Force production and mechanical accommodation during convergent extension

Convergent extension (CE) is a major contributor to the morphogenetic movements that physically shape the early vertebrate embryo. During CE, dorsal embryonic tissues progressively deform themselves and elongate the whole embryo along the anterio-posterior (AP) axis while narrowing in the mediolateral (ML) direction (Keller et al., 2003). From a mechanical perspective, the degree of deformation must be proportional to active forces, which drive the tissue deformation, and must be inversely proportional to passive tissue mechanical properties such as stiffness, which resist deformation. Previous studies indicated that the stiffness of the dorsal tissues increased eight- to tenfold over the course of CE (Zhou et al., 2009); yet, these same tissues maintain a nearly constant elongation rate. Another finding from that same study revealed that both whole embryos and dorsal tissue isolates cultured in a Rho kinase (ROCK) inhibitor could also maintain constant elongation rates indistinguishable from controls. These findings raised a number of questions concerning the regulation of force and mechanical resistance during CE and whether the two processes are coupled.

Mechanical considerations such as these suggest that the bulk tissue elongation forces should match the changes in tissue stiffness (Davidson et al., 2009). The coordination of force production with the local mechanical environment could be accomplished through a variety of mechanisms, ranging from purely mechanical feedback to mechanosensing and signaling pathways (Schwartz and DeSimone, 2008; Zhou et al., 2010; Miller and Davidson, 2013). Alternatively, dorsal axial tissues might ignore signals from their external mechanical environment and generate the same amount of force regardless of the stiffness of the rest of the embryo. Given the importance of these issues to the successful outcome of morphogenesis, it is surprising that the molecular or physical processes that balance force-to-mechanical resistance, or provide mechanical accommodation during morphogenesis, are largely unknown.

To understand whether force production and mechanical resistance are coupled requires the ability to quantify force production within the embryo. Yet, few direct mechanical measurements of force production by embryonic tissues are available to test whether force production is independent of the local microenvironment or whether mechanosensing and signaling feedback networks operate. Thus, in order to understand how force and stiffness are coordinated in multicellular tissues we first need to reliably quantify these forces.

Several techniques have been used to directly measure the bulk forces generated during morphogenesis. In one study, a fiber optic system was demonstrated that could measure the elongation force of a Keller sandwich explant made from Xenopus laevis frog dorsal marginal zone tissues (Moore, 1994). Another study used a pair of parallel wires glued to the superficial ectoderm to measure tension forces within the early neural plate of Ambystoma mexicanum embryos (Benko and Brodland, 2007). The forces needed to stall neural fold closure in two amphibian species, Triturus alpestris and A. mexicanum, have been estimated with magnetically manipulated steel ‘dumb-bells’ (Selman, 1955,, 1958). All of these biophysical approaches require dedication of specialized equipment, micro-manipulated optical fibers, thin wire force transducers or magnetically controlled steel dumb-bells to measure force production within a single embryo or tissue explant for extended periods of time. Such approaches provide insights into the physical constraints of morphogenesis but are not well suited to complex analyses of the molecular and mechanical coordination of force production during morphogenesis.

For these reasons we developed a new tool to measure tissue-scale force production and investigated the mechanical control of elongation during convergent extension.

We report here the development and application of a reliable technique to measure force production by converging and extending dorsal tissues microsurgically isolated from gastrulating X. laevis embryos. Using this technique, we reveal cryptic changes in force production that balance altered tissue stiffness and suggest that mechanical feedback at the tissue level is in part responsible for robust convergent extension movements. Furthermore, we demonstrate that the notochord plays at best a minimal role in driving axial elongation, whereas the primary contributors to force production during elongation are the neural plate, the medial-most paraxial mesoderm and more posterior dorsal tissues.

