The origin of rigidity in disordered materials is an outstanding open problem in statistical physics. Previously, a class of 2D cellular models has been shown to undergo a rigidity transition ...controlled by a mechanical parameter that specifies cell shapes. Here, we generalize this model to 3D and find a rigidity transition that is similarly controlled by the preferred surface area S0: the model is solid-like below a dimensionless surface area of s 0 S 0 V ¯ 2 3 5.413 with V ¯ being the average cell volume, and fluid-like above this value. We demonstrate that, unlike jamming in soft spheres, residual stresses are necessary to create rigidity. These stresses occur precisely when cells are unable to obtain their desired geometry, and we conjecture that there is a well-defined minimal surface area possible for disordered cellular structures. We show that the behavior of this minimal surface induces a linear scaling of the shear modulus with the control parameter at the transition point, which is different from the scaling observed in particulate matter. The existence of such a minimal surface may be relevant for biological tissues and foams, and helps explain why cell shapes are a good structural order parameter for rigidity transitions in biological tissues.
Abstract
Active matter with local polar or nematic order is subject to the well-known Simha-Ramaswamy instability. It is so far unclear how, despite this instability, biological tissues can undergo ...robust active anisotropic deformation during animal morphogