (Alt et al., 2017; Misra et al., 2016; Aigouy et al., 2010; Le Garrec et al., 2006). To capture the large features spanning thousands of cells, one typically turns to elastic models where AB polarity is implicit and epithelia is presented as a 2D elastic sheet (Hannezo et al., 2014; Etournay et al., 2015; Hufnagel et al., 2007; Nagai and Honda, 2009; Aliee et al., 2012; Nagai and Honda, 2001).

We developed a theoretical approach that, with only a few parameters, bridges cellular and organ scales by integrating both types of polarity. A main difference to earlier approaches is that a cell’s movement is coupled to how its AB polarity and PCP are oriented relative to each other and relative to neighbor cell polarities. In other words, in our approach, the adhesion strength between neighbor cells is modulated by the orientation of their polarities. We find that polarity enables complex shapes robust to noise but sensitive to changes in initial and boundary constraints, thus supporting that morphological stability and diversity are emergent properties of polarized cell populations. Lumens, folds, and stable tubes emerge as a result of energy minimization. We make testable predictions on morphological transitions in pancreatic organoids, tubulogenesis, and sea urchin gastrulation. Our approach illustrates the evolutionary flexibility in the regulatory proteins and networks, and suggests that despite differences in proteins between organisms, the same core principles may apply.

Model

There are three key elements that allow us to bridge the scale from cellular level to macroscopic stable morphologies.

(1) Cells are approximated by point particles

Cell–cell adhesion is modeled by repulsive and attractive forces acting between cell centers. This allows a substantially gain in computation time compared to vertex-based models where cell surface adhesion is explicitly considered (Alt et al., 2017). The potential for pairwise interaction between two interacting neighbors, i and j, separated by distance r_ij is

(1) $V_{ij}=e^{-r_{ij}}-S{e}^{-r_{ij}/\beta },$

where the first term corresponds to repulsion, and the second term to attraction (see Figure 2A). For a pair of non-polar cells, the strength of attraction S = 1. β >1 is the parameter that sets how much longer the attraction range is compared to repulsion. We set β = 5 throughout the paper, but our results and conclusions are consistent for smaller β. The main results are also not sensitive to the exact choice of the potential, thus for example the higher power in the exponential,

(2) $V_{ij}=e^{-{\left(r_{ij}\right)}^4}-S{e}^{-{\left(r_{ij}/\beta \right)}^4},$

give qualitatively similar results (see Figure 2—figure supplement 1A). The potential energy of a cell is the sum of pairwise neighbor interactions

(3) $V_i={\sum }_j^{}{V}_{ij}.$

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(2) Cells interact with (a subset of Voronoi) neighbors

Interacting neighbors of cell i are selected from a subset of cells sharing a Voronoi surface. The subset is limited to the nearest neighbors j which are closest to the midpoint between i and j (Figure 2B–C). This constrain effectively corrects for the finite volume associated with point particles and assures that two cells will not interact if the line of sight between their centers is separated by a surface of a third cell. Without this constraint, the macroscopic morphologies collapse. However, our results are robust to replacing the line of sight constraint with full Voronoi and a cut-off distance for attraction force (Figure 2—figure supplement 1B).

(3) Cell–cell adhesion depends on the orientation of polarity

To capture directional adhesion, we set the strength of attraction, S, to be dependent on the relative orientation of the polarities in each of the cells. We assume that AB polarity and PCP are orthogonal, and that the polarities of one cell align with the polarities of its neighbor cells. Mathematically, we introduce unit vectors, ${\displaystyle {\hat{p}}_i}$ and ${\displaystyle {\hat{p}}_j}$ representing AB polarity, and ${\displaystyle {\hat{q}}_i}$ and ${\displaystyle {\hat{q}}_j}$ representing PCP for cell i and j, respectively. We set

(4) ${\displaystyle S={\lambda }_1{S}_1+{\lambda }_2{S}_2+{\lambda }_3{S}_3}$

where λ₁, λ₂, and λ₃ are the strengths of the different polarity terms. We require that

(5) ${\displaystyle {\lambda }_1+{\lambda }_2+{\lambda }_3=1,}$

to satisfy the constraint that perfectly aligned cells always have a steady state distance of 2 cell radii (for β = 5). To capture that in an epithelial sheet, AB polarities align parallel to each other and tight adherens junctions form in the plane perpendicular to AB polarity, we introduce the quadruple product

(6) ${\displaystyle S_1=\left({\hat{p}}_i\times {\hat{r}}_{ij}\right)\cdot \left({\hat{p}}_j\times {\hat{r}}_{ij}\right).}$

This makes two interacting cells with AB polarity maximally attracted (S₁ = 1) if the two apical sides are next to each other. On the other hand, if apical side of one cell is next to the basal side of another cell, the two cells will be maximally repulsing (S₁ = −1), see Figure 2A.

