Live 3D imaging and mapping of shear stresses within tissues using incompressible elastic beads

The cohesion and morphogenesis of living tissues require coordinated processes at the cellular scale, based on changes in cell number, size, shape, position and packing (Heisenberg and Bellaïche, 2013; Guirao et al., 2015). These rearrangements are possible because cells can generate and exert mechanical stresses on their surroundings, or conversely feel the stresses and transduce them into biological signals. The complete process is thus regulated under the dual control of genetics and mechanics, which mutually feed back to each other, and drive the growth and shape of tissues (Desprat et al., 2008). Hence, the impact of mechanics on tissue fate and organization is considerable, either for healthy organisms during embryo development (Krieg et al., 2008; Le Goff et al., 2013; Heisenberg and Bellaïche, 2013; Hiramatsu et al., 2013; Hamada, 2015; Herrera-Perez and Karen, 2018), or in pathological conditions (Wells, 2013; Delarue et al., 2014b; Angeli and Stylianopoulos, 2016). Quantitative studies about the role of mechanical constraints in morphogenesis and development benefit from a precise and quantitative knowledge of the spatial distribution of mechanical stresses, from the subcellular scale to the tissue scale, and of its temporal evolution.

In the past decades numerous methods have been developed in order to achieve in situ stress measurements, using different and complementary techniques; for reviews see Sugimura et al. (2016); Campàs (2016); Roca-Cusachs et al. (2017); Gómez-González et al. (2020). To summarize, these techniques can be classified into approximately four categories: (1) external contact manipulations, including micropipettes (Mitchison and Swann, 1954; Hochmuth, 2000; Von Dassow et al., 2010; Guevorkian et al., 2010), microplates (Desprat et al., 2005; Mitrossilis et al., 2009; Tinevez et al., 2009; Mgharbel et al., 2009), atomic force microscopy (AFM) indentation (Butt et al., 2005; Elkin et al., 2007; Xiong et al., 2009; Franze, 2011; Lau et al., 2015) and traction force microscopy (TFM) (Nier et al., 2016); (2) manipulations using light, comprising laser ablation (Rauzi et al., 2008; Bonnet et al., 2012; Porazinski et al., 2015) and optical tweezers (Neuman and Nagy, 2008; Bambardekar et al., 2015) – and also by extent magnetic tweezers (Hosu et al., 2003; Tanase et al., 2007; Mazuel et al., 2015); (3) non-contact optical imaging, in which one can find birefringence (Nienhaus et al., 2009; Schluck and Aegerter, 2010) and stress inference (Chiou et al., 2012; Ishihara et al., 2013; Brodland et al., 2014; Roffay et al., 2021); and (4) embedded local sensors, from fluorescence resonance energy transfer (FRET) at the molecular scale (Grashoff et al., 2010; Borghi et al., 2012) to microsensors at the cell scale (Campàs et al., 2014; Dolega et al., 2017; Mongera et al., 2018; Mohagheghian et al., 2018; Lee et al., 2019; Träber et al., 2019).

The latter technique based on microsensors is quantitative, barely perturbative and suitable to monitor tissue stresses at the scale of a cell or a group of cells. Two main avenues have already been explored. The pioneer article (Campàs et al., 2014) used incompressible liquid droplets to measure the shear stress tensor, which is the most important stress component to understand how anisotropic forces govern tissue morphogenesis. The droplet synthesis and manipulation, as well as the data analysis, is a tour de force. Liquid microdroplets, made of fluorocarbon oil, were injected in aggregates of mesenchymal cells in living mandible explants. Coating the oil surface with a biocompatible surfactant enables droplet insertion in the tissue. The mechanical stresses exerted by the surrounding cells modify the droplet shape, and the deviation from the average stress normal to the droplet surface can be calculated from its local curvature, according to Laplace's law. Calibration of the liquid/tissue surface tension enables direct measurements of the three-dimensional (3D) components and orientations of the shear stress tensor. The same group proposed an alternative use of liquid droplets as actuators to probe the tissue mechanical properties (instead of stress), and applied it to zebrafish embryos (Mongera et al., 2018).

Other groups have favored elastic beads because they are easier to produce, tune, calibrate, manipulate, insert in tissues and analyze. The stress exerted on a solid sensor by surrounding cells can be deduced from its deformation, provided that the elastic moduli are determined through an independent calibration. Inspired by Matrigel pressure sensors (Monnier et al., 2016), sensors have been prepared using polyacrylic acid (PAA) hydrogels (Dolega et al., 2017; Lee et al., 2019; Träber et al., 2019), the Young modulus of which can be tuned from 60 to 4000 Pa (Lee et al., 2019), or alginate gels (Mohagheghian et al., 2018), and injected in cell aggregates or zebrafish embryos. Water can flow in and out of a hydrogel, making it compressible. In principle this method yields access to the total stress, that is, simultaneously the compression stress (including osmotic pressure contributions) and shear stress. The rest state for each sensor without stress can be determined at the end of the experiment by lysing the cells; once the compression stress is determined, the shear stress can be estimated by subtraction (Mohagheghian et al., 2018).

