Differential proliferation regulates multi-tissue morphogenesis during embryonic axial extension: integrating viscous modeling and experimental approaches Free

Differences in growth rate and biophysical properties have been proposed to be influential parameters for shaping biological forms since the last century (Thompson, 1917). In vertebrate embryos, tissue morphogenesis occurs early during development, when the body transforms from a round shape (disk or sphere, depending on the species) to a characteristic elongated sphere where the longest axis defines the anteroposterior direction (Bénazéraf, 2019; Mongera et al., 2019). This extension involves the coordinated shaping of the different tissues that compose the embryonic body, such as the neural tube (NT), which gives rise to the central nervous system, the paraxial mesoderm (PSM), which gives rise to the vertebrae and muscles, and the notochord (NC), an axial mesodermal tissue that secretes diffusible molecules involved in the patterning of both of these tissues to specify a variety of different cell types. Located in the tail of an embryo, posterior to these elongating tissues, lies an embryonic territory that contains an undifferentiated pool of progenitor cells known as neuromesodermal competent cells (NMCs) (Binagui-Casas et al., 2021; Guillot et al., 2021; Romanos et al., 2021; Tzouanacou et al., 2009). These cells self-renew, undergo specification and migrate into the elongating neural tube and paraxial mesoderm, hence contributing to the extension of these tissues. Besides the addition of progenitor from the posterior zone (PZ), multiple cell processes are implicated in posterior tissue extension; for example, cellular rearrangements such as convergent extension by intercalation have been shown to participate in the posterior extension of different posterior tissues in different species (Keller et al., 2000; Shindo, 2018). A gradient of non-directional motility within the paraxial mesoderm has also been shown to drive tissue extension in bird embryos (Bénazéraf et al., 2010). The transition of posterior high cell motility to low motility in the anterior part of the tissue is linked to a fluid-to-solid phase transition, which is proposed to play a key role in axial extension (Mongera et al., 2018). Even though the relative importance of the action of each posterior tissue on the whole elongation process remains to be elucidated, experiments showing that posterior PSM deletion slows down elongation more than deletion of other tissues indicate a pivotal role of this tissue in this process (Bénazéraf et al., 2010). Further studies recently proposed that paraxial tissue can generate forces to push on axial tissue, thus sustaining the extension process (Xiong et al., 2020). Overall, the process of posterior extension is defined by a specific and complex choreography in which tissues grow differently and slide along one another while elongating (Bénazéraf et al., 2017). The neural tissue and the notochord, in particular, are moving towards the posterior part of the embryo faster than the paraxial mesoderm. Because axial extension occurs in growing embryos, it raises the question of whether differential growth or differential cell proliferation plays a role in this process. The answer differs significantly when looking at different species of vertebrate embryos. In zebrafish, the posterior extension does not seem to solely rely on growth, as it does not involve a massive change in tissue volumes and because cell cycle mutants do not display obvious truncation (Steventon et al., 2016; Thomson et al., 2021; Zhang et al., 2008). On the contrary, in catfish, mouse and chick embryos, volumetric growth occurs in the developing tails (Bénazéraf et al., 2017; Steventon et al., 2016). It has been shown that bird or mouse posterior embryonic tissues are actively proliferating, with nearly all cells being engaged in the cell cycle; this possibly explains most of this volumetric growth (Minchington et al., 2023; Molina and Pituello, 2017). However, measures of cell cycle durations in bird embryos have shown significant variations among posterior tissues (Bénazéraf et al., 2017). For example, the average cell cycle duration is shorter in the paraxial mesoderm compared with the axial mesoderm or the neural tube. Although the role of cell proliferation in axial extension has been proposed to be minimal over a short-time duration in bird embryos (Bénazéraf et al., 2010), whether it might influence multi-tissue morphogenesis over longer-time scales has not been specifically addressed.

In the past two decades, several modeling approaches have been developed to enhance our comprehension of the underlying processes that govern posterior axial extension. They successfully recapitulate and bring insight into many crucial biological aspects of this complex morphological process. This includes the roles of FGF8 signaling on cell migration, posterior morphogenesis and patterning (Harrison et al., 2011; Baker and Maini, 2007; Bénazéraf et al., 2010; Regev et al., 2022), the influence of cell adhesion on cell motion and tissue biophysical properties (Das et al., 2017; Dray et al., 2013; Mongera et al., 2018), as well as the putative mechanical coupling between posterior tissues (Xiong et al., 2020). However, most of the current axial extension modeling approaches primarily center on the role of the paraxial mesoderm, neglecting a comprehensive exploration of how other developing tissues might influence the elongation process. Additionally, these approaches frequently disregard the significance of cell proliferation, placing greater emphasis on growth attributed primarily to cell injection from the progenitor zone.

