Multi-scale model

To investigate whether reproducibility is a by-product of complex morphogenesis, we constructed a deliberately simplified model of morphogenesis. To capture the noisy dynamics of real morphogenesis, we employed the Cellular Potts Model (CPM), which uses a Metropolis algorithm to simulate stochastic cell motion and cell shape dynamics [34,35]. The CPM models the development of a group of cells on a two-dimensional square grid (250 × 250 pixels). The population of cells on the grid represents a developing tissue, which we term a “morphology”. Each cell consists of a collection of neighbouring pixels on the grid (Fig 1A). Pixels not occupied by cells represent the medium (Fig 1A), which is akin to an extracellular matrix or fluid [36]. Cell motion occurs through stochastic extensions and retractions of cell boundaries driven by pixel copy processes at these boundaries (Fig 1A; see Methods). Pixels on the grid are chosen in a random order with replacement for copy attempts. The unit of time is the number of pixel copy attempts equal to the total number of pixels on the grid, hereafter referred to as a developmental time step (DTS). The probability that a pixel copy occurs is determined by the minimisation of free energy arising from cell-cell adhesion, cell-medium adhesion, cell shape and cell size, as described later.

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Fig 1. Multi-scale model of morphogenesis evolution.

(A) Three neighbouring cells on a CPM grid. A cell consists of one or more pixels. Each cell is coloured by its “cell state” defined by the concentrations of all its proteins converted to Boolean values (the medium is represented by white pixels). Pixels with alternating stripes indicate pixel copies at cell boundaries. Each cell contains a genome that encodes transcription factors (TFs; circles, squares, triangles), adhesion proteins (sticks with a lock or key) and membrane tension proteins (not shown). Arrows indicate regulation of gene expression by TFs (arrow head for activation, blunt head for inhibition). Double harpoon arrows indicate diffusion of morphogens (membrane-permeable TFs). (B) Adhesion proteins facilitate the binding of cells to each other via a lock and key mechanism, or to the surrounding medium (not shown). (C, D) Illustration of shape complexity measurements for two morphologies. (C) depicts the measurement of how much the morphology deviates from a perfect circle (black), with the centre of the circle being the morphology’s centre of mass. Black arrows at equidistant intervals around the circle mark directions where the morphology’s shape deviates from the circle, with longer arrows contributing to a higher score. (D) depicts the measurement of inward folding using double sided arrows, with more arrows contributing to a higher score. The cells are all coloured grey to emphasise that only the shape, not cell states, determine the complexity score. (E) A population consists of 60 morphologies (three depicted). Morphologies undergo a developmental phase on separate CPM grids for 12,000 DTS, and then a reproduction phase, where morphologies with complex shapes are selected. Reproduction can occur without mutation (black arrows) or with mutation (orange arrow), with mutation determined probabilistically. Mutations change the topology of the GRN (dashed orange arrow), with the example showing a change from inhibition of gene y by gene x to activation of gene y by gene x. (F, G) Five-generation moving average of population fitness (shape complexity) over two separate evolutionary simulations. The morphologies above the plots are a trace of ancestors, with each morphology depicted at the end of its development (12,000 DTS) from six different generations over the course of evolution. The generation number of each morphology is shown to its left. “Evolved morphologies” of each simulation are shown on the far right.

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To simulate the development of a morphology, the CPM is run for 12,000 DTS, which is approximately the minimum time it takes for a fast-growing morphology to reach the edge of the grid. Each morphology starts as a circular shape consisting of 64 cells of approximately 75 pixels each (Fig 1E). After initialisation, cells grow if they are mechanically stretched and shrink if they are mechanically squeezed (see Materials and methods). Cell stretching and squeezing are induced by adhesion to neighbouring cells and the extracellular medium, although they can also occur stochastically. When a cell reaches a size of 100 pixels, it divides into two daughter cells of approximately equal size. The plane of cell division aligns with the minor axis of the mother cell, thereby minimising the elongation of the resulting daughter cells. Linking cell division to stretching is a simplified mechanism of growth reflecting the mechano-sensitive cell division observed in several developmental contexts [37–39].

To model basic cell mechanics, we equip cells with genes encoding three kinds of proteins that are nearly universally present in animal cells during development: adhesion proteins, membrane tension proteins and transcription factors. Adhesion proteins modulate the adhesiveness of cells to neighbouring cells and the extracellular medium (Fig 1A and 1B). These proteins are modelled as either locks or keys. The adhesion energy between two neighbouring cells is proportional to the number of complementary lock-key pairs they express. Similarly, the adhesion energy of a cell to the medium is proportional to the number of medium-adhesion proteins expressed by that cell (Materials and methods). Differential adhesion energies drive the rearrangement of cells to maximise contact with other cells or the medium to which they strongly adhere (small energy) and minimise contact with those to which they weakly adhere (large energy), thereby inducing directed cell motion. Adhesion proteins also influence cell fluidity, i.e., the rate of cell rearrangements. When the free energy arising from cell-cell adhesion is low, cells behave more fluid-like (e.g., mesenchymal cells). When this energy is high, cells behave more solid-like (e.g., epithelial cells) [40]. Differential adhesion is incorporated into our model, as it is known to drive several processes essential for real morphogenesis, such as cell sorting, formation of tissue boundaries and cell migration [2,41].

Membrane tension proteins make the cell shape less deformable by energetically constraining the length of its dynamically determined longest axis, irrespective of the cell’s orientation. By making the cell shape less deformable, membrane tension proteins suppress cell motion. The constraint on the length of the longest axis increases with the number of expressed tension proteins (Materials and methods). These proteins model an increase in cell membrane tension, which occurs in real cells by the accumulation of actin filament stress [42]. To maintain the simplicity and broad applicability of the model, we do not implement polarised cell tension or contractility, properties required for processes like invagination.

To model gene regulation, we couple each CPM cell to gene expression dynamics, as previously done [29,30,43–46]. Each cell is equipped with transcription factors (TFs), that form a gene regulatory network (GRN) with its adhesion and membrane tension proteins. A GRN is a graph consisting of nodes representing proteins and edges representing TF-mediated activation or inhibition of gene expression (Fig 1A). Concentrations of proteins within a cell are determined by numerically integrating a set of ordinary differential equations given by the GRN (Materials and methods). To model cell-cell signalling, a minority fraction of TFs (hereafter called morphogens) diffuse between cells and into the medium (Fig 1A; Materials and methods). These morphogens model those encountered in real development, such as Wnts and BMPs [19,47,48]. Morphogens allow different cells to express different proteins even though all cells within a morphology have identical genes and GRNs.

Besides morphogen gradients, another way to induce differential protein expression in real development is the asymmetric distribution of proteins between cells [49]. To model this, we distribute the concentrations of two non-morphogen TFs asymmetrically between cells at the initial 64-cell stage. We restrict one of these TFs to 32 cells left of the vertical centre line and the other to 32 cells below the morphology’s horizontal centre line. Thus, the cells in each of the four sectors in the spheroid each have unique concentrations of these two TFs (illustrated in Fig 1E), although whether this asymmetry persists over development depends on the gene regulatory network.

To visualise the model, cells are distinguished based on the tension or adhesion proteins they express by assigning each cell a state defined as a vector of Boolean values, where each Boolean value indicates whether a protein is expressed or not (Materials and methods). All cells with the same state are shown with the same colour on the CPM grid (Fig 1). Cell states are only used to visualise and analyse model outcomes and do not play any role in model dynamics.

To simulate the evolution of morphogenesis, we generated an initial population of 60 morphologies (Fig 1E), with each assigned a different randomly generated GRN. Each morphology develops on a separate CPM grid. We applied a genetic algorithm to select for shape complexity by measuring inward folding and deviation of the morphological shape from a circle at the end of the 12,000 DTS (Fig 1C and 1D; see Materials and methods). The 15 morphologies with the highest shape complexity reproduce four times to populate the next generation, and their GRNs undergo a single mutation with a probability of 50%. A mutation changes the regulatory effect of one TF on one gene, such as causing a TF to switch from inhibiting the expression of a gene to activating its expression (see Fig 1E). The GRNs used in our main set of simulations are comprised of nine transcription factors (including three morphogens), 15 adhesion proteins and two membrane tension proteins. Gene duplication and deletion do not occur. To broadly explore the types of morphogenesis our model evolves, we also run simulations where morphologies start as a rectangular initial shape with proteins asymmetrically distributed along the longest axis of the rectangle (S1 Fig), simulations where morphogen diffusivity mutates alongside GRN mutations (S1 Fig), simulations with different selection pressures (S2 Fig; see Materials and methods) and simulations with other numbers of genes (S3 Fig).

Reproducibility is not an intrinsic property of complex morphogenesis

To investigate whether morphogenesis is intrinsically reproducible, we conducted 126 independent evolutionary simulations of our model starting from a circular initial condition. We ran each simulation for at least 2.5 × 10³ generations, which is usually sufficient to reach a plateau in fitness (S4 Fig). We then identified the fittest morphology (i.e., most complex shape) from the final generation (hereafter referred to as an “evolved morphology”) from each simulation. To ensure that we were analysing the reproducibility of complex shapes, we removed 36 evolved morphologies that did not reach an arbitrary threshold of complexity (our results are similar when a different threshold is used; S5 Fig). The 90 morphologies above this threshold each display a different shape (S4 Fig), with the evolution of two representative morphologies illustrated in Fig 1F and 1G, termed morphology-1 and morphology-2, respectively.

We examined whether the 90 evolved morphologies with complex shapes display reproducible morphogenesis by repeatedly simulating their development with different pixel copy orders (60 replicates per morphology, hereafter referred to as developmental replicates). To quantify reproducibility, we measured a “reproducibility score”, which indicates how geometrically similar the morphologies of replicates are to each other (see Materials and methods). The reproducibility score depends on the geometry, size and time taken to generate the morphology but is invariant to reflection, rotation and translation of the morphology. The distribution of reproducibility scores across all 90 evolved morphologies is bimodal (bimodality coefficient = 0.69, Fig 2A), with 19 morphologies in the upper mode, which we term highly reproducible (including morphology-2; Fig 2D shows four other examples), 65 morphologies in the lower mode, which we term poorly reproducible (including morphology-1; Fig 2E shows four other examples), and six that are between the two modes, which we term intermediately reproducible (see S8 Fig for information about intermediately reproducible morphologies). The morphologies in the lower mode also tend to have higher variation in shape complexity than those in the higher mode, supporting our assertion that they are less reproducible (S9 Fig). The bimodal distribution implies that evolved morphologies consist of a mixture of two populations with distinct properties.

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Fig 2. Both highly and poorly reproducible morphologies evolve in response to selection for shape complexity.

(A) Reproducibility scores against shape complexity scores for the 90 morphologies that reached a threshold of shape complexity. Circles indicate morphologies with a single strongly connected component (SCC) (mean reproducibility = 52.0%, n = 65); filled triangles indicate morphologies with multiple SCCs with unidirectional transitions between them (mean reproducibility = 72.1%, n = 24). The blue diamond is a morphology with multiple SCCs without unidirectional transitions (S6 Fig). Numbered arrows refer to the morphologies in panels B, C, D, and E. (B, C) Simplified cell state spaces for (B) morphology-1 and (C) morphology-2. Node colours correspond to cell state colours depicted in the morphologies to the left each of cell state space. Arrows are cell-state transitions. Cell states are partitioned into strongly connected components (SCCs, coloured boxes). Node sizes depict cell state frequency over all of development; node colours correspond to cell states from morphology-1 and morphology-2, respectively (shown to the left of each state space). See S7 Fig for the state spaces without pruning of nodes and edges. (D) Four morphologies that are highly reproducible. (E) Four morphologies that are poorly reproducible. (F) Four highly reproducible morphologies from simulations where morphologies evolved with morphogens mutating (for more information see S1 Fig).

https://doi.org/10.1371/journal.pcbi.1014361.g002

Highly and poorly reproducible morphologies have distinct cell-state transition dynamics

To understand how differences in reproducibility between morphologies arise, we investigated how these differences covary with shape complexity or the number and spatial distribution of cell states. While we found that differences in reproducibility could not be explained by shape complexity (see S1 Appendix, S10 Fig) or the number of cell states (S11 Fig), we noticed a stark difference in the spatial distribution of cell states between highly and poorly reproducible morphologies, as depicted by the distribution of cell colours in the morphologies shown in Fig 2. Specifically, in highly reproducible morphologies, each cell state is localised to a specific morphological domain. By contrast, in poorly reproducible morphologies, cell states are unlocalised, reappearing all over each morphology.

We asked whether differences in spatial distributions of cell states in highly and poorly reproducible morphologies are due to differences in cell-state transition dynamics, i.e., how and which cells transition between states. To answer this question, we recorded all cell-state transitions of all cells in each evolved morphology to generate a “cell state space” for each morphology, which is defined as a graph consisting of nodes representing cell states and edges representing all possible transitions between cell states (see Materials and methods). Since each cell state space contains numerous nodes and edges, we simplified the state spaces in two steps. First, we pruned infrequently observed cell states and cell-state transitions. Second, we split the cell state space into strongly connected components (SCCs), where an SCC is a set of cell states for which there exists a transition pathway (direct or indirect) from any cell state to any other cell state within that set. We found that almost all (64 out of 65) poorly reproducible morphologies had only a single SCC, as illustrated for morphology-1 in Fig 2C. In contrast, all 19 highly reproducible morphologies and most (five out of six) intermediately reproducible morphologies had cell state spaces containing multiple SCCs, as illustrated for morphology-2 in Fig 2B. Moreover, most (24 out of the 25) morphologies with multiple SCCs had at least one SCC that unidirectionally transitioned to another SCC (Fig 2A; see S6 Fig for details about morphologies without unidirectional SCC transitions). The difference in reproducibility between morphologies with a single SCC and those with multiple SCCs is statistically significant (p < 10⁻¹², two-tailed t-test), suggesting that the presence of multiple SCCs is involved in morphogenetic reproducibility.

To investigate why there is a relationship between the number of SCCs and reproducibility, we examined the relationship between cell states and cell motions, given that cell motion drives morphogenesis and differences in cell motions across developmental replicates drive reproducibility. To examine cell motions, we visualised the velocity of every cell at multiple time points during the development of evolved morphologies across multiple developmental replicates (here using morphology-1 and morphology-2 as examples). The velocity plots reveal two ways in which morphology-1 (one SCC) undergoes morphogenesis. The first way is “bifurcations”, in which a cluster of moving cells splits into two clusters (Fig 3A and S2 Movie). We hypothesised that bifurcations are induced by the transition of moving cells to stationary cells at the cluster tip. While observing cell trajectories suggested that these transitions cause bifurcations, the complex interactions between morphogen signalling, adhesion, and cell division make it difficult to isolate causal mechanisms through observation alone. Thus, we tested if moving-to-stationary transitions cause bifurcations by artificially converting the states of three moving cells to stationary states at the cluster tip. We performed this conversion by extracting the protein and morphogen profiles of an arbitrary stationary cell with the same morphology, and overwriting the states of three adjacent moving cells with these extracted profiles. The result shows that this conversion induced a bifurcation (S12 Fig), supporting the hypothesis. In the two developmental replicates shown in Fig 3B, a bifurcation occurs in replicate-1 but not in replicate-2, indicating bifurcations are prone to stochasticity.

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Fig 3. Highly reproducible morphologies have moving and dividing cells that undergo unidirectional transitions to non-moving and non-dividing cells.

(A, B) Two developmental replicates of morphology-1 (A) and morphology-2 (B) are depicted after 2,600, 4,000, 8,000 and 12,000 DTS, showing a difference in their reproducibility. Dashed arrows in (B) indicate the presence (replicate-1) or absence (replicate-2) of a bifurcation in collective cell motion; asterisks indicate protrusions. Vector plots show the displacement of the centre of mass of each cell during 2,000 DTS at each respective time point, with colours indicating magnitude (the lighter, the larger). (C) Average cell momentum magnitude for each SCC from the 25 morphologies with multiple SCCs. Momentum is the distance travelled by a cell per DTS multiplied by its size in pixels. Black arrows indicate unidirectional transitions between SCCs. Grey lines connect SCCs from the same morphology that do not have unidirectional SCC transitions between each other. Filled orange triangles are SCCs from morphologies that have unidirectional SCC transitions. All transitory SCCs are excluded (see Materials and methods for information about transitory SCCs). Blue diamonds are SCCs from the highly reproducible morphology that does not have unidirectional SCC transitions. The numbered morphologies correspond to those from Fig 2D. (D) Stacked bar charts showing the proportion of developmental time spent in each cell state and the proportion of cell divisions undergone by each state across all cells during developmental replicate-1 of morphology-1 and development replicate-1 of morphology-2. Diagonal lattices are pruned states. (E) The rate at which cells divide per developmental time when their state belongs to an upstream SCC (left) or a downstream SCC (right). Each data point represents an SCC from (C). Black lines connect upstream SCCs to their counterpart downstream SCCs. Boxes show medians and interquartile ranges (IQR); the downstream SCC box is tiny because most division rates are either very low or 0. Numbers on top of the box plots are median cell division rates.

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The second way in morphology-1 (one SCC) undergoes morphogenesis is “protrusions”, where a new cluster of moving cells forms (Fig 3A). We hypothesised that protrusions are induced by the onset of collective motion from previously stationary cells. To test this, we converted the states of three stationary cells to moving cell states using the same conversion protocol described previously. The result shows that this conversion induced a protrusion (S12 Fig), supporting the hypothesis. In the two developmental replicates shown in Fig 3B, two cell groups protrude in replicate-1 around 8,000 DTS, whereas three cell groups protrude in replicate-2 around 12,000 DTS in different locations from replicate-1. This result indicates that protrusions are also prone to stochasticity.

Together, these results indicate that morphogenesis in morphology-1 is driven by noise-prone transitions between moving and stationary states, which reduces reproducibility. Because all states belong to the same SCC, these transitions are possible and can be noise-prone. Evolution likely selects for more noise-prone transitions because it promotes morphogenesis, thereby increasing shape complexity and overall fitness. These results are not limited to morphology-1: noisy transitions between moving and stationary states appear to drive morphogenesis in most of the poorly reproducible morphologies we evolved, even for those with vastly different mechanics, such as “finger” formation via cell death (S6 Fig, S5 Movie). Consequently, these noisy transitions are a fundamental cause of poor reproducibility in our model.

We hypothesised that moving and stationary states are partitioned into separate SCCs in highly reproducible morphologies, thereby preventing noisy transitions between them. We first tested this hypothesis on highly reproducible morphology-2 (two SCCs). The velocity plots for morphology-2 reveal that cells in states belonging to one of its two SCCs collectively move downwards and radiate slightly outwards, like a travelling wave (Fig 3A; S1 Movie). Cells in states belonging to the other SCC show little motion. These position-dependent collective cell motions generate a “cap” of moving cells and a “stalk” of stationary cells forming in the wake of the moving cap (Fig 3A). These observations suggest that moving and stationary states are not only dynamically partitioned into distinct SCCs as we hypothesised, but also spatially partitioned into distinct domains in morphology-1.

To confirm that different SCCs have different cell-motion properties across all highly reproducible morphologies, we measured the momentum of cells in each SCC (Materials and methods). The result shows a significant disparity (mean 6.7-fold difference) in the average magnitude of cell momentum between SCCs across all highly reproducible morphologies (Fig 3C), indicating that moving and stationary states are indeed partitioned into separate SCCs. Moreover, when there are unidirectional transitions between SCCs (24 out of 25), the transitions are always from high momentum upstream SCCs to low momentum downstream SCCs (black arrows in Fig 3C). These results indicate that noisy transitions from stationary to moving cells (for instance, those that lead to protrusions in morphology-1) do not occur in highly reproducible morphologies because stationary cell states cannot transition to moving cell states, which is likely a contributing factor towards their higher reproducibility.

We wondered, mechanistically, what causes the difference in mobility between cell states in different SCCs. To answer this, we compared the cell-cell adhesion energy among cells from upstream SCCs to the cell-cell adhesion energy among cells from downstream SCCs, since cell-cell adhesion energies are the key component of cell mobility in our model. We found that the cell-cell adhesion energy was much lower among those from upstream SCCs (mean 1.39) than those from downstream SCCs (mean 3.51), with no overlap between groups (S13 Fig). This difference in cell-cell adhesion energies means that cells in upstream SCCs behave more fluid-like, whereas cells in downstream SCCs behave more solid-like.

Since moving cells in upstream SCCs do not disappear despite transitioning to stationary cells in downstream SCCs, we thought moving cells might be dividing rapidly. To test this, we compared how often cells in moving versus stationary SCCs divide across all morphologies with multiple SCCs (Fig 3D and 3E). We found that cells in moving SCCs divide a median of 49.6 times more frequently than those in stationary SCCs. Together, these asymmetries in cell motion, adhesion and division indicate that highly reproducible morphologies establish a “morphogenetic division of labour”, whereby domains of moving, dividing cells “shape” morphologies whilst stationary cells “maintain” the morphologies shaped by moving cells.

Since our model has specific initial conditions, genetic parameters, and selection pressures, we sought to confirm that the complex, reproducible morphogenesis driven by morphogenetic division of labour is robust to these variables, rather than evolving as an artefact of our specific simulation setup. To this end, we ran simulations of our model with different initial shapes (S1 Fig), simulations where morphogen diffusivity can mutate (S1 Fig), simulations with different selection pressures (S2 Fig) and simulations with different numbers of genes (S3 Fig). We found that morphogenetic divisions of labour that confer complex yet reproducible morphogenesis evolved in all of these kinds of simulations. Moreover, morphogenetic divisions of labour evolved in 80–90% of simulations when we modified the selection criterion to indirectly select morphogenesis that undergoes directional motion (which indirectly selects for reproducibility; Materials and methods and S2 Fig), indicating that morphogenetic divisions of labour are easy to evolve from different initial GRNs. Given that these morphogenetic divisions of labour resemble progenitor-cell systems found in animal development [2], where progenitor cell states irreversibly differentiate into one or a few differentiated cell types, we hereafter define upstream SCCs as progenitor-cell types and downstream SCCs as differentiated-cell types. We refer to unidirectional transitions from progenitor to differentiated as cell differentiation. We use these terms to denote the hierarchical and functional roles of cell states within our model, acknowledging that they represent a simplified abstraction of more complex regulatory and epigenetic states found in real animal cells.

Regulated cell-motion transitions at domain boundaries enables complex yet reproducible morphogenesis

We wondered how morphogenetic divisions of labour increase morphogenetic reproducibility despite frequent cell differentiation from moving states (progenitor cells) to stationary states (differentiated cells), which cause poor reproducibility in evolved morphologies with only a single SCC, because these transitions are prone to noise. We noticed that this differentiation occurs at a steady rate and only at the boundaries between domains of progenitor and differentiated cell types, suggesting that differentiation does not reduce reproducibility because it is tightly regulated. For example, in morphology-6, which has two progenitor cell types denoted type-1 and type-2 (Fig 4A), progenitor cells initially differentiate exclusively at the boundary between the type-1 and type-2 progenitor cells and, subsequently, exclusively at the boundary between progenitor cells and differentiated cells (S3 Movie). Both progenitor-cell domains are present very early in morphology-6 development (i.e., just after the initial 64-cell stage). Whether a cell starts at type-1 or type-2 depends on the concentration of the TF distributed asymmetrically across the horizontal centre line. Since type-1 and type-2 stem cells start in distinct domains, the spatial layout of progenitor and differentiated cells ends up mirroring the topology of the cell state space, with differentiated cells consistently forming between the progenitor-cell domains (Fig 4A; see S7 Fig for other examples).

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Fig 4. Mechanisms underlying progenitor-cell differentiation and motion.

(A) Development and cell state space of morphology-6, showing two progenitor-cell types, one differentiated-cell type, and a transitory SCC (see Materials and methods for information about transitory SCCs). The morphology is shown after 2,000, 6,000 and 12,000 DTS. (B) Contours showing the concentrations of the three morphogens (x, y and z) overlaid on morphology-6 after 9,000 DTS. Each contour joins points of equal concentration of the same morphogen. (C) Schematic depicting type-1, type-2 and differentiated cell domains from (B), with progenitor-cell motion (grey arrows) and differentiation (black arrows) indicated. Vertical dashed lines indicate cell-type boundaries. Below, morphogen concentrations along the cross-section line are plotted, along with the sums of cell protein concentrations for cells along the cross-section (each cross is one cell). (D) Polar plots of momentum magnitude by angle of motion for each cell type summed over all cells over the 12,000 DTS of morphology-6 development. (E) Development of type-1, type-2 and differentiated cells in isolation. Polar plots show distributions of cell momentum as in (D). (F) Development and cell state space of morphology-11, showing a progenitor-cell type, a differentiated cell-type and a transitory SCC. The morphology is shown after 2,000, 6,000 and 12,000 DTS and its state space. (G) Contours showing the concentrations of morphogens y (green) and z (orange, morphogen x is hidden for visibility) overlaid on morphology-11 after 8,000 DTS. (H) A schematic depicting progenitor and differentiated cell domains for morphology-11, with progenitor-cell motion (grey arrows) and differentiation (black arrows) indicated. The combined presence of morphogens y and z induces differentiation of progenitor cells. (I) Polar plots of momentum magnitude by angle of motion for progenitor and differentiated cell types summed over all cells over the 12,000 DTS of morphology-11 development shown in (F). (J) Development of progenitor and differentiated cells from morphology-11 in isolation. Polar plots show distributions of cell momentum as in (I).

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Boundary-localised differentiation suggests that progenitor-cell differentiation is spatially regulated. To determine how this regulation is achieved, we focused on morphology-6 as it has two separate examples of progenitor-cell differentiation. We hypothesised that boundary-localised differentiation is caused by morphogen-mediated interactions between progenitor and differentiated cells, as morphogens provide a means to spatially regulate gene expression. To test this, we made a contour plot of morphogen concentrations for morphology-6 (Fig 4B). The plot shows that the concentrations of different morphogens abruptly change at the boundaries between progenitor and differentiated cells (dashed lines in Fig 4B). To determine whether gene expression responds to these changes in morphogen concentrations, we plotted the sum of protein concentrations (excluding morphogens) for cells along a cross-section of morphology-6 (Fig 4C). The plot shows that this sum changes abruptly wherever morphogen concentrations change. These results suggest that a specific morphogen profile induces differentiation, with differentiated cells producing this profile and thus localising differentiation to the boundary between progenitor and differentiated cells. To directly test this, we isolated each progenitor-cell type from the other two cell types, thereby removing the effect of differentiated cells on morphogen profiles. We isolated cell types by creating morphologies in which all cells in the initial spheroid were set to the state that is most frequently observed for each of the progenitor-cell types (e.g., the green cell state shown in Fig 4A for type-1 progenitor cells). We developed these morphologies for 12,000 DTS, and found that neither type-1 nor type-2 progenitor-cell types differentiate (Fig 4E). To test the robustness of our results, we repeated the above analyses on morphology-11, which undergoes a different kind of morphogenesis from morphology-6—progenitor cells spread around the surface of the morphology before differentiating like epiboly [50] (Fig 4F, S4 Movie). We found that morphology-6 progenitor cells differentiate in response to the combined exposure to two types of morphogens (Fig 4G and 4H), and that progenitor cells do not differentiate when isolated from differentiated cells (Fig 4J). These results support the hypothesis that differentiated cells induce progenitor-cell differentiation, thus localising differentiation to the boundaries between cell types.

We next asked whether the above findings—that differentiated cells induce progenitor-cell differentiation—are generalisable to all morphologies with progenitor-cell differentiation. To answer this, we isolated each of the 30 progenitor-cell types from the 24 morphologies with progenitor-cell differentiation from all other cell types and tested whether the isolated progenitor cells differentiate through the same method as described in the previous paragraph. We found that the great majority (26 out of 30) do not differentiate when isolated from other cell types (S14 Fig shows why the four exceptions do not counter our hypothesis). This result, along with the observation that every progenitor-cell type always differentiates at the boundary shared with differentiated cells (Figs 3A and 4), indicates that differentiated cells induce progenitor-cell differentiation.

The spatial consistency of progenitor-cell differentiation is crucial but not sufficient for complex yet reproducible morphogenesis. The other critical factor is the motion of progenitor cells in a consistent direction, as this motion creates the complex shapes that are conserved across developmental replicates. Given that progenitor-cell differentiation is induced by differentiated cells, we hypothesised that the directionality of progenitor-cell motion is also induced by differentiated cells. To test this, we determined how progenitor cells’ motion depends on the presence or absence of differentiated cells by measuring the magnitude of progenitor cells’ momentum as a function of the angle of momentum, either in normal development (differentiated cells present) or when each progenitor-cell type is isolated (differentiated cells absent). The result shows that cell momentum is strongly directional (i.e., anisotropic) in normal development, whereas it is radially symmetrically distributed (i.e., isotropic) when cells are isolated (Fig 4 shows results for morphology-6 and morphology-11; see S14 Fig for other morphologies). We quantified this difference in anisotropy by calculating the ratio of the variance in momentum magnitude across angle to the mean momentum magnitude across angle (Materials and methods). We found that anisotropy of progenitor-cell motion decreased an average of 65-fold across the 30 progenitor-cell types when progenitor cells were isolated (Fig 5). To understand mechanically how this anisotropic motion occurs, we analysed the expression of adhesion proteins across morphologies with progenitor-cell differentiation, given that our model is primarily adhesion-based. We found that across all morphologies with progenitor-cell differentiation, progenitor and differentiated cells consistently differed in either their adhesion to one another or their adhesion to the medium (S13 Fig). However, the exact mechanism relating differential adhesion to anisotropic motion depends on the morphology (e.g., morphology-6 vs morphology-11 in Fig 4). Together, these results indicate that interactions between domains of progenitor and differentiated cells (differentiation, signalling, adhesion) transform independently “simple” domains, which are trivially reproducible, into complex ones without sacrificing reproducibility.

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Fig 5. Interactions between cell types drive anisotropic progenitor-cell motion.

(A) Motion anisotropy of progenitor cells within morphologies (orange) and isolated (black) plotted against average cell momentum magnitude (n = 30 cell types). Grey lines connect a progenitor-cell type within a morphology to that progenitor-cell type in isolation. (B) The same as (A), but for differentiated cells (n = 26 cell types). (C) The same as (A), but for cells from morphologies with only one SCC (n = 65).

https://doi.org/10.1371/journal.pcbi.1014361.g005

Cellular potts model

Our model extends a Cellular Potts Model (CPM) introduced by Hogeweg (2000) [29] by adapting an implementation of the Tissue Simulation Toolkit [68,69]. The CPM dynamics are driven by pixel copying, where repeated random sampling of pixels on grid determines the location of these copies. For each chosen pixel, a random neighbouring pixel from its Moore neighbourhood is selected as the recipient of the pixel copy. Whether the pixel copy is accepted depends on its effect on the system’s energy, denoted as H, represented by

(1)

Here, H encompasses the total surface energy accumulated from cell-cell adhesion (J_ij) and cell-medium adhesion (J_im), which are both determined dynamically as functions of protein concentrations (explained later). The index i is of pixels at cell boundaries. The index j is of pixels neighbouring i that are occupied by a different cell from i. The index m is of pixels neighbouring i that are occupied by the medium. Every pixel on the grid has an associated value, , that represents either the cell that occupies that pixel () or the medium (). Each cell () with current size (in pixels) is constrained to size with parameter ( for all simulations). The longest axis (in pixels) of each cell is constrained to length with . , and are determined dynamically (explained later).

If a pixel copy attempt increases H, it is accepted with probability , where is the change in H made by a pixel copy, and T is a temperature-like parameter that we arbitrarily fixed to T = 3. Otherwise (), the pixel copy attempt is always accepted. One developmental time step (DTS) is complete when the number of pixel copy attempts equals the number of pixels on the grid.

Development

A morphology starts as a single cell that undergoes six divisions equally spaced in time over the first 300 DTS. During the first 1,500 DTS (referred to as an equilibration phase), and for all cells. After the equilibration phase, J_ij and J_im are determined by protein concentrations in the cells (described later). At the beginning of the equilibration phase, is set to the initial size of each of the 64 cells and does not change throughout the equilibration phase. After the equilibration phase, cell growth and shrinkage can occur through modulation of the target cell size, . increases when a cell is stretched, and decreases when a cell is squeezed, as follows. When , is updated such that . When , is updated such that . When the cell volume reaches or exceeds a threshold (i.e., ), the cell undergoes division perpendicular to its longest axis. One daughter cell retains the same index as the mother cell, while the other daughter cell is assigned a new index . After division, the target area of the daughter cells’ are and . Protein concentrations remain unchanged upon cell division. If , the cell dies.

Gene regulatory network and morphogens

We model protein concentration dynamics using a system of ordinary differential equations that correspond to the reaction-only component of the Reinitz and Sharp model [70]. In each cell , there are N_genes genes, indexed by p, that each encode a protein whose intracellular concentration is denoted by . With the exception of

(which are morphogens, described later), the following equation determines the change in over time t:

(2)

with one unit of t being one DTS. The first term on the right-hand side represents the increase in due to gene expression, and is a sigmoidal function that depends on transcription factor regulation, f_p(x) (described subsequently), with a maximum production rate a and large (=20). The second term represents protein decay with rate b. We set a = b for all p. This ensures that when a gene is activated (f_p(x) > 0), the production term dominates and . Conversely, when a gene is inhibited (f_p(x) < 0), the production term is negligible compared to the decay term and . Thus, the expression dynamics are a switch-like response where genes transition between “on” or “off”, i.e., equilibrates at 1 if the gene is constantly expressed or 0 if it is constantly not expressed. We assume that the timescale of gene expression is slower than the timescale of CPM dynamics. Thus, we set a and b to small values (specifically, 6.25 × 10⁻³). Numerical integration of Equation 2 occurs with via the Euler method (we use a subscript because Equation 4, described subsequently, is numerically integrated at a different interval). We chose this value of instead of a smaller value to improve computational speed. The value of can be large because the rate parameters a and b are very small. The initial concentrations are for transcription factors (except for the two that are asymmetrically distributed, which are either or depending on at the four-cell stage), and for all other proteins (described later). Integration of Equation 2 begins after the four-cell stage is reached (100 DTS), which is when the asymmetric distribution of transcription factors (TFs) occurs.

Function f_p(x) in Equation 2 sums the regulatory effects of N_TF TFs (including morphogens) as follows:

(3)

where is the regulatory effect of TF encoded by gene on the expression of gene p (). The TF encoded by activates the expression of p if , inhibits if and has no effect if . The parameter sets the base level of gene expression. for all simulations, so protein concentrations equilibrate at 0 when they are not regulated by any TFs.

To model cell-cell signalling, the first N_morph () of the N_TF TFs are morphogens. These morphogens diffuse between cells and into the surrounding medium. The concentration of morphogen p on pixel i on the grid, , is determined by the following coupled ODE:

(4)

where D is a diffusion constant, is a production rate, is a decay rate and is the Heaviside step function that evaluates to one if at pixel i (the pixel is occupied by a cell) and zero if (the pixel is medium). The concentration represents a signalling protein that activates the expression of morphogen p. The concentration of signalling proteins is determined by Equation 2. The operator is the Laplacian acting on , which, in this context, is subtracted from the average of in its von Neumann neighbourhood (i.e., four nearest-neighbouring pixels) divided by a space step (dx)², with dx = 1/250 (250 is the length of the square grid in pixels). Numerical integration for Equation 4 occurs with . Equation 4 is subject to the initial condition for all i and p. The pixels at the boundary of the grid are subject to for all t and p. The constants , and for all p were used for the 126 simulations outlined in the main text. We chose these values for two reasons. The first is so that the maximum concentration is approximately 1 and thus similar to the concentrations of TFs per Equation 2. The second is so that the characteristic diffusion length, , is similar to that of a paracrine morphogen, such as Wnts, which signal only to nearby cells [48,71]. This characteristic diffusion length is five pixels, which is approximately the diameter of a cell. Since regulates genes in each cell , we average over all i with the value to obtain . However, we also ran 44 simulations where D_p and mutate (S1 Fig and Materials and methods Section Evolution), resulting in each morphogen having a unique characteristic diffusion length. During the equilibration period (DTS < 1500), the parameters and are multiplied by eight in order to speed up the time taken for morphogens to reach a steady state. The faster timescale does not have an effect on morphogenesis because there is no GRN-driven dynamics during the equilibration phase, as described later.

For the simulations presented in the main text, we fixed and . In S1 and S2 and S3 Figs, where we explore variations in network size, the specific values for N_morph and N_TF are explicitly stated.

Adhesion and membrane tension proteins

Cells encode adhesion proteins constituting N_pairs pairs of lock-and-key proteins that determine cell-cell adhesion and N_med medium proteins that determine cell adhesion to the extracellular medium as well as cell fluidity (explained subsequently). While a simpler homotypic adhesion system, such as that mediated by E-cadherin [72], could have been used, we chose this asymmetric lock-and-key system to provide a more flexible adhesion code to allow a larger array of evolvable interactions between cell states. Each lock protein has a complementary key protein to which it can bind. When two cells are in contact, the adhesion energy between them (J_ij in Equation 1) decreases with the number of expressed pairs of compatible locks and keys (described subsequently). Each pair of lock and key reduces adhesion energy by the same amount. The adhesion energy to the extracellular medium (J_im in Equation 1) decreases with the number of medium proteins expressed by a cell. Medium proteins have graded adhesion strengths (described subsequently). All J_ij and J_im variables used in our model are positive because cell disintegration is favoured when they are negative. When J_ij and/or J_im are small, the energy barriers to cell rearrangements are low and cells behave more fluid-like. When J_ij and/or J_im are high, the energy barriers to cell rearrangements are high and cells behave more solid-like.

Adhesion protein concentrations are Booleanised to an ON or OFF state for adhesion energy calculations (i.e., ON if else OFF). Specifically, J_ij between neighbouring pixels i and j that belong to different cells is:

(5)

where if the k-th lock in the cell of pixel i and the k-th key in the cell of pixel j are both ON and otherwise . Similarly, J_im between pixels i and m belonging to a cell and the medium, respectively, is:

(6)

where if the k-th medium-adhesion protein is ON, and otherwise . and for all simulations. In S16 Fig, we show that the morphogenesis of evolved morphologies is not destroyed by changes to these parameters.

Cells encode N_tens membrane tension proteins that make the cell shape less deformable by energetically constraining it to an elliptic shape. Consequently, cells become more solid-like when they express membrane-tension proteins. Cell shapes are defined as ellipses that have a major axis of length constrained to a target length of with (see Equation 1). increases with the number of expressed membrane tension proteins. The major axis is defined as the longest dimension of the cell, irrespective of its orientation. Because the energetic constraint applies to this dynamically identified major axis without preference for a specific direction, there is no inherent polarity to the cell’s contraction. The orientation and length of the major axis of a cell are re-evaluated after each pixel copy attempt involving that cell. Membrane tension proteins are first Booleanised to an ON or OFF state by the same method used for adhesion proteins. For simulations in the main text, . When one length protein is ON in cell , is set to ; when two are ON, is set to (the numbers 1.65 and 1.25 are arbitrarily chosen). To implement the presence or absence of a length constraint depending on whether membrane tension proteins are expressed, we set if either membrane tension protein in cell is ON, otherwise .

For the simulations presented in the main text, we fixed , and . In S1 and S2 and S3 Figs, where we explore variations in network size, the specific values for N_pairs, N_med and N_tens are explicitly stated.

We calculated surface tensions for the 24 morphologies that evolved progenitor-cell differentiation in the simulations presented in the main text. These calculations determined the surface tension for three distinct interfaces: (i) between each progenitor-cell type and the medium, (ii) between each differentiated-cell type and the medium, and (iii) between progenitor and differentiated cells. We determined surface tensions, by the following formula:

(7)

where is the adhesion energy arising between cell types and , while and are adhesion energies arising from contact between two cells of the same type, and , respectively. The parameter T is the temperature-like parameter governing the Metropolis algorithm, which scales all energy barriers in the CPM (T = 3 for all simulations). For each surface tension calculation, we used the most frequently observed cell state to represent its corresponding cell type. The value of the surface tension between a cell type and the medium (m) predicts the morphology of cell clusters [34]. If , cell clusters will minimise their surface area and form circular aggregates. If , cells maximise their contact with the medium and disperse from each other. As , the interface with the medium becomes energetically neutral, and the shape of cell clusters is less constrained by adhesion.

Evolution

To simulate the evolution of morphogenesis, we established an initial population of 60 morphologies with each assigned a different GRN and developing on a separate CPM grid. Each GRN is specified by 234 values representing the regulatory effects of all transcription factors, as described in Equation 3 (nine transcription factors regulate 26 genes including themselves). The values of the GRNs assigned to the initial population are randomly generated according to the following probabilities: , , , and . These probability choices are arbitrary. The 15 morphologies with the highest shape complexity reproduce four times to populate the next generation (the definition of shape complexity is described in the next paragraph). Upon reproduction, there is a 50% chance that one of the values in the GRN mutates. The specific that mutates is chosen at random with equal probability. The mutation alters the value of , independent of its current value, according to same probabilities used to generate the assigned to the initial population. Gene duplication and deletion do not occur.

In simulations where morphogen diffusivity mutates (S1 Fig), the values of the diffusion constant, D_p (see Eq. 4) start as a random value taken from a normal distribution between and . We also change the secretion constant, (see Eq. 4), to match changes in D_p. Specifically, the value of is equal to:

(8)

where is the minimum value of and the second term makes increase with D_p, modulated by the constant . The purpose of increasing with D_p is to ensure that the maximum concentration of morphogen in Equation 4 remains close to one. Only D_p, not , affects the characteristic length of morphogen diffusion at equilibrium, as described in Materials and methods Section Gene regulatory network and morphogens. Upon reproduction, there is a 25% chance that one of the D_p values mutates, with the choice of p occurring randomly with equal probability. The mutation alters the current value of D_p by multiplying it by , where is a random number picked from a normal distribution of mean zero and standard deviation 0.25.

Our measure of shape complexity takes inspiration from an algorithm that quantifies the complexity of two-dimensional shapes [73]. Specifically, we measure shape complexity by summing two measures: the deviation of morphology from a circle (denoted by z₁; S17 Fig) and the degree of inward folds (denoted by z₂; S17 Fig), as described below.

The deviation of a morphology from a circle z₁ is defined by the following equation:

(9)

where r_c is the hypothetical radius of the morphology if its mass were to be redistributed into a perfect circle, and is the maximum distance from the centre of mass of the morphology to any pixel in the direction specified by angle (one pixel corresponds to one unit mass). The notation indicates an average taken over all angles , where is discretised into 360 degrees for computation.

The degree of inward folds (z₂) measures the number and length of concave regions of the morphology. To identify these regions, we begin by drawing horizontal parallel lines across the CPM grid spaced one pixel apart, resulting in a total of 250 lines. Next, we located all segments along the lines that intersect the extracellular medium in between cells. The segments were discarded if they did not exceed a minimum length of 20 pixels to filter out inward folds due to stochastic cell boundary fluctuations. The above procedure was repeated by tilting the 250 parallel lines at 12 evenly-spaced angles across the range . z₂ is defined as the square root of the total number of the located segments (z₂ is independent of the lengths of the retained segments).

We defined shape complexity as . The weights of 2 and 2.5 were selected to ensure that the maximum value of either term is approximately 100 for shapes that are realistically achievable within this model, ensuring that neither term dominates the fitness criterion. In order to filter out noise, shape complexity is taken as an average of 10 evenly-spaced measurements over the last 1,000 DTS. If cells lose physical contact with other cells during development (see S16 Fig for examples), we assign the morphology a fitness of 0 as our quantification of shape complexity is not designed to handle multiple shapes on the same grid. The contributions of deviation of a circle and inward folding to the shape complexity of each morphology is shown in S17 Fig.

We implemented an alternative fitness criterion to determine where progenitor-cell-based morphogenesis is evolutionary accessible (see S1 Appendix and S2 Fig), which has two stages. In the first stage, the fitness criterion is the displacement of a morphology’s centre of mass, measured in pixels, from the start to the end of the 12,000 DTS, denoted z₃. Once the average fitness across all morphologies in a population exceeded 15 pixels, the fitness criterion transitioned to the second stage. In the second stage, the fitness criterion is (. The weightings of each criterion were selected so that the maximum value of each is approximately 100 for shapes that are realistically achievable within this model, ensuring that neither term dominates the fitness criterion.

To determine whether an evolutionary simulation succeeded or failed, we used fitness thresholds. The thresholds were applied to the morphology that recorded the most complex morphology in the final generation of the simulation. In the simulations selecting only for shape complexity, we applied a threshold of 70 after averaging the shape complexity over 60 developmental replicates in order to account for variance in the complexity score. However, this threshold does not affect our key findings, as there are still highly reproducible morphologies with progenitor-cell differentiation below the threshold (see S5 Fig). In the simulations selecting for both shifting centre of mass and shape complexity, the fitness threshold we used to determine whether an evolutionary simulation succeeded was whether the second stage was reached.

Cell state and state space

The cell state is an n-dimensional Boolean vector, where n is the total number of adhesion and membrane tension proteins (although the results do not change when TFs are included in the vector as well). Each element in this vector is the concentration of each protein Booleanised to either ON or OFF (as described previously for lock, key and tension proteins). Each Boolean vector is assigned a single colour on the CPM grid. These colours are chosen arbitrarily, and each morphology has its own distinct colour set. While this approach means that the same cell state might appear in different colours across morphologies, it is uncommon to encounter identical cell states in different evolved morphologies. The cell state of each cell is determined after every numerical integration step of Equation 2. A change in a cell’s Boolean vector corresponds to a cell state transition.

To generate the cell state space, we recorded all cell state transitions for all cells after 6,000 DTS from the beginning of development. Although it makes no qualitative difference when the recording of transitions begins, starting at 6,000 reduces the appearance of “transient” cell states [74], such as the cell state corresponding to the initial conditions of cell proteins, which makes it easier to visualise the cell state space. Cell state transitions are recorded from 10 developmental replicates in order to reduce the effect of noise on cell state space creation. The state space is the directed graph, where the nodes are the cell states and the edges are the transitions. S7 Fig shows two examples of these directed graphs. To simplify the graph, we prune rare transitions (those that occur less than five times across all cells per developmental replicate) and rare cell states. To identify rare cell states, we first count the number of cells in each state after each integration step of Equation 2 to obtain a frequency distribution of cell states. Rare cell states are those that have a frequency of less than 1% from the 10 developmental replicates. However, even without any pruning of cell state transitions and cell states, the cell state space of 22 out of 24 morphologies designated as having progenitor-cell differentiation still exhibit irreversible differentiation (S16 Fig shows one that does not follow this rule).

We used a depth-first search algorithm to identify a graph’s strongly connected components (SCCs). Irreversible differentiation occurs when there is weak connectivity between SCCs (i.e., a path exists from SCC u to SCC v, but not v to u). SCCs that do not have incoming paths from any other SCCs (source components) are designated as progenitor-cell types. SCCs that have no outgoing paths to any other SCCs are designated as differentiated-cell types.

Reproducibility score

We measured the morphogenetic reproducibility of morphologies in a rotation-, reflection- and translation-invariant manner, as follows. We first replayed the development of a morphology 60 times with different random seeds. We then performed pairwise comparisons of all developed morphologies ( comparisons). For each pair of morphologies, we computed morphological similarity scores between the two CPM grids on which morphologies developed (denoted by A and B). We computed the morphological similarity scores (denoted Jac) over many rotations, reflections and translations of grid B relative to a fixed grid A to find the maximum possible similarity between them (denoted Jac_max).

To calculate Jac, grids A and B are transformed from Cartesian to polar coordinates, as follows. The polar coordinate system is discretised into 250 × 360 pixels. Each pixel in is mapped to the pixel closest to in A or B, where r is the distance from the midpoint of grid A or one of 25 equally-spaced locations in grid B (thus, multiple pixels in the polar coordinate system can be mapped to the same pixel in Cartesian coordinates). Let and be one if the mapped pixel at a given () coordinate in A or B belongs to a biological cell and 0 otherwise. We then calculated the Jaccard index, , which measures the similarity between grids and , as follows:

(10)

where is the number of pixels in the polar coordinate system with a value of one (i.e., the pixel is occupied by a cell) on both and , where is the Kronecker delta. The multiplication by r accounts for the fact that the length of an arc drawn by an increment in increases as r increases. Similarly, is the number of pixels in the polar coordinate system in which or (or both). Thus, when and are exactly the same, . Next, we shift all values of to , corresponding to a one-degree rotation in Cartesian coordinates, and compute again. We repeat these one-degree rotations 360 times, computing for each. Next, we invert all values of to , which corresponds to a reflection of grid B, and repeat the 360 one-degree rotations again, computing for each. Furthermore, we repeated these 720 computations for 25 equally-spaced translations of grid B (as mentioned previously), achieved by shifting the location on grid B chosen to be r = 0 for the polar coordinate system. Translations occur in steps of five pixels at a time to make up a five-by-five square. The maximum recorded across all rotations, reflections and translations for each pairwise comparison is the maximum possible similarity, denoted Jac_max. The reproducibility score for a morphology is the average Jac_max across all pairwise comparisons.

Momentum and anisotropy

We defined the momentum, , of cell at time t (where t is in DTS) as the distance travelled by the cell’s centre of mass between t − t_w and t multiplied by the cell’s mass at , with unit mass represented by one pixel:

(11)

where is the cell’s centre of mass, ) is the cell’s mass at t, and t_w is the waiting time. We set t_w to 500 in order to average out the influence of stochastic membrane fluctuations and cell divisions on the centre of mass of the cell, thereby capturing the true mobility of cells. If a cell divides during the waiting time and the values of the parent and daughter cells differ, Equation 11 is modified by subtracting the position of the parent cell’s centre of mass () from the daughter cell’s centre of mass (), where is the index of the parent cell and is the index of the daughter cell.

In order to separate cell momentum by SCC, we first connected each recording of cell momentum to the state of the cell at . We then assigned each momentum recording into an SCC depending on that cell state. To create the polar plots of cell momentum magnitude (Fig 4D and 4E), we categorised each momentum measurement assigned to an SCC into one of 36 bins based on its direction. Each bin encompasses momentum measurements within an angular width of 10°. To measure anisotropic motion (Fig 5), we calculated the variance across these 36 bins. To account for total momentum, we divided the variance by the mean momentum across the 36 bins, equal to the dispersion index. The anisotropy value for an SCC is the average of the dispersion indices across five developmental replicates.