To combine the chemical and mechanical advantages of incompressible liquid droplets, namely fluorescence, functionalization and accurate shear stress measurements, with the ease of tuning and using compressible solid beads, we developed a pipeline of techniques based on incompressible solid beads of a diameter comparable with the cell size. The material must exhibit a well-defined elastic behavior, with a Young modulus comparable with the one of the surrounding tissue (order of magnitude 10³ Pa), in order to get a measurable deformation under physiological stresses, the expected order of magnitude of which is, for example, 10² Pa in zebrafish development (Mongera et al., 2018). Coating of the sensor surface is necessary to make them biocompatible and to make their insertion non-perturbative. To observe the sensor's deformation and get a precise 3D reconstruction of its shape, stable fluorescent labeling is also needed.

Our choice fell on polydimethylsiloxane (PDMS), which is an elastic elastomer, with a Young modulus adjustable down to a few hundred Pa (Hobbie et al., 2008). Production of small droplets of PDMS polymerizable mixture, having a fixed diameter, can be easily controlled through a microfluidic device. We introduce a novel method to bind the elastomer to a fluorescent dye through covalent bonds, leading to a stable, homogeneous and high intensity fluorescence. Coating the PDMS with cell adhesion proteins should also be possible, as mentioned in the literature (Toworfe et al., 2004; Schneider et al., 2011). The sensors can be embedded in the tissue in a non-perturbative way, and their deformation can be followed over minutes or hours. For 3D image analysis, we implemented an active contour method algorithm to determine the shape of the deformed sensors. This method leads to direct and accurate measurements of the 3D components and orientations of the shear stress tensor. This contrasts with the use of compressible hydrogel sensors, which are mainly sensitive to the local pressure inside the tissue, and the shear stress is deduced by subtraction from the total stress. When only the shear stress measurement is required, using incompressible PDMS sensors is easier and more precise.

The technique was successfully tested in two different systems, in vitro and in vivo. First, reconstituted cell aggregates were chosen as a tumor model, for which it is well known that mechanical constraints have a major influence on the organization and fate (Delarue et al., 2014b; Northcott et al., 2018). Moreover, it is relatively easy to produce aggregates with embedded sensors, which makes it a privileged system to validate the method. Second, we investigated the distribution of mechanical constraints in the prechordal plate (PPl) of the zebrafish embryo during its development. Indeed, based on in vitro observations, it has been postulated that anisotropies and heterogeneities of mechanical stresses are present in the PPl, and are of importance to guide its migration (Weber et al., 2012; Behrndt and Heisenberg, 2012). However, owing to the lack of appropriate tools, the existence of such anisotropies could not be directly tested so far. The implantation of our mechanical sensors in this system could help to definitively decide between the different models actually disputed.

In both cases, in vitro and in vivo, we report here results concerning the spatial repartition of the shear stresses, providing clear demonstration of the usability and potential of these new sensors.

As shown in the Materials and Methods (Strain-stress relationship), determining in situ the local stress tensor requires calibration of the sensor shear modulus μ. Its value was determined by two methods, first at the macroscopic scale with a commercial rheometer, and secondly in situ at the sensor scale with a custom-made setup allowing uniaxial compression of aggregates.

A rheometer (ARES G2) was used to follow the evolution of the elastic moduli of the PDMS preparation during its gelification. The polymerizable mixture was introduced in either plate-plate or cone-plate geometry, and maintained at a constant temperature of 60°C or 80°C, and the storage and loss moduli G′ and G″ were measured every 15 min, in the range 0.1 Hz<f<10 Hz. A typical gelification curve is shown in Fig. S1: after a transient increase over the course of ∼1 h, G′ gradually tends towards a plateau, with the final value reached after ∼ 12 h at 60°C or ∼ 3 h at 80°C. The same behavior is observed for G″.

At any given stage of gelification, G′ was found independent of the excitation frequency f, and G″ increased approximately linearly with f, which corresponds to a Kelvin-Voigt behavior. Moreover, at the end of the gelification plateau, the ratio G′/G″ was found to be of the order of 10 at f=10 Hz for a standard gel composition. Thus, when the PDMS gel is submitted to a static (or very slowly varying) stress, it may be considered as a purely elastic solid, and it is legitimate to confound its static shear modulus μ with its storage modulus G′ extrapolated at f=0 Hz.

