Abstract
As developing tissues grow in size and undergo morphogenetic changes, their material properties may be altered. Such changes result from tension dynamics at cell contacts or cellular jamming. Yet, in many cases, the cellular mechanisms controlling the physical state of growing tissues are unclear. We found that at early developmental stages, the epithelium in the developing mouse spinal cord maintains both high junctional tension and high fluidity. This is achieved via a mechanism in which interkinetic nuclear movements generate cell area dynamics that drive extensive cell rearrangements. Over time, the cell proliferation rate declines, effectively solidifying the tissue. Thus, unlike wellstudied jamming transitions, the solidification uncovered here resembles a glass transition that depends on the dynamical stresses generated by proliferation and differentiation. Our finding that the fluidity of developing epithelia is linked to interkinetic nuclear movements and the dynamics of growth is likely to be relevant to multiple developing tissues.
Main
Cells within developing tissues reorganize at the same time as tissue growth takes place. The extent and dynamics of cell rearrangements can substantially change during tissue development^{1,2}, reflecting solid–fluid transitions in the physical properties of tissues. In most cases, these transitions have been proposed to result from alterations in cell density, cell motility, internal myosin and/or cadherinmediated adhesion forces at cell junctions, or external mechanical forces^{2,3,4,5,6,7,8}. Cell rearrangements have also been shown in theory and in some experimental situations to depend on active stresses within tissues, such as the ones generated by cell division^{9,10,11,12}. Yet, in many cases, the dynamics of cell rearrangements and the factors that control them are poorly understood.
The spinal cord of amniotes develops from a flat epithelial sheet—the neural plate—that folds to form a closed neural tube^{13}. These morphogenetic changes are accompanied by cell intercalations and convergent extension, which are mediated by planar cell polarity and actomyosindependent contractility of the apical adherens junctions, as well as basolateral protrusive activity^{14,15}. However, whether these are the only factors contributing to cell rearrangements in the neuroepithelium is an open question. Furthermore, the quantitative dynamics of cell rearrangements during development remain unclear. Here we use highly resolved clonal analysis to measure the rate of cell rearrangements in the mouse neuroepithelium over time, thus inferring the longterm rheological properties of the tissue. We further propose a theoretical framework for how active stresses generated during tissue growth contribute to cell rearrangements.
Cell rearrangements decline over time
To quantitatively measure cell rearrangements in the neural tube without the risk of perturbing the native mechanical environment of embryo growth in utero, we used clonal labelling to track how the positions of daughter cells that are initially adjacent change with respect to each other over time (Fig. 1a). A key aspect to achieve reliable tracing is the sparseness of labelling. Mosaic analysis with double markers (MADM)^{16,17} is a twocolour labelling system known for its sparseness. Therefore, we used Sox2–CreERT2induced MADM recombination (Fig. 1a and Supplementary Fig. 1a,b) to label cells with very low probability—we detected between one and five clones per spinal cord (Fig. 1c).
We induced MADM clonal labelling by injecting pregnant mothers with tamoxifen at embryonic days E8.5, E9.5 and E10.5, and harvested the embryos 24 h later. The cytosolic fluorescent reporters allow the labelled cells to be detected at the apical surface of the epithelium (Fig. 1b,d). In addition, immunostaining for the tight junction marker ZO1 allows us to segment individual cells and precisely determine the number and neighbour relationships of the labelled cells.
We focused our analysis on clones in the dorsal (pD) and intermediate (pI) progenitor domains, which span more than half the D–V length of the neural tube (Fig. 1b). The mean clone sizes of MADM clones decline from 4.1 ± 0.3 cells per clone at E8.5 to 2.1 ± 0.1 cells per clone at E10.5 (Supplementary Fig. 1c). This reflects a twofold decline in the tissue growth rate (from 0.087 ± 0.009 h^{−1} to 0.046 ± 0.004 h^{−1}), which is consistent with previous estimates^{18} (Supplementary Fig. 1d and Methods). The clone size distribution at E8.5 further shows that 2, 4 and 8 cell clones are the most abundant, indicating that cells divide up to three times and without substantial progenitor loss (Supplementary Fig. 1e). At E9.5 and E10.5, larger clones are progressively underrepresented, consistent with a longer cell cycle length and loss of progenitors due to terminal differentiation at these stages. Together, these observations indicate that the MADM clones accurately reflect the dynamics of tissue growth.
We next analysed the clonal shapes to estimate the extent of cell rearrangements. In many tissues, such as the Drosophila wing disc or mouse skin, uniform tissue growth with minimal cell rearrangements results in the formation of coherent clones^{19,20}. By contrast, cell rearrangements cause clone fragmentation, where subsets of labelled cells are surrounded by nonlabelled neighbours. We, therefore, used the number of fragments per clone as a readout of cell rearrangements. To exclude the effects of clone size, we measured the fragments for clones of a given size. The number of fragments linearly depends on the clone size for small clone sizes (≤4 cells) for which reliable statistics can be obtained (Fig. 1e). This allows us to define the fragmentation coefficient ϕ as the slope of a linear fit to the number of fragments as a function of clone size (for clone sizes ≤4 cells). We found that MADM clones labelled at E9.5 and E10.5 had very few fragments, corresponding to ϕ = 0.11 (95% confidence interval (CI) of 0.08 and 0.15) and 0.25 (95% CI of 0.21 and 0.30), respectively. By contrast, clones labelled at E8.5 were highly fragmented with ϕ = 0.61 (95% CI of 0.55 and 0.68) (Fig. 1e).
Consistent with their higher fragmentation, clones labelled at E8.5 had dispersed at a larger maximum distance from the clone centroid, namely, 10.2 ± 1.4 μm, whereas clones labelled at E9.5 and E10.5 dispersed up to 3.3 ± 0.4 and 3.0 ± 0.8 μm, respectively (Supplementary Fig. 1f). The dispersal of cells was nearly isotropic with respect to the clone centre, with the exception of clones in the motor neuron progenitor (pMN) domain, which have a larger A–P/D–V aspect ratio compared with clones in other domains at E10.5 of development (Supplementary Fig. 1g). This effect is consistent with previous observations and is related to the differentiation dynamics in the pMN domain^{21}. Altogether, these results indicate that cell rearrangements occur frequently before E9.5 and significantly decline at later stages.
Tissue fluidity at high junctional tension and contractility
To investigate how the high extent of cell rearrangements at early developmental stages is achieved, we used a twodimensional vertex model of the apical surface of the neuroepithelium^{21,22}. In this model, polygonal cells change neighbours by a process called T1 transition, in which an edge initially shared between two adjacent cells collapses and subsequently reforms in a different orientation, leading to the separation of the cells. The packing geometry of cells in vertex models depends on the normalized tension (\({{{\bar{\varLambda }}}}\)) and normalized contractility (\({{{\bar{\varGamma }}}}\)) parameters. In the classical vertex model formulation^{22}, cells have a constant target area. By contrast, in our model, the target area depends on the cell cycle time^{21}. This reflects the fact that cells in pseudostratified neural epithelia undergo interkinetic nuclear movements (IKNMs) during the cell cycle. In these movements, the position of the nucleus along the apicobasal axis of cells may affect the apical cell surface area. To verify that the IKNM effect we implemented in the model reflects the actual apicalarea cell cycle dynamics in the tissue, we measured the distribution of cell areas as a function of cell cycle phase at E8.5 and E10.5. To do this, we used short (20–30 min) EdU pulse labelling to mark the Sphase nuclei, 2 h EdU pulse to mark the G2 nuclei and phospho–histone 3 staining to mark cells undergoing mitosis (Fig. 2a and Methods). We combined EdU/pH3 immunostaining with sparse mosaic cytosolic tdTomato labelling to identify individual cell bodies, and with ZO1 immunostaining to measure the apical surface areas that correspond to specific nuclei. This analysis confirmed that the position of the nucleus relative to the apical surface changes with the cell cycle at both E8.5 and E10.5, reflecting the fact that the nuclei undergo IKNM (Fig. 2b). Furthermore, consistent with the model, we found that cells in mitosis have more than twofold higher mean apical cell area than cells in S phase, whereas the mean apical areas of cells in S and G2 phases were similar (Fig. 2c).