This study aims to understand the role of differential growth and tissue interactions in coordinating the extension of the posterior tissues of the vertebrate embryo. We seek to identify key cell and tissue properties involved in this process. By adopting a purely mechanical viewpoint, excluding assumptions about external signals and responses, we have developed a new theoretical framework using continuum models to examine the influence of proliferation, injection and viscosity on interacting posterior tissues. Our 2D multi-tissue model, based on partial differential equations (PDEs), integrates biophysical properties contributing to tissue dynamics. Building on previous works (Degond et al., 2022), we consider tissue growth driven by cell proliferation and cell ingression from the progenitor zone.

The main outcome of this study is that, contrary to previous reports, cell proliferation, rather than cell injection from the progenitor zone, is the most crucial parameter in coordinating axial extension and shaping of posterior tissues. Our novel modeling approach for multi-tissue development not only highlights the role of differential tissue growth but also emphasizes the reciprocal mechanical influence that growing interacting tissues have on the shape of one another and on elongation dynamics.

We aimed to develop a model describing the spatio-temporal dynamics of the PSM and neural tube densities. In the model, variations in cell densities are driven by tissue-specific cell proliferation rates (8.75 h. for the PSM and 10.83 h. for the NT) and cell ingression from the PZ as described in the literature. The PZ is modeled as a regressing injection zone along the anteroposterior axis, located at the posterior tip of the modeled tissues (Fig. 1A,B; see supplementary Materials and Methods Section 1.2, Fig. S2). It participates in adding new cells at the posterior tip of both the NT and the PSM. Recent data (Bénazéraf et al., 2017; Mongera et al., 2018; Xiong et al., 2020) indicate that modeling posterior tissues as independent growing tissues, i.e. without mutual interactions, is not sufficient to explain how these tissues expand over time, and suggest that intra-tissue and inter-tissue mechanics impact tissue shape and dynamics. Therefore, we used a PDE-based model to account for the mechanical effects within and between neighboring tissues by considering different tissue velocities, which are strongly linked to tissue viscosity (Fig. 1C). This model integrates tissue-specific cell proliferation rates (see supplementary Materials and Methods Section 1.2; Fig. S1), cell injection, densities and viscosities to collectively influence pressure, tissue densities and tissue deformations. We set up the model with some of the NT already formed, flanked by two stripes of PSMs on either side (see supplementary Materials and Methods Section 1.3, Fig. S3). These tissues are constrained laterally by the intermediate/lateral plate mesoderm and anteriorly by the somites (structures of higher cell density), constituting the fixed limits (walls) of the model. To make our model as close as possible to the dynamics of the vertebrate embryo, we collected parameters from the literature such as cell density, tissue viscosity, inter-tissue friction and cell proliferation rates (Fig. 1D). The main parameter that was missing was the injection rate of new cells from the PZ feeding the growing tissues. To estimate this parameter, we analyzed time-lapse movies of transgenic quail embryos (Bénazéraf et al., 2017) and measured the average cell motion vectors within specific zones delimiting the interface between the PZ and the NT, as well as the PZ and the PSM in our model. We applied these motion vectors to the tissue density measured in these zones to estimate the flux of cells across these two inter-tissue boundaries (Fig. 1E, Figs S6 and S7). Our data show that, for a representative coronal tissue section of 10 μm, 10 cells/h on average enter the PSMs from the PZ, whereas 5 cells/h enter the neural tube. With the different parameters we collected and measured, we developed a biologically based theoretical framework that allows the modeling of growth and shape of posterior tissues during bird embryonic development.