At the end of the polymerization plateau, G′ does not evolve any more, and its value is retained in the following as the value of the shear modulus μ_b for bulk polymerized PDMS. This value strongly depends on the mixture composition. It is close to 500 Pa for the ratio of crosslinker to PDMS m_cross=0.0160 m_PDMS, but reaches 1000 Pa for m_cross=0.0170 m_PDMS. We noticed that the final value of μ_b depends also, but in a lesser extent, on the crosslinker and inhibitor concentrations, and on the temperature set for gelification. However, once fully polymerized, the shear modulus μ_b of the elastomer remains constant and does not depend on the operating temperature.

To compare the shear modulus μ_d of small spherical sensors with that of bulk PDMS μ_b, we used a dispersion of the polymerizable mixture in water: a small amount of this mixture was added to water and vigorously shaken for a few seconds to make a coarse emulsion. The suspension was left to buoy up at room temperature, during a time lapse τ, and centrifuged until complete droplets coalesced. The supernatant was then sampled, and baked in the rheometer in the same conditions as bulk PDMS, i.e. at 80°C for ∼ 3 h. The evolution of G′ during polymerization was similar to the one of the bulk, but the final plateau value μ_d was always found to be smaller than the bulk value μ_b measured for the non-emulsified mix having the same composition. The ratio μ_d/μ_b reached a stable plateau value ≈0.43 when τ≥24 h. We interpreted this observation by assuming that a small amount of a mix component, likely the crosslinker, may diffuse out of the PDMS emulsion droplets and dissolve in the surrounding water. As seen above, a small variation of the crosslinker concentration is enough to induce a significant change in the final mechanical properties of the gel. As the microsensors are made from small droplets suspended in water before polymerization, this effect has to be considered for a proper calibration of their mechanical properties. In practice, we decided to measure μ_b with the rheometer for every batch of bulk mix polymer used to make microsensors, and then to apply a constant corrective factor in order to get an estimate of the final shear modulus μ_d of the spherical elastic sensors: μ_d=0.43μ_b.

For this calibration, we used a custom-made uniaxial rheometer, allowing us to apply either a controlled force or a controlled deformation to a cell aggregate (Desprat et al., 2005, 2006; Mitrossilis et al., 2010). To summarize, the cell aggregate may be squeezed between two glass plates, a rigid one and a flexible one acting like a cantilever. The plates are actuated by two piezoelectric stages. A feedback loop maintains the extremity of the flexible plate, on the aggregate side, at a fixed position, whereas its other extremity is free to relax with time. This allows recording of the evolution of the force F(t) exerted on the aggregate, at constant aggregate deformation. F(t) is calculated from the flexible plate deflection, knowing its rigidity k=81.2 nN/μm.

Practically, we selected a CT26 aggregate of 100-200 μm diameter containing a sensor localized close to the aggregate center, and we seized it between the two plates of the rheometer (Fig. 1A,A′). We then applied a step motion to the rigid plate to squeeze the aggregate (Fig. 1B), while the flexible plate extremity close to the aggregate was regulated at constant position. From this initial instant we recorded the relaxation of the flexible plate's deflection during 15 to 30 min, and thus the time evolution F(t), while the aggregate deformation remained constant. Simultaneously, we imaged the shape of the sensor in its median plane (Fig. 1B,B′). Two or three successive squeezings and relaxations were operated on the same aggregate. From these relaxations we inferred at any time t the force F(t) exerted on the aggregate, and the deformations ε_zz(t) and ε_rr(t) of the sensor, respectively, in the compression direction and perpendicular to it.

Fig. S2 shows an example of the time evolution of σ_a(t), ε_zz(t) and ε_rr(t), immediately after a compression step on an aggregate. Fig. 1C reports the relationship between the sensor deformations ε_zz(t), ε_rr(t) and the average stress σ_a(t) for the same set of data. During the first 30 s, both the average stress and the sensor deformation relax rapidly with time (yellow points on Fig. S2 and Fig. 1C). It is likely that this regime is dominated by rapid relaxation processes, such as cytoplasm viscoelasticity, intercellular adhesion remodeling and cell rearrangements, and that the stress components are heterogeneous in the aggregate during this time lapse. Over a longer period, however, the relaxation slows down and one expects that hypothesis 4 of our model, namely in-plane spatial homogeneity of σ_zz, becomes verified. In this regime (red and blue points on Fig. S2 and Fig. 1C), σ_a(t) linearly varies with ε_zz(t), according to the model prediction given by Eqn 4. Hence, from the linear fits shown in Fig. 1C, one can extract the values of the effective shear modulus μ_e=790±160 Pa and of the aggregate surface tension γ_ag=9±2 mN/m for this particular experiment. Note that we experimentally measure ε_zz≈− 2ε_rr at any time, which is consistent with the incompressibility of the sensor and with the cylindrical symmetry assumptions.