In other epithelia, fluctuations in the levels of myosin activity at cell junctions cause variation in the line tension and edge lengths of cells on a timescale of seconds to minutes^{23}. We reasoned that a similar effect might occur in the neuroepithelium. We, therefore, introduced linetension fluctuations in the model as an Ornstein–Uhlenbeck process. To this end, we introduced a noise term in \({{{\bar{\varLambda }}}}\), drawn from a Gaussian distribution with characteristic deviation time σ and temporal correlation time τ (Methods). Increasing values of σ shifted the distribution of edge length fluctuations in the simulations (Fig. 2d). Hence, to obtain an experimental estimate of σ, we performed shortterm live imaging of ZO1–GFPexpressing neuroepithelia at E8.5 and E10.5 of development (Methods and Supplementary Video 1). Although this procedure requires neuraltube dissection and the tissues can only be maintained live for 1–2 h, this approach provides an estimate of the variations in edge lengths that occur on shorter timescales. We observed that the distribution of edge length deviation during a 20 min interval corresponds most closely to simulations with σ = 0.02 at both E8.5 and E10.5 (Fig. 2d); hence, we used this value of σ in our subsequent analysis.
To determine the model parameters that reproduce the experimentally observed clone fragmentation, we performed a systematic screening of the intermediate region of the \({{{\bar{\varLambda }}}}\)–\({{{\bar{\varGamma }}}}\) parameter space, where the network configuration is expected to be the most similar to epithelial tissues^{22,24} (Supplementary Table 2). We used a proliferation rate of 0.09 h^{−1} (equivalent to a cell cycle length of ~8 h), which corresponds to the experimentally measured value at E8.5 (ref. ^{18} and Supplementary Fig. 1d). We traced clones in silico for 16 h, which corresponds to the duration of Cre activity in experiments (Supplementary Fig. 1b and Methods). The model revealed that the fragmentation coefficient varies across the (\({{{\bar{\varLambda }}}}\),\({{{\bar{\varGamma }}}}\)) parameter space (Fig. 3a). In particular, ϕ changes nonmonotonically along the \({{{\bar{\varLambda }}}}\) axis: it decreases, reaches a local minimum and subsequently increases with increasing values of \({{{\bar{\varLambda }}}}\) (Fig. 3a). To qualitatively capture these differences, we defined an arbitrary threshold value of ϕ = 0.3, which subdivides the parameter space into three subregions. We refer to these as regions A, B and C (Fig. 3a). Regions A and C have high fragmentation (ϕ ≥ 0.3) and high T1 transition rate, whereas region B has low fragmentation (ϕ < 0.3) and low T1 transition rate (Fig. 3a,b and Supplementary Videos 2–4). To further characterize the differences between regions, we compared the profiles of the selfoverlap function^{25,26}, which quantifies the fraction of cells that remain within approximately a cell radius of their relative initial position in the tissue. We found that these profiles are distinct in regions A and C compared with region B (Extended Data Fig. 1). Differences in the shape of the selfoverlap function have been associated with glassy dynamics in vertex models^{26}, suggesting that the differences between regions A, C and B represent transitions between fluidlike and solidlike states. Previous studies of vertex models have revealed that a densityindependent fluidtosolid phase transition^{1,7,22} characterized by a change in cell shape index occurs in a similar position in the parameter space to the transition between regions A and B that we observe. By contrast, the high rate of T1 transitions in region C has not been previously observed and is surprising, given that the ground state of the model in this region is solid^{22,24}.
The high fragmentation coefficient that we observed at E8.5 (Fig. 1e) is consistent with both high fragmentation regions A and C. Hence, more than one mechanism, captured by either region A or C, could explain how the high fragmentation rates are achieved at early developmental stages. To distinguish potential mechanisms and understand how fragmentation is achieved in the E8.5 neural tube, we compared the cell shapes in simulations of regions A versus C (Fig. 3c–f and Supplementary Fig. 2). Several firstorder descriptors of cell shapes (Supplementary Table 3) were similar between regions A and C. For instance, these regions were characterized by high cell shape index and low packing order, measured by the fraction of hexagons, which are indicators of tissue fluidity^{7} (Fig. 3d,e). By contrast, a subset of cell shape descriptors differed between regions A and C. These included the coefficients of variation (CV) of the cell area, perimeter and elongation, as well as the arearatio slope (Supplementary Fig. 2). The most striking difference between regions A and C was that only region C had high cell area CV, whereas in region A, the cell areas were nearly uniform (Fig. 3f).
Comparisons of cell shapes between model and experimental data have been used to infer the mechanical parameters of tissues^{27,28}. Therefore, to determine the (\({{{\bar{\varLambda }}}}\), \({{{\bar{\varGamma }}}}\)) parameter region characteristic of the E8.5 neuroepithelium, we immunolabelled the tight junctions in E8.5 neural plates and segmented the cell shapes (Fig. 3g). We found that for most cell shape descriptors, the best correspondence between data and simulations is in region C (Extended Data Fig. 2). A simultaneous comparison of a set of several descriptors confirmed that the best match to the experimental data is in region C (Fig. 3h). This suggests that the high fluidity of E8.5 epithelium is achieved in the regime of high junctional tension and contractility characteristic of region C. This is consistent with observations that the maintenance of high junctional tension is needed for proper neural tube closure at early developmental stages^{29,30}.
IKNM fluidizes the neuroepithelium
High fluidity in region C has not been previously observed; hence, we investigated how the high level of cell rearrangements in this region arises. Because the implementation of an IKNM effect and \({{{\bar{\varLambda}}}}\) noise are distinct features of our model, we first compared how the rate of T1 transitions depends on these features (Fig. 4a). In the absence of any cell divisions and noise, T1 transitions are not observed. In the absence of cell divisions, in the presence of only junctional noise with σ = 0.02, the rate of T1 transitions in region C was also zero, similar to what is expected from the solid ground state of the network in this parameter region. The implementation of cell division by IKNM without \({{{\bar{\varLambda }}}}\) noise resulted in a low T1 rate (<0.1 cell^{−1} h^{−1}). A classical implementation of cell divisions without an IKMN effect, but with linear cell area increase during the cell cycle, both with or without \({{{\bar{\varLambda}}}}\) noise, also resulted in a low T1 rate (<0.1 cell^{−1} h^{−1}) (Fig. 4a and Supplementary Fig. 3a,b). By contrast, the IKNM effect and \({{{\bar{\varLambda}}}}\) noise together increased the T1 rate to 0.37 ± 0.02 cell^{−1}h^{−1} and resulted in levels of clone fragmentation that are comparable with the experimentally observed value. This indicates that junctional noise on a timescale of minutes and fluctuations induced by IKNM (on a longer timescale of minutes to hours) cooperate to induce an increase in T1 rates that effectively fluidizes the tissue.
To further explore how IKNM is associated with T1 transitions, we analysed the quartets of adjoining cells undergoing T1 transitions in simulations. This revealed that T1 quartets have a distinct distribution of mean cell areas compared with random quartets of cells (Supplementary Fig. 3c). In particular, T1 quartets have, on average, one large cell and three smaller cells. Consistent with this distribution, a fraction of T1 transitions were followed by cell division of the largest cell in the simulations (Supplementary Fig. 3d). Nevertheless, the majority of T1 events did not coincide with cell divisions and could be either preceded or followed by cell divisions (Supplementary Fig. 3e). Consistent with this, we observed examples of T1 transitions occurring before cell division, after cell division, coincident with division or in the absence of cell division in shortterm liveimaging experiments of mouse embryos expressing ZO1–GFP (Supplementary Fig. 4a–d). Furthermore, cell divisions that we observed in timelapse imaging of neural epithelia mosaically expressing membranelocalized GFP (n = 17 dividing cells; Extended Data Fig. 3) were not associated with the separation or rearrangement of daughter cells within at least 30 min after cytokinesis. In addition, treatment with calyculin A, which leads to basolateral enrichment of Factin (Supplementary Fig. 4e) and has been shown to increase junctional stability at mitosis and prevent direct daughtercell separation on cytokinesis^{11} (Methods), does not affect clone fragmentation in the neural tube (Supplementary Fig. 4f). Altogether, these observations suggest that cell rearrangements are not driven by the mitotic cell or its daughter cells in a direct cell autonomous manner.
To further investigate how IKNM influences cell rearrangements, we asked if the high cell area heterogeneity in the presence of IKNM (Fig. 4b) is sufficient to account for the increased cell rearrangements in region C. To address this possibility, we simulated a tissue without IKNM, in which the cell area grows linearly during the cell cycle, but with target cell areas drawn from a random distribution with CV comparable with the experimentally measured one (linear + A0 noise condition (Methods)). These simulations show that increasing the target cell area heterogeneity is not sufficient to increase the rate of T1 transitions (Fig. 4b,c). An alternative possibility is that the specific kinetics of cell area increase during the cell cycle generated by IKNM leads to a higher rate of T1 transitions. Consistent with this idea, the apical target area that increases exponentially over the cell cycle can generate increased T1 transitions. Furthermore, the sharper the increase in exponential growth rate, the higher is the area heterogeneity and higher is the rate of T1 transitions (Fig. 4a,b). Altogether, this analysis suggests that the specific cell area dynamics during the cell cycle, that is, the sustained low cell area during interphase and rapid increase at mitosis, are crucial for cell rearrangements in region C.
Consistent with the model, our experimental data from EdU and pH3labelling experiments show that cells in G2 have similar apical areas to cells in the S phase, but lower than cells in mitosis (Fig. 2c). This argues against a linear increase in cell area during the cell cycle and suggests that the apical cell area rapidly increases during cell division. Timelapse imaging of neural epithelia expressing membrane GFP confirmed that the subapical cell area increases several times within less than 60 min before cytokinesis (Extended Data Fig. 3). These kinetics are similar to the rapid increase in apical area observed before cell division in simulations (Fig. 4c). Altogether, these data support the results of the model and indicate that the kinetics of cell area variability that fluidizes the neuroepithelium is driven by the cell cycle and reflects the changing apicobasal nucleus position during IKNM.
Cell cycle dynamics influence cell rearrangements
Despite the presence of IKNM throughout development, the extent of clone fragmentation declines after E8.5, which raises the question of how this change is regulated. One possibility is that the mechanical parameters (\({{{\bar{\varLambda}}}}\) and/or \({{{\bar{\varGamma}}}}\)) change over time, such that the tissue ends up in the solidlike region B at later stages. To test this possibility, we performed laser ablation of individual cell junctions in E8.5 and E10.5 neural tubes which expressed ZO1–GFP (Fig. 5a,b, Supplementary Videos 6 and 7 and Methods). We observed no significant difference in the initial recoil velocity of vertices following laser ablation between the two developmental stages, suggesting that the active tension at these stages is similar (Fig. 5c). Further supporting this conclusion, an analysis of the cell shapes in neuroepithelia from E9.5, E10.5 and E11.5 embryos revealed that the experimentally observed cell shapes are consistent with parameter values characteristic of region C (Supplementary Fig. 5). These results suggest that changes in \({{{\bar{\varLambda}}}}\) and \({{{\bar{\varGamma}}}}\) are not the major factors underlying the change in tissue fluidity over time.
The impact of IKNM on clone fragmentation revealed by our model suggests that the cell division rate could be critical for regulating the extent of cell rearrangements by controlling the level of active stresses that generate fluctuations in the tissue. Between E8.5 and E10.5 of development, the proliferation rate decreases and terminal differentiation commences^{18}, which lowers the net tissue growth rate by about twofold (Supplementary Fig. 1d). To test whether this could lead to tissue solidification, we lowered the proliferation rate in the vertex model simulations from 0.09 to 0.03 h^{−1}. This resulted in a strong decline in the fragmentation coefficient of clones throughout most of the (\({{{\bar{\varLambda}}}}\),\({{{\bar{\varGamma}}}}\)) parameter space (Fig. 5d and Supplementary Fig. 6a). In region C, ϕ declined by about twofold and a corresponding decline in T1 rates was observed (Supplementary Fig. 6b), whereas the cell area CV was reduced to a lesser extent and remained significantly higher than in region B (Supplementary Fig. 6c). This reduction in fragmentation coefficient in the model is reminiscent of the experimentally observed reduction in ϕ (Fig. 1e), suggesting that the decreasing rate of proliferation over time is a key driver of the decline in cell rearrangements.
This analysis predicts that artificially lowering the proliferation rate would lead to a lower level of cell rearrangements. To test this, we induced Confetti clones at E7.5 and then cultured the embryos from E8.5 in the presence of cell cycle inhibitors lmimosine or aphidicolin for 42 h (Fig. 5e and Supplementary Fig. 7a). As expected, these treatments resulted in reduced mean clone sizes compared with control embryos (Fig. 5f and Supplementary Fig. 7b). Crucially, a comparison of the inhibitortreated with vehicletreated control embryos showed that for a given clone size, the number of fragments per clone was significantly reduced in both lmimosine and aphidicolintreated conditions (Fig. 5g and Supplementary Fig. 7c). These results are in agreement with the model prediction and confirm that the proliferation rate has a profound influence on the extent of cell rearrangements in the neuroepithelium.
Besides the rate of proliferation, the overall rate of tissue growth can also be affected by cell loss. From E9.5 to E10.5 of development, terminal differentiation in the pMN domain results in the loss of progenitors from the neuroepithelium and also contributes to lowering the growth rate in this domain^{18}. To test the effect of progenitor cell loss by terminal differentiation on cell rearrangements, we modelled cell loss in silico by randomly assigning a zero target area to a fraction of cells (Methods). These simulations showed that cell loss leads to a decrease in clone fragmentation throughout the (\({{{\bar{\varLambda }}}}\),\({{{\bar{\varGamma }}}}\)) parameter space (Supplementary Fig. 7d). Consistent with this prediction, Confetti clones in the pMN domain have significantly lower fragmentation coefficients than clones in the pD domain, where the differentiation rate is lower in this time interval (Supplementary Fig. 7e). This indicates that in addition to decreasing the proliferation, the increasing terminal differentiation rate also contributes to reduce the extent of cell rearrangements in the neural epithelium over developmental time. Altogether, these observations strongly suggest that the increasing solidification of the mouse neuroepithelium over time is controlled by the observed changes in cell cycle dynamics over time.
Changes in the tissue growth rate have previously been linked to alterations in anisotropic growth and tissue morphogenesis^{21,31}. To understand the consequences of reduced proliferation and therefore cell rearrangements for tissue morphogenesis, we treated E8.5 embryos with aphidicolin for 8 h. This resulted in striking changes in the shape of the neural plate, where the treated embryos had an increased ratio of anterior–posterior to dorsoventral length of the neural plate (Supplementary Fig. 8). This result is consistent with previous predictions of our model^{21} and indicates that there is an inherent link among tissue fluidity, growth rate and tissue shape.
Conclusion
Morphogenetic processes have been recently linked to transitions in the material properties of tissues^{32}. Here we demonstrate that in the mouse neural tube epithelium, there is a significant decline in tissue fluidity around E9.5 of development. Our data suggest that this decline resembles a glass transition, controlled by changes in active stresses within the tissue. We show that in the neuroepithelium, active stresses are generated by IKNM during the cell cycle. Consequently, the proliferation rate determines the extent of cell rearrangement and tissue fluidity.
Previous studies have shown that tissue rheology can change in the absence of noise or fluctuations^{3,4,33,34}. In many cases, transitions in fluidity have been linked to changes in cell density or in mechanical properties, such as cell adhesion, cortical tension and contractility^{2,3,5,6,33,35,36}. Tissue fluidity can also decline as a result of T1 delay times or nonlinear junction behaviours^{37,38,39}. By contrast, the role of cell divisions, apart from a few experimental examples^{3,10,11} and theoretical predictions^{9,40,41,42,43}, has been largely underappreciated. Slowing down of growth is a hallmark of development and has been measured in multiple tissues^{44,45,46}. Our findings, therefore, suggest that increases in tissue rigidity over developmental time could be a natural consequence of the cell cycle dynamics in many tissues.
IKNM is the characteristic of many epithelia^{47,48,49}, and hence, our finding that IKNM fluidizes the epithelia might be relevant to other tissues besides the neural tube. Our analysis indicated that IKNMs exert their effect on cell rearrangements by affecting the apical surfacearea kinetics of cells, which results in large cell area variation. Interestingly, in the Drosophila wing disc, the presence of smallerthannormal mutant cells has been shown to induce clonal fragmentation^{19}. Yet, in our analysis, imposing ‘static’ cell area variation did not lead to high clonal fragmentation, indicating a distinct mechanism where the kinetics of cell area changes are crucial for epithelial rearrangements.
Methods
Experiments
Mouse strains and generation of clones
All the animal procedures were performed in accordance with the relevant regulations and were approved under the license BMWFW66.018/0006WF/V/3b/2016 from the Austrian Bundesministerium für Wissenschaft, Forschung und Wirtschaft. The following strains were previously described: MADM11^{TG} and MADM11^{GT} (ref. ^{17}), Rosa26–Confetti (Brainbow2.1 (ref. ^{50})), Sox2–CreERT2 (ref. ^{51}), mTmG (ref. ^{52}), R26–ZO1–GFP (ref. ^{53}), Rosa26–tdTomato (ref. ^{54}). To generate MADM clones in MADM^{TG/GT} transheterozygous Sox2–CreERT2expressing embryos, MADM11^{TG/TG} mice were bred to MADM11^{GT/GT} and Sox2–CreERT2/+, and pregnant females were injected with 3 mg per mouse of tamoxifen. To generate Confetti clones, heterozygous Sox2–CreERT2 mice were bred to heterozygous Rosa26–Confetti and pregnant mothers were injected with 0.75 mg per mouse of tamoxifen. Tamoxifen stock was prepared fresh in sunflower oil.
The first time point where we observe labelled cells is 8 h after tamoxifen injection (Supplementary Fig. 1b), reflecting the time it takes for the nuclear translocation of Cre and subsequent onset of reporter expression. Thus, the time of Cre activity in the 24 h tracing experiments (Fig. 1) is considered to be 16 h.
Immunohistochemistry, EdU incorporation and imaging
For E9.5 and later stages, embryos were bisected along the roof plate before fixation and along the floor plate before immunostaining. Embryos were fixed in 4% paraformaldehyde and subsequently in methanol. Primary and secondary antibody incubations were 24 h each. Washes in phosphatebuffered saline with 0.1% Tween were 10 h each. The brachial region was flat mounted with grease spacers between slide and coverslip. Primary antibodies used were mouse antiZO1 (Invitrogen, 1:90), goat antiOlig2 (R&D Systems, 1:100), sheep antiGFP (AbD Serotec, 1:1,000), rabbit antiRFP (Rockland, 1:2,000), mouse antiNkx2.2 (DSHB, 1:20), rat antipH3 (Sigma,1:1,000), goat antiSOX2 (R&D Systems, 1:100), rabbit antiBrachyury (Abcam, 1:100). Secondary antibodies used were donkey antimouse Alexa Fluor 647 and donkey antigoat FITC (Jackson Immuno, 1:250), donkey antirabbit Cy3 and donkey antirat Cy3 (Jackson Immuno, 1:1,000), donkey antisheep FITC (Jackson Immuno, 1:250).
For MADM clone analysis, embryos were immunostained against ZO1, RFP, GFP and Nkx2.2. Clones located within 25 μm dorsal to the Nkx2.2 domain boundary were considered to be pMN clones. For Confetti clone analysis, embryos were immunostained against ZO1 and Olig2. To stain actin filaments, the following steps of the protocol were modified: embryos were fixed in 4% paraformaldehyde overnight, methanol fixation was omitted and Alexa Fluor 488 Phalloidin (Thermo Fisher Scientific, 1:100) was added together with the secondary antibody.
For EdUlabelling experiments, Sox2–CreERT2 mice were bred to ROSA26–tdTomato. Pregnant mice were intraperitoneally injected with 3 mg tamoxifen at E6.5 and with 0.5 mg EdU in phosphatebuffered saline (stock, 2.5 mg ml^{–1}) on the day of the experiment. For Sphase labelling, mice were sacrificed 20 min after EdU injection at E8.5 and 30 min after injection at E10.5. To label cells in G2, mice were sacrificed 2 h after EdU injection. The embryos were dissected, fixed and immunostained against ZO1 and RFP, as described above. Subsequently, the incorporated EdU was detected using the Alexa Fluor 488 ClickiT EdU imaging kit and protocol (Invitrogen, C10337).
Imaging was performed using a 40×/1.3 numerical aperture oil objective on an LSM880 inverted confocal microscope. Images of the apical surface capturing the entire dorsoventral length of the epithelium were acquired through tile scanning with Z slices 0.8 μm apart. The tiles were configured in the form of a grid and overlapped 10%. Subsequently, the tiles were stitched using the BigStitcher plugin in Fiji version 2.9 (ref. ^{55}).
Mouse embryo culture and inhibitor treatments
To combine clone tracing with mouse embryo culture and inhibitor treatments, heterozygous Sox2–CreERT2 mice were bred to Rosa26–Confetti. To induce sparse labelling (see the ‘Clone identification and fragmentation coefficient estimation’ section), pregnant mothers were injected with 0.75 mg per mouse of tamoxifen at E7.5. After 24 h, at E8.5, embryos were dissected and cultured with their yolk sac intact in temperaturecontrolled roller culture^{56} (5% CO_{2} and 20% O_{2}). The embryo culture medium consisted of 1:1 rat serum: dissection medium^{57} (Gibco DMEM/F12 without phenol red (Thermo Fisher), 10% Gibco foetal bovine serum (Thermo Fisher), 1× penicillin–streptomycin (Sigma)). To perturb proliferation, embryos were cultured in the presence of 210 μM lmimosine (Sigma) or 800 nM aphidicolin (Sigma) for 42 h. Calyculin A (Merck Millipore) was used at a final concentration of 0.6 nM for 42 h of culture. After culture, the embryos were harvested and processed for imaging, as described above.
Laser ablation
Embryos heterozygous or homozygous for R26–ZO1–GFP were collected at E8.5 and E10.5 of development. To perform laser ablation, whole E8.5 embryos or dissected flatmounted E10.5 brachial neural tubes were immobilized for live imaging in glassbottom dishes (Ibidi) in embryo culture medium (see ‘Mouse embryo culture and inhibitor treatments’ section) supplemented with 8 mg ml^{–1} fibrinogen (Millipore). Thrombin (0.5 U μl^{–1}, Sigma Aldrich) was added to form a fibrin gel^{18}. The samples were kept in an environmental chamber with 5% CO_{2} at 37 °C. Laser ablation was performed on an Andor spinningdisc system with inverted Axio Observer Z1, CApochromat 63×/1.2 waterimmersion objective (Carl Zeiss) using a 355 nm pulsed UVA nanolaser (Teem Photonics) at 1.8–1.9% laser power with 25 pulses (2 shots µm^{–1}) at 1,000 Hz. Junction ablation was performed approximately at the centre of the cell edge between two vertices along a 2 µm line oriented along the A–P axis. Edges oriented dorsoventrally and located at intermediate dorsoventral positions were selected for the experiment. Images were collected with 250 ms exposure time and frame rate of 1.5 s. To determine the recoil velocities, the positions of the vertices were manually tracked over time in Fiji. The initial recoil velocity was defined as the distance between the vertices at t1 minus the distance at t0 (before the cut), divided by the time interval between t0 and t1 (1.5 s).
Live imaging
To image the apical surface of the neuroepithelium at the level of tight junctions, embryos heterozygous or homozygous for ZO1–EGFP were collected at E8.5 and E10.5. Whole E8.5 embryos and bisected E10.5 brachial neuraltube regions were positioned on 35 mm glassbottom dishes (Ibidi) in an embryo culture medium (see the ‘Mouse embryo culture and inhibitor treatments’ section) and immobilized with coverslips on silicon grease spacers. Imaging of the ZO1–GFPlabelled apical surfaces was performed on an inverted LSM800 confocal microscope using a 40×/1.2× water objective. The Z stacks were acquired with Z slices 0.75 μm apart for a period of about 1–2 h.
Imaging of the cell membranes (marked by membrane GFP) at the subapical level was performed using E10.5 Sox2–CreERT2/+, mTmG/+ embryos of mothers injected with 1 mg tamoxifen 24 h before dissection. Brachial neuraltube regions were dissected and prepared for imaging the same way as that for laser ablation experiments. The Z stacks were acquired with Z slices 0.7 μm apart.
Data analysis
Clone identification and fragmentation coefficient estimation
Images were processed in Fiji. Labelled progenitor cells were manually marked at their apical surface at the level of ZO1 staining. Fragments were defined as groups of adjacent cells that share an edge or a vertex. Clones were defined as groups of labelled progenitor cells in close proximity of each other (<25 μm to the nearest labelled cell).
In the case of MADM clones, G2 recombination followed by the X segregation of chromosomes in mitosis produces a GFP and RFPexpressing daughter cell (Fig. 1a). Both GFP and RFPexpressing cells were included in defining a clone. G2–Z segregation produces an unlabelled and a GFP/RFPcoexpressing daughter cell. G1 recombination produces a GFP/RFPcoexpressing cell. In our dataset, we found an increasing proportion of GFP/RFPcoexpressing cells over time (Supplementary Fig. 1a), which correlates with the increasing relative G1 duration over time^{18}. This suggests that the majority of GFP/RFPcoexpressing clones result from G1 recombination. We, therefore, also included GFP/RFPcoexpressing clones in the analysis. MADM clones and the fragments they contain were manually identified from images.
In the case of Confetti clones, only the RFP, YFP and CFP reporters, which can be detected at the apical surface, were used for the analysis, whereas clones labelled by the nuclear GFP were excluded. The sparseness of labelling in the experiments was as follows: 322 ± 38, 366 ± 39 and 235 ± 36 cells mm^{–2} for CFP, RFP and YFP, respectively (mean ± standard error of the mean (s.e.m.) for 54 images is given). To identify Confetti clones, the cell coordinates were recorded and subsequently analysed using a custombuilt Python script, similar to the one reported elsewhere^{18}. CFP, RFP and YFP channels were separately analysed. The labelled cells were assigned to the same fragment if the distance between them was <5 μm and to the same clone if they were <25 μm apart. These assignments were consistent with the visual identification of fragments and clones, except in occasional cases where the clone size was unreasonably large. Labelled postmitotic neurons that have delaminated from the neural epithelium were excluded from the analysis.
We determined the fragmentation coefficient ϕ by fitting f = ϕs + b to the respective dataset, where f is the mean number of fragments for a given clone size, s is the size of the clone in cells and b is an offset parameter that is chosen in such a way that the line crosses through the point (1, 1), reflecting the fact that singlecell clones have one fragment by definition. For MADM clones analysed 24 h after tamoxifen injection, reliable statistics could be obtained for clones with four cells or less; hence, only these clone sizes were used for estimating ϕ. In mouse embryo culture experiments, Confetti clones were analysed 64 h after tamoxifen injection. In this case, reliable statistics could be obtained for clone sizes of ≤8 cells and these were used to estimate ϕ.
Growth rate estimation
The growth rate of MADM clones k_{g} was inferred from the mean clone size s as k_{g} = ln(s)/∆t , where ∆t = 16 h is the time interval of Cre activity in the experiments (Supplementary Fig. 1b–d and Methods).
Spread and anisotropy of clones
To estimate the spread of clones, the coordinates of cell centres in a clone were used to determine the clone centroid. The maximum spread of the clone was quantified as the distance between the clone centroid and furthest cell centre. To estimate the mean maximum spread for a given developmental stage, clones of all sizes were taken into account (including singlecell clones).
Clone anisotropy was quantified by drawing a bounding rectangle around the clone, using the apical cell outlines, marked by ZO1, to demarcate the cells. Images are always oriented so that the vertical axis is aligned with the tissue D–V axis. The aspect ratio of the clone is then given by the D–V to A–P side lengths of the bounding rectangle. Note that quantifying the clone shape at the apical surface, rather than the cell bodies or nuclei, avoids potential artefacts of tissue mounting, where the clone shape could be affected by the misalignment of the apical and basal surfaces of the neural epithelium.
To quantify the selfoverlap function in simulations, we adapt the definition from other work^{25,26} for a growing tissue. The selfoverlap function is defined as \(Q_\mathrm{s}\left( t \right) = \frac{1}{N}\mathop {\sum }\nolimits_{k = 1}^N w\left( {\left {\widetilde {r_k}\left( t \right)  \widetilde {r_k}\left( 0 \right)} \right} \right)\), where \(\widetilde {r_k}\left( t \right)\) is the position of the centre of mass of the kth cell at time t, w is a window function that gives 1 for \(\left {\widetilde {r_k}\left( t \right)  \widetilde {r_k}\left( 0 \right)} \right \le \tilde R_{\mathrm{cell}}\) and 0 otherwise, and N is the number of cells. The \(\tilde R_{\mathrm{cell}}\) value is the characteristic length that corresponds to the initial cell radius in the reference frame defined below. To correct for growth, we consider every cell trajectory in a reference frame that is centred at the initial position of that cell and normalized by the fastest growing dimension. In practice, we apply the following two steps: (1) for every newborn cell with position (r_{D–V}, r_{A–P}) at time t = 0, we shift the corresponding cell trajectory to start in (r_{D–V}, r_{A–P})→(0, 0); (2) we then normalize the cell position over time by dividing its position by the D–V and A–P extensions of the growing tissue with the correction from the anisotropic growth, that is, \(\widetilde {r_k}\left( t \right) = (\frac{{r_{\rm{DV}}}}{{L_{\rm{DV}}}},\frac{{r_{\rm{AP}}}}{{L_{\rm{AP}}}}\langle\frac{{L_{\rm{AP}}}}{{L_{\rm{DV}}}}\rangle)\), where L_{D–V} and L_{A–P} are the respective tissue dimensions at time t, and \(\langle\frac{{L_{\rm{AP}}}}{{L_{\rm{DV}}}}\rangle\) is the average A–P/D–V ratio for a given simulation. The \(\tilde R_{\mathrm{cell}}\) value is defined as \(\frac{{\sqrt {\langle A\rangle }}}{2}/L_{\rm{DV}}\), where 〈A〉 is the average cell area at time t = 0, and L_{D–V} is taken at t = 0.
