Synthetic symmetry breaking and programmable multicellular structure formation

Summary

During development, cells undergo symmetry breaking into differentiated subpopulations that self-organize into complex structures^1–5. However, few tools exist to recapitulate these behaviors in a controllable and coupled manner^6–9. Here, we engineer a stochastic recombinase genetic switch tunable by small molecules to induce programmable symmetry breaking, commitment to downstream cell fates, and morphological self-organization. Inducers determine commitment probabilities, generating tunable subpopulations as a function of inducer dosage. We use this switch to control the cell-cell adhesion properties of cells committed to each fate^10,11. We generate a wide variety of 3D morphologies from a monoclonal population and develop a computational model showing high concordance with experimental results, yielding new quantitative insights into the relationship between cell-cell adhesion strengths and downstream morphologies. We expect that programmable symmetry breaking generating precise and tunable subpopulation ratios and coupled to structure formation will serve as an integral component of the toolbox for complex tissue and organoid engineering.

Graphical abstract

eTOC

A stochastic genetic recombinase switch was used to model symmetry breaking in mammalian cells and regulate cadherin expression to generate predictable 3D multicellular structures.

Introduction

Symmetry breaking and self-organization are fundamental principles driving the development of multicellular organisms^1,2,12,13. From a genetically identical population of cells emerges a spatially organized and complex tissue structure with hundreds of cell types of distinct function¹⁴. In order to generate the intricate and diverse tissue architectures found within multicellular organisms, cells must first undergo symmetry breaking events that divide the homogenous population into differentiated subpopulations, and those subpopulations then self-assemble to generate the correct morphology^5,15–17. Molecular mechanisms have been elucidated for these events in early embryos of some model organisms such as nematode¹⁸, sea urchin¹⁹ and frog²⁰, which all show distinctive asymmetric patterning of morphogens starting at the one-cell stage. In these organisms, such patterning dictates differentiation into the trophectoderm and inner cell mass. In contrast, mammalian embryos do not show distinct morphogen patterning before differentiation into the trophectoderm and inner cell mass. This lack of easily identifiable determinants of cell fate in the early mammalian embryo has made it more challenging to identify the exact mechanisms driving initial symmetry breaking and subsequent self-organization, and hence these remain unclear⁵. Engineering simpler model systems that recapitulate essential elements of these complex biological events can support our understanding of the basic mechanisms governing these processes^21–24.

Additionally, recapitulation of symmetry breaking and self-organization processes to form distinct cell populations with complex spatial relationships is crucial for organoid engineering, an emerging field that seeks to create synthetic organ-like structures for transplantation, drug screening, or laboratory study ^22,25–28. By controlling cell differentiation and morphogenesis externally, through modulating growth factors or culture conditions, or internally, through genetic engineering of stem cells, researchers have shown that cells in a dish can be coaxed into organizing into complex multicellular forms that mimic some of the structure and function of the organ itself²⁷. The central challenges of organoid development currently include (1) inability to generate the full repertoire of cell types from a single source, (2) inability to control multicellular organization in high resolution and concurrently with differentiation, and (3) the inability to mature beyond a fetal developmental stage²⁷.

Tools to predictably break symmetry in a homogenous cell population and drive the formation of multicellular patterning would be valuable for addressing these limitations. Although there are several well-characterized mechanisms for symmetry breaking found in nature, including cytoskeleton polarity, lateral inhibition, and stochastic transcription factor heterogeneity^1,12,29–33, currently none of these processes are easily controlled in the complex organoid environment, where many cell type lineages are asynchronously developing in tandem, and where exogenous manipulation of essential cellular organelles like the cytoskeleton and pleiotropic effectors like native transcription factors could interfere with the natural signaling pathways at work in organoid cell function. Synthetic genetic circuits represent an attractive way to control symmetry breaking processes while functioning orthogonally to natural systems^21–24. To guide symmetry breaking and morphogenesis, we present a synthetic genetic circuit that demonstrates cell-autonomous and tunable symmetry breaking into two daughter subpopulations and self-assembles into a variety of three-dimensional structures that are determined by the fraction of cells in each subpopulation and the adhesive properties of each cell state.

To achieve irreversible commitment to mutually exclusive daughter states, we use two orthogonal recombinases that each excise DNA between their unique recognition sites. Cells begin in a homogenous parent state without recombinase expression, and then we induce expression of two recombinases that cut at distinct recognition sites. These recognition sites are arranged such that a recombination event models a commitment event to a permanent daughter state in our system, because recombinase excision irreversibly removes the recognition sites required for further recombination. In nature, symmetry breaking events occur at various ratios, including differentiation of the inner cell mass of the early embryo into epiblast and primitive endoderm fates at a roughly 60%−40% split³⁴, and differentiation of the pancreas where 90–95% of pancreatic progenitor cells commit to the exocrine lineage as opposed to endocrine^35,36; thus, we aimed to create a model system in which the probabilities of each daughter state could be precisely tuned. We decided to create a circuit in which small molecule inducer concentrations modulate the probabilities of each daughter state. We used two small molecule-inducible transcriptional systems to drive expression of one recombinase protein each. Within a single cell, the expression level of both recombinases dictates the probability of recombination to each of the two daughter states. Some cells remain in the parent state throughout the experiment, especially those induced with low inducer concentrations causing low probabilities of recombination to both daughter fates. Thus, over a population of cells, of each recombinase, this probabilistic and autonomous commitment process creates a stochastic distribution of parent and daughter states.

Next, we aimed to guide self-organization among the daughter subpopulations resulting from synthetic symmetry breaking. Many symmetry breaking events, especially those during embryonic development, drive subsequent changes in morphology^17,37. While advances in bioprinting and hydrogels have enabled a significant amount of control over cell sorting within organoids in vitro^26,38, these techniques are generally limited to setting up the initial conditions for patterning with few options to control cell sorting dynamically throughout organoid development. In order to drive morphological self-organization of the daughter subpopulations, we connected daughter state commitment to expression of specific cadherin cell adhesion proteins.

Cadherins are a superfamily of glycoproteins that are involved in homotypic cell-cell adhesion, and are essential for holding cells together and creating tissue boundaries during development^39,40. These proteins generally participate in homotypic adhesion, but also show some heterotypic adhesion to other classes of cadherins. Previous synthetic morphogenesis studies have combined several preexisting cell populations with differential cadherin expression to generate autonomously sorted structures, including an inner sphere with an outer layer of cells^9,41, segmented and separate populations³⁹, and maze-like and intertwined populations^42,43. A synthetic model coupling cell-cell contact to cadherin expression via the SynNotch platform²⁴ was able to demonstrate symmetry breaking in a homogenous population and induce self-assembly into a two-layer structure. This model is particularly relevant for symmetry breaking events that depend on cell-cell communication, such as lateral inhibition resulting from Delta-Notch signaling²⁹, but may not be useful for certain symmetry breaking and differentiation events in a developing organoid, where target cell populations may be dispersed throughout the organoid and not in physical contact. Additionally, such systems would require significant further engineering to allow precise control over the ratios of subpopulations produced from the symmetry breaking process and would require a new circuit to be optimized for every desired ratio. Thus, we sought to create a complementary model using small molecule induction to control symmetry breaking cell-autonomously and generate subpopulations at precisely defined ratios.