In case of planar polarization, we define

(7) ${\displaystyle S_2=\left({\hat{p}}_i\times {\hat{q}}_i\right)\cdot \left({\hat{p}}_j\times {\hat{q}}_j\right),}$

which makes the attraction maximal if the PCP of two cells are parallel to each other and perpendicular to their AB polarities. In addition, we assume that similarly to AB polarity, two cells with PCP are maximally attracted if their PCP are parallel and cells have the same kind of pole (e.g. Vangl-enriched) next to each other,

(8) ${\displaystyle S_3=\left({\hat{q}}_i\times {\hat{r}}_{ij}\right)\cdot \left({\hat{q}}_j\times {\hat{r}}_{ij}\right).}$

We later show that this assumption makes neighbor exchange on a sheet possible and results in CE. However, unlike with tight junctions, preferred directional adhesion with PCP is not as well established. Cells adhere to each other by membrane proteins assembled in adherens junctions just below the apical surface. Both proteins regulating adherens junctions, for example Smash (Beati et al., 2018), as well as adherence proteins forming adherens junctions can be planar polarized, for example Bazooka, E-cadherins (Simões et al., 2010; Tamada et al., 2017; Levayer and Lecuit, 2013; Warrington et al., 2013; Aigouy and Le Bivic, 2016). These indirectly support our assumption of anisotropic, planar polarized adhesion.

The motion of the cells and their polarities are calculated assuming overdamped dynamics

(9) ${\displaystyle \frac{{d\overline{r}}_i}{dt}=-\frac{d{V}_i}{d{\overline{r}}_i}+\eta ,}$

(10) ${\displaystyle \frac{{d\overline{p}}_i}{dt}=-\frac{d{V}_i}{d{\overline{p}}_i}+\eta ,}$

and

(11) ${\displaystyle \frac{{d\overline{q}}_i}{dt}=-\frac{d{V}_i}{d{\overline{q}}_i}+\eta ,}$

where the ${\displaystyle {\overline{p}}_i}$ and ${\displaystyle {\overline{q}}_i}$ differentiation takes into account the rotation of polarity vectors, and η is a random uncorrelated Gaussian noise. In practice, we implemented the model in a MATLAB script (available in the Materials and methods section), where we use the Euler method. We perform the differentiation along the polarity by differentiating along all three cartesian coordinates (see Model details in the Materials and methods section). After each time step, we normalize the updated polarity vectors. The above differentiation does not include the change in partners when neighborhood changes. This is treated as a non-equilibrium step where potential energy can increase (Equation 3). Biologically, this is similar to cells spending biochemical energy as they rearrange their neighborhood.

The point particle approximation has been utilized earlier for modeling non-polar cell adhesion in early blastocyst (Krupinski et al., 2011), slug formation in amebae (Dallon and Othmer, 2004), and PCP organization in primitive streak formation (Newman, 2008). The main novelty of our approach is the dynamical coupling of cell positions and polarity orientations (Equation 6–8).

We have recently introduced effective representation of AB polarity, and showed that it is sufficient for capturing spherical trophectoderm in the early blastocyst (Nissen et al., 2017). Expanding on that work, we here explore how AB polarity supports diverse yet stable and complex morphologies.

Stable complex shapes emerge from randomly polarized cell aggregates

Adult organismal shapes are stable over long time, maintaining sizes and relative positions of lumens and folds, despite continual local damages and cell renewal. To test if cellular polarity could enable such stability in time and to random local perturbations, we first performed a series of tests with AB polarized cells (Figure 3 and Figure 3—video 1).

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When starting a bulk of cells with AB polarities pointing randomly, an initial rapid expansion (Figure 3A–C) stabilizes into a complex morphology of interconnected channels (Figure 3C–E). The shape remains unchanged for at least 10 times longer than the initial expansion (Figure 3C–E). The stability of the shape is illustrated by the time evolution of the average energy per cell (Figure 3F) that after an initial fast drop converges to a constant value. As expected, this value is higher than the energy of a hollow sphere (yellow dot in Figure 3F) – a configuration obtained if we start with radially, instead of random, polarized cells and preserve radial polarization at all times. The observed behavior is not sensitive to the shape of the potential (Figure 2—figure supplement 1A) but is sensitive to how the neighborhood is defined (Figure 2—figure supplement 1B–D). Rerunning the simulation in Figure 3 with different initial conditions results in a different stable shape (Figure 2—figure supplement 1E and Figure 3—figure supplement 1).

The macroscopic features of the shapes are robust to noise (Figure 2—figure supplement 1E–F and Figure 3—figure supplement 1). While the shapes emerging under high and low noise are not identical, the relative position and sizes of the majority of channels and lumens are preserved. The changes caused by noise stem from perturbations during initial expansion stage. If the same level of noise is applied after the system reached stable state, after time t = 10000, noise does not cause any major macroscopic changes (Figure 2—figure supplement 1G). The obtained shapes have self-sealing features, as an initial cut and unwrapping of a section of a surface refolds and seals back into the original morphology (Figure 3—figure supplement 2A–C). Furthermore, the shapes (Figure 3D–E) are also robust to overall growth (Figure 3—figure supplement 2D–F) retaining the same macroscopic features, just scaled to a larger size. Robustness to noise and cell proliferation further support the link between polarity and stability of morphologies, for example organ shapes, as they expand from infant to adult.

The final shapes are robust to noise but sensitive to initial and boundary conditions

The orientation of polarities in a subpopulation of cells may be set by the environment that the cells are embedded in, for example signaling molecules deposited into extracellular matrix can influence orientation of the AB polarity (Overeem et al., 2015) or signals from neighboring cells of a different type can orient PCP (Chu and Sokol, 2016). We will refer to these constraints as boundary conditions.

To investigate sensitivity to boundary conditions, we consider three cases where polarities are fixed at all times and point either radially out from the center of mass (Figure 4A), radially out from a central axis (Figure 4B), or pointing away from a central plane (Figure 4C). As anticipated, the difference in symmetries of boundary conditions results in a sphere, a cylinder, or two parallel planes (Figure 4—video 1). At the same time, in these symmetric cases, the differences in initial conditions but without imposed boundary conditions are not sufficient to generate different structures; they all converge to the nested ‘Russian doll’-like hollow spheres (Figure 4D). In development, this highlights the importance of the neighboring tissues for defining boundary conditions.

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Our results thus support the idea that polar adhesion enables stable and robust macroscopic shapes. The closest biological parallels would be the complex luminal morphologies emerging in reaggregation experiments on for example Hydra (Seybold et al., 2016) or in vitro culture of purjunkie brain cells (Muguruma et al., 2015). Together with our simulations, these experiments highlight how stable and complex morphologies can develop in non-proliferating populations from cell rearrangements alone.

Folding by pressure or rapid proliferation result in different fold-morphologies

Transitions from spheres to folded shapes are ubiquitous in development. Folds are an important part of in vivo organ development, and the composition of cell types in the folded organoids is closer to that in real organs (Greggio et al., 2013). To date, it is unclear what drives the transition from spheres to folded lumens. One possibility is that it is driven by the mechanical properties of the matrigel that effectively may place the growing organoid under pressure. Alternatively, data from 3D brain organoids suggests that the rapid cell proliferation leads to the emergence of surface folding (Li et al., 2017).

The simplicity of our tool allows to explore both of these scenarios. To model dividing cells, we pick a random cell from the entire population and introduce a new daughter cell with inherited polarity direction placed in a random location a half cell radius away from the mother cell. This event introduces dynamic perturbation by locally increasing cell density and requires some time to relax back to equilibrium. If proliferation is slow, and the time between two cell divisions anywhere in the system is longer than relaxation time of the whole system (the time it takes to reach equilibrium), the system approaches global equilibrium and will expand as a sphere. However, if proliferation is increased, the system will be pushed out of equilibrium and folds will emerge (Figure 5A, Materials and methods). In more quantitative terms, our forces are such that a single cell can move up to 0.2 cell diameter per time unit. For cells that divide every 1000 time units, the transition to non-equilibrium buckling happens when the system has grown to about 5000 cells (Figure 5—video 1).

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As cells divide faster, our simulations predict a transition from a smooth spherical shell to an increasingly folded structure with multiple pronounced folds, in line with the observation of brain organoids proliferating at different rates (Li et al., 2017). In comparison with the model for cortical convolutions by Tallinen et al., 2016 in which folding is a result of expanding cortical sheet adhered to the non-expanding white matter core, our mechanism does not require a bulk core. Instead folds emerge in a fast-expanding sheet when the growth is faster than the global relaxation to dynamical equilibrium.