We first verified that our model can reproduce the tissue growth and elongation features of living embryos. We observed that, after 20 h, the model reproduces tissue shapes comparable with those observed in vivo at stages HH10 to HH11 (10-15 pairs of somites) (Fig. 2A,B, Movie 1). Tissues form and elongate posteriorly, and the cell density distributions show increasing gradients from posterior to anterior, as described in vivo (Bénazéraf et al., 2010, 2017). We measured the elongation rate and found it equal to 128 μm/h (dashed line in Fig. 2B), which is within the range described in the literature (Bénazéraf et al., 2010; Regev et al., 2022). We plotted the velocity profiles within each tissue to investigate whether the model recapitulates the global tissue dynamics observed in vivo. As in the embryo, we observed that the PSM and the NT display an anteroposterior velocity-increasing gradient (Fig. 2C, Fig. S10B). Interestingly, the vectors located in the most posterior part of the tissue display medial-to-lateral biases that support the appearance of vortices in cell motions, as has been observed in vivo by quantification of rotational tissue movements clockwise and counterclockwise on the right and left PSM, respectively (Banavar et al., 2021; Bénazéraf et al., 2017; Lawton et al., 2013). The analysis of these rotational movements validates the presence of clockwise and counterclockwise movements in the PSM in our model (Fig. 2D). Furthermore, in line with previous reports suggesting that pressure from the PSM is a driving force during posterior elongation (Michaut et al., 2022 preprint; Xiong et al., 2020), we noticed that within the anterior part of the NT, the velocity vectors located close to the interface with the PSM point inwards, indicating a response to the PSM compression (Fig. 2C). By estimating potential pressure in the model, we found out that PSM pressure is globally higher than that in the NT, with a significant increase close to the interface with the NT (Fig. 2E, blue arrow; Fig. S10A), and with an anteroposterior decreasing pressure gradient in both tissues (Fig. 2E). To test whether this compression between tissues could be related to differential local tissue expansion, we analyzed the velocity divergence as a readout of expansion in our tissues (Fig. 3F). Interestingly, we found a posterior-to-anterior decreasing gradient of divergence in the PSM (green arrow) with a divergence that is globally higher than in the NT (blue arrows), as described in vivo. To test whether our model can reproduce the inter-tissue sliding (Bénazéraf et al., 2017), we quantified the differential motion between the PSM and the NT (i.e. sliding) (Fig. 2G and supplementary Materials and Methods Section 2.1, Fig. S4). We defined and computed the sliding as the difference in the average NT and PSM velocities along the anteroposterior axis. Our analysis shows a significant (positive) sliding in the mid-to-anterior part of the tissues, reaching a maximum value of 12 μm/h, with this sliding slowly decreasing posteriorly. The velocity of the NT is then much higher than that of the PSM, which causes the NT to slide past the PSM, as is the case in vivo. Overall, our results show that our continuum model successfully captures multiple aspects of the tissue dynamics, including rotational movements, expansion/compression profiles and tissue sliding, all mirroring the in vivo processes, indicating that mesoscale growth and mechanics are an important part of the multi-tissue morphogenesis taking place in bird embryos.

As our model effectively recapitulates the NT and PSM dynamics, we wanted to test its ability to capture the dynamics between other posterior tissues. More specifically, we sought to model the growth dynamics of the notochord (NC) as, similarly to the NT, this tissue is known to physically interact with the paraxial mesoderm, yet it exhibits distinct growth and dynamic properties (Fig. 3A) (Bénazéraf et al., 2017). Indeed, the NC has an average cell cycle duration of 28 h, which is much slower than other posterior tissues (8.75 h for the PSM and 10.83 h for the NT). Despite this discrepancy, it displays the highest posterior velocity during axial elongation, making it the tissue that slides the most with the PSM (Bénazéraf et al., 2017). To test this apparent paradox, we needed to estimate the flux of new cells from the PZ into the NC to evaluate NC growth fully. Using the same image analysis method, we did not observe any influx of cells coming from the PZ to the NC, indicating that, at the analyzed stages, cells of the progenitor zone do not contribute or contribute only minimally to the NC growth (Fig. 3B). Therefore, we set the cell flux equal to zero for the NC in our model. The NC viscosity was estimated to be close to that of the NT, as both tissues are epithelial tubes, and cell proliferation in the NC was set to 28 h. We simulated the axial elongation of the PSM and the NC, and deduced a NC elongation rate of 150 μm/h. Consistent with biological observations, the numerical simulation shows that the notochord maintains its elongated shape with a subtle anterior thinning in comparison with the simulated neural tube under the same conditions (Fig. 3C,D). When comparing their velocity, we found an anteroposterior increasing gradient in both the NC and the NT, with the NC velocity being remarkably higher than that of the NT (Fig. 3E). As a consequence, our model shows a significant sliding along the anteroposterior axis between the PSM and the NC, with the NC reaching much higher velocities and sliding past the PSM (Fig. 3F). In contrast to the PSM/NT sliding, the PSM/NC sliding is significantly greater, mirroring the observed differences observed in vivo for these tissues (Bénazéraf et al., 2017). Our results show that our model can reproduce the NC dynamics by adapting the parameters specifically for the NC.