In this work, we develop a tunable stochastic cadherin switch allowing monoclonal cell lines to autonomously subdivide into two controllable daughter subpopulations. The ratio of the subpopulations generated by this subdivision is controlled by the concentrations of small molecule inducers. The two daughter subpopulations each express a different cadherin protein, inducing differential adhesion between subpopulations. The resulting mixed subpopulations autonomously assemble, resulting in a wide variety of three-dimensional morphologies from these initial monoclonal cell populations. We developed a three-dimensional particle-based model of cell sorting which generated morphologies that correlated well with experimental results. We then used the computational model to systematically explore homotypic and heterotypic parameter regimes that extended beyond our experimental observations, which shed insight on how these adhesion strengths influence self-assembly in tissues more generally.

Results

We developed a genetic circuit that uses a race condition⁴⁵ between recombinases to induce a probabilistic and cell-autonomous commitment to one of two daughter fates. Each cell begins in a homogenous parent state (S₀), and upon recombinase expression and DNA excision, may commit to one of two daughter states (S₁ and S₂) (Figure 1A). Some cells remain in the original state for the duration of the experiment, especially those exposed to low inducer concentrations. The circuit takes advantage of pairs of recombinase recognition sites placed in a mutually exclusive way such that the DNA excised between flanking complementary recognition sites includes sites required by the competing recombinase^46–48 (Figure 1B). Therefore, the circuit can only change states once, such that the daughter state depends on which recombinase completes processing first. The resulting circuit creates a race condition in which transitioning from an EBFP⁺ state to an EYFP⁺ or mKate⁺ cell state is a stochastic process primarily dependent on intracellular concentrations of both recombinase proteins (Figure 1C). A schematic of all cell lines used is shown in Supplementary Figure 1.

Figure 1: Overview of symmetry breaking circuit.

A) When the circuit is activated, a single cell can commit unidirectionally from the initial uninduced state (S₀) to either S₁ or S₂ states (top). If the circuit is activated in a monoclonal, homogenous population of S₀ cells, we induce an independent, irreversible commitment to S₁ or S₂ in each cell, resulting in a heterogenous mix of subpopulations (bottom). Each cell state is defined by expression of a fluorescent protein as well as, in later versions of the circuit, a unique cadherin protein, which induces self-assembly into multicellular structures. During induction, some cells may remain uninduced in the initial state throughout the experiment. B) Transiently transfected recombinases bind their respective sites, attP and attB (triangles marked “B” and “P”; yellow = φC31, red = Wβ) in the genomically integrated reporter switch and recombine these sites as shown (L indicates the recombined site). This results in genomic rearrangement and an irreversible change in cell state, shown here with fluorescent proteins. C) Competition between recombinases φC31 and Wβ generates a stochastic and cell-autonomous process of fate commitment. By tuning expression of the two recombinases, we can produce a wide variety of subpopulation ratios. D) Plasmids constitutively expressing φC31 and Wβ are transfected into reporter cell lines with integrated fluorescent reporter circuit, as shown in Figure 1B, at different ratios to observe the fraction of cells in each state 72 hours post-transfection. The average fractions for three replicates within the subpopulation of cells negative for the S₀ marker EBFP are shown, with error bars indicating standard deviation between the fractions. We chose to focus on EBFP⁻ cells so as to increase the likelihood that the cells considered had undergone a recombination event. All conditions with non-zero recombinase transfection had 1000 EBFP⁻ cells or greater per replicate. Since the fraction of EBFP⁻ cells was smaller in the zero recombinase transfection versus the other conditions (about 0.5% versus about 5%), fewer EBFP⁻ cells were analyzed: at least 250 per replicate. We hypothesize that most of the EBFP⁻ cells in the zero recombinase transfection condition had undergone epigenetic silencing rather than leaky recombinase activity, since the vast majority of these cells remained negative for S₁ and S₂ fluorescent markers. The bar charts show the fraction of cells that were EYFP⁺ and/or mKate⁺ within the EBFP-negative population of cells that had switched out of the initial EBFP⁺ state. The leftmost bar shows that very few EBFP⁻ cells express EYFP or mKate in the absence of transfected recombinase. Across all transfected conditions, about 20% of cells were triple negative for EBFP, EYFP and mKate, indicating epigenetic silencing of the fluorescent reporters. 2–4% of EBFP⁻ cells with zero recombinase transfection were positive for mKate, which is possibly due to spontaneous recombination in this population. A small fraction (1–3%) of cells were double positive for both EYFP and mKate in conditions with transfected φC31 recombinase, indicating a possible rare double integration of the reporter circuit. In the case of rare double integration of the reporter circuit, double positive cells for mKate and EYFP arise when one integration transitions to an mKate⁺ state while the other transitions to an EYFP⁺ state. A potential explanation for this phenomenon is that φC31 has some recombination activity at the Wβ recombinase sites, which may act as pseudo sites⁴⁴. Raw data is shown in Supplementary Figures 2–3.

We first tested if recombinases with mutually exclusive sites could be used to generate symmetry breaking of a single monoclonal population into two distinct subpopulations at different ratios determined by the addition of small-molecule inducers. To test this, we genomically integrated into CHO-K1 landing pad cells⁴⁹ a single copy of a transcriptional unit with fluorescent protein coding sequences separated by recombinase cleavage sites and terminators (Figure 1B, Supplementary Figure 1A). In the absence of recombinases, the vast majority of cells express EBFP and not EYFP or mKate (Supplementary Figure 2). Expression of recombinases can cause restructuring of the DNA sequence. Wβ recombinase can flip out the EBFP coding sequence and subsequent terminator so that the cell expresses mKate alone. φC31 recombinase can flip out EBFP, mKate, and subsequent terminator so that the cell expresses EYFP alone. We transfected constitutive φC31- and Wβ-expressing plasmids into this reporter cell line at different ratios and observed the resulting cell states in the subpopulation of cells negative for the S₀ marker EBFP (so as to increase the likelihood that the cells considered had undergone a recombination event). We found that cells subdivided into subpopulations at ratios reproducibly controlled by the concentrations of recombinase expression (Figure 1D and Supplementary Figure 3). We noted that about 20% of the EBFP⁻ cells transfected with recombinase were also negative for EYFP and mKate, which we hypothesize is due to epigenetic silencing of the integration likely prior to transfection (since epigenetically silenced cells are enriched for in the EBFP⁻ population analyzed). We also observed that a small fraction (<3%) of EBFP⁻ cells were double positive for EYFP and mKate, indicating possible rare double integration of the reporter circuit. 2–4% of EBFP⁻ cells with zero recombinase transfection were positive for mKate, indicating potential spurious recombination in a small fraction of cells.