Two gels of slightly different compositions have been tested. Within our experimental accuracy, no significant difference can be detected between the value of ⁠, averaged over N assays in different aggregates, and the value of μ_d measured at the macroscopic scale for a coarse emulsion made out of the same gel (see Table 1). These results do not allow us to isolate the contribution of capillary effects in the effective shear modulus μ_e, according to Eqn 2. Either this contribution is smaller than expected, or the determination of μ_d and μ_e is not accurate enough to measure the difference between them. We will hereafter take μ_d as the reference value for the effective elastic shear modulus of the sensors.

From the uniaxial compression of aggregates we can also infer the surface tension γ_ag between the aggregate and the culture medium. The values range from 3 to 12 mN/m for different aggregates. Although the dispersion is important, the order of magnitude corresponds to the expected one.

Thirteen aggregates, containing deformable sensors located at different positions, were imaged with a two-photon microscope. We analyzed the shape of 17 sensors. To compare the results, we define a dimensionless position r=r_c/R_a as the ratio of the distance r_c from the aggregate center to the sensor center, over the distance R_a from the aggregate center to the aggregate edge in the direction of the sensor. This definition takes into account the fact that the aggregate might be not spherical but slightly ellipsoidal.

The results are gathered in Fig. 2A. Each sensor is set at its reduced position r, and is represented by an ellipse showing its deformation projected in the (x, y) plane of the image. As the actual deformations are small (<10%), they were multiplied by a factor of four on the scheme to be visible. The main components of the associated shear stress are represented as red bars of length proportional to the stress amplitude. In most cases, one of the main axes remains close to the Oz optical axis, which justifies the projection in the (x, y) plane.

By looking at the stress orientation and amplitude, one retrieves several pieces of information. First, the in-plane main axes of the sensors are mostly aligned along the radial and orthoradial directions of the aggregate referential. Fig. 2B represents the distribution of angles between the radial direction and the direction of the longer axis of the sensor (blue: sensors showing a difference in half axes larger than the estimated accuracy threshold 0.5 μm; red: other sensors). This distribution is non-uniform and indicates that the sensors are principally compressed in the orthoradial direction. Second, the component σ_zz, represented by a color code in Fig. 2A, is always positive and ranges between 0 and 250 Pa (see Discussion).

In principle, the evolution of this stress map can be followed in time. We were able to image the spreading of some aggregates deposited on the bottom plate of the Petri dish, over the course of a few hours, by taking stacks every 15 min. In most cases, the axes orientations and half-lengths of the sensors remained stable with time, within experimental accuracy. Longer recordings would be necessary to see an evolution, and to follow the aggregate spreading process until its term.

In vivo, the spatial distribution of mechanical stresses, their inhomogeneities and their local anisotropy play a determinant role in the morphogenesis process, as they directly influence cell polarization and migration. For example, it was established in vitro that Xenopus PPl cells can be polarized by application of a mechanical stress of a few Pa (Weber et al., 2012). The PPl is a group of cells that are the first to internalize on the dorsal side of the embryo, at the onset of gastrulation. During gastrulation, they migrate in the direction of the animal pole, followed by notochord precursors, as shown in Fig. 3A-D (Kimmel et al., 1995; Solnica-Krezel et al., 1995). Based on this observation, it was proposed that migration of the PPl is guided in vivo by the existence of stress anisotropies within the tissue, used by cells as directional cues (Weber et al., 2012; Behrndt and Heisenberg, 2012). Our sensors are well suited to measure the 3D stress anisotropy and to map the stress in the PPl. Conversely, the PPl appeared as a good model to demonstrate their in vivo capabilities.

The PPl and the notochord cells are labeled in the Tg(gsc:GFP) line, which was used in these experiments. Sensors were implanted in the PPl of seven different embryos. As an example, Fig. S3 shows a mesoscale view of the PPl containing a sensor, and enlarged images on the same sensor presenting a visible deformation. Some sensors could be followed over time, by taking images every 30 s or 60 s. An overview of the full dataset is shown in Fig. S4. We report in Fig. 3E,F a selection of 12 measurements, at different stages of gastrulation, from 60% to 85% of epiboly. The common effective shear modulus of all the sensors was μ_d=430 Pa. To analyze the stress spatial distribution, the PPl was divided into nine zones (a grid of front/middle/rear and left/center/right), as shown in Fig. 3E. For legibility, the projection of the shear stresses in the PPl (x, y) plane is drawn as an ellipse for each sensor, whereas the projection along the perpendicular axis z (confounded with the observation axis) is represented by a color code. In the PPl plane, x is the direction of the PPl progression and y the perpendicular direction. All the stress components lie in the range −60 to +60 Pa, with approximately equal distribution between positive and negative values.