To quantify the forces driving convergent extension we developed a method to measure force using an agarose gel as a force sensor. Agarose has been used to culture cells and tissues and its mechanical properties have been extensively characterized (Tokita and Hikichi, 1987; Ross and Scanlon, 1999; Normand et al., 2000; Balgude et al., 2001; Gordon et al., 2003; Chen et al., 2004; Zeng et al., 2006). We embedded microsurgically isolated X. laevis embryonic tissues that include the dorsal anlagen (referred to here as the dorsal isolate) in ultra-low-gelling temperature agarose in a fluid state at room temperature (RT) (Fig. 1A). Cooling the fluid agarose to 14.5°C causes the agarose to solidify around the dorsal isolate. Immediately after gelling, the solid agarose immobilizes the explant, but as the dorsal isolate generates force it pushes on and deforms the surrounding gel along the AP axis (Fig. 1B). Agarose holds the tissue in place and acts as a force sensor. To determine forces generated by the extending dorsal isolate we calculated the forces needed to displace the gel (Fig. 1C). To calculate those forces we first obtained a map of gel displacement by tracking the movement of small fluorescent beads embedded in the gel (Fig. 1D). The beads closer to the AP ends of the tissue moved more than 20 µm (Fig. 1E), whereas beads further from the tissue showed little displacement. From the displacement field and the mechanical properties of the agarose gel obtained using a rheometer we computed the stress field surrounding the elongating tissue. In order to validate the operation of the agarose gel as a force sensor we carried out a series of tests (see supplementary material methods), confirming the precision of our strain measurements (supplementary material Fig. S1) and the accuracy of the stress calculations (supplementary material Fig. S2).

In order to confirm that the agarose gel was not simply tracking tissue displacements but rather reporting forces produced by the explant, we investigated the changing stress levels as an embedded tissue elongated. Confocal stacks of bead positions were collected at 1-h intervals and their displacements calculated (Fig. 2A). From these displacements we calculated stress in the gel (Fig. 2B), maximal displacement of the gel at the AP ends (Fig. 2C), mean stress (Fig. 2D) and stress profiles (Fig. 2E). The stresses at the anterior and posterior ends of the explants are directed normally to the surface of the gel. The mean stress was computed as the average stress value of stress immediately surrounding the anterior or posterior ends of explants. Furthermore, high AP-directed stresses at the AP ends of the explant are mainly compressive (balancing tensional stresses in the gel are typically low but can be observed at the mediolateral face of the explant, see supplementary material Fig. S6A). These observations reveal the explant extending and coming to equilibrium against the constraining agarose gel (see supplementary material Movie 2). To further confirm the ability of the gel to capture the time dependence of tissue elongation forces we measured the forces generated by converging and elongating dorsal isolates with stiffness-calibrated glass needles (supplementary material Fig. S3). We further confirmed the gel sensor with animal cap explants that are not known to undergo elongation, and found that these tissues did not produce observable stress (supplementary material Fig. S4). These tests demonstrate that gel stress equilibrates with the forces produced by the explant.

Dorsal isolates undergoing CE deform the agarose gel at both anterior and posterior ends (Fig. 3A-C) and lose contact with the gel along the mediolateral sides (Fig. 3B,B′). The displacement of the gel is reported by registration of two images, one taken shortly after the explant is immobilized and one 4 h later (Fig. 3D). We assumed there were no stresses present in the gel at the start of the experiment and calculated stress within the gel in its final, deformed state. From the bead displacement maps and the viscoelastic properties of agarose gel we used a commercially available finite element (FE) solver to compute the equivalent or von Mises stress [σ; Fig. 3E; see Fischer-Cripps (2007)]. Von Mises stress, which we subsequently refer to as ‘stress’, is commonly used to indicate the total stress present in a material. However, we can also represent the forces produced by elongating dorsal isolates with strain energy density (supplementary material Fig. S5A) or maximum principal stress (supplementary material Fig. S5B-D), or directly calculate the strain in the gel (supplementary material Fig. S6).

Embedded dorsal isolate tissues extend for more than 4 h in the gel and produce a maximum stress (σ_max) up to 7 Pa (Fig. 3E) along the mediolateral axis. The maximum stress always colocalizes with the mediolateral midline of the dorsal isolate (Fig. 3E), suggesting that the notochord and surrounding paraxial mesoderm tissues extend faster and produce larger forces compared with the other tissues. We routinely found that posterior ends produce a higher, more medially focused pattern of stress than that of the anterior end (Fig. 3E,F), even though we found no significant difference comparing the maximum or mean stress between the two ends. We note that stresses, unlike net forces at the ends, do not need to balance, as the areas and distribution of stresses are not uniform.