Next, in order to generate a wide variety of subpopulation ratios without the need to engineer a new cell line for each ratio, we developed a cell line which reliably generates specific subpopulation ratios as a function of small molecule inducer concentrations. We used PiggyBac transposase to integrate a plasmid encoding transcription factors rtTA3, PhlF-ABI, and VPR-Pyl1, which mediate induction of inducible transcriptional units by doxycycline (Dox) and abscisic acid (ABA), and another plasmid encoding inducible recombinases φC31 and Wβ, into a CHO landing pad line⁴⁹ to create a master polyclonal transcription factor and inducible recombinase expression cell line (Supplementary Figure 1B). Next, we integrated at the Rosa26 landing pad site a construct encoding the output of the tunable switch, specifically fluorescent reporters with recombinase target sites, similar to the one in the cell line shown in Figure 1B (Supplementary Figure 1C). The resulting cell line contains a circuit in which orthogonal inducers Dox and ABA regulate expression of Wβ and φC31, respectively. Induction of these recombinases can trigger a recombination event in the switch output from S₀/EBFP expression to S₁/mKate or S₂/EYFP expression. (Figure 2A). Increasing the concentration of each inducer increases the transcription rate of a given recombinase gene, leading to a higher concentration of recombinase protein within the cell. This in turn increases the probability a recombinase will be bound to a pair of recombinase sites and catalyze a change in cell state. In this way, inducer titrations shift the probability of individual cells in the population switching to a specific state.

Figure 2: Symmetry breaking circuit with fluorescent reporters.

A) Circuit diagram of small molecule-induced recombinases switching the fluorescence reporter state. Addition of Dox induces expression of Wβ, which flips the circuit into an mKate-high state, while ABA addition induces φC31, which flips the cell into an EYFP-high state. Commitment to a particular state is achieved by the corresponding recombinase excising the indicated DNA segment. B) A Gaussian Mixture Model (GMM) was used to classify cells into three single-positive populations (Supplementary Figure 3). 3D graphs of selected ABA and Dox concentrations colored according to GMM classified single positive subpopulations: S₀/EBFP (blue, parent state), S₁/mKate (red), and S₂/EYFP (yellow). Data in B-C was generated from a monoclonal line derived via single cell sorting following PiggyBac and landing pad integration of the circuit in A. C) Heatmap of cell state distribution after induction of the small moleculeresponsive circuit shown in Figure 2A with varying combinations of concentrations of ABA and Dox to activate expression of φC31 and Wβ, respectively. Adding both small molecules causes cells to stochastically switch to one of the two daughter states as a function of recombinase expression levels. 72 hours post-induction, we performed flow cytometry to quantify expression of fluorescent reporters. Average fractions of three replicates are shown, where each analyzed condition for each replicate contained at least 2800 cells. Raw data can be found in Supplementary Figures 4–15, and data on the concordance between replicates can be found in Supplementary Figures 16–17. Color scale is linear and indicates the percentage of cells in each state.

We added varying concentrations of each small molecule inducer to cells with integrated fluorescent reporter and performed flow cytometry to quantify the distribution of cell states after 72 hours of induction. We analyzed this data by fitting a Gaussian mixture model (Supplementary Figure 4 and Supplemental Movie 1) to the fluorescent distributions to categorize cells into three subpopulations, each of which contained cells positive for a single fluorescent reporter. 3D scatter plots of flow cytometry data for four example conditions are shown in Figure 2B. The percentage of cells classified into each subpopulation under each inducer condition is shown in Figure 2C, and flow cytometry data for each well is shown in Supplementary Figures 5–16.

We observed a wide variety of subpopulation ratios, which were finely tunable and dependent on inducer concentrations. Subpopulation ratios showed high concordance between replicates (Supplementary Figure 17), with a mean standard deviation across all channels for each well of 1.7% of the population (Supplementary Figure 18). At lower concentrations of Dox and ABA, we observe a large proportion of cells remaining in the initial EBFP⁺ state. At these concentrations, we hypothesize transcription of both recombinases is weak and their corresponding concentrations are similarly low. This results in recombination being a low-probability event, and the majority of cells remain in the EBFP⁺ state. As the concentration of each inducer increases, the probability of changing to the corresponding cell state increases. With intermediate to high doses of each inducer in the same well, we were able to generate subpopulations with similar numbers of EYFP⁺ and mKate⁺ cells. For example, at 500 ng/mL Dox and 50 μM ABA, our circuit generates 57% single positive mKate⁺ cells and 33% EYFP⁺ single positive cells. By increasing ABA to 150 μM, we can instead generate 45% mKate⁺ and 47% EYFP⁺ cells. If we instead keep ABA at 50 μM but increase Dox to 2000 ng/mL, our circuit symmetry breaks into 68% mKate⁺ and 25% EYFP⁺ cells. We also induced symmetry breaking that skewed highly towards one subpopulation, demonstrating up to 80% mKate⁺ single positive cells.

We next sought to generate a diverse set of morphological shapes from a homogenous monoclonal population, building on our ability to generate subpopulations at precisely controlled ratios. Starting with the master polyclonal transcription factor and inducible recombinase cell line, we coupled specific cadherin protein expression to expression of the S₁ and S₂ fluorescent reporters mKate and EYFP in the daughter subpopulations, generating the stochastic cadherin switch circuit shown in Figure 3A (Supplementary Figure 1B, 1D). From the initial S₀/EBFP⁺ state without cadherin expression, induced commitment to one of the two daughter state results in expression of one of two possible cadherins, which initiates differential adhesion between the subpopulations and subsequent self-organization. Following induced recombination, resulting yellow cells express cadherin 1 (E-cadherin) and resulting red cells express cadherin 6 (K-cadherin). Both of these cadherin proteins have been experimentally characterized to bind with fairly strong affinity to cadherin proteins of the same respective subtype, and expression of each of these proteins in mammalian cells has been shown to induce cellular aggregation^39,50. We continued to use CHO-K1 cells as a chassis, as they have no endogenous cadherin expression and only limited self-aggregation using other pathways and thus are commonly used to study cadherin behavior^51,52. While implementing the circuit in cell lines more relevant to applications such as regenerative medicine, including embryonic stem cells, may be an interesting direction for future work, isolating the effects of induced cadherin aggregation is conflated by the cells’ endogenous multicellular structure formation and resulting phenotypic changes^53,54. In contrast, our choice of CHO cells as a chassis enables the engineered cells in the parent EBFP⁺ state to retain the low adhesive state of wild-type CHO cells, which tend to form loose aggregates in three-dimensional culture⁹. As a result, multicellular structure formation observed in downstream experiments can be attributed to exogenous cadherin expression⁴².

Figure 3: Competing recombinase circuit to control cadherin expression.

A) Circuit diagram of a tunable stochastic cadherin switch, which is composed of small molecule-induced recombinases φC31 and Wβ switching cell states defined by co-expression of fluorescence reporters and cadherins. ABA and Dox induction result in expression of E-cadherin (encoded by Cdh1, yellow) and K-cadherin (encoded by Cdh6, red), respectively. B) Montages of images showing induction of the tunable stochastic cadherin switch shown in A with varying combinations of concentrations of ABA and Dox (three replicates). Concentrations of inducers determine the fraction of the population that switches to a given state. Cells sort based on the expression of the cadherin they express, and a variety of shapes can be generated. Increased EBFP fluorescence was detected in structures formed from induction with the highest Dox inducer concentrations, which we hypothesize may be due to autofluorescence from dead cells due to Dox cytotoxicity⁵⁵ and/or hypoxia at the center of tightly packed shapes. Scale bars are shown below each replicate. Brightfield microscopy of shapes can be found in Supplementary Figure 21. Coverage of the 3D space of all possible cell state proportions can be found in Supplementary Figure 22. C) Close-up of images highlighting classes of shapes generated in 3B. Shapes include (left to right) a loose monolayer with distinct clusters, single cores from both daughter states, single-core structures with an outer layer of clusters, and adjacent large cores.