Segmentation of cell shapes
Apical surfaces immunostained for ZO1 of the E8.5, E9.5, E10.5 and E11.5 neural tubes were segmented using the Tissue Analyzer plugin^{58} in Fiji. The cell outlines were automatically identified and manually checked for correctness. This plugin provided the description of a polygonal mesh including the vertices and edges of cell outlines as well as the number and identity of cell neighbours. The cell area was calculated using a standard formula for the area of the ngon and cell perimeter as a sum of the length of polygon edges. The cell elongation was calculated as done elsewhere^{21,27}.
In the EdU and pH3labelling experiments, the distance between the nucleus centre and ZO1labelled junctional level was measured using Imaris 9.1 (Oxford Instruments) from confocal Z stacks. To associate EdUlabelled nuclei with the corresponding apical surfaces of cells, cells that expressed cytosolic tdTomato and therefore allowed tracing the cell body were used for the analysis.
Cell areas in the EdU experiment (Fig. 2) and membraneGFPexpressing embryos in live imaging (Extended Data Fig. 3) were quantified using Imaris 9.1. For this, the cells were segmented using a watershed function. Subsequently, the cell area was estimated from the segmented cell volume within the relevant Z slice divided by the voxel depth. For the EdU experiments, the relevant Z slice is the one marked by the ZO1 expression. For the membraneGFPexpressing embryos, the cell areas were quantified at the Z position corresponding to 2.1–2.8 μm below the apical cell membrane.
To quantify the variation in edge length over time, timelapse images of ZO1–GFPexpressing neural tubes were segmented in Fiji in the same way as the fixed images. To avoid large fluctuations in edge length that arise as a result of cell divisions, T1 transitions or segmentation errors, we only quantified the edges that could be tracked throughout the duration of the experiment and were at least one cell away from the cells undergoing large fluctuations.
Simulations
Vertex model description and implementation
The vertex model used in this study is based on another work^{21} and was implemented here using Python 3.7. Briefly, the following energy function is minimized in every simulation step:
where α = 1,…, N_{c} enumerates all the cells; i = 1,…, N_{v} enumerates all the vertices; K_{α} is the elasticity coefficient; A_{α} is the area of cell α; \(A_\alpha ^0(t)\) is the preferred area of cell α at time t; Λ_{ij} is the linetension coefficient associated with the cell edge between i and j of length l_{ij}; and Γ_{α} is the contractility coefficient of cell α with perimeter L_{α}. We assume that the parameters are the same for each cell (K_{α} = K, Γ_{α} = Γ) and for each edge (Λ_{ij} = Λ) if no noise in the line tension is considered. The preferred cell area \(A_\alpha ^0(t)\) is a piecewise linear function reflecting the effect of IKNM on the apical cell area in the four phases of the cell cycle (G1, S, G2 and M; also see below). Adopting the same notation as previous studies^{21,22}, we use the normalized parameters as \({{{\bar{\mathrm \Lambda }}}} = \frac{{\Lambda }}{{K\left( {A^0} \right)^{3/2}}}\) and \({{{\bar{\mathrm \Gamma }}}} = \frac{{\Gamma }}{{KA^0}}\), where A^{0} is the average target area during the cell cycle.
The motion of vertices is determined from the firstorder kinetics: \(\frac{{\mathrm{d}r_i}}{{\mathrm{d}t}} =  \frac{1}{\mu }\frac{{\partial E}}{{\partial r_i}}\), where r_{i} is position of vertex i and μ is the drag coefficient. Tissue growth was considered to be anisotropic with drag coefficients μ′ and μ″ in the D–V and A–P directions, respectively^{21}.
The following changes were made in the current version of the model:
Implementation of junctional noise
We considered that fluctuations in the internal line tension follow an Ornstein–Uhlenbeck process, namely, \(\frac{{\mathrm{d}{{\Lambda }}_{ij}}}{{\mathrm{d}t}} =  \frac{1}{\tau }\left( {{{\Lambda }}_{ij}  {{\Lambda }}_{ij}^0} \right) + {{{{\xi }}}}_{{{{{ij}}}}}\left( t \right)\), where ξ_{ij}(t) is white, uncorrelated noise with ‹ξ_{ij}(t) = 0› and \(\langle\xi _{ij}\left( t \right)\xi _{kl}\left( {t^\prime } \right) \rangle= \frac{{2\sigma ^2}}{\tau }\delta _{ik}\delta _{jl}\delta (t  t^\prime )\). We used the following discretization^{23}:
where Δt is a time step used in the simulation, τ is the linetension correlation time, σ is the intrinsic linetension deviation, \(\Lambda_{ij}^{0}=\Lambda\) is a reference line tension that corresponds to the line tension without noise and \(\overline {\xi _{ij}} \left( t \right)\) is drawn from the Gaussian distribution N(0, 1).
Implementation of T1 and T2 transitions
The T1 transition is defined elsewhere^{21}. In particular, when an edge between two neighbouring cells is shorter than a predefined small length l_{T1}, this edge is replaced with a new edge that is perpendicular to the old edge and has a length l_{new} = 1.01l_{T1}. Using this definition, we observed that for negative line tension or in the presence of linetension fluctuations, immediately after a T1 transition, the new edge can shrink instead of extending, thus leading to a reverted T1 transition. This can occur multiple times at a given edge, and hence, we call this an oscillatory T1 transition.
One strategy to partly mitigate the occurrence of oscillatory T1 transitions is to increase the l_{new}/l_{T1} ratio^{59}. However, particularly in region A, the oscillatory T1 transitions are generic and increasing l_{new}/l_{T1} does not result in decreasing the number of oscillatory T1 transitions. Therefore, we approached this instead by keeping track of the oscillatory T1 transitions and subtracting them from the overall count of T1 events in our statistics. More specifically, we track how many T1 transitions occurred for every edge, using the dictionary data structure in Python. If repeated T1 transitions occurred n_{T1} times between time t_{0} and t_{nT1}, their contribution to the T1 unique rate, namely, T1_{UNQ}, was considered to be 1/n_{T}_{1} for times between t_{0} and t_{nT1}.
T2 transitions are defined in another work^{21}. In particular, cells in which the area becomes very small have shrinking edges. This results in sequential T1 transitions, which finally lead to a doublesided cell with zero area. Such cells are removed from the simulation by merging the two vertices that delimit the doublesided cell into one vertex. The last T1 transition that results in a doublesided cell is counted as a T2 transition, and is not included in the overall number of T1 transition events. All the T1, T1_{UNQ} and T2 rates are estimated in time windows of Δt = 2 h by the dividing number of respective events with the average number of cells in this time window. The T1 rate reported in the main text as well as in Figs. 3b and 4a and Supplementary Fig. 6b is defined as the T1 unique rate.