Although the growth of the NC (cell proliferation and progenitor flux) is limited compared with the PSM and NT, its elongation rate is relatively fast. Our model underscores the role of the interplay between growth dynamics and mechanical effects in driving the elongation of the NC. Paradoxically, the limited growth of the NC facilitates the conversion of pressure from its surrounding tissues into posterior movement. Indeed, regardless of whether NC proliferation is increased, a situation that would correspond to the control PSM/NT situation (Fig. 2B), or PSM proliferation is decreased, as in the PSM/NT simulation with inhibited PSM cell proliferation (Fig. 4A, middle panel), both scenarios correspond to a slower extension of the NC. Overall, our model demonstrates its versatility by successfully capturing various dynamic properties observed in vivo across multiple posterior tissues.

One key question in the vertebrate embryo axial extension field is the relative influence of cell injection versus tissue-specific cell proliferation. To understand the effect and the role of cell proliferation versus cell injection, we built and compared two scenarios (numerical simulations): in the first scenario, we completely inhibited cell proliferation in the PSM, by setting the corresponding parameter to zero; in the second scenario, we completely inhibited the entry of new cells from the PZ into the PSM. We compared these simulations and looked at specific properties of tissue dynamics (Fig. 4A). We analyzed the elongation rates in the two scenarios and found that inhibiting cell proliferation resulted in a severely reduced elongation rate (reaching 64 μm/h), compared with inhibiting cell injection, which yielded an elongation rate of 128 μm/h that was comparable with the control case (Fig. 4A,B). As the global shapes of the tissues exhibit significant differences between the two scenarios, we quantified the tissue shape and density at the final time. By measuring the width of the NT in each scenario, we observed that, in the absence of cell proliferation in the PSM, the NT gains in width all along the anteroposterior axis at the expense of the PSM, which appears thinner compared with the control or compared with the samples without cell injection in the PSM (Fig. 4A,C). This observation is consistent with the fact that, in the absence of cell proliferation, we noted little to no pressure in the PSM (see Fig. S10C), which normally participates in applying lateromedial forces on the NT and contributes to its posterior extension rather than its widening. Next, we analyzed the effects of inhibiting either PSM cell proliferation or cell injection on the inter-tissue dynamics. Our analysis of the posterior extension of tissues showed that, when inhibiting cell proliferation in the PSM, velocities of both the PSM and the NT strictly diminish along the anteroposterior axis compared with the uninhibited case (Fig. S9A,B). These results suggest that cell proliferation promotes the motion of both tissues and supports the mechanical role of PSM in NT elongation. We next quantified the tissue sliding and noted that, in the absence of cell proliferation (Fig. 4D, pink line), inter-tissue sliding is decreased anteriorly (Fig. 4D, blue arrow), and significantly increased at the most posterior positions along the anteroposterior axis (green arrow) in comparison with the control (gray line). When we inhibited PSM cell injection, both PSM and NT velocities were slightly reduced but, more importantly, they were relatively similar to the control case along the anteroposterior axis (Fig. S10D), which yielded a very low difference in sliding in comparison with the control case. However, the main difference is most noticeable in the posterior position, where the injection is taking place (Fig. 4D, orange versus gray line). Accordingly, we observed a difference in the posterior NT morphology, which was slightly shorter in the no-cell injection case compared with the control (orange arrow in Fig. 4A). This highlights a very localized effect of cell injection on overall tissue morphology. Therefore, our model suggests that PSM cell proliferation orchestrates the dynamics and extension of both the NT and PSM, influencing tissue shaping and morphological changes along the anteroposterior axis, in contrast to the localized impact of cell injection.