From this polyclonal population containing the tunable stochastic cadherin switch circuit, we used single cell sorting to generate three monoclonal cell lines. For shape formation experiments, 1000 cells were added to an ultra low adhesion round bottom well for each condition and induced with the noted Dox and ABA concentrations. Structures were imaged after 96 hours to reduce residual EBFP expression, which was especially notable at the 72-hour mark for cells at low to medium inducer combinations. Cells were not prohibited from growing and dividing throughout the course of the experiment, although contact inhibition may hinder cell division, especially for cells located near the core of 3D structures. Figure 3B shows the result of small molecule induction for 96 hours for three replicates of one of the monoclonal cell lines. At low ABA & Dox, most cells remain in the low adhesion blue state with small numbers of cadherin-expressing red and yellow cells sorting into small clusters, similar to results observed by Tordoff et al⁹. As ABA increases and Dox remains low, the number of yellow cells increases and forms a central sphere with some small clusters of red satellite cells, and as Dox increases but ABA remains low, red cells form a large central sphere. In the intermediate states, a wide variety of structures are observed, largely composed of a central single large cluster surrounded by smaller clusters. We observed three main classes of shapes that were reliably formed across the three replicates at specific induction levels: those with a loosely adherent blue background and yellow “polka dot” clusters, those with a yellow central core and adjacent red clusters, and those with a red central core and adjacent yellow clusters. These classes of shapes are highlighted in Figure 3C. We also observed additional complex patterning, such as structures with a single inner core surrounded by an outer core, multiple inner cores surrounded by a single outer core, multiple inner cores each surrounded by an outer core, and interspersed cores, all from a homogenous initial population.

From this experimental data, we were able to formulate design rules for the programmable generation of the main classes of structures. To produce a blue background with yellow polka dots, cells should be exposed to zero Dox induction and very low (1 μM) ABA induction. In order to produce a central yellow core, cells should be exposed to zero Dox induction and higher ABA induction. The number of red clusters then depends on the ABA concentration. Moderate concentrations (5–10 μM) tend to generate three to four small red clusters, whereas high concentrations such as 150 μM tend to generate one to two red clusters. In contrast, to produce a central red core, cells should be induced with low ABA concentrations, such as 3 μM, and Dox concentrations 50 ng/mL or greater. The number of adjacent yellow clusters is controlled by the Dox concentration—from three to four (50 ng/mL Dox) down to one or two (250–500 ng/mL Dox) and then zero (2000 ng/mL Dox). These guidelines are summarized in Supplementary Table 1.

Supplementary Figures 19–20 show the dynamics of multicellular structure formation for six representative inducer combinations. At 24 hours, cell lines exposed to high inducer combinations begin to aggregate into tighter clusters than those exposed to very low inducer concentrations, indicating initial cadherin expression, and faint mKate and EYFP can be detected in a significant proportion of cells (although initial cell state transitions can be difficult to observe due to residual EBFP expression). Between 24 and 48 hours, mKate and EYFP expression levels increase and EBFP decreases in mKate⁺ and EYFP⁺ populations. EYFP⁺ and mKate⁺ cells begin to cluster with other cells of the same type, especially in wells with higher inducer concentrations. In the cell lines that eventually form a single central core cluster, in the time period between 48 and 72 hours, smaller clusters merge to form one central cluster with adjacent clusters of the opposite cell type. At 72 hours, the majority of multicellular structures have reached a quasi-stable state, and thus between 72 and 96 hours we did not observe major changes in multicellular structures. However, clusters did continue to condense and increase in fluorescence intensity between 72 and 96 hours, while also reducing residual EBFP expression in mKate⁺ and EYFP⁺ cells. We also explored shape formation in the other two monoclonal cell lines (Supplementary Figure 23) and noted that different cell lines require slightly different inducer concentrations to generate similar shapes, pointing to a role for copy number of recombinase and transcription factors in the dynamics of the circuit.

To move towards model-guided design of self-assembling shape formation, we built a 3D particle-based model to simulate cell sorting^56–58 using differential adhesion and repulsion forces between cells to update each cell’s position at a fixed velocity (Figure 4A–B). Each experimental well condition was recapitulated in silico by initiating the corresponding simulation with 1000 cells in the parent state (blue) with no explicit cadherin expression. Throughout the simulation, the cells may transition to the Cadherin 1-expressing (yellow) or the Cadherin 6-expressing (red) states (Supplementary Movie 2). These transitions were modeled as a Poisson process with an average transition rate calculated from the final ratio of cell states in each experimental well at 96 hours. In accordance with experimental observations of an initial period without cell state transitions, likely due to the dynamics of inducing gene expression and resulting recombinase activity, the simulations skip the first 24 hours and then model cell state transitions between 24 and 96 hours. In the first 24 hours, cells essentially remain inert in the S₀ state and thus do not require explicit modeling.

Figure 4: In silico model of shape formation.

A) Schematic of model which updates cell position at each time step according to forces acting upon the cell. B) Forces included in the model: cell-cell adhesion, steric repulsion, and random noise. C) Results from simulations of 1000 cells randomly seeded in accordance with subpopulation ratios estimated from experimental results from Fig 3B, modeled with cell state transitions and resulting multicellular structure formation occurring between 24 and 96 hours. Scale bars are shown below each replicate. D) Example comparisons between experimental data shown in Fig. 3C (top row) and simulation results (bottom row) for various inducer combinations. E) Comparison between experimental and simulation results of average cluster number and cluster size from three experimental and simulation replicates. Color scales are linear.

Initially, we estimated all model parameters from the literature, including calculating cell-cell adhesion constants from surface plasmon resonance measurements of protein disassociation constants as in Katsamba et al³⁹ (Supplementary Methods). This model produced multicellular structure predictions that were visually different than the experimental results, predicting formation of a single yellow central core in the vast majority of inducer combinations despite formation in only ~20% of experimental cases (Supplementary Figure 24). We analyzed cluster characteristics of both the experimental and initial simulation results and once again observed notable differences (Supplementary Figure 25). Our initial estimation of cell-cell adhesion had only considered the protein-protein adhesion strength of each cadherin. However, cell-cell adhesion is a product of complex factors involving cadherin protein binding kinetics, cadherin protein surface expression levels, post-translational modifications to the cadherin protein, and signaling pathways induced by cadherin binding, and thus we postulated that considering only the protein-protein adhesion strength of each cadherin in combination with the estimated number of surface cadherins did not accurately estimate the total cell-cell adhesive forces. We next performed a parameter sweep to determine appropriate values for the cell-cell adhesion constants and for the noise magnitude (Supplemental Methods). The resulting optimized simulation produced results that are visually similar to the experimental results (Figure 4C–D, Supplementary Figure 26). The cluster number and cluster size metrics of the experimental and simulation results also showed strongly similar trends (Fig 4E).