Using our gel force sensor, we are also able to quantify stress produced by tissue thickening (Keller et al., 2008) along the dorso-ventral (DV) axis by simply changing the orientation of the dorsal isolate in gel. In the case described above, the dorsal isolate was positioned in the gel, with the ML axis and AP axis aligned with the plane of the confocal section (Fig. 3A). Alternatively, by positioning the dorsal isolates with their DV axis and AP axis aligned with the plane of the confocal section (Fig. 3G), we can measure displacement and stress in the gel produced by elongation and dorsal ventral thickening (Fig. 3H and I, respectively). We found that DV thickening produced stresses comparable to AP elongation. By checking the displacement map, we found that gel deformation along the AP axis was focused along the anterior and posterior axes (Fig. 3H and supplementary material Fig. S7B), whereas deformation by thickening was located along dorsal and ventral faces (Fig. 3K and supplementary material Fig. S7B). There was minimal gel displacement along the DV direction at anterior and posterior ends (supplementary material Fig. S7), caused by either tissue elongation or thickening. Positioning a dorsal isolate so that the ML and DV axes were aligned with the confocal plane (Fig. 3J) revealed tissue thickening forces produced at the midpoint of the AP axis (Fig. 3K,L). By positioning dorsal isolates in different orientations we are able to extend our two-dimensional (2D) stress maps to visualize the three-dimensional (3D) distribution of stress produced during CE.

With a robust tool to measure stress production we investigated the regulation of AP elongation forces by myosin II contractility. Dorsal axial tissue explants elongate ∼40% over 5 h without any physical constraints (Fig. 4A,B). As a previous study had shown that the ROCK inhibitor Y27632 greatly reduces tissue stiffness along the AP axis (Zhou et al., 2009), we expected that it would also affect rates of convergent extension; however, we found no significant difference of elongation rates (Fig. 4B). To test whether force production was coupled to tissue stiffness, we embedded dorsal isolates in agarose gel with Y27632 and found that their elongation rate was reduced (Fig. 4A,B), and that they produced lower maximal stress (Fig. 4C,D; mean σ_max of control and Y27632-treated tissues were 5.0±1.6 Pa and 1.4±0.7 Pa, respectively) and lower mean von Mises stress (<σ>; Fig. 4G). Thus, we conclude that lowered ROCK and myosin activity coordinately reduces both bulk stiffness and bulk force production, thus allowing unconstrained dorsal isolates to elongate at the same rate as untreated dorsal isolates. Surprisingly, reduced force production after ROCK inhibition is only revealed when isolates are challenged to elongate against the mechanical constraints of the agarose gel.

From previous studies of mechanical feedback in single cell mechanics [e.g. Discher et al. (2005)] we suspected that increasing the stiffness of mechanical microenvironment would increase the elongation forces generated by dorsal isolates. Our gel-based force sensor allows us to change the mechanical environment by simply altering the concentration of the embedding agarose gel. To determine the effects of mechanical environment on the force production of dorsal axial tissues, we measured force production by explants embedded in agarose gels with three different concentrations (0.6%, 0.9% and 1.2%), the elastic moduli of which were typically 30, 200 and 500 Pa, respectively. We found that dorsal isolates embedded in 200 or 500 Pa gels generated greater stress than those embedded in 30 Pa gels (Fig. 5A-C). Both the σ_max and <σ> produced by dorsal isolates in the 0.6% gel were significantly lower than those in stiffer gels (Fig. 5D-F). Comparisons between the stress produced in 0.9% and 1.2% gels are challenging, as stiff gels introduce higher background noise, due to the combination of high stiffness gel and small inaccuracies in measuring gel displacements. Thus, dorsal isolates can generate larger stresses when faced with a stiffer microenvironment. These results suggest that mechanical feedback produces greater stress to overcome larger mechanical constraints of the surrounding gel.

Many sources of mechanical feedback have been proposed to regulate embryogenesis, ranging from intra-molecular changes in conformation to multi-protein mechanochemical signaling networks (Mammoto and Ingber, 2010; Miller and Davidson, 2013). However, to understand how such a feedback network might operate we sought to model the basic mechanics of a converging and extending tissue surrounded by a passively compliant material. Using an FE model we simulated a dorsal isolate with a morphology sampled from a representative explant (Fig. 5G) embedded within an agarose gel (Fig. 5H). To mimic the mechanics of convergent extension we implemented a mediolaterally oriented contractile stress (see Materials and Methods) and calculated the effects of AP elongation on stress in the surrounding gel. The qualitative pattern of von Mises stress in the model matches stress patterns of dorsal isolates embedded in agarose gels. We could match the stress magnitude to observed levels by adjusting the maximum contractile stress to 15 Pa (compare Fig. 5I with Fig. 3E). Next, we increased the model stiffness of the surrounding gel to 200 and 500 Pa and calculated the stress produced by the same internally generated mediolateral stress (Fig. 5J,K). We found that stress profiles across the AP face of simulated dorsal isolates were similar to those observed experimentally (compare simulated stress in Fig. 5L with measured stresses in Fig. 5D). Thus, increased stress production in response to increased physical constraint can be attributed to the maintenance of constant levels of mediolateral contraction within the elongating tissue.

As dorsal isolates confined in gels do not extend as much as ‘free’ isolates we tested whether the internal architecture of axial and paraxial tissues were altered after gel confinement. To check the architecture of dorsal axial tissues for irregular development, we fixed unconfined and gel-bound isolates, stained fibronectin, and collected confocal sections. The projections of fibronectin fibrils show that dorsal isolates embedded in stiffer gels were wider and included a curved notochord with a ‘knob-shaped’ posterior end (supplementary material Fig. S8A-C) compared with dorsal isolates cultured without mechanical restriction (supplementary material Fig. S8D). (Note: other aspects of morphogenesis in dorsal isolates, including neural tube closure, are not perturbed in gel-confined explants or gel-confined whole embryos; see supplementary material Movie 1.) Knob-ended notochords could have been caused by shear movements between the notochord and adjacent paraxial mesoderm (Wilson et al., 1989; Keller et al., 1992), and suggested that the shear stresses between the notochord and adjacent paraxial mesoderm might play a role in generating force during CE. Curved notochords also suggested dorsal axial tissues might undergo Euler buckling as they extend and narrow within the confines of the agarose gel.

The formation of a knob at the posterior end of the notochord suggested that notochord shear or extrusion might be responsible for the asymmetric stress patterns observed at the posterior end of elongating dorsal isolates and that the notochord contributes to force production. To test the contribution of the notochord to force generation, we compared the forces produced by dorsal isolates from which the notochord was excised with force produced by mock-operated dorsal isolates. The mock control and notochord-less dorsal isolates (Fig. 6A) were microsurgically prepared as described previously (Zhou et al., 2009) and their architecture confirmed by confocal sections of stained fibronectin fibrils (Fig. 6A′,B′). We found that notochord-less dorsal isolates embedded in 30 Pa gel generated a similar magnitude and pattern of force compared with mock control dorsal isolates (Fig. 6C); the mean values of σ_max produced by mock control and notochord-less dorsal isolates were 4.5±1.1 Pa and 4.1±1.7 Pa, respectively (Fig. 6C). We found no significant difference in σ_max or <σ> between mock control and notochord-less dorsal isolates (Fig. 6D,E). Thus, the notochord is unlikely to contribute to the magnitude and patterning of the forces driving tissue elongation during late gastrulation and early neurulation.

To test the relative contribution of other tissues within the dorsal isolate we created a series of explants from pieces of the dorsal isolate. After we confirmed the capacity of the explants to elongate we measured their force production capacity. First, we compared the stress produced by axial and paraxial (medial-notochord-medial; MnM) and lateral plate mesoderm (LL) using explants described previously (Zhou et al., 2009). MnM explants produced significantly higher σ_max and <σ> than the LL explants (supplementary material Fig. S9). Next, we compared the force production of anterior and posterior halves with intact dorsal isolates (Fig. 7A). We note that anterior and posterior halves elongate to the same degree (supplementary material Fig. S10; Movie 3). Anterior halves produced significantly lower σ_max and <σ> than the full dorsal isolate, whereas stress production by the posterior halves could not be distinguished from either anterior halves or the full dorsal isolate (Fig. 7B,C). In both cases the fragments of the dorsal isolate that contain the most paraxial mesoderm produce the greater stress. However, the dorsal isolate consists of both neural and mesodermal tissues, and both tissues are known to elongate in isolation (Elul et al., 1997; Poznanski et al., 1997; Keller et al., 2000). As the neural plate in Xenopus is only two cells thick, we utilized a neural plate ‘sandwich’ (NS) explant created from two neural plates isolated from embryos at mid- to late-gastrula stages (st. 11.5-12; Fig. 7D). The two explants were held together at their deep cell interface for 30 min until free-epithelial edges resealed. NS explants elongate at the same rate as dorsal isolates (supplementary material Fig. S11; Movie 4) and are easily confined within agarose force-reporting gels. However, due to the thin form of the NS they frequently buckle within the constraining gel after elongating more than 3 h. Thus, we restricted analysis to NS stresses produced within 2 h after embedding. We found that the NS explant generated levels of stress similar to those produced by the dorsal isolates (Fig. 7E-H; P=0.4 and P=0.6 for σ_max after 1 and 2 h, respectively). With their large contribution to the cross-section of dorsal tissues and their similar levels of stress production, we conclude that the stress production by paraxial mesoderm and prospective neural tissues contribute equally to elongation within dorsal tissues.

To investigate force production and mechanical accommodation during convergent extension we developed an innovative method to reliably measure the elongation force generated by dorsal tissues between late gastrula and early neural tube stages. The agarose gel force sensor enables analysis of force production during CE and might be used to measure force production by other elongating tissues, of which there are many in animal development. The non-adhesive gel-based sensor has several advantages over the previous methods for measuring forces produced during morphogenesis: (1) a gel can hold a tissue without drift as the tissue extends and changes its shape; (2) measurement of forces produced by multiple explants in a single gel allows higher throughput than cantilever-based methods; and (3) the gel force sensor can be adapted to measure a wide range of forces by tuning the elastic modulus of the agarose gel. Furthermore, more complex hydrogels can be formulated with ECM protein fragments or custom time-release growth factors to instruct programs of morphogenesis (Lee and Mooney, 2001). Although our technique is designed to measure uniaxial elongating forces of dorsal tissues, gel-based force sensors could also be used to measure forces produced by other tissues that extend or grow irregularly.

We applied the gel-based sensor to measure the elongation force produced by dorsal axial tissues. Physical constraints of the gel can prevent some natively occurring large-scale movements, e.g. blocking some aspects of tissue folding, and also induce tissue buckling that would not normally occur. For instance, we routinely observed bending by thin neural plate sandwiches, reminiscent of Euler buckling, as these tissues elongated. Dorsal tissues generated stresses of 5.0±1.6 Pa on the surrounding gel, whereas animal cap tissue explants generated stresses of 0.3±0.1 Pa. The production of stress by a dorsal isolate is often asymmetric; the posterior faces of dorsal isolates produced slightly higher stress than the anterior end due to the smaller cross-sectional area of the posterior end where it contacts the agarose gel (Fig. 3C) (Wilson et al., 1989; Keller et al., 1992). Physical principles dictate that elongation forces must balance at the two ends but stresses do not need to balance. As force is the product of stress and area, the posterior directed stress can be larger than the anterior due to differences in the distribution of stress or differences in the surface areas of the two faces.

