Necking and failure of constrained 3D microtissues induced by cellular tension

Significance

Tension generated by molecular motors is fundamental to many processes such as cell motility, tissue morphogenesis, tumor growth, metastasis, and fibrosis. Cell-derived tension depends strongly on the mechanical environment. The rigidity of the medium in which the cells are found has been shown to affect cell division, differentiation, and apoptosis. In this work, through a combination of experiments and mathematical models, we find that cell-derived tension can be of sufficient magnitude to cause failure of complex-shaped microtissues. We show that failure can be controlled by varying the amount of extracellular matrix in the tissue and by tuning the stiffness of the constraining structures. Our results can aid the design of stable tissue constructs for applications in regenerative medicine.

Abstract

In this paper we report a fundamental morphological instability of constrained 3D microtissues induced by positive chemomechanical feedback between actomyosin-driven contraction and the mechanical stresses arising from the constraints. Using a 3D model for mechanotransduction we find that perturbations in the shape of contractile tissues grow in an unstable manner leading to formation of “necks” that lead to the failure of the tissue by narrowing and subsequent elongation. The magnitude of the instability is shown to be determined by the level of active contractile strain, the stiffness of the extracellular matrix, and the components of the tissue that act in parallel with the active component and the stiffness of the boundaries that constrain the tissue. A phase diagram that demarcates stable and unstable behavior of 3D tissues as a function of these material parameters is derived. The predictions of our model are verified by analyzing the necking and failure of normal human fibroblast tissue constrained in a loop-ended dog-bone geometry and cardiac microtissues constrained between microcantilevers. By analyzing the time evolution of the morphology of the constrained tissues we have quantitatively determined the chemomechanical coupling parameters that characterize the generation of active stresses in these tissues. More generally, the analytical and numerical methods we have developed provide a quantitative framework to study how contractility can influence tissue morphology in complex 3D environments such as morphogenesis and organogenesis.

Cytoskeletal tension is fundamental to many cellular processes such as cell motility (1), cytokinesis (2), tissue morphogenesis (3–6), and remodeling, as well as pathologic processes such as tumor growth, metastasis (7), and fibrosis (8). Moreover, it is becoming increasingly clear that tissue-engineering strategies can benefit from a fundamental understanding of cytoskeletal tension (3–6). A number of cells such as fibroblasts, cardiomyocytes, epithelial cells, endothelial cells, and smooth muscle cells develop significant contractile forces as part of their physiological function (9–12). One important function of cellular forces is to act on the surrounding extracellular matrix (ECM), align the matrix, and reorganize tissues as they form and develop. Most biological tissues experience multiaxial loading in vivo, often with complex boundary constraints. The mechanical stresses that arise from these constraints will lead to spatial variations in contractility, which in turn can influence the morphology of the tissue.

To quantitatively study the influence of mechanical constraints and matrix stiffness on contractility, we have recently developed a microelectromechanical systems-based technology to generate arrays of microtissues embedded within 3D micropatterned matrices that mimic the biomechanical environment that cells see in a native tissue. In our experiments, microcantilevers simultaneously constrain contraction of cardiac microtissues (CMTs) and report forces generated by the CMTs in real time (Fig. 1). Depending on the stiffness of the cantilevers and the density of the collagen matrix, the tissue was found to either fail by thinning and necking at the center or attain a stable, yet significantly strained morphology (6).

Fig. 1.

Influence of microcantilever stiffness and matrix composition on static contractility and the cross-sectional area of CMTs. (A) Side view showing a linear band of CMT held by cantilevers. (B) Temporal evolution of CMTs constructed in 0.5 mg/mL fibrin and 1.0 mg/mL collagen gels and tethered to rigid (k = 0.45 mN/mm) cantilevers. Shown are representative images depicting the time course of contracting CMTs. Red arrowheads indicate the locations of necking and failure at day 7. (C) Cross-sectional area and (D) force evolution for various pillar stiffnesses and collagen densities. The bar charts with error bars (standard deviations of 20 CMTs) represent experimental observations, while the curves are fits from simulations. (E–G) Contracted morphology of tissues with different pillar stiffnesses: (Upper) experimental observations along with the shapes from simulations and (Lower) finite element mesh showing relative stretch ratios along the tissue.