The dynamics of simulated and experimental multicellular structure formation are shown in Supplementary Figure 27. As described above, the simulations skip the first 24 hours and perform cell state transitions and cell-cell adhesion between 24 and 96 hours of simulated time. With the simulated initial delay, we observed a high degree of concordance between the simulated and experimental dynamics. Both experimental and simulated conditions showed initial cell state transitions in a significant fraction of cells between 24 and 36 hours. Simulated cells tended to coalesce into a single central cluster around 60 to 72 hours, similar to the 48 to 72 hours observed in experimental wells. Almost all simulated conditions produced structures that were fairly stable between the 84- and 96-hour time points, with minor clusters merging in a few cases, which was also observed in experimental results. Overall, simulated shape formation followed the same dynamics of shape formation seen in the experimental wells as exemplified by the timing of initial state transitions, core cluster formation, and achievement of a stable state.

We were also interested in understanding how the timing of the simulated Poisson commitment process might affect the multicellular structures formed. To that end, we repeated simulations with the same parameters, but with all cells starting in their final S₀, S₁, or S₂ states without state transitions over the course of the simulation. The simulated multicellular structures are visually quite similar to the experimental results and to the simulation results in which cells transition states during the simulated time (Supplementary Figure 28). Analysis of cluster size and number was also highly similar between the three cases (Supplementary Figure 29). We conjecture that because in the conditions simulated with dynamic cell state transitions 30–70% of the simulated cells that undergo a state transition within between 24 and 96 hours are already committed at the 48 hour mark, the timing differences between the two types of simulations did not greatly impact the final shape formation.

We then decided to exploit the simplicity of the cell-cell interaction rules in our model in order to explore how cell-cell adhesion may affect multicellular structure formation. In nature, cadherin proteins contribute to a wide range of homophilic and heterophilic cell-cell adhesion affinities, but it is difficult to fine-tune these parameters independently in order to understand the effects of cell-cell adhesion strengths on multicellular structure formation. As a model system, we simulated symmetry breaking and shape formation while varying cell-cell adhesion affinities to explore how cell types may exist in ‘salt-and-pepper’ highly mixed states, or alternatively in large and distinct clusters, which both occur during various stages of development (for example, cells transition from a high degree of mixing in the trophectoderm to distinct clustering in the endoderm and mesoderm states). We hypothesized that the heterotypic adhesion strength would have a significant impact on the intermixing between the daughter subpopulations. Starting from the model that reproduced our experimental results, we systematically varied the heterotypic adhesion between daughter populations, as well as the ratios of the different daughter states, while holding homotypic adhesion strengths constant (Supplementary Figures 30–31). We noted a sharp transition between segregated and intermixed daughter populations as a function of heterotypic adhesion strength, although it appeared largely independent of the proportion of the two daughter states (Supplementary Figure 31A, green line). We next varied both the homotypic and heterotypic adhesion strengths and observed that when heterotypic adhesion strength is below 70% of homotypic adhesion strength, daughter populations will segregate into distinct clusters. However, when heterotypic adhesion strength was increased to 75% of homotypic adhesion or higher, the populations were intermixed throughout the duration of the simulation (Supplementary Figure 31B, pink line). As heterotypic adhesion strength was increased towards this threshold, the number of clusters adjoining the central cluster decreased, such that in many cases simulations yielded two adjacent large clusters for heterotypic strengths just below 70%. This finding has implications for the evolution of mammalian cadherin proteins. The first protocadherin protein evolved in pre-metazoan organisms, possibly for capture of bacterial prey⁵⁹. This single primitive protein underwent duplication and mutation events that generated multiple cadherin subtypes, which promoted differential adhesion between cells and allowed for multicellular structure formation^60,61. Our simulation results indicate that, as duplicated cadherin proteins evolutionarily diverged, mutations between them which even slightly weakened heterotypic adhesion strength while maintaining homotypic adhesion strength may have allowed cell sorting into distinct subpopulations.

Strong homotypic adhesion in a mixed population is important for the formation of a central core cluster, as we observed in the majority of experimental wells. However, interestingly, our simulations indicated that increasing homotypic adhesion between red cells beyond wild-type parameters eventually led to red cells being unable to form a central cluster and instead remaining in several scattered clusters (Supplementary Figures 32–33, blue line in Supplementary Figure 33). We hypothesize that above a certain homotypic adhesion strength and with physiologically relevant heterotypic adhesion strength and cell velocities, cells are kinetically trapped in their positions for the duration of the experiment and are unable to reach and merge with nearby clusters of the same cell type⁹. Interestingly, this behavior could be “rescued” by either increasing cell velocity or strengthening heterotypic adhesion. In Supplementary Figure 34, we repeat the simulation in Supplementary Figure 32 with increased cell velocity, and we find that cells are able to form single central red cores even at very high red-red homotypic adhesion strengths. In Supplementary Figure 31B (orange line), we show that for wells containing 50/40/10% R/Y/B cell proportions, we observed a single core cluster (either red or yellow cells) when heterotypic adhesion was 12.5% of homotypic adhesion strength or greater, and multiple clusters for both cell types when heterotypic adhesion strength was below this threshold. However, this threshold was influenced by the proportions of each cell type, as shown in Supplementary Figures 30–31, and appears to be the product of a complex interplay between homotypic adhesion strength, heterotypic adhesion strength, and ratios of cell subpopulations. Therefore, we postulate that homotypic and heterotypic cell-cell adhesion strengths have evolved to “thread the needle” between forming loose or intermixed aggregates and promoting cell-cell adhesion so strongly that they prevent cell motility.

Model-guided design based on careful selection of cadherin proteins or similar adhesion molecules would be a powerful tool for achieving multicellular morphologies that emulate biological structures such as those relevant to organoid engineering. Homophilic and heterophilic protein interaction strength and cellular adhesion strength have been characterized for a number of cadherins^50,62. Additionally, several mutations in cadherin proteins have been characterized to increase or decrease the homophilic and/or heterophilic adhesion strength^63,64. This model could inform the selection of existing cadherin proteins to achieve desired shapes, or in the case where no existing cadherin proteins suffice, the model could guide the evolution of designer cadherin proteins with specific non-natural homotypic and heterotypic adhesion strengths to achieve novel structures. Overall, we posit that this model based upon simple rules of cell-cell interaction may serve as a viable way to learn and predict which experimental parameters correspond to which cell-sorting behavior and emergent aggregate organization.

Discussion

How many ways can cells “flip a coin” to determine their downstream fate? A number of mechanisms for induction of symmetry breaking have been described in the literature. Symmetry breaking events studied thus far are either cell-autonomous or dependent on multi-cell interactions, and are based on mechanisms such as stochastic gene expression⁶⁵, cell-cell communication⁶⁶, cell-intrinsic physical asymmetry¹², and relative positioning of cells¹⁵ as factors that can affect a cell’s fate commitment. Synthetic model systems can provide insights as to the properties of symmetry breaking decisions and serve as important tools for tissue engineering and organoid development. Here we present a stochastic recombinase genetic switch, which we use to model cell-autonomous irreversible commitment to symmetry breaking driven by an intercellular stochastic process. Molecular mechanisms for many natural symmetry breaking processes, especially those in mammalian development such as blastomere formation and commitment of the blastomere to inner cell mass and trophectoderm fates, still have not been elucidated^13,15,67. Our system serves as a simple model of one possible mechanism of these and other symmetry breaking processes, where commitment to one of two daughter states is a stochastic process driven by expression levels of competing intracellular regulatory proteins.