To try to reconcile our data with the fact that inhibition of cell proliferation in the PSM on short-time ranges has been shown to have limited effects on extension (Bénazéraf et al., 2010; Michaut et al., 2022 preprint), we explored the temporal dynamics of the influence of proliferation on PZ regression in our model. To monitor the axis elongation rate, we measured the evolution of the PZ posterior displacement over time (Fig. 4E, see supplementary Material and Methods Section 1.2, Fig. S2). Strikingly, from 0 to 4.5 h, the position of the PZ was very similar between the different considered cases (control, PSM cell proliferation null and PSM cell injection null). Over longer time scales, the differences in the PZ evolution became more significant, with the control having the most posterior position and the samples with no PSM cell proliferation having the least posterior position. Altogether, our results show the predominant growth effect of PSM cell proliferation compared with cell injection. Cell proliferation plays a vital role in maintaining tissue shape and density, and promoting tissue velocity and posterior movements. It also affects inter-tissue dynamics, i.e. when absent, tissues experience less anterior sliding and, unexpectedly, more posterior sliding. Our analysis further reveals the strong coupling of the PSM and NT dynamics, i.e. when tissue-specific biological properties of one tissue are deregulated, it dramatically affects the other. Finally, our model suggests that the role of cell proliferation on tissue dynamics prominently takes hold with a specific latency as it starts manifesting clear effects only after at least 5-6 h.

To validate the predictions of our model concerning the long-term effects of differential cell proliferation rates in multi-tissue morphogenesis, we experimentally inhibited cell proliferation in the PSM of transgenic quail embryos by electroporating a plasmid encoding the cell cycle inhibitor CKI p27 in precursors of the paraxial mesoderm in stage HH5 embryo. Because with our electroporation technique we obtain a transfection rate of 50% or higher in the PSM (Bénazéraf et al., 2010), inhibiting the cell cycle progression by artificially overexpressing p27 is expected to significantly slow down the average cell cycle duration of the whole tissue. To have access to global tissue dynamics, p27 expression vector or control plasmids were electroporated in H2B-mCherry transgenic quail embryos (Movie 2), and time-lapse imaging movies were recorded between stages HH9 and HH11, a time window when most electroporated cells are located in the PSM tissues. First, we verified that we effectively reduced the number of proliferating cells by executing EdU staining on the electroporated embryos. Analysis shows that significantly fewer PSM cells electroporated with p27 have incorporated EdU after a 1 h pulse compared with control cells (10% for p27 embryos versus 35% in control empty vectors) (Fig. 5A). Because we injected the p27 expression vector in progenitor cells before they enter the PSM, we wanted to examine whether cell injection from the PZ was deregulated in the p27 electroporated embryos. To do so, we analyzed transgenic quail embryo movies by measuring the injection rate of p27 electroporated embryos versus empty vector control embryos (described previously). We observed no significant difference in global cell injection in either the PSM or the NT (Fig. 5B). Therefore, our experimental condition corresponds to a situation in which proliferation is inhibited in the PSM tissue without significantly affecting the flux of cell injection from the PZ. We then analyzed tissue morphology and dimensions of embryos electroporated with the p27 and control vector. Analysis revealed that p27 electroporated embryos were shorter than control embryos; accordingly, the length of the PSM was found to be reduced in the p27-electroporated embryos (Fig. 5C,D). Because our model predicts that inhibition of cell proliferation in the PSM affects NT morphology, we quantified the width of the NT at different locations along the AP axis and found that the NT widens along the axis (Fig. 5C,E). We analyzed axis elongation and tissue motion in the movies of p27 or control vector electroporated transgenic quail embryos to study the tissue dynamics due to the inhibition of cell proliferation in the PSM. In line with the decrease in PSM size, we observed that elongation rates were decreased in the p27 condition compared with empty vector controls (45 versus 90 μm/h) (Fig. 5F). To assess whether tissue velocity is altered when PSM proliferation is diminished, we tracked cell motion in the NT and the PSM in the p27 electroporated embryo and compared it with control embryos along the AP axis (Figs S8 and S9C-D). We found that the velocity of the PSM is globally reduced along the anteroposterior axis in p27 electroporated embryos in comparison with the control embryos. We also found that the NT velocity of the p27-electroporated embryo is also globally reduced, especially anteriorly, and that posterior velocity is similar to that of the control embryos. By comparing the PSM and NT inter-tissue sliding, we found that, anteriorly, sliding is reduced in p27 electroporated embryos in comparison with the control embryos (Fig. 5G, blue arrow). In contrast, posteriorly, the p27 electroporated embryos had a significantly higher sliding (Fig. 5G, green arrow). Strikingly, our model accurately predicted the observed difference in sliding and tissue velocity behaviors in conditions where cell proliferation was inhibited in the PSM (Figs S8 and S9, compare Figs 4D and 5G). In conclusion, our experimental approach shows that reducing the proliferation in the PSM leads to a decrease in the elongation of the PSM and the NT, and affects the shaping of the NT, highlighting the non-tissue autonomous mechanical action of the PSM on the NT. These results also validate the predictive power of our model to highlight the strong inter-tissue mechanical coupling during axis elongation.