From Fig. 3E, no evident correlation emerges between the sensor location in one of the nine zones of the PPl and the stress orientation and amplitude in the same zone. However, in Fig. 3F, the value of the shear stress amplitude ⁠, averaged over the left/center/right zones and over the different epiboly stages, is compared at the front (N=3) and at the middle (N=7) of the plate. The difference is significant and is a first indication that stress gradients exist in the PPl.

Concerning the time evolution of the stress, we were able to follow the stress components for seven sensors, between 15 and 30 min, at different stages of epiboly. They did not show any significant changes, except for one event which we describe in detail in Fig. S5.

In both experiments, in vitro and in vivo, we have demonstrated that our pipeline of techniques, based on the use of PDMS elastic microsensors embedded in living tissues, can be used to locally determine the amplitudes and orientations of the shear stress components, to map them across the tissue, and to retrieve their temporal evolution.

In vitro, for freely spreading cell aggregates, the order of magnitude of the shear stress amplitude typically lies between 10 and 100 Pa, consistent with other measurements in similar systems (Lucio et al., 2017; Lee et al., 2019; Mohagheghian et al., 2018). Moreover, Fig. 2C shows that the deviator stress amplitude σ increases from the aggregate center to its edge, and that the stress component along the orthoradial direction is larger than along the radial direction. On the other hand, according to our observations, the optical axis Oz systematically coincides with one of the main shear stress axes, with a positive value of the shear (extension). A possible explanation would be that, besides the applied geometrical correction due to light refraction (see Materials and Methods section ‘Active contour method’), light diffusion in the tissue may also affect the quality of the image, especially at large depth inside the tissue (≥100 μm). If this is the case, the systematic elongation of the sensor in the z-direction could be an artefact related to the imaging method. Fortunately, this does not affect our conclusions concerning the radial/orthoradial privileged orientations in the (x, y) plane, nor the variations of the stress amplitude with reduced distance r.

It is interesting to compare these results with those reported in the literature. In the pioneering work by Delarue et al. (2014a), after submitting reconstituted cell aggregates to an external pressure, it is observed that the cell density increases from the aggregate periphery to its center. The model developed in that paper attempts to correlate this density increase to the local cell polarization: the cell elongation evolves, on average, from the radial direction near the center to the orthoradial direction close to the aggregate boundary. Actually, this differs from our own observations: our sensors are mainly oriented in the radial direction at the aggregate periphery, and the cell polarization is expected to be in the same direction, following the stress anisotropy. Besides the fact that the cell lines are different in the two experiments (BC52 versus CT26), one should keep in mind that an important external pressure is intentionally applied in Delarue et al. (2014a) in order to modify the aggregate organization and the cell growth conditions.

In another paper (Lee et al., 2019), hydrogel sensors are embedded in spherical aggregates made of HS-5 fibroblasts. In this paper two components of the full stress tensor were measured versus the distance to the aggregate center, respectively in the radial and orthoradial directions in the observation plane. Both their amplitudes were between −400 and −1500 Pa. Their average value represents the isotropic part of the stress: it is negative and thus corresponds to a compression. The difference between the two components, which is the local shear stress, is ∼±100 Pa, comparable with the accuracy of their measurements. This order of magnitude is similar to ours. Moreover, they observe that both stress components increase from the aggregate edge, to reach a maximum value, and then decrease again towards the aggregate center. Although the two experiments give access to different quantities (total stress in their case, shear stress in ours), the behaviors observed in both situations are consistent.

In vivo, in the PPl of the zebrafish embryo, we were able to measure both the amplitude and orientation of the main stress components at different stages of epiboly. The retrieved values are of the expected order of magnitude, 10² Pa, and the sensitivity is of the order of 10¹ Pa. The shear stress amplitude appears to be larger at the center of the PPl than at its front, which supports the hypothesis that stress gradients exist in the PPl and might play an active role in the PPl migration. A systematic survey of the stress amplitude and orientation in the different PPl regions is needed to confirm this hypothesis. Concerning time evolution, we could follow the stresses only on a short time interval compared with the gastrulation characteristic time. This might explain why we observed only one event exhibiting a measurable time evolution. Similar events have to be identified before discussing their possible relevance in the full migration process. Another important issue, which was not investigated here, concerns the stress distribution in the direction orthogonal to the PPl plane. Indeed, one suspects that the friction of the PPl over the neuroectoderm might play an important role in the migration process (Smutny et al., 2017). Thus, further investigations will also have to include the vertical position of the sensor inside the PPl as one of the relevant parameters of the problem.