Programmable symmetry breaking represents a major step in the engineering of self-organizing organoid systems. Induction of transcription factor expression can be a powerful approach to guide cells towards mature and differentiated states³⁸, but in order to generate differentiated cells at appropriate ratios, it would be useful to induce transcription factor expression in cell populations that match the desired ratio of the differentiated populations. Our synthetic symmetry breaking switch would be useful for inducing differentiation in organoids. Some symmetry breaking processes relevant to organoid maturation generate cells committed to daughter fates at roughly equal proportions, and others are skewed processes that produce daughter subpopulations of highly disparate ratios. Thus, a model system to provide insight on mechanisms of these processes should be both flexible in the subpopulation ratios it can generate, and easily tunable to a desired ratio. Our tunable symmetry breaking circuit can reliably generate subpopulation ratios that range from evenly split to up to 80% skewed towards a single subpopulation. By modulating concentrations of small molecule inducers, we are able to generate subpopulations at gradually shifted ratios. Such a bifurcation switch would be useful to generate specific ratios of cell-types important for organ function, for example hepatocellular and biliary cells in the developing liver⁶⁸ or the differentiation of a subset of stem cells into Paneth cells in intestinal organoid development⁶⁹.The precise control that our switch affords over the ratios between daughter subpopulations should allow it to be useful in engineering a wide variety of organoids.

Our system is also able to link symmetry breaking processes to adhesion-driven self-organization and achieves complex patterning, such as two-layer structures with a single core or multiple cores, demonstrating that symmetry breaking and cell sorting in tandem can be tuned to achieve a wide variety of three-dimensional morphologies. The asymmetry, polarization, clustering, and budding demonstrated in the shapes generated by this circuit (Figure 3B) are reminiscent of the events of early embryogenesis and organogenesis. These programmable structures could also be used as scaffolding to build more complex organ structure for the generation of organoids and organs for transplantation. However, further characterization of the morphologies formed when this system is engineered in other cell types with varying degrees of endogenous adhesion protein expression would be prerequisite for these organoid engineering applications. Our system was able to elucidate design rules for the generation of multicellular structures with a central core and various numbers of adjacent clusters. Our three-dimensional simulation of multicellular adhesion-based sorting recapitulates our observed experimental structure formation, suggests quantitative hypotheses about how homotypic and heterotypic cellular adhesion strengths “thread the needle” to achieve desired multicellular shapes, and allows for large-scale and efficient screening of desired multicellular shape formation. In order to control multicellular organization even more precisely and generate more complex structures, future systems will likely need to more specifically control the spatial organization and timing of cells committing to each daughter state. Additionally, aggregate size has been shown to be a crucial determinant of adhesion-mediated cell sorting⁴³; further characterization of this system with greater or fewer than the 1000 cells seeded here may yield additional insights and enable the generation of even more diverse multicellular structures. In particular, experiments in which a far smaller number of cells is seeded (~10–50 cells) could be useful in determining the design rules of cadherin-mediated cell sorting in blastocyst formation, since experiments in synthetic embryos have indicated that differential adhesion between cells expressing different cadherins plays a crucial role in this self-assembly process⁴⁰. We expect programmable symmetry breaking and structure formation to serve as a useful model of embryogenic processes, as well as an integral component of the toolbox for complex organoid engineering.

STAR methods

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Ron Weiss (rweiss@mit.edu).

Materials Availability

Plasmids generated in this study have been deposited to Addgene (Addgene IDs 204489–204496).

Data and Code Availability

• All raw data flow cytometry and image data has been deposited at Zenodo and is publicly available as of the date of publication. DOIs are listed in the key resources table.

• All original code has been deposited at Zenodo and is publicly available as of the date of publication. DOIs are listed in the key resources table.

• Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Key resource table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Antibodies
Bacterial and virus strains
Biological samples
Chemicals, peptides, and recombinant proteins
Critical commercial assays
Deposited data
Experimental images and flow cytometry data	This paper	https://github.com/Weiss-Lab/synthetic-symmetry-breaking, version of record deposited at Zenodo at: https://doi.org/10.5281/zenodo.8067436
Experimental models: Cell lines
cNW2.2: reporter cell line described in SI Fig 1A to generate data shown in Fig 1D & SI Fig 2	This paper	N/A
mCE40m4: colors-only symmetry breaking cell line described in SI Fig 1C to generate data shown in Fig 2B–C & SI Fig 3–17	This paper	N/A
cNW1m4: colors + cadherins symmetry breaking & shape formation cell line described in SI Fig 1D to generate data shown in Fig 3B–C, Fig 4D, and SI Fig 18–21	This paper	N/A
cNW1m3: colors + cadherins symmetry breaking & shape formation cell line generating data shown in SI Fig 23 (top row)	This paper	N/A
cNW1m6: colors + cadherins symmetry breaking & shape formation cell line generating data shown in SI Fig 23 (bottom row)	This paper	N/A
Experimental models: Organisms/strains
Oligonucleotides
Recombinant DNA
pJT87 (Wβ expression)	This paper	Addgene ID 204489
pJT88 (φC31 expression)	This paper	Addgene ID 204490
pJTT002 (colors-only switch responsive to recombinases used in cell line shown in Figure 2)	This paper	Addgene ID 204493
aCE89 (ABA-ON φC31, Dox-ON Wβ)	This paper	Addgene ID 204494
aCE90 (ABA-ON φC31-PEST, Dox-ON Wβ-PEST)	This paper	Addgene ID 204495
aCE106 (PhlF-NES-ABI, NLS-VPR-Pyl1, & rtTA3 expression)	This paper	Addgene ID 204496
aCE120 (colors-only switch responsive to recombinases used in cell line shown in Figure 1)	This paper	Addgene ID 204491
aCE121 (colors& cadherins switch responsive to recombinases used in cell line shown in Figures 3–4)	This paper	Addgene ID 204492
Software and algorithms
Analysis code for experimental images & flow cytometry	This paper	https://github.com/Weiss-Lab/synthetic-symmetry-breaking, version of record deposited at Zenodo at: https://doi.org/10.5281/zenodo.8067436
Python ABM library	This paper	https://github.com/kemplab/pythonabm, version of record deposited at Zenodo at: https://doi.org/10.5281/zenodo.8007463
3D cadherin ABM to simulate symmetry breaking & multicellular structure formation	This paper	https://github.com/kemplab/cho-adhesion-model, version of record deposited at Zenodo at: https://doi.org/10.5281/zenodo.8007620
ImageJ	Schneider et al.72	https://imagej.net/ij/index.html
MATLAB		https://www.mathworks.com/products/matlab.html
Other

Experimental model and subject details

Cell lines

CHO-K1 cells (female) with a monoallelic integrated LP in the putative Rosa26 locus⁷⁰ were maintained in Dulbecco’s modified Eagle’s medium F12 (DMEM/F-12) supplemented with 10% FBS and 1% MEM Non-Essential Amino Acids (Gibco). Cells with an integrated circuit were maintained with selection added to the media of puromycin (10μg/mL), blasticidin (30μg/mL), and zeomycin (1000μg/mL). Cell lines were not authenticated. All cell lines used in the study were grown in a humidified incubator at 37 °C and 5% CO2.