While we find that the external pressure is not necessary for folding, pressure alone can also drive folding (Figure 5B, Materials and methods). However, this scenario contradicts the observation that pancreatic organoids can grow as spheres or folded morphologies in gels with the same stiffness but different media composition (Greggio et al., 2013).

In principle, both scenarios may contribute to folding, but visually the fold morphologies are different. To differentiate between the two, we have quantified the final folded structures in terms of their local minima (Figure 5, Materials and methods, see also Figure 5—video 1). Our simulations predict that in the pressure-driven case, the number of local minima will reach an upper limit as organoids increase in size (Figure 5B). In the case of out-of-equilibrium proliferation, new folds can continue forming as organoids grow (Figure 5A). Increased proliferation causes more and shallower folds. These folds are different than obtained with pressure which causes fewer but deeper minima. Quantitatively, both the depth and the horizontal extension of the folds are about double as large with pressure than with growth-induced folding (Figure 5—figure supplement 1).

PCP enables convergent extension and robust tubulogenesis

Despite the numerous evidences supporting the role of PCP in tubulogenesis, it remains unresolved whether oriented cell division or the extent of CE controls tube length and width (Karner et al., 2009; Carroll and Yu, 2012). It is also debated if it is important for the tubes to maintain regular shape, or if it is only important for tube initiation and growth (Kunimoto et al., 2017).

The simplicity of our approach allows us to address these questions by introducing cell–cell interactions through PCP. This term favors front-rear cell alignment in the interaction potential with only two additional parameters: the strength of the orientational constraint of AB polarity with respect to PCP, λ₂, and the strength of PCP, λ₃ (see Equations 4–8). For simplicity, we focus on the stability (ability to maintain regular diameter over time) and tube morphogenesis in systems without cell division.

Inducing PCP in a spherical lumen leads to two significant events. First, independent of initial orientation, after some transient time PCPs becomes globally ordered and point in direction parallel to an emerging equator that self-organizes around the sphere (inset in Figure 6). This arrangement has the lowest energy. Second, cells start intercalating along the axis perpendicular to PCP orientation, gradually elongating the lumen (Figure 6—video 1). During intercalations, cells exchange their neighbors through T1-like transitions as reported experimentally, Figure 6—figure supplement 3 (Nishimura et al., 2012; Sanchez-Corrales et al., 2018). The intercalations along the axis continue until the force balance between AB polarity and PCP is restored at a new equilibrium. Thus, our model predicts that the strength of PCP (λ₃) relative to AB polarity (λ₁) determines the width and the length of the tube (Figure 6). We obtain similar results if we constrain PCP to always remain in the apical plane and thus does not allow PCP to reorient AB polarity (Figure 6—figure supplement 1, Materials and methods). Note, that this result is very different in nature from the tube presented in Figure 4B as both AB polarity and PCP can now reorient in each cell at any time.

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These results support the observations that stable tubes can emerge without cell proliferation. In addition, when first the tube is formed, loss of PCP does not lead to cyst formation as recently shown by Kunimoto et al. (2017). However, localized cysts could result if the lumen is initialized with varying strength of PCP along the axis (Figure 6—figure supplement 2).

Two polarities are sufficient to explain major features of sea urchin gastrulation

Currently invagination in neurulation and gastrulation is understood and quantitatively modeled as a process driven by changes in cell shapes or the mechanical properties of cells with AB polarity (Rauzi et al., 2013; Tamulonis et al., 2011; Misra et al., 2016; Hočevar Brezavšček et al., 2012). This process is often assumed to be driven by apical constriction and decoupled from the eventual tube formation and elongation. However, emerging data suggests that PCP drives both invagination and tube elongation (Nishimura et al., 2012; Croce et al., 2006; Long et al., 2015) likely because apical constriction is controlled by PCP (Ossipova et al., 2014; Nishimura et al., 2012; Croce et al., 2006) and cell intercalations, similar to those in CE, contribute to invagination (Nishimura et al., 2012; Rembold et al., 2006; Sanchez-Corrales et al., 2018).

To probe the limits of our approach, we investigated if AB polarity, PCP, and boundary conditions reminiscent of posterior organizer (Loh et al., 2016) are sufficient to recapitulate the main stages of sea urchin gastrulation: invagination, tube formation, elongation (by CE), and finally merging of the gastrula tube with the pole opposite to invagination site.

The current understanding is that sea urchin gastrulation consists of primary invagination, driven by swelling of the inner layer of extracellular matrix beneath invaginating cells (Lane et al., 1993), and formation of a ring of bottle cells due to apical constriction (Kimberly and Hardin, 1998), and secondary invagination where tube elongates due to CE (Lyons et al., 2012). PCP is necessary for both invagination, possibly through its effect on apical constriction in bottle cells (Nishimura et al., 2012), and tube extension (Croce et al., 2006).