In this study, we have developed and applied a new theoretical framework to study the influence of different types of growth and tissue mechanical properties on inter-tissue interactions and tissue shaping. We applied this model to axial extension, a complex multi-tissue morphogenetic process conserved between different vertebrate embryo species. Using continuum modeling and hydrodynamic physics, our new framework predicts how different growth parameters influence multi-tissue dynamics. We found that the proliferation rate is a prominent yet underestimated feature of long-term multi-tissue dynamics, influencing both tissue extension and inter-tissue sliding. Our model efficiently predicts and quantifies inter-tissue mechanical effects, revealing how one tissue can shape its neighboring tissue. Indeed, we observed the mechanical impact of the PSM on the NT, suggesting that NT shape is influenced by more than only inherent tissue-specific properties. Finally, we validated the predictions of our model by demonstrating the influence of the higher PSM cell proliferation rate, compared with other tissues, on multi-tissue shaping, dynamics and axial extension.

To calibrate our model, we needed to characterize the injection rate from the progenitor region into the PSM, the NC or the NT. Although the cellular contribution from the PZ to the PSM or the NT has been visualized and/or demonstrated using different fate-mapping techniques, at different developmental stages and in different species (Cambray and Wilson, 2002; Catala et al., 1996; Guillot et al., 2021; McGrew et al., 2008; Tzouanacou et al., 2009), it was still unclear how many cells entered the tissue per unit time. We have estimated the fluxes of cells entering the NT, PSM and NC from the PZ in the quail embryo using nuclei tracking to calibrate our model and enhance its fidelity to the vertebrate embryo. The fluxes obtained are of a similar order to those in the zebrafish embryo (Banavar et al., 2021). The analysis of our model indicates that the role of this injection alone is minimal compared with the role of cell proliferation and tissue mechanical interactions. Hence, the substantial contribution to elongation arises more from the proliferative activity of newly entered cells, which generate significant offspring, rather than the injection itself. This is consistent with the fact that the deletion of the progenitor region does not have immediate consequences on axis elongation speed compared with the deletion of the caudal PSM (Bénazéraf et al., 2010). However, the predominant role of cell proliferation in multi-tissue axis extension contrasts with previous data supporting the view that cell proliferation has little to no role in axis elongation (Bénazéraf et al., 2010; Michaut et al., 2022 preprint). This discrepancy might be explained by distinct experimental procedures used to inhibit cell proliferation so far. Inhibiting DNA synthesis, or cell division, which indeed affects proliferation, may have only a limited impact on cell growth in terms of volume gain. Indeed, our observation is that when inhibiting proliferation using aphidicolin treatment (DNA replication inhibitor), cells are larger than in control embryos, suggesting that tissue growth might still be present in treated embryos (Fig. S11). A second explanation could lie in the time scale of the analysis, as the effect of inhibiting cell proliferation has been assessed over a rather shorter time scale (6-8 h) in these previous works. Our model predicts that the effects of inhibiting cell proliferation become visible only after at least 8 h of treatment. This delayed appearance of phenotype highlights how cellular and mechanical processes at the microscopic level evolve over time, and result in noticeable changes at the macroscopic scale. Interestingly, a recent study suggests that controlling cell density could be a crucial feedback mechanism in regulating elongation speed (Lu et al., 2024 preprint). Finally, it is important to note that the relative influence of cell proliferation in axis elongation may be dependent upon the developmental stage. Our analysis specifically concerns the interval encompassing stages HH8 to HH12, corresponding approximately to the junctional neurulation process (Dady et al., 2014). It is plausible that, in earlier developmental stages, particularly during primary neurulation, wherein gastrulation events are taking place, the influence of progenitor fluxes may have a more important impact on axial elongation than at later stages.