Method details

Plasmid construction

Plasmids were constructed using a modified version of the hierarchical MoClo system⁷¹. Mouse Cdh1 and Cdh6 coding sequences were ordered from IDT without Type IIS restriction sites and inserted into L0 destination vectors. L1 expression vectors were assembled from L0s containing the different components for each transcriptional unit. L2 vectors were assembled from L1 vectors by Golden Gate cloning using SapI, into either a modified PiggyBac backbone compatible with SapI-based Golden Gate, or into a landing pad payload backbone⁴⁹ similarly modified to work with SapI-based Golden Gate.

Genomic integration

Cell transfections were performed using Lipofectamine 3000 (Invitrogen) with 1.1:1.1:1 Lipofectamine 3000:P3000:DNA ratio by weight in 24-well or 96-well format. Complexes were prepared in Opti-MEM (Gibco). 150,000–200,000 cells or 1,000 cells were plated on the day of transfection into 24-well or 96-well plates respectively in DMEM culture media without antibiotics and analyzed by flow cytometry after 72 hours. The transcription factor and inducible recombinase plasmids were integrated using 150,000 cells in a reverse transfection in a 12-well plate with 1μg plasmid expressing the PiggyBac integrase, and 0.5μg of each of the plasmids to be integrated to create the master polyclonal transcription factor and inducible recombinase expression cell line. Selection (zeocin and blasticidin) started after 48 hours and continued for 7 days. The cell-state expressing plasmids were integrated into the master polyclonal transcription factor and inducible recombinase expression cell line using Bxb1 integration into the landing pad site⁴⁹ in a 24-well plate seeded with 30,000 cells the day before transfection, transfecting 250 ng each of the circuit to be integrated and Bxb1. Selection (puromycin in addition to the other selection markers) was started after 24 hours.

Flow cytometry and single-cell sorting

Cell fluorescence was analyzed with LSR Fortessa flow cytometer, equipped with 405, 488, 561, and 637 nm lasers (BD Biosciences). The following laser and filter combinations were used to evaluate the fluorescent proteins used in transient transfection experiments for color controls in this study: EBFP, 405nm laser, 450/50 filter; EYFP, 488nm laser and 515/20nm filter; mKate, 561nm laser, 582/42 filter. After 7 days of selection post-landing pad integration, EBFP⁺/EYFP⁻/mKate⁻ cells were sorted to single cells using a BD FACS ARIA equipped with 405, 488, 561 and 640nm lasers. 4 monoclonal lines were generated for the fluorescent reporters only circuit, and 3 monoclonal lines for the fluorescent reporters and cadherins circuit. The following laser and filter combinations were used to evaluate the fluorescent proteins used in single-cell sorting: EBFP, 405nm laser, 450/40 filter; EYFP, 488nm laser, 530/30 filter; mKO2, 561nm laser, 582/15 filter.

Structure formation assay

To make 3D aggregates, a monoclonal cell line expressing the circuit to be tested was trypsinized. Cells were counted and diluted, and 1,000 cells were seeded in each well of a 96-well ultra-low attachment (ULA) round bottom plate. Media containing inducer and selection (to maintain the circuit) was added to each well and the cells were centrifuged at 300 x g for 5 min. to bring the cells to the bottom of the well. Cells were incubated with inducer for 96 hours and then imaged.

Microscopy and image analysis

Aggregates were imaged in the EVOS inverted digital fluorescence microscope from Advanced Microscopy Group (AMG).

Initial image processing was done with ImageJ⁷². Images were systematically brightness and contrast enhanced using the same parameters for all images within a figure unless otherwise noted. Then, cluster analysis was done in MATLAB to convert each red and yellow fluorescence image for each condition into a binary mask, filtering out low intensity pixels (<0.12 in the yellow channel, <0.07 in the red channel). The number and average area of clusters were computed for each condition. Raw images and ImageJ analysis macros will be made available on GitHub.

Data Analysis

Analysis of flow cytometry data and cluster statistics for experimental and simulated images were performed in MATLAB. Flow cytometry analysis was performed with a custom pipeline⁷³. Raw cytometry data and MATLAB analysis scripts will be made available on GitHub.

Mechanistic Model Description

A 3D agent-based model (ABM) was built using a previously established differential adhesion model⁹ with several modifications. Cells are modeled as rigid spheres of three cell types: blue, red, and yellow cells all with equal cell radius $\sigma$. The three cell types vary in adhesion strength, and are in ascending order: parent state as blue, Cadherin 6-expressing as red, and Cadherin 1-expressing as yellow, with yellow-yellow adhesion the highest. To match experimental conditions, the model was initialized with 1000 cells consisting solely of parent state blue cells with a fixed probability of transitioning to a cadherin expressing (red, yellow) state determined by the final fluorescence intensity expression of the corresponding Dox/ABA dosage experiment, $p_{totalcellscommitted}$. For computational efficiency, our simulations skip the first 24 hours in which we do not model cell state transitions, and model cell state transitions and resulting structure formation between 24 and 96 hours. The fixed transition probability at each 10 second time step $(x)$ is back calculated from the final fluorescence intensity expression (over 72 hours of simulated time, there are 72 * 60 * 6 = 25,920 time steps):

$${(1-x)}^{25,920}=\left(1-p_{totalcellscommitted}\right)$$

When a cell is indicated to transition to a cadherin expressing state, the probability of choosing the red or yellow states is a Bernoulli random variable that is also determined by the fluorescence intensity expression. The probability of choosing one of the two states is equal to the experimental intensity expression ratio of the respective cells over the total non-blue intensity expression. Cell locations are used for local cell interaction calculations, and are updated each time step (10 simulated seconds), at a fixed velocity of 0.6 * $\sigma$ (cell radius). The direction of each cell’s velocity vector is determined by the normalized sum of all forces acting upon each cell. These forces are Brownian motion, a gravity field, and the local cell-cell interactions of homotypic and heterotypic adhesion and steric repulsion.

All cells experience Brownian motion and a gravity field designed to mimic cell behavior in an ultra-low binding U-shaped bottom well. To represent the curvature of the well, the magnitude of the gravity field is dependent on the xy planar distance $\left(\left|r_{i,xy}\right|\right)$ from the center of the simulation space. $U_g$ represents the strength of the force of gravity and R is the radius of the well.

$$f_i=-U_g{r}_i\frac{\left|r_{i,xy}\right|}{R}$$

The presence of the pairwise forces is dependent on the distance $r_{ij}$ between two cells. At $\left|r_{ij}<1\right|$, two cells are overlapping, and each cell experiences a strong $\left(U_{hc}={10}^4\right)$ steric repulsion force to remove overlap of the rigid bodies.

$$f_{ij}=-U_{hc}\frac{r_{ij}}{\left|r_{ij}\right|}$$

At $\left|1<r_{ij}<3.2\ast \sigma \right|$, cells experience attraction due to cell-cell adhesion. This is described by

$$f_{ij}=U_{ij}\left(r_{ij}-r_e\right)\frac{r_{ij}}{\left|r_{ij}\right|}$$

Where $U_{ij}$ is the pairwise cell type specific adhesion parameter. There are 6 cell-type specific adhesion parameters, of which 3 are for the homotypic $\left(U_{bb},U_{rr},U_{yy}\right)$ adhesion forces, and 3 are for the heterotypic $\left(U_{br},U_{by},U_{ry}\right)$ adhesion forces. The equilibrium distance of intercellular interaction is denoted as $r_e$. The isotropic noise of the attraction force is represented with a Gaussian noise vector ${\xi }_{ij}$ with magnitude $\alpha$. At $\left|r_{ij}>3.2\ast \sigma \right|$, the cells are sufficiently far and do not interact. As cells sufficiently distant from each other do not influence each other, local interactions are identified through a fixed radius search of 3.2 * $\sigma$ for nearby cell bodies (as optimized in Tordoff et al⁹) and then subsequently divided into adhesion and repulsion forces. The list of parameters used for the model is included in TABLE 2. A representative simulation is shown in Supplementary Movie 2. Code generating all simulations can be found at https://github.com/kemplab/cho-adhesion-model.