We find no significant difference between stress produced by mock-operated and notochord-removed explants. A tissue must meet certain requirements to generate force for axis elongation: certain length, width and thickness. Both the neural epithelium and paraxial mesoderm meet those requirements. By contrast, the notochord after stage 12 has completed mediolateral cell intercalation and presents too small a cross-sectional area to generate large forces. The most parsimonious interpretation of these results is that notochord does not contribute significantly to the force production. We might speculate that the embryo compensates for the removal of notochord; however, such a new mechanism would require added levels of mechanosensing and response by paraxial tissues, which we are not able to exclude. We would not dispute that the notochord plays a mechanical role to straighten the tail-bud stage embryo but that a mechanical role during axis elongation is unlikely.

Previous biomechanical analyses of gastrula stages in Xenopus embryos (Beloussov et al., 2006; Zhou et al., 2009,, 2010; von Dassow et al., 2010, 2014; Luu et al., 2011) have suggested that mechanical feedback mechanisms operate during development to allow robust morphogenetic movements. Force production in dorsal tissues is needed to deform both those tissues and the rest of the embryo. Once removed from the embryo, forces acting within the dorsal isolate still carry out work to deform the dorsal isolate. It is tempting to suggest that all stiffness in the embryo is produced by mechanical contractility and that reduction in force production seen with ROCK inhibition exactly matches reduction in stiffness. Such coupling might operate cell-autonomously, for instance by coupling force production to tissue stiffness via myosin II contractility. This mechanism is qualitatively compatible with our observation that elongation forces of dorsal tissues were significantly decreased when myosin II contractility was reduced. However, myosin II contractility does not account for all stiffness within embryonic tissues; disruption of contractility by Y26732 reduces Young's modulus by 40-60% at stage 16 (Zhou et al., 2009), and we observe an 80% reduction in stress production. Furthermore, it is unclear whether feedback or mechanical coupling operates over longer distances, for instance coordinating ML tensile stress within adjacent neural and paraxial mesoderm cells when these tissues differ in stiffness or as tissues stiffen during the course of gastrulation. A wide range of mechanosensing and mechanotransduction pathways have been identified in cultured adult cells [e.g. by Discher et al. (2005)], yet few of these feedback systems have been rigorously tested in developing embryos. How these mechanical processes are triggered and coordinated, and how ‘mechanical’ information is sensed and passed from cell to cell in the embryo remains unknown.

These studies raise several interesting questions regarding the roles of mechanical adaptation and accommodation in morphogenesis. There has been little agreement in the field of mechanobiology over the use of the term ‘adaptation’. Many usages simply reflect the observation of changes in force production in response to an altered mechanical environment. A more stringent definition of adaptation would require identification of mechanosensing pathways that coordinate a defined mechanical response in the embryo. In this study, we observed increased force production by dorsal isolates in response to an increasingly stiff surrounding gel. However, after developing a rather simple mechanical model we discovered that increased force production did not necessitate complex mechanosensing by embryonic tissues but could be understood as a simple accommodation of a tissue that generates constant ML tension. It is possible that mechanical adaptation might occur during development but simpler mechanisms of mechanical accommodation need to be excluded. Criteria for assessing processes that maintain robust morphogenesis must include time-dependent changes in geometry such as deformation analysis. As deformation rates depend crucially on force production and mechanical properties, efforts seeking to identify potential adaptive pathways during morphogenesis must perturb these underlying mechanics in a controlled manner and must be capable of evaluating effects on both force production and mechanical properties. Analyses of morphogenesis that limit their focus to deformation rates alone might overlook processes that provide regulation and feedback that enable robust morphogenesis.

There are technical and conceptual limitations of our force measurement method. Technically, due to optically opaque tissue we were unable to track gel deformation in all three dimensions. Equilibrium stresses we report represent maximum stall forces. In several cases, the maximum strain in the gel reached up to 20%, which reduces the accuracy of our analysis. Lastly, whereas we can formulate an agarose gel with a bulk elastic modulus to match the residual Young's modulus of the embryo, we cannot formulate a gel to mimic the viscoelastic properties of embryonic tissues. Gels do not undergo plastic deformation, and thus the elastic solid mechanical properties of agarose limits the ultimate degree of elongation for all tissues embedded in such gels.