Motivated by these observations, we set boundary condition such that PCP of the invaginating cells are oriented around the anterior-posterior (top-bottom) axis, and are always in the apical plane. This constrain on PCP orientation allows for CE. While this particular configuration is not documented, it is consistent with observed effects of WNT orienting PCP within the apical plane (Humphries and Mlodzik, 2018). Second, we simulate the combined effect of bottle cells (Figure 2—figure supplement 2) and bending by swelling of the extracellular matrix by applying an external force, F, on AB polarity (see Materials and methods). This force gradually reorients AB polarity away from the anterior-posterior axis, thus leading to bending of the epithelial sheet. The effect is maximal for cells closest to the anterior-posterior axis. This external force is a phenomenological description aiming at capturing the observed effects of how change in AB polarity results in tissue bending and does not aim at capturing mechanisms driving the reorientation of AB polarity.

As a result, cells start to rearrange, the bottom flattens (Figure 7B) and bends inward (Figure 7C). Subsequently, the CE-driven by PCP causes the invaginated cells to rearrange, tube elongates, and merges with the top of the sphere (Figure 7F and Figure 7—video 1). In line with experimental observations, the tube elongates due to cells moving in the tube (Martik and McClay, 2017).

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The contribution of extracellular matrix swelling is specific to sea urchin invagination. As bottle cells alone are sufficient to drive invagination in other systems, we have tested a cell-intrinsic scenario where AB polarities of bottle cell neighbors prefer to be tilted towards each other (by e.g. modifying the potential in Equation 6 to capture AB polarities as in Figure 2—figure supplement 2, unpublished results). This also resulted in successful invagination.

Several observations suggest that radial intercalation movements towards the center of invagination may drive tissue bending (Sanchez-Corrales et al., 2018; Panousopoulou and Green, 2016; Rembold et al., 2006; Chung et al., 2017). When tested in our model, PCP alone (omitting external or cell-intrinsic reorientation of AB polarity) results in exvagination and tube elongating outside of the sphere (this is more energetically favorable than extending inwards). A similar, exogastrulating, phenotype was observed in a PCP mutant (Long et al., 2015). While the similarity may be accidental, it is possible that this mutation abrogates PCP-driven apical constriction (Ossipova et al., 2014; Nishimura et al., 2012) and since cellular apices face outside, abrogating apical constriction eliminates the bias in direction of tube formation. We have also tested the consequences of apical constriction (reorienting AB polarity) in the model without PCP and while this was sufficient for invagination, the cavity remained spherically symmetric and failed to form a tube.

Despite the stunning diversity and complexity of morphologies, the same concepts seem to emerge across organismal development. One of them is the link between local, cellular, and global, organs/whole organism, symmetry breaking. We know, from experimental and, to a lesser degree, theoretical work that cellular polarity is essential for forming axis, complex folded sheets, and interconnected tubes (Figure 1—figure supplement 1). What we do not know is why are these shapes so stable, and where do the differences between species and organs come from. To understand the differences, we typically compare genes or gene regulatory networks, thus limiting our understanding to analog processes within a few related species.

Throughout the paper, we use the Euler method to integrate the ordinary differential equations stated in the Model section with dt set to 0.1 or 0.2. Lower dt values will give qualitatively similar results but with increased simulation time. Higher dt values, will result in a collapse of the presented morphologies. The time unit is arbitrary, and the same throughout the paper.

The noise parameter η = 10⁻⁴ where nothing else is stated. Lower noise values will give more smooth simulations, while η on the order of 10⁰ will result in collapsing shapes (see also Figure 2—figure supplement 1E–F).

All data generated or analyzed during this study are included in the manuscript and supporting files. MATLAB code to reproduce or generate new data is added as a supplementary zip file together with a MATLAB script to visualize the data.

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Author details

1. Silas Boye Nissen

   1. Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
   2. StemPhys, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing—original draft, Project administration, Writing—review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-9473-4755

2. Steven Rønhild

   Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

   Contribution

   Software, Writing—review and editing

   Competing interests

   No competing interests declared

3. Ala Trusina

   1. Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
   2. StemPhys, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing—original draft, Project administration, Writing—review and editing

   For correspondence

   trusina@nbi.ku.dk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-1945-454X

4. Kim Sneppen

   Center for Models of Life, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing—original draft, Project administration, Writing—review and editing

   For correspondence

   sneppen@nbi.dk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-9820-3567

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