Table 2.

Parameter values used in model.

Symbol	Parameter Description	Value
$\sigma$	Cell radius	5 μm
$v$	Cell velocity	0.6 * 𝜎 μm/time step
$U_{bb}$	Blue-blue cell adhesion strength	0.1 pN/μm
$U_{rr}$	Red-red cell adhesion strength	3 pN/μm
$U_{yy}$	Yellow-yellow cell adhesion strength	4 pN/μm
$U_{br}$	Blue-red cell adhesion strength	0.1 pN/μm
$U_{by}$	Blue-yellow cell adhesion strength	0.1 pN/μm
$U_{ry}$	Red-yellow cell adhesion strength	0.1 pN/μm
$U_g$	Gravity field strength	0.2 pN/μm
$U_{hc}$	Hardcore repulsion strength	103 pN/μm
$\alpha$	Magnitude of Brownian motion vector	10 pN
$r_e$	Equilibrium intercellular distance	10.1 μm
$R$	Well radius	650 * $\sigma$

Model Implementation

We implemented the 3D cadherin ABM using the PythonABM library. PythonABM is an installable library that provides a framework for building efficient ABMs in Python. This framework stores agent values through a series of NumPy arrays such that highly-parallelizable methods can distribute agent tasks across multiple processors for efficient, independent computations. Through the Numba library, PythonABM uses CPU/GPU parallelization to accelerate a fixed-radius neighbor search method that determines the neighboring agents, located within some distance, of each agent. Pairwise neighbor connections between agents are stored in index-based graphs from the iGraph library, which follows the indexing used to store agent values. PythonABM also includes default OpenCV support for 2D imaging of the simulation space at each step, and recently, 3D imaging has been introduced.

The framework provides a simple user-interface for starting new or continuing existing simulations, though there is built-in support for running simulations directly on the command-line. Users can specify simulation parameters through either command-line arguments, YAML template files, or directly in the model code. PythonABM includes multiple methods for initializing agent values such as importing data from CSV files. Moreover, PythonABM, by default, automatically saves agent values to CSVs at each step of a simulation along with an image of the simulation space.

PythonABM is hosted on Python’s package index, PyPI, such that the library can be installed through the command-line. For more information, PythonABM’s documentation describes the complete installation process and provides an example script for starting to use the framework. PythonABM is available at https://github.com/kemplab/pythonabm or https://pypi.org/project/pythonabm/.

To handle the computational load of 3D simulation, we developed a backend physics engine in Python to parallelize operations. The simulations were run for 72 simulated hours on the Phoenix High Performance Computing (HPC) Clusters provided by the Partnership for an Advanced Computing Environment (PACE) at the Georgia Institute of Technology, Atlanta, Georgia, USA. Typical simulations ran for ~6 hours.

Parameter Sweep:

Several parameters were derived from the literature: cell radius, well radius, alpha (random motion vector), equilibrium distance, and hardcore repulsion^9,74,75. We first attempted to estimate some cell-cell adhesion strengths from the literature. Wild-type CHO cells do not express cadherins^51,52, and mouse E- and K-cadherin have been demonstrated not to demonstrate binding in surface plasmon resonance experiments³⁹. Therefore, all adhesion strengths except $U_{rr}$ (red-red) and $U_{yy}$ (yellow-yellow) were initially set to zero. We next initially estimated $U_{rr}$ and $U_{yy}$ to be 7.3 pN/μm and 0.80 pN/μm, respectively, using the equations and binding constants from Katsamba et al³⁹. The results of these initial simulations, which are shown in Supplementary Figures 24–25, do not correlate well with experimental observations. Due to the highly complex nature of cell-cell adhesion strengths, including cadherin protein binding kinetics, cadherin protein surface expression levels, post-translational modifications to the cadherin protein, and signaling pathways induced by cadherin binding^76,77, we identified the literature-estimated adhesion parameters as a likely source of error. We then performed a parameter sweep to identify which combination of the different cell-type specific adhesion parameters $U_{yy}$, $U_{ry}$, etc, the attraction constant $U_g$, and the fixed velocity $v_{ij}$ best matched experimental results. During the parameter sweep, we first optimized $v_{ij}$ for consistent aggregation and sorting consistent with results from the previous 2D model⁹, considering values between 0.5 – 5 μm/time step (10% - 100% of cell diameter) in increments of 0.5, on simulations using cell state ratios from three representative inducer combinations: 2000 ng/mL Dox 150 μM ABA, 100 ng/mL Dox 50 μM ABA, and 500 ng/mL Dox 5 μM ABA,. $U_{bb}$, $U_{br}$, and $U_{by}$ were all set to 0.1 pN/μm to correspond to the low adhesion state observed in wild-type CHO cells. Next, we explored homotypic adhesion constants from 0–10 pN/μm in increments of 0.5, heterotypic adhesion constant 0–3 pN/μm in increments of 0.5, and gravitational constant 0–4 pN/μm in increments of 0.5. These parameters were tested combinatorially and sequentially on the three representative inducer conditions using a grid search and screened visually, and then viable parameter combinations were tested on cell state ratios corresponding to all inducer combinations in the experimental matrix. Analysis on number of clusters and cluster sizes for all inducer combinations, as described in the Methods section, was used to determine the optimal parameter combination.

Quantification and statistical analysis

Quantification was performed using custom code in MATLAB, which has been made publicly available at DOIs listed in the key resources table. Statistical analysis is described in figure legends. Number of samples are indicated whenever appropriate.

Supplementary Material

1. Supplementary Movie 1: Gaussian Mixture Model (GMM) to identify cell populations.

Related to Figure 2. Distribution of fluorescence as measured by EBFP (Pacific Blue-A), EYFP (FITC-A) and mKate (PE-A). MATLAB’s built-in GMM fitting function (fitgmdist) was used to cluster a combined population of all cells from all wells in a replicate into three populations. For ease of viewing, these plots show 10,000 cells randomly sampled from cells from all wells of each replicate.

2. Supplementary Movie 2: Timelapse of simulated symmetry breaking and self-assembly.

Related to Figure 4. 1000 blue cells were simulated undergoing symmetry breaking and self-assembly for 72 hours for the case of 50 μM ABA and 10 ng/uL Dox as described in the Methods and Supplemental Methods sections. The final images after 72 simulated hours are shown in Figure 4.

Highlights.

• Inducers trigger a genetic circuit to symmetry break at programmable ratios.

• Programmable symmetry breaking regulates cell-cell adhesion to generate 3D structures.

• A computational model predicts self-organizing 3D structures formed by the circuit.