To conclude, we have assembled a pipeline of techniques which meets all the requirements to quantitatively map in 3D and in real time the local shear stresses in living tissues, with a sensitivity in the order of 10 Pa. In complement to existing techniques, it appears to be a valuable tool to investigate the role of mechanical constraints in morphogenesis and development.

The summary of requirements is: the sensors must have a size comparable with the cell size, and be easily fabricated and manipulated in large quantities; they must be biocompatible and conveniently embedded in living tissues; their rigidity has to be homogeneous and close to that of the tissues; they can be fluorescently labeled in a homogeneous and stable way; their 3D shape can be precisely reconstructed with minimal effort; the effective shear modulus of the sensors can be reliably calibrated in situ; the shear stress tensor can be decoupled from compression stress and directly derived from the 3D deformations of the sensor using linear elasticity; shear stresses of the order of 10² Pa can be measured with 10 Pa sensitivity. To satisfy simultaneously all these constraints, we introduce the following pipeline of techniques.

The microsensors were made out of a silicon elastomer similar to usual PDMS. The polymerizable mix preparation is first dispersed into liquid droplets of ∼30 μm in size, thanks to a microfluidic circuit, and afterwards polymerized at 80±1°C. Briefly, the main component of the elastomer is PDMS. It is mixed with a polymeric hydrosilane (methylhydrosiloxane-dimethylsiloxane copolymer) that acts as a crosslinker via hydrosilylation of the vinyl ends of PDMS (Fig. S6A). The ratio of crosslinker to PDMS must be carefully controlled (m_cross=1.60×10⁻²m_PDMS) to achieve the desired shear modulus after polymerization. The hydrosilylation reaction is initiated using Karstedt's catalyst (m_catal=4.286×10⁻³m_PDMS). A divinylic inhibitor (diallyl maleate) is also added (m_inhib=0.8571×10⁻³m_PDMS) to slow down the reaction kinetics. The as-prepared mixture needs to be stored at 4°C and should be used within a few days or else the crosslinking reaction significantly moves forward (see supplementary Materials and Methods for more details).

Dispersing the polymerizable mix into spherical droplets of homogeneous size was carried out using a custom-made microfluidic circuit, by a classical flow focusing method (Kim et al., 2007). The dispersed phase (polymer mix) meets the carrier phase (water) at a four-channels crossing, and the resulting droplets suspension is collected at the output. The rectangular channel section was ∼50×20 μm². The injecting pressures of both phases was finely tuned and regulated by a Fluigent controller, in order to get a steady-state dripping instability and a constant droplet diameter. This diameter was tuned between 20 and 40 μm, but was stable for a given droplet batch: the diameter distribution of droplets is quite monodisperse within the same production (Fig. 4). The monodispersity is not necessary for shear stress measurements. In the present work, the monodispersity facilitates the calibration. Moreover, it allows identification of possible optical artefacts during image analysis (see Results): we discarded images of beads with a measured radius outside of the expected range.

The droplet polymerization into spherical elastic beads is achieved by baking the suspension at 80°C for 3 h. The final gel was very soft, sticking easily and irreversibly to any wall, including inert surfaces such as Teflon or silanized glass. Thus one must avoid the contact of the droplets with any solid surface during and after polymerization. To achieve this, the beaker containing the suspension was placed on a turntable during baking, rotating at about one turn per second. As the gel is less dense than water, buoyancy makes the polymerized beads spontaneously concentrate at the center of the meniscus of the water free surface. Then, the concentrated suspension containing ∼10⁴ beads per ml can be collected with a micropipette, aliquoted in Eppendorf tubes and immediately stored in a freezer at − 20°C. The beads remain stable for weeks at this temperature, and are thawed before immediate use.

In order to observe the sensors embedded in the tissue, a dye must be added to the elastomer. The objective is to get a homogeneous fluorescent signal, with high enough intensity to be able to visualize the sensors with usual 3D fluorescent microscopy techniques (confocal, spinning-disk and two-photon microscopies) and to precisely reconstruct their 3D shape. Several hydrophobic dyes did not lead to satisfying labeling: either the dye could not be homogeneously dispersed in the polymer mix (fluorescein diacetate), or it was partially released in the water solution surrounding the beads, so that the fluorescent signal rapidly decreased with time (Nile Red and Cryptolyte™). We also attempted to label the elastomer with quantum dots (QDs) dispersed in the mixture. However, despite some specific coating to make them hydrophobic, QDs remained partially aggregated and the dispersion was not complete.