It has also been demonstrated that monodispersed cells can organize themselves into 3D microtissues in the absence of an ECM or scaffold (3, 4, 13, 14). To examine the role of mechanical constraints on contractility and morphology of these microtissues, building on the method to form complex shapes (3, 4), here we designed hydrogel molds to consistently control the generation of tension and failure spatially. The molds consisted of recesses in the shape of two toroids connected by a central trough, thus forming a loop-ended dog bone (Fig. 2). When fibroblasts settled into the recess bottoms and began self-assembling, the resulting dog bones were constrained by and suspended between the recessed pegs. The connecting rod between the pegs narrowed significantly, ultimately failing in the middle, leaving two tori microtissues. Furthermore we found that treating the cells with Y-27632 (3), a well-known inhibitor of Rho kinase-mediated contraction, led to tissues that were stable against necking, while treating the tissues with TGF-β led to more rapid narrowing and failure compared with untreated controls. These observations clearly show that the intrinsic contractility of microtissues in constrained geometries leads to morphological instabilities that are strong enough to cause failure of the tissues.

Fig. 2.

Experimental observations (A) and 3D finite element simulations (B) of necking and failure of loop-ended dog bone-shaped (tissue self-assembled in recesses consisting of two toroidal shapes surrounding circular pegs connected by a central trough) microtissues. The inner and outer radii of the tori are 400 μm and 1.2 mm, respectively; the distance between the centers of the tori is 9 mm; and the width of the connecting rod is 800 μm. Red arrowheads indicate the locations of necking and failure. (B) Failure simulated using a 3D finite element model (failure occurs near the midpoint of the connecting rod). Contours of axial stretch are also given in the figure.

Why do constrained microtissues narrow and fail catastrophically? And, why does failure depend on the stiffness of the boundaries and the density of the matrix? Answers to these questions can help us understand how tissues regulate their contractility temporally and spatially in response to mechanical cues. In this paper, we develop a quantitative framework to model morphological evolution and failure in 3D microtissues and derive a criterion to determine the stable and unstable behavior as a function of geometric and biochemomechanical material parameters. We show that stress-dependent active contraction gives rise to a unique type of necking instability that is distinct from other elastic instabilities hitherto studied. By comparing the observed morphologies with the predictions of our model, we are able to determine the chemomechanical material parameters that govern active response in different types of tissues.

Experiments on 3D Microtissue Constructs

Mechanics and Stability of Engineered Cardiac Tissues.

To examine the role of mechanical constraints on dynamic contractility and morphology of engineered cardiac tissues, we generated microscale constructs of cardiac cells within collagen/fibrin 3D matrices in microfabricated arrays of wells within a polydimethylsiloxane (PDMS) mold as described previously (6). Two T-shaped cantilevers incorporated within each template constrained the contracting matrix to form a linear band that spanned across the top of the pair of cantilevers (Fig. 1A). The spring constant of each cantilever could be controlled by altering the ratio between PDMS and curing agent, whereas the mechanical properties of the ECM could be modulated by varying the collagen density. Remodeling and compaction of the matrix by the cells led to a steady reduction of construct size (Fig. 1B), while the contractility of the CMTs increased, leading also to thinning at the center of the linear band between the cantilevers. By day 7, the width of the tapered central part of the tissue was significantly smaller than the rest of the tissue, subsequently leading to failure.