Cell removal from the tissue through differentiation is implemented similar to another work^{21} with the additional requirement that if the cell was randomly selected for differentiation, the linetension coefficients Λ_{ij} for this cell are no longer fluctuating and have assigned a positive value of Λ_{ij} = 0.2, which fosters shortening of all the edges of this cell.
Cell lineage tracing
To efficiently analyse in silico clonal populations, the complete information about cell lineage, that is, daughtercell identifiers and division times, are stored. For the analysis of clone fragmentation in silico, we used all the clones per simulation and ten independent simulations per parameter set.
Parameters of the model
The used parameters are summarized in Supplementary Table 2. The default proliferation rate in the model k_{p} = 0.09 h^{−1} has been chosen to match with the experimentally observed tissue growth rate (Supplementary Fig. 1d): it results in mean sizes of simulated clones of three to four cells (depending on the exact value of \({{{\bar{\Lambda }}}}\) and \({{{\bar{\varGamma }}}}\)), which is similar to the clone sizes observed in injection at E8.5 (Supplementary Fig. 1c). The specific proliferation (k_{p}) and differentiation (k_{n}) rates used in the simulations are given in the corresponding figure legends. The critical area A_{C} has been set to 27 µm^{2} so that the range of cell areas in the simulations (Supplementary Fig. 2a) is comparable with the range of areas measured in the experiments (Supplementary Fig. 5b). Furthermore, A_{C} = 27 µm^{2} results in a very close agreement between the mean edge length in simulations and in the E10.5 experimental data (1.47 ± 0.01 μm and 1.43 ± 0.01 μm (mean ± s.e.m.), respectively; Fig. 2d). The length of a simulation step Δt = 0.29 s has been chosen such that the model has high temporal resolution and includes on the order of 10^{5} simulation points per cell cycle (10^{5} points correspond to 8 h). The units of force are arbitrary. Every data point across the (\({{{\bar{\Lambda }}}}\),\({{{\bar{\varGamma }}}}\)) parameter space was obtained by pooling together cells from ten independent simulations for a given set of parameters \(({{{\bar{\Lambda }}}},{{{\bar{\varGamma }}}},k_\mathrm{p},k_\mathrm{n})\).
Vertex model initialization and in silico clone tracing
The vertex model is initiated with a regular hexagonal lattice of ten rows with ten cells per row. In the initial simulation phase, the tissue grows for 16 h with \({{{\bar{\Lambda }}}}\) = −0.184, \({{{\bar{\varGamma }}}}\) = 0.07, k_{p} = 0.09 h^{−1}, k_{n} = 0 h^{−1} and σ = 0. In our default simulations, we used distinct drag viscosities for the D–V and A–P dimensions, resulting in rectangular tissues with A–P/D–V length ratios of <1 at the end of the simulations (Supplementary Videos 2–4). After the initial simulation phase, the number of cells is 460 ± 19 (mean ± standard error), the time is set to 0 and the parameters are modified to the target simulation parameters. The tissue is then allowed to grow with the target simulation parameters for 8 h. Subsequently, clones are labelled and clonal populations of cells are tracked for 16 h.
Analysis of cell area heterogeneity and kinetics
To investigate the area variability of cells undergoing a T1 transition, we defined quadruplets of neighbouring cells, designated as A, B, C and D, where A and B share a common edge and C and D do not (Supplementary Fig. 4c). We further define the cell names based on the cell area such that area of A < area of B and area of C < area of D. If the common edge between A and B shrinks below l_{T1}, a T1 transition takes place, as a result of which A and B are no longer adjacent, whereas C and D become new neighbours. For comparison, ‘random’ quadruplets are generated by randomly finding A and B cells separated by a common edge, and finding cells C and D that are adjacent to A and B, but not to each other. Note that because the polygonal mesh has no rosettes, that is, each vertex has three edges associated with it, the assignment of a quadruplet to an edge is unique.
Cell area kinetics during cell cycle and cell division
The IKNM is approximated as a linear combination of two terms, one corresponding to a linear increase in cell volume and the other interpolating for the change in apical cell surface as a function of the age of a cell:
where g_{α} is the growth rate of the cell α, Δt = t – t_{new} is the age of the cell that divided at t_{new} and ρ_{α}(Δt) is a piecewise linear function representing the apical–basal position of the nucleus as a function of Δt and cell cycle phase^{21}. This function equals to zero in the S phase of the cell cycle in which the nucleus stays basal, and its value is 1 during mitosis when the nucleus is apical. The exact form of ρ_{α}(Δt) is defined elsewhere^{21}. The growth rate g_{α} is drawn from a normal distribution with mean equal to 1/t_{T}, where t_{T} is the total cell cycle time, and standard deviation σ_{g} = 0.45/t_{T}. Negative growth rates are not allowed. In the simulation, the proliferation is defined as λ = ln(2)/t_{T}. The cell divides when the cell is in the M phase, that is, \(\frac{{{{\Delta }}t}}{{t_\mathrm{T}}}\) > 0.9, and the cell volume exceeds a critical value, namely, \(A_\alpha ^0\left( t \right)\) > A_{C}. The cell divides by introducing a new edge that splits this cell into two daughter cells that enter the next cell cycle^{21}.
To study how the cell area kinetics affects T1 transitions, we use different forms of \(A_\alpha ^0\left( t \right)\) and different levels of linetension noise σ. In Fig. 4, different conditions for area kinetics are defined as follows:
‘N/A’, no division, σ = 0.02
‘IKNM’, \(A_\alpha ^0\left( t \right)\) as in equation (3), σ = 0;
‘Linear’, cell divisions without IKNM (that is, ρ_{α}(Δt) = 1 in equation (3)), σ = 0.02;
‘IKNM’, default condition with \(A_\alpha ^0\left( t \right)\) as in equation (3) and \({{{\bar{\mathrm \Lambda }}}}\) noise as in equation (2) with τ = 37 s, σ = 0.02;
‘Linear + A0 noise’, apical–basal position of nucleus in equation (3) is replaced with ρ_{α}(Δt) = z, where z is drawn from a uniform distribution ranging from 0 to 2; σ = 0.02;
‘exp λ’, ‘exp 2λ’, ‘exp 4λ’, a linear increase in cell area is replaced with an exponential increase, that is, \(A_\alpha ^0\left( t \right) = \frac{1}{2}({{{\mathrm{exp}}}}(g_\alpha {{\Delta }}t) + 1)\); λ, 2λ and 4λ correspond to g_{α} = 1, 2 and 4, respectively; σ = 0.02.
Comparison between model and data
To estimate σ from the experimental data, simulations were adjusted to generate 48 frames every 25 s, so that the timescale is comparable with the liveimaging experiment and similar to the correlation time τ = 37 s. Simulations were initialized for 16 h with initialization parameters followed by 8 h with target parameters and defined magnitude of noise (σ = 0, 0.01 and 0.02). After that, the edges were tracked and filtered to avoid segmentation errors or fluctuations due to divisions and T1 transitions influencing the outcome. In particular, the edges were selected only if the two cells connected by an edge did not divide or undergo a T1 transition for the tracking interval of 48 frames. Furthermore, only the tracked edges that shared their two vertices with other tracked edges were analysed.
For every edge, the relative edge length is defined as \(\tilde l=l/\bar l\), where l is the edge length at a particular time and \(\bar l\) is the mean edge length throughout the time interval. The standard deviation of the relative edge lengths \({{{\mathrm{std}}}}(l/\bar l)\) over time changes with the absolute edge length (Fig. 2d) in a manner that depends on the value of σ. This was used to compare the simulations with the experimental data for different σ values.