For efficient fluorescent labeling of the elastomer, two challenges needed to be overcome: the dye had to easily disperse into the polymerizable mix and, once dispersed and after curing, should not escape the meshwork of the gel and leak into the surroundings. Our strategy entailed attachment of the organic dye to the crosslinker via a parallel hydrosilylation reaction (Fig. S6B). The dye therefore needed to bear a vinyl terminal group. Isothiocyanate-bearing fluorophores can be conveniently modified through quantitative C-N bond formation. We selected Rhodamine B isothiocyanate because of its emission in the red and condensed it with allylamine to give a vinyl-terminated Rhodamine B analogue (Fig. S6C). The compound was then added to PDMS in a molar ratio of one fluorophore for 1000 PDMS strands (see supplementary Materials and Methods for more details).

Fig. 5 shows some examples of brightfield and fluorescent images of single sensors suspended in water, embedded in a CT26 reconstituted aggregate or implanted in a zebrafish embryo. Fig. 5B′ is a 3D reconstruction obtained with ImageJ software, where the deformation from spherical shape is clearly visible. The fluorescent signal is homogeneous in the bead volume and the contrast is high enough to detect the sensor border (see Active contour method, below). Moreover, we checked that the fluorescence in the elastomer remains stable over several days. Thus this labeling technique fulfills all the requirements for further quantitative image analysis.

CT26 cells, stably transfected with Lifeact-GFP (a gift from Danijela Matic Vignjevic, Institut Curie, Paris, France) were cultivated in T75 flasks at 37°C in 5% CO₂, in DMEM culture medium completed with 10% (v/v) fetal bovine serum (FBS) and 1% antibiotics (penicillin-streptomycin; ThermoFisher, 15140122), and passaged every 3 days. For the preparation of aggregates, confluent cells were detached using 5 ml of a solution containing 0.05% of trypsin in buffer (ThermoFisher, 25300054). Incubation was limited to ∼1 min, in order to form cell leaflets and avoid complete cell dispersion. In parallel, elastomer microsensors were functionalized by adding 1 ml of fibronectin solution in PBS (50 μg/ml) to 1 ml of freshly thawed bead suspension. The final suspension was left to incubate for 1 h at 37°C, and then directly added to ∼10 ml of detached cells suspension without further rinsing. Aggregates containing inserted beads were prepared in Petri dishes on which the cell and bead suspension was deposited, placed on an orbital agitator (∼50 rotations/min) and left to grow for at least 24 h in an incubator at 37°C. To get spherical aggregates, it is suitable to let them grow for at least 48 h. The diameter of the obtained aggregates lies between 100 and 500 μm. With few exceptions, they contain at most one bead per aggregate.

Embryos were obtained by natural spawning of Tg(-1.8 gsc:GFP)ml1 adult fishes (Doitsidou et al., 2002). All animal studies were approved by the Ethical Committee No. 59 and the Ministère de l'Education Nationale, de l'Enseignement Supérieur et de la Recherche under the file number APAFIS#15859-2018051710341011v3.

Embryos were grown at 28.5°C until reaching shield stage (6 h post fertilization). Embryos were then processed as explained in Boutillon et al. (2018). Using a large glass needle (35 μm opening) mounted on a pneumatic microinjector (Narishige IM-11-2) under a fluorescence-stereo microscope, a sensor was inserted in the shield of an embryo, which expresses GFP in the Tg(-1.8 gsc:GFP)ml1 line. Transplanted embryos were then incubated at 28.5°C until reaching the desired stage between 60% and 85% of epiboly (6.5 to 8 h post fertilization). Embryos were then selected for the presence of the sensor in the PPl. Here, this represented ∼50% of transplanted embryos: in the other embryos, the sensor was either not in the PPl or had been expelled out of the embryo. Selected embryos were mounted in 0.2% agarose in embryo medium on the glass coverslip of a MatTek Petri dish (Boutillon et al., 2018) and placed on an inverted TCS SP8 confocal microscope (Leica SP8) equipped with an environmental chamber (Life Imaging Services) at 28°C and an HC PL APO 40×/1.10 W CS2 objective (Leica). Imaging parameters were set to acquire the whole sensor (z-stack) in less than 15 s, to minimize displacement due to the migrating neighboring cells.