X. laevis embryos were obtained by standard methods (Kay and Peng, 1991), fertilized in vitro, dejellied in 2% cysteine and cultured in 1/3× MBS (Sive et al., 2000) at 14.5-21°C to stage 16 (Nieuwkoop and Faber, 1967). Before creating explants, vitelline membranes of embryos were removed with forceps (Fine Science Tools) and transferred in DFA media [Danichik's For Amy; Sater et al. (1993)]. Dorsal axial tissues, neural sandwiches and animal cap tissues were microsurgically dissected from embryos using hair loops and hair knives. To stain fibronectin fibrils, explants embedded in agarose gel were fixed in 3% TCA in 1× PBS (Davidson et al., 2004), stained with mAb 4H2 (Ramos and DeSimone, 1996) against Xenopus fibronectin (1:500) and visualized with a rhodamine-conjugated goat anti-mouse IgG antibody (Jackson ImmunoResearch). After staining, the dorsal axial tissues were dehydrated in methanol and cleared in Murray's clear (Davidson et al., 2004). Single optical sections and z-series of explants were collected with a confocal laser scan head (SP5, Leica Microsystems) mounted on an inverted compound microscope (DMI6000, Leica Microsystems) using image acquisition software (LASAF, Leica Microsystems). Average projection and reslicing of z-series stacks was obtained with ImageJ (v. 1.38, Wayne Rasband, NIH). Experiments involving X. laevis embryos were performed under a protocol approved by the University of Pittsburgh Institutional Animal Care and Use Committee (PHS Assurance Number: A3187-01).

To measure the tissue elongation forces, dorsal axial tissues were embedded in non-adhesive agarose gel. Briefly, ultra-low-gelling temperature agarose (type IX-A; Sigma) was dissolved in DFA solution at 65°C and cast in a 13×10×6 mm chamber. The molten gel was cooled to RT and remained liquid. Red FluoSpheres (580/605; absorption/emission wavelength in nm; Invitrogen) were evenly dispersed in the liquid solution as markers to track the gel deformation. Tissue explants were prepared and allowed to heal for 20 min to clear debris. The explants were then rinsed in fresh media and transferred to the liquid gel at RT. Once tissues were positioned to allow more than 1 mm separation between explants, walls of the chamber and upper surface of the gel, the chamber was moved to a 14.5°C incubator to chill for 20-30 min. Embryos and isolated tissues continued to develop through stage-specific milestones such as neural fold fusion (supplementary material Movie 1).

The mechanical properties of the agarose gels were measured to extract parameters needed to model stress production in a FE model. We estimated the maximum local strain in the gel domain induced by extending dorsal axial tissues to be less than 20% over 5 h; the agarose gel is therefore modeled as a linear viscoelastic material (Findley et al., 1989; Normand et al., 2000). We measured the viscoelastic properties of the agarose gel by performing oscillatory shear flow tests using a rheometer (AR2000; TA Instruments). The bulk elastic modulus G′ (or storage modulus) and viscous modulus G″ (or loss modulus) were measured in frequency sweep shear mode over 0.1 to 100 rad/s. For viscoelastic parameters, the measured elastic modulus over this range of frequencies was fitted with a two-mode linear viscoelastic model using Mathcad software (v14, PTC), and then the extracted parameters (long-term elastic modulus, elastic modulus at each mode and its corresponding relaxation time) were used to build a Prony series to model the viscoelastic material in FE model (Zeng et al., 2006). The long-term elastic modulus was typically 30, 200 and 500 Pa for gels with concentration of 0.6%, 0.9% and 1.2%, respectively. The Poisson's ratio of our agarose gels was assumed to be 0.5, based on a previous study (Normand et al., 2000).

To track the deformation in the gel domain, we adapted an algorithm which registers two images by deforming one image of beads to match the other (Sorzano et al., 2005). Red fluorescent beads (0.2 µm, 1 µm and 15 µm diameter) were suspended in gel and scanned confocally using a 10× objective. A short confocal z-stack was collected near the dorsal ventral mid-plane every 2 h and maximal projection of each z-series stacks was obtained using ImageJ (NIH). Two maximal projection images of each explant at different time points were aligned (Thevenaz et al., 1998) and analyzed by the bUnwarpJ plug-in for ImageJ, which reported the deformation between the two images as a vector field.