To compare the cell shapes between vertex model simulations and experimental data, we use the cumulative distance between the model and experimental data including the following nondimensional descriptors D ∈ {p_{0}, ϵ, α, hex, p_{0CV}, ϵ_{CV}, A_{CV}, P_{CV}} (Supplementary Table 3). For each descriptor D, we calculate a difference \(D_{{\Delta }} = \left {\overline {D_{\mathrm{exp}}}  \overline {D_{\mathrm{sim}}} } \right/\widehat {\sigma _D},\), where \({\overline {D_{\mathrm{exp}}}}\) is the mean value of D obtained for all the cells in the simulation for a given set of parameters \(({{{\bar{\Lambda }}}},{{{\bar{\varGamma }}}},k_\mathrm{p},k_\mathrm{n})\) at final time, \(\overline {D_{\mathrm{sim}}}\) is the mean value of D estimated for the segmented data at a specific developmental stage and \(\widehat {\sigma _D}\) is the standard deviation of \(\left {\overline {D_{\mathrm{exp}}}  \overline {D_{\mathrm{sim}}} } \right\) over all the samples. The cumulative distance Δ_{tot}/n is then defined as the sum of differences D_{Δ} for all the descriptors, normalized to the number of descriptors, that is, \(\frac{{{{\varDelta }}_{\mathrm{tot}}}}{n} = \frac{1}{{{{{\mathrm{number}}}}\;{{{\mathrm{of}}}}\;{{{\mathrm{descriptors}}}}}}\mathop {\sum }\limits_D D_{{\Delta }}\).
The z score for a given cell shape descriptor is defined as \(z = \left( {\overline {D_{\mathrm{sim}}}  \overline {D_{\mathrm{exp}}} } \right)/\sigma _{\overline {D_{\mathrm{exp}}} }\), where \(\sigma _{\overline {D_{\mathrm{exp}}} }\) is the standard deviation of the mean \(\overline {D_{\mathrm{exp}}}\) estimated for different experimental images (Supplementary Table 1). In Extended Data Fig. 2, we report the absolute value of z score.
Software and code
The vertex model code used in this study is available via GitHub at https://github.com/mpzagorski/vertex_model_python_3. The code is modified from another work^{21} to include an adaptation to Python 3.7, implementation of junctional noise, handling oscillatory T1 transitions and cell lineage tracing (Methods). Custom code in Mathematica 12.1 (Wolfram) was used to analyse the results of the vertex model, estimate fragmentation coefficient, cell shape descriptors, and comparison between simulation and experimental data. Supplementary Videos 2–4 are generated using custom Mathematica 12.1 code by postprocessing the results of the vertex model simulation.
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The vertex model code used in this study is available via GitHub at https://github.com/mpzagorski/vertex_model_python_3. Other code supporting the analysis in this paper is available from the corresponding authors on reasonable request.
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Acknowledgements
We thank S. Hippenmeyer for the reagents and C. P. Heisenberg, J. Briscoe and K. Page for comments on the manuscript. This work was supported by IST Austria; the European Research Council under Horizon 2020 research and innovation programme grant no. 680037 and Horizon Europe grant 101044579 (A.K.); Austrian Science Fund (FWF): F78 (Stem Cell Modulation) (A.K.); ISTFELLOW postdoctoral program (A.S.); Narodowe Centrum Nauki, Poland SONATA, 2017/26/D/NZ2/00454 (M.Z.); and the Polish National Agency for Academic Exchange (M.Z.).
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L.B.M. performed the experiments and quantified and analysed the data. A.S. performed and analysed the laser ablation experiments. M.Z. performed the computational analysis. L.B.M., A.S., E.H., M.Z. and A.K. designed the experiments. A.K., E.H. and M.Z. analysed and interpreted the results. A.K. conceived the project and wrote the paper.
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Extended data
Extended Data Fig. 1 Selfoverlap function for regions A, B, C.
Q_{s}(t) quantifies the fraction of cells that remain within a characteristic distance (approximately one cell radius) after time t, after the displacement due to growth has been accounted for (Methods). Thus, fewer cells have changed their relative position in the tissue in region B compared to regions A and C. Single simulation for each region at \({{{\bar{\mathrm \Gamma }}}}\) = 0.12 and varied line tension \({{{\bar{\mathrm \Lambda }}}}\) = −0.711 (Region A, n = 702 cells), \({{{\bar{\mathrm \Lambda }}}}\) = −0.393 (Region B, n = 2546 cells), and \({{{\bar{\mathrm \Lambda }}}}\) = −0.074 (Region C, n = 1805 cells). Only cells that have not divided within 8 h are considered.
Extended Data Fig. 2 Comparison of cell shape descriptors between experimental data at E8.5 and simulations.
Difference (absolute Zscore values) between experimental data at E8.5 and simulations in units of standard deviations of the experimental data separately for each cell shape descriptor (defined in Table S3) across the (\({{{\bar{\mathrm \Lambda }}}}\),\({{{\bar{\mathrm \Gamma }}}}\)) parameter space (see Methods). Low values indicate good agreement between simulation and experimental estimates. The dashed lines correspond to ϕ = 0.3 and delineate regions A, B and C (see Fig. 3A).
Extended Data Fig. 3 Cell area of neural progenitors at cell division.
A. Snapshots from confocal time lapse imaging (Supplementary Video 5) of neuroepithelial cells expressing membrane GFP at E10.5. The first subapical zsection (shown) was used for tracing and quantification. Asterisk marks a dividing cell and its daughters. Scale bars, 5 μm. B. Mean cell area over time quantified from timelapse movies such as the one shown in A. n = 17 cells from 3 embryos, shaded regions 95% confidence intervals. Time = 0 is the first time point where two distinct daughter cells were detected. Dividing cells rapidly increase their apical area prior to mitosis.
Supplementary information
Supplementary Information
Supplementary Figs. 1–8 and Tables 1–3.
Timelapse imaging of ZO1–GFPexpressing neural tube at E8.5. Cells were segmented and bonds were tracked to estimate the variation in edge length over time (Fig. 2d).
Simulations performed in parameter region A (\(\bar{\varGamma}\) = 0.12 and \({{{\bar{\Lambda }}}}\) = −0.711). Example clones are displayed in different colours. k_{p} = 0.09 h^{−1}, k_{n} = 0 h^{−1}.
Simulations performed in parameter region B (\({{{\bar{\mathrm \Gamma }}}}\) = 0.12 and \({{{\bar{\Lambda }}}}\) = −0.393). Example clones are displayed in different colours. k_{p} = 0.09 h^{−1}, k_{n} = 0 h^{1}.
Simulations performed in parameter region C (\({{{\bar{\mathrm \Gamma }}}}\) = 0.12 and \({{{\bar{\Lambda }}}}\) = −0.074). Example clones are displayed in different colours. k_{p} = 0.09 h^{−1}, k_{n} = 0 h^{1}.
Confocal timelapse imaging of the apical surface of a neural tube that mosaically expresses membrane GFP at E10.5. The first subapical z section (shown) was used for tracing and quantification.
Laser ablation experiment at E8.5 of development. Anterior, left; dorsal, up. Scale bar; 10 μm. Time reported in seconds.
Laser ablation experiment at E10.5 of development. Anterior, left; dorsal, up. Scale bar, 10 μm. Time reported in seconds.
Supplementary Data 1
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BocanegraMoreno, L., Singh, A., Hannezo, E. et al. Cell cycle dynamics control fluidity of the developing mouse neuroepithelium. Nat. Phys. (2023). https://doi.org/10.1038/s4156702301977w
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DOI: https://doi.org/10.1038/s4156702301977w
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