To simultaneously image the tissue and the sensors embedded inside it, several techniques were used. Sensors in suspension in water were imaged using a spinning-disk microscope (Andor Revolution CSU X1, mounted on an Olympus IX 81 inverted microscope equipped with a 40× water immersion objective), in order to check their sphericity and the good quality of their fluorescence (intensity and homogeneity) (Fig. 5A). Two-photon microscopy was used for the visualization of reconstituted cell aggregates. Experiments were carried out at the multiphoton facility of the ImagoSeine imaging platform (Institut Jacques Monod, Université de Paris, France). The aggregates were deposited in a Petri dish, in a chamber regulated at 37°C, and observed for up to 12 h under a 20× water immersion objective at the early stage of their adhesion to the bottom plate. For the rhodamin dye, the excitation laser was tuned at λ=840 nm and the emitted light was collected through a dichroic mirror at λ≥585 nm. Lifeact-GFP of CT26 cells was excited at λ=900 nm and the fluorescence was collected at λ≤585 nm. Image stacks were recorded along the optical axis every 0.5 μm, with a lateral resolution down to 0.1 μm/pixel (Fig. 5B). A confocal microscope (Leica SP8) was used to image the PPl of the zebrafish embryos. The sample was maintained at 28°C. Image stacks (40× water immersion objective) were recorded at regular time intervals (30 s to 1 min) at different stages of the epiboly, between 60% and 85%. As the PPl is migrating at a velocity up to 2 μm/min, the acquisition time for a whole stack must be smaller than 15 s to avoid drift in the images; therefore, images were recorded every 2 μm. The excitation laser was tuned at λ=498 and 550 nm and the emitted light was collected between 507-537 nm and 569-673 nm (Fig. 5C).

Once the 3D contour of the sensor has been determined from the images, a renormalization factor r=0.935 must be applied to the shape of the sensor along the optical axis (z) direction. This factor takes into account a geometrical correction due to light refraction through the tissue/PDMS interface, which acts like a spherical diopter between two media of different optical indices (1.35<n₁<1.40 for the living tissue; n₂≈1.44 for PDMS). The factor r was calibrated in situ, by comparing the shape of hard non-deformable spherical sensors (μ∼2×10⁴ Pa) with their reconstructed image. Calibrations were performed both in cell aggregates and in zebrafish embryos, leading to the same value r=0.935±0.02, which was retained in the following.

With this active contour method, we estimate that we can detect a sensor deformation if the difference between two half axes is at least equal to 0.5 μm, which represents the accuracy of our measurements.

In this section we describe a model to calculate all the components of the stress tensor at any point of an aggregate submitted to a uniaxial compression, from which we derive the expression of the deformation of an elastic incompressible sensor embedded in the aggregate.

The other components σ_zr(r, z) and σ_rr(r, z)=σ_φφ(r, z) can also be written after some long but straightforward calculations, using Eqns 21, 22 and the boundary condition Eqn 19 to calculate σ₁(z).

In this model, B has a homogeneous value over the volume of the sensor, but it may vary over time. Indeed, a displacement step, which imprints a constant deformation to the aggregate, is applied to the rigid plate at t=0. Immediately after the step, the stress in the aggregate is inhomogeneous but it rapidly evolves, within a few minutes, through different relaxation mechanisms, to become homogeneous again at the end of relaxation. B(t) is therefore a function of time which must relax towards zero at infinite time. To interpret our data concerning σ_a(ε), we discard the first instants of the relaxation, and we only consider the longer time limit, for which we expect the stresses to be homogeneous again, allowing us to make the approximation B=0. Within this assumption, Eqn 34 exactly reduces to Eqn 4 used in the main text. To comfort this assumption, we also performed the data analysis by taking a non-zero value for B, leading to parabolic variations for σ_zz as a function of r. Assuming, for example, that σ_zz vanishes at the aggregate edge (r=R₁, z=0), from Fig. 1C one retrieves μ_e=670 Pa, instead of μ_e=790 Pa in the case B=0. Considering all other sources of uncertainty, the results do not appear to be significantly different.

We acknowledge Alain Richert, Gaëtan Mary, Magid Badaoui, Alain Ponton and Sandra Lerouge for their assistance and advice in experimental handling, Hélène Delanoë-Ayari for critical reading of the manuscript, and Mathieu Roché and Julien Dervaux for fruitful discussions. We acknowledge the ImagoSeine facility at Institut Jacques Monod, especially Xavier Baudin and Orestis Faklaris, and the Polytechnique Bioimaging Facility, for assistance with live imaging on their equipment.

The peer review history is available online at https://journals.biologists.com/dev/article-lookup/doi/10.1242/dev.199765