When dorsal isolates are embedded and elongate in an agarose gel, they can deform the gel in three possible directions: along the AP, ML and DV axes. By changing the orientation of the dorsal isolate within the gel, we found minimal displacement along the DV direction at either anterior or posterior ends, indicating that the gel was primarily under compression in the AP direction. Furthermore, dorsal isolates regularly extended only in the AP direction; thus, we simulated tissue elongation as a 2D plane stress problem. We then constructed a 2D FE model to compute the stress field in the gel based on the displacement field and the viscoelastic properties of the gel. Briefly, 2D FE meshes were generated with the free software TRIANGLE (version 1.6). Triangular elements with same areas and refined angles are constructed based on the real position of each bead reported by the bead tracking algorithm (Sbalzarini and Koumoutsakos, 2005). Three-node linear plane stress triangle elements (CPS3) were used for simulations. In our FE model, the gel domain is assumed to be isotropic and linear viscoelastic. The displacement of the gel was applied as a load through boundary conditions. The FE mesh, extracted viscoelastic parameters of the agarose gel, initial conditions and displacement boundary conditions of the gel served as input for the commercial FE software package ABAQUS (Dassault Systems), and the stress distribution in the agarose gel surrounding a sample was computed. The von Mises stress of each node immediately surrounding the anterior and posterior ends of tissues was extracted from the ABAQUS output files. The σ_max and <σ> were calculated for statistical analysis and color contours, plotted based on average values within the elements. Typically, we found that stresses at the anterior and posterior ends of the explant were directed normally to the surface of the gel. Furthermore, all stresses in the gel were compressive.

The 2D numerical simulations were performed with custom-made nonlinear FE analysis software. An image of a representative tissue explant and gel was used to generate the FE mesh using Cubit (v14.0, Sandia) mesh generation software. A simulated explant was placed within gel (2.5×2.5 mm) and both domains were discretized with four-noded quadrilateral elements. The explant was discretized with 2149 nodes and 2064 elements, whereas the gel domain was discretized with 5147 nodes and 4995 elements. A special four-noded interface element (Maiti and Geubelle, 2006; Nittur et al., 2008) was used at the interface between tissue explant and gel to account for the contact and sliding between these two domains as well as separation during the course of tissue elongation. Both the gel and explant were considered nearly incompressible (Poisson's ratio of 0.45) and neo-Hookean. The active stress in the FE model acts along the mediolateral axis. For the explant, we kept the material property (elastic modulus) constant with a value of 30 Pa (Zhou et al., 2009). Active contractile stress within the simulated tissue was held constant as the gel stiffness was increased from 30 to 200 and 500 Pa. To simulate force production within the explant we used a time-ramped contractile stress of σ_act (x,t)=σ₀t/t_max with σ₀=15 Pa along the medial lateral plane for all simulations. The magnitude of σ₀ was adjusted so that stress at the anterior and posterior face of the simulated explant matched the stresses observed in 30-Pa gels. Distribution of von Mises stress in the gel at the posterior face along the ML plane near the interface between explant and gel was obtained at time t=t_max when the contractile stress achieved its maximum value of 15 Pa.

Data were drawn from at least three separate experiments performed in triplicate. The data are presented as mean±s.d. and analyzed using SPSS version 16.0 statistical software. Two-way ANOVA, which includes treatment and clutch as fixed and random factor, respectively, was used to calculate the statistical difference of σ_max and <σ> between treatments. A P-value <0.05 was considered significant.

We would like to thank members of the Davidson lab, especially Drs Sagar Joshi, Hye Young Kim and Michelangelo von Dassow for their comments and support during these studies. Progress would not have been possible without the technical assistance of Ms Lin Zhang. We thank Dr Sachin Velankar for training and use of his AR2000 Rheometer.