The PSM has been proposed to be a major driver of axis elongation, leading to several agent-based models to explain its elongation (Bénazéraf et al., 2010; Regev et al., 2022; Xiong et al., 2020). As we scale up to the tissue level, the computational expense of such models increases, favoring PDE-based models. Although PDE models are valuable in understanding embryonic development, they mainly focus on PSM elongation (Baker and Maini, 2007; Baker et al., 2008; Degond et al., 2022; Harrison et al., 2011; Regev et al., 2022; Ross et al., 2015). Addressing both intra- and inter-tissue mechanical interactions is crucial due to the role of mechanical forcesin axis elongation. In our model, which focuses on elongation with expanding tissues, we performed a preliminary parameter sensitivity analysis to evaluate the importance of variations in biophysical properties relative to differences in proliferation (Fig. S10E and see supplementary Materials and Methods Section 2.2, Fig. S5). The analysis results reveal that PSM and NT viscosity stand out as the second most critical parameters (after proliferation) in influencing the morphogenetic process of elongation. Aligned with this analysis, recent research has highlighted the crucial role of tissue viscosity, particularly the graded viscosity within the PSM of the zebrafish embryo, in the fluid-to-solid transition along the antero-posterior axis, which determines the shape of the tissue and its ability to elongate unidirectionally (Banavar et al., 2021; Mongera et al., 2018). It is worth noting that our model system diverges from zebrafish, where tissue growth seems globally less prominent during posterior body development, potentially explaining some of the disparities between the two models (Steventon et al., 2016). Our continuum model, incorporating short- and long-distance effects within and between highly proliferative tissues, reproduces embryonic tissue dynamics, including cell vortices resulting from our choice for the velocity viscous equation (Degond et al., 2022). Distinguishing tissues by growth potential helps predict interactions such as sliding and mutual shaping. Our analysis shows that, for proper shaping and elongation, it is crucial for a growing tissue to face resistance from its neighboring tissue, which creates reciprocal pressure effects. Therefore, the geometric arrangement of interacting tissues is essential for inter-tissue force transmission and axis elongation. Specifically, both PSMs must surround the NT to channel its elongation. Expanding our model to a 3D multi-tissue framework could elucidate the necessity of this geometric arrangement and allow us to investigate mechanical relationships between PSM, NT and NC tissues, highlighting the significance of tissue interfaces. The fact that tissues slide relative to each other indicates the importance of the nature of the physical interface between tissues. Several works have emphasized the role of the extracellular matrix, which accumulates at the boundaries between tissues, as an important player in the process of axis elongation. For example, it has been shown that, in mouse or zebrafish embryos, cell and ECM interaction is involved in posterior tissue shaping and elongation (Dray et al., 2013; Girós et al., 2011; Guillon et al., 2020). In particular, integrin alpha 5 and alpha V morpholino injection in zebrafish leads to a bent NC, suggesting that, if the link between developing tissue is impaired, one tissue can deform the other in a non-physiological manner (Dray et al., 2013). This study suggests that the interface between differential growing posterior tissues has to allow them to slide against one another. Interestingly, in other developmental contexts, such as gut morphogenesis, and because tissues are physically attached, they can deform, bend or coil in a physiological manner instead of sliding. In this case, the tube and the mesentery, which are attached and having different growing rates, cause the gut to coil (Savin et al., 2011).

This study presents a previously unreported multi-tissue modeling framework in the context of morphogenesis. Although our investigation primarily focuses on vertebrate embryo elongation, the versatility of this model extends its applications to various developmental systems. Indeed, despite potential variations in specific mechanisms reflected by different parameter values, the inherent generality of the model makes it adaptable to diverse contexts. Moreover, our model exhibits potentially more general applications for tissues displaying differential viscosity and growth characteristics, as is the case for tumors (both within the tumor or with respect to its environment), therefore opening the way to exploring cancer dynamics and predicting behavior. (Perthame and Vauchelet, 2015; Dębiec and Schmidtchen, 2020). Our model serves as a benchmark in multi-tissue modeling within developmental biology. Its minimal assumptions on tissue dynamics provide a unique approach to comprehending large-scale dynamics and morphologies. This distinctive feature highlights the versatility of our model as a robust framework for investigating a broad spectrum of developmental processes and extends its relevance to the broader field of systems in biology.

The USC aviary and local French hatchery (les cailles de Chanteloup) provided wild-type quail embryos (Coturnix japonica). PGK1:H2B-chFP quail lines have been described previously (Huss et al., 2015) and are maintained in the USC aviary. The embryos were staged according to Ainsworth et al. (2010) and Hamburger and Hamilton (1951). Embryos were cultured ex ovo with filter paper on albumen agar plates using the EC (early chick) technique (Chapman et al., 2001). For global proliferation treatment, stage HH6-7 embryos were positioned on agar albumen plates containing either DMSO or aphidilcolin as described previously (Bénazéraf et al., 2010) and cultured for 24 h.