Interestingly, the density of the collagen/fibrin matrix impacted both the shape and the cross-sectional area of the tissue, as well the tensile force it exerted on the cantilevers. We observed that the tension increased when the collagen density was increased from 1.0 to 2.5 mg/mL (Fig. 1D), reaching plateau values of 8.0 ± 2.2 and 11.4 ± 1.7 μN, respectively. However, this higher tension in the CMTs constructed from 2.5 mg/mL collagen was not sufficient to overcome the increased rigidity of the denser collagen matrix, thus leading to better stability against necking (Fig. 1F). In addition to the density of the ECM we examined the influence of the spring constant of the cantilevers. The static tension was higher between flexible cantilevers (k = 0.20 N/m) than CMTs between rigid cantilevers (k = 0.45 N/m) at day 4 (Fig. 1D). Nevertheless, the significantly lower beating contractility of the CMTs tethered to flexible cantilevers (Fig. 1G) led to better stability against necking. Thus, CMTs appear to be more stable against necking when the density of the ECM is increased and when the stiffness of the supporting cantilevers is decreased. In addition, Hansen et al. demonstrated that inhibiting ECM degradation was necessary to avoid failure in CMTs (15). These results confirm that cardiac tissues require high ECM densities to be stable over weeks.

Stability of Loop-Ended Dog-Bone Fibroblast and Hepatocyte Tissues.

To study the morphological evolution and stability of constrained fibroblast tissues, we produced loop-ended dog bone-shaped microtissues from normal human fibroblasts (NHFs), using the self-assembly technique (without preformed ECM) described previously (3). Briefly, we cast nonadhesive hydrogel micromolds consisting of 2% (wt/vol) agarose in phosphate-buffered saline from PDMS micromold negatives. A suspension of NHFs fluorescently labeled with CellTracker Green 5-chloromethylfluorescein diacetate was then added to the inner chamber of each micromold. Over the next 20 min, the NHFs would settle into the recesses of the micromold and begin to self-assemble into microtissues consisting of two toroids connected by a rod. These loop-ended dog-bone microtissues were constrained at each end by micromold pegs rising through the lumen of each toroid. Upon self-assembly, the microtissue would compact as the individual NHFs adhered to one another. This, in turn, generated tension within the microtissue, manifested by thinning and elongation in the rod portion of the tissue. This thinning would eventually lead to necking and failure, at which point the resulting free ends of the rod would immediately contract to either toroid to relax tension within the tissue. Fig. 2A and SI Appendix, Fig. 12A, Upper and Lower depict the morphologies of microtissues which thinned near the center of the rod, at one side of the rod, and at two points along the rod, respectively.

SI Appendix, Fig. 11 depicts the evolution of the cross-sectional area near the center of the dog-bone microtissue. We find that the width decreases rapidly at all positions along the rod portion of the microtissue during the initial few hours after seeding. During the narrowing phase the majority of cells in the constrained connecting rod elongated dramatically to lengths over 30 times their starting diameters. In addition, after failure of the connecting rod, the resulting toroid microtissues would tighten around the constraining posts and thin, to the extent that they would often fail again at a single point, resulting in a curved, noncontiguous rod (SI Appendix, Fig. 12C). Altogether our results provide a detailed description of the morphological changes associated with the generation and relaxation of tension within a constrained microtissue. Elongation and failure of the microtissue were not observed in self-assembling unconstrained rods of similar size. The above experiments were repeated with rat hepatocyte (H35) cells which are much less contractile than NHFs. While cell–cell adhesions still drove the formation of dog-bone tissues, the H35 dog bones did not develop significant tension or fail and were found to be stable for at least a week (SI Appendix, Fig. 10).

Three-Dimensional Mechanosensing Model

To quantitatively analyze the complex morphological evolution described in the experiments above, we designed a three-element model that consists of two passive elastic elements to represent the elasticity of the series and parallel elements and an active contractile element, as shown in Fig. 3A. The stiffness of the tissue, in general, is expected to have components that act in series and in parallel to the active element. The series component, (E_C), which accounts for the stiffness of the actin filaments on which the myosin motors move and the stiffness of the cell–cell contacts, is added in series with the active element (representing the contractility of the tissue). The sum of the stiffnesses of the ECM and some passive contributions of the cell that act in parallel with the active element is denoted by . With position vectors to a material point in the initial and current configurations given by (SI Appendix, section 6) and , respectively, the total deformation gradient tensor is expressed as , where . Since the parallel element deforms in parallel with the cells, their deformation gradients are the same. , where the subscripts P, A, and C are used to denote the contributions from the parallel passive elasticity, actomyosin activity, and the series passive elasticity, respectively. Note that here the deformation gradient of the cells is multiplicatively decomposed into active and passive components .

Fig. 3.

(A) Three-dimensional mechanosensing model: (Upper) schematic of the constrained tissue (green) and the ECM (magenta) and (Lower) three-element model for the active tissue, where tissue stiffness (green), including cell stiffness (light green) and cell–cell contact stiffness (dark green), are modeled using a hyperelastic constitutive relation (with tangent stiffness ) in series with the an active contractile element (red) that obeys a Hill-like stress vs. strain-rate relation. These components act in parallel with the ECM, whose tangent modulus is denoted by (pink). Elastic boundaries are modeled as springs (black) with spring constant . (B) The modified Hill force–velocity model for contraction of nonmuscle cells in contractile tissues (16). Black lines represent actin bundles and red lines with heads denote myosin motors. Blue and red arrows indicate stress fiber contraction and extension, respectively. The curve shows the variation of the normalized contraction rate as a function of normalized stress for active tissues. The initial increase in the magnitude of stress is due to recruitment of actin filaments to form stress fibers; once these fibers are formed (as shown schematically by the oriented network on the top), the stress vs. strain-rate relation resembles that of a sarcomere. The physical origin of the necking instability in contractile tissues is shown schematically on the bottom. When a small perturbation is applied to the shape, such that the cross-sectional area of the point E (or C) is smaller (or greater) than the unperturbed area, increase (or decrease) in active stress relative to the stall stress leads to greater stretching (or contraction) of the tissue. This in turn leads to a further decrease (or increase) in the cross-sectional area and necking of the tissue. The green dashed line denotes the slope of the strain rate vs. stretch curve at the stall point (K).

To relate the stress at any material point to deformation, we have to consider constitutive laws for each of the mechanical components. The total stress at a material point is the sum of the passive stress from the series and parallel components, . Note that since the contractile element and the passive tissue element transmit the same forces (Fig. 3A), the active stress experienced by the actomyosin system is the same as the passive stress in the series component of tissue, . For the passive strain-hardening response of the series and parallel elements, we use neo-Hookean hyperelastic stress–strain relations as described in SI Appendix, section 2.

Recent experiments on fibroblasts (12) subject to uniaxial loading have shown that their active stress vs. strain-rate response obeys the classic Hill relation (16) (SI Appendix, Eq. 1) at large stresses, while a deviation is observed at small applied stresses. According to the Hill relation (which applies to muscle-like sarcomere structures) the contraction rate is largest in the absence of any applied stress, monotonically decreases with increasing tensile stress, and eventually vanishes when the applied stress equals the stall stress (SI Appendix, Fig. 1). However, for nonmuscle cells considered in this work, the density of stress fibers depends on the level of applied mechanical stresses due to the dependence of the biochemistry of actin polymerization and myosin phosphorylation on mechanical stresses (17–19); at low stress levels, the density of stress fibers and (hence the contractile strain) increases with increasing stress, eventually saturating at large enough stress levels. This is clearly illustrated by experiments on spreading of nonmuscle cells on deformable substrates—the forces the cells exert on the substrate through their contractile machinery initially increase as the stiffness of the substrate is increased, saturating on substrates of very large stiffness (11, 12). As we show in SI Appendix, section 1.2, when the coupling between biochemistry and mechanical stresses is considered, we obtain a curve that resembles the Hill curve with a monotonic decrease at large stresses close to the stall stress, but a nonmonotonic relation at small stresses (Fig. 3B). It has also been seen that stress-fibers only form in nonmuscle cells in the presence of anisotropic stresses – for example, cells in collagen gels develop structural anisotropy and a high density of fibers under uniaxial loading, whereas isotropic stress fields do not lead to dense aligned fibers (20). Therefore, in our 3D large-deformation analysis, we assume that the active deformation rate depends only on the active deviatoric stress, but not on the pressure;

where and are the maximum values of the stall stress and contractile strain rate, respectively, and is the equivalent active deviatoric stress in which . The function is plotted in Fig. 3B. Note also that the above equation (Eq. 1) ensures that deformation due to active stresses are volume conserving. While more complex relations (that include compressibility effects) for active strain-rate response can be used (21), as we show in SI Appendix, section 1.2, the main features of necking and failure are captured with the simple form that we use here. Using the conditions for mechanical equilibrium and Eq. 1, we have developed a finite element method to model the 3D morphological evolution of the tissue. The key features of the necking instabilities observed in our experiments can be understood quantitatively using the linear stability analysis that we consider next.

Stability of Constrained Microtissues: Linear Analysis and Failure Maps

To study the stability of the tissue due to spatial perturbations in shape or contractility, we first consider the case of the tissue of initial uniform cross-sectional area suspended between flexible posts of stiffness k_b and initially separated by distance , shown in Fig. 3A. When the tissue (assumed to be incompressible) contracts and equilibrates, the cross-sectional area in the deformed configuration can be related to the local stretch by (the overline symbol is used to denote quantities in equilibrium). As we show in SI Appendix, section 2.4, equilibrium stretch and tension can be solved using the following equations obtained from the constitutive equations and the conditions for force balance in the tissue and at the boundaries:

where is the uniaxial constitutive relations for the parallel element, and is the force exerted by the tissue and the ECM on a post. If the initial area of cross-section is a constant, as we show in SI Appendix, section 2.4, the steady-state stress that develops in the tissue is approximately given by

where is the tangent parallel stiffness at zero strain. Note that this stress increases with increasing stiffness of the posts and decreasing parallel stiffness/ECM density . The maximum stress that is achieved in the limit of infinitely stiff posts and vanishing parallel stiffness is the characteristic stall stress of the tissue, .

Next, as in the case of necking analysis for passive materials (22), we apply an elastic perturbation to the tissue such that the cross-sectional area is altered by a small amount from the equilibrium area (Fig. 4A) and study the conditions under which these small perturbations can grow in an unstable manner leading to necking. For a tissue whose overall elastic response is incompressible, the perturbation in the area is accompanied by variations in stretch, such that . By using the hyperelastic constitutive laws for the parallel element and Eq. 1 and applying the boundary conditions at the posts (SI Appendix, section 2), we find that the perturbation evolves according to the equation

where is the slope of the contractile strain vs. stress curve (the green dashed line in Fig. 3B) and and are the tangent series and parallel stiffnesses in equilibrium, respectively. For the tissue to be stable, the right-hand side of the above equation should be negative. The instability using the two-element model without a series component can be directly derived from Eq. 4 as ; the tissue is stable as , so the series passive components are not essential to obtain the instability. Using the expression for the steady-state stress in the tissue in SI Appendix, Eq. 17, we have plotted a phase diagram that shows the regime of stability of the tissue as a function of two parameters, the scaled tangent parallel stiffness, , and the scaled stiffness of the posts, , in Fig. 4E. We find that the tissue is unstable when the scaled parallel and boundary stiffnesses are small and large, respectively. In particular, when the tissue is fully constrained, the tissue is only stable if the tangent parallel stiffness is larger than the stall stress, . This suggests that in the absence of any parallel component, the active response of the constrained tissue renders the tissue unstable.

Fig. 4.

Schematics of perturbations in the initial cross-sectional area, (A); contractility, (B); and ECM density/parallel stiffness, (C) along the tissue. Colors in C (blue to red) represent variation in ECM density/parallel stiffness (low to high). (D) The morphology of the tissue is stable when and thins in an unstable manner, indicating the onset of necking, when . (E) Stability phase diagram plotted as a function of the scaled boundary stiffness and the scaled parallel stiffness . Light blue and magenta represent stable and unstable regions, where the perturbations in A–C grow in an unstable manner as the phase boundary is approached.

The physical origin of the instability of the tissue constrained by rigid posts can be qualitatively understood from Fig. 3B—here we consider the evolution of two regions, one whose area of cross-section is larger than the equilibrium value and the other whose area is smaller. Since the stress in the former (latter) case is smaller (larger) than the stall stress, we can see that the tissue contracts (stretches) in the former (latter) case. This in turn leads to further compression (stretching) in the regions whose perturbed area is larger (smaller) than the steady-state value, thus leading to the growth of the instability. The key physical feature that leads to the instability is the positive slope of the Hill curve (K) at the isometric point. When the ECM density is sufficiently large, the elastic response of the ECM counteracts the effects of the active response and is able to stabilize the tissue against necking. These results from the linear stability analysis qualitatively explain why the fibroblast tissues with no ECM fail much more easily by necking compared with the cardiac tissue constructs with large densities of ECM. Furthermore, the evolution of the perturbation depends on the steady-state stress, , which in turn depends on the stiffness of the posts. Decreasing the stiffness of the posts also leads to a decrease in stress and renders the tissue more stable, in agreement with our experiments.

While we have considered elastic perturbations to a uniformly contracted state, inhomogeneities in the seeding of the tissue can lead to variations in the initial cross-sectional area of the tissue , the number density of the cells (and hence the maximum stall stress, ), and the density (and hence the stiffness, ) of the ECM. These variations in turn will lead to variations in cross-sections and stretches along the length of the tissue in equilibrium (Fig. 4D). Using the prefix δ to denote variations in the chemomechanical material parameters, we show in SI Appendix that the variation in stretches along the tissue can be written as

The above result shows that for a given density of ECM, the variations in stretches increase with increasing contractility and hence or ; the variations diverge when , indicating the onset of necking (Fig. 4D). No physically admissible solutions can be found when (SI Appendix, sections 2.2 and 2.3). Note that the conditions for stability found for the variations in the initial material parameters are the same as the conditions found for the elastic perturbations in shape, illustrating the general applicability of these criteria.

Three-Dimensional Finite Element Simulations of Tissue Morphology Evolution and Failure

To model the evolution of the tissue for large deformations, we implemented the hyperelastic constitutive equations and Eq. 1 in a user material model in the finite element package ABAQUS (23). The details of the user material implementation are given in SI Appendix. Simulations of the evolution of the morphology of the fibroblast tissue are shown in Fig. 2B and SI Appendix, Fig. 12B. Here we start by applying perturbations in shape (amplitude 5% of the thickness) to an otherwise symmetric dog-bone shape at different positions along the length of the connecting rod. Although the magnitudes of the initial perturbations are too small to be seen early on (Fig. 2B, Upper), the perturbations grow in amplitude with time. Our simulations are able to faithfully reproduce all aspects of the evolution of the perturbations including thinning and extension at points where the perturbations are applied and thickening at regions close to the peg. When the perturbations are applied at two points (SI Appendix, Fig. 12B, Lower), one close to the end and the other near the middle, thinning is observed at both points initially, although the point at the center eventually thins much more rapidly during the later stages of evolution, in agreement with observations. By fitting the computed cross-sectional area along the connecting rod with experimental measurements (SI Appendix, Fig. 11), we are able to quantify all of the parameters in our constitutive model. We find that the initial series stiffness, , the initial parallel stiffness, , and the maximum rate of active contraction, . The rate of active contraction we have deduced from the simulations of tissue morphology is close to the value measured for single fibroblasts held between flexible plates (12), lending strong support to our analysis. Since the parallel stiffness in these tissues is only a small fraction of the stall stress, the failure of the tissue observed in the experiments is consistent with the phase diagram in Fig. 4E. We observe greater rates of contractility in tissues treated with TGF-β, confirming that TGF-β enhances actomyosin activity. The failure of toroidal ends after the failure of the central band is discussed in SI Appendix, section 9.

In the case of the CMTs, we carried out simulations to model the evolution of the tissue from day 0 to day 7 and monitored both the force exerted by the tissue on the cantilevers and the shape of the tissue (Fig. 1 C and D). On day 3, we applied an isotropic contractile strain of 1% to mimic the formation of cell–cell contacts and the compaction process. By fitting six sets of measured tissue force and area vs. time curves (Fig. 1 C and D) to our simulations (using the least squares method described in SI Appendix), we determined the following material parameters: series stiffness, ; maximum stall stress, ; maximum rate of active contraction, ; and the initial parallel stiffness, for collagen and for collagen, respectively. The initial stiffness of the ECM increases with deformation following the neo-Hookean constitutive law. Our simulations clearly explain why the tissue is much more stable to necking (Fig. 1 E–G) when the ECM density is large and the cantilevers flexible.

We also find that the parameters we have obtained here are consistent with properties of cardiac tissues reported in the literature. There are 58% cardiomyocytes and 42% nonmyocytes (mostly fibroblasts) in our CMTs (6). The Young’s modulus of fibroblast tissue has been reported to be in the range of (24) and the Young’s modulus of myocardium is around (25). The value we obtained for series stiffness, , lies between the stiffness of cardiomyocytes and fibroblasts. Also, the stiffness of heart muscle has been reported to be at the beginning of diastole (26), consistent with the values we obtained using our fitting procedure. The elastic modulus of fibroblast microtissues has been reported as , depending on the chemical treatment (27). The mechanical properties of collagen–fibrin cogels (0–2.1 mg/mL collagen and 0–3.0 mg/mL fibrin) have been measured experimentally and fitted using a nonlinear stress–strain relation (28). Pure collagen (2.1 mg/mL) possesses a much higher stiffness and a larger degree of nonlinearity than pure fibrin (3.0 mg/mL), suggesting that the mechanical behavior of a collagen/fibrin ECM is dominated by the response of collagen. The Young’s modulus at small strains has been reported to be 0.2 kPa for 2.1 mg/mL pure collagen; the values we obtain ( for collagen and for collagen) are therefore in this range.

Discussion and Future Work

In passive materials, necking is generally understood on the basis of the classic Considère instability (29), which provides a criterion for failure based on passive elastic response. According to this criterion, necks form in strain-softening materials under tension when regions of the material with reduced area (and hence larger stresses) rapidly elongate. This occurs if the material does not sufficiently strain harden to compensate for the increased stresses in narrowed regions. This model however cannot explain the necking observed in the active tissues that we consider here because (i) it is well established that the passive response of the cytoskeletal components of the cells as well as the tissues themselves is generally of a strain-hardening nature (30) and because (ii) this model does not account for the active response of the material. As we have shown here, necking can be observed even in active materials with a passive strain-hardening response. Models have also been developed to include the role of actomyosin activity using continuum and discrete approaches (21, 31–37), but none of these models account for the coupling between geometry, stress, and the actomyosin activity responsible for the dramatic evolution of morphology observed in constrained microtissues (Figs. 1 and 2). The predictions of our model are also consistent with the observations of necking and failure in constrained reconstituted actomyosin gels made using skeletal muscle myosin II-thick filaments, actin and α-actinin (38). Interestingly, unconstrained gels were found to be stable, indicating that the observed instability is caused by the mechanism proposed in this work.

Three-dimensional tissue constructs have served as in vitro models of tissue morphogenesis and wound healing, and, more recently, designs such as honeycombs and toroids have been proposed as strategies for drug delivery and in vivo implantation (4). In natural tissues, actomyosin activity is significantly influenced by whether the cells see a homotypic or heterotypic environment (39). We have recently shown that the self-assembly forces exerted by cells in the mixed heterotypic cell environment are significantly greater than the forces in their respective homotypic environments. Extension of the work described here to study the stability of tissue constructs with multiple cell types of different stiffness would be of fundamental relevance to tissue engineering as well as developmental biology because natural tissues are not composed of a single cell type. They are a mix of cell types and there are few, if any, means to quantify these interactions in 3D. In natural tissues, cells also assemble a dense meshwork of ECM proteins, including collagen, fibronectin, tenascin, and other proteins. Understanding how forces arising from contractility influence the formation of these proteins and how increased stiffening of the tissue due to these proteins regulates contractility can provide insights into the mechanisms of branching of blood vessels and bronchial tubes, shaping of organs, and embryogenesis (40, 41).