A mouse P27 construct was acquired through Addgene (plasmid 15192). We collected stage HH5-7 quail embryos. Empty or p27 vectors (2-5 μg/μl) were microinjected between the vitelline membrane and the epiblast (Iimura and Pourquié, 2008) in the anterior part of the primitive streak and surrounding epiblast, in a region that contains paraxial mesoderm precursors (Iimura et al., 2007). The electrodes were positioned on either side of the embryo, and five pulses of 5.2 V, with a duration of 50 ms, were carried out at a time interval of 200 ms. The embryos were screened for fluorescence and morphology. Proliferation rates were assessed by EdU staining (Click-iT EdU Alexa Fluor 647 Imaging Kit, Thermo Fisher Scientific, C10340).

For live-imaging experiments, embryos were cultured at 37°C in culture imaging chambers (Lab Tek chambered #1 coverglass slide, Thermo Fisher Scientific) pre-coated with a mix of albumen agar (Chapman et al., 2001). Embryos were imaged using an inverted 780 Zeiss microscope using confocal excitation with 20×/0.8, or 25×/0.8 objectives from stages HH9 to HH11. For time-lapse imaging, several adjacent xyz fields of view were stitched together post-collection. Images were taken every 5-10 µm in the z-axis with a time resolution ranging from 5 to 6 min. The Spot and track functions of Imaris were used to localize nuclei in the 3D image dataset and to extract nuclei coordinates (x,y,z,t) (details on sliding quantification and referential used can be found in the section ‘Cell velocity and sliding mesurements in the vertebrate embryo'). The data were then analyzed using Excel, MATLAB or Python. Cell size measurement was estimated as the distance between two PSM nuclei on confocal section of DAPI-stained embryo.

The mathematical model was simulated using a finite volume scheme on a staggered grid. All the numerical simulations were performed using Matlab 2022a [see the supplementary Materials and Methods Section 1 ‘Mathematical model’ for detailed explanation of the PSM-NT model setup (model equations, model hypotheses, initial condition, tissue separation property) and for the numerical setting; and for the model for the PSM and the notochord]. For detailed computations of the model outputs (tissue velocities, sliding and elongation) see the supplementary Materials and Methods Section 2 ‘Quantification of model outputs’ (this section also includes the details of the sensitivity analysis performed on the model parameters). The Matlab code is available at https://github.com/MicheleRo/multi-tissue-viscous-model.

We used quail embryos with DAPI staining to measure tissue densities in the PSM, NT and notochord with Imaris software (using the Spot function). In each tissue, we measure the total number of cells inside regions that were 100 µm in length, spanned the full width of the tissue and 10 µm in depth in the PSM, NT and notochord. These cross-sections were taken in the mid-depth of the tissue. Measurements were taken in three locations along the antero-posterior axis: anterior (at the level of the somite), mid (800 µm downwards of the somite) and posterior (800 µm downwards of the middle location).

We used live movies of transgenic quail embryos to quantify cell velocity within each tissue (NT and PSM) and inter-tissue sliding. First, we changed the reference frame from the laboratory frame to the frame of the last formed somite. This can be achieved by computing the average velocity of cells within the last formed somite and shifting the reference frame to correspond to the average velocity computed. To obtain cell velocities in each of the PSMs and in the NT, we first created a reference axis along the antero-posterior axis, specifically going from the last formed somite, denoted y_min, to the end of the neural tube, denoted y_max. We define this distance as d=y_max−y_min. We then detected and tracked cells (of diameter 8 µm) in five regions of size 0.1 d×0.1 d×10 μm³ equally spaced along the antero-posterior reference axis [y_min, y_max], over 1 h of live imaging. Cell velocities were averaged in each spot, and the velocities at each spot were plotted per embryo (Fig. S9C,D). Finally, to obtain the sliding, for each embryo, we subtracted the average NT velocities obtained from the PSM average velocities, and averaged the resulting sliding to obtain the plot. The same computations were carried out for the transgenic quail embryo electroporated with p27.

The authors thank the Benazeraf team members, particularly Cathy Soula and Myriam Roussigné, for their assistance and comments on the work. We also thank the LITC and image analysis platforms of the CBI, as well as the PLATIM platform of the ENS Lyon. Special thanks are extended to David Huss for his valuable assistance in transgenic quail husbandry.

The peer review history is available online at https://journals.biologists.com/dev/lookup/doi/10.1242/dev.202836.reviewer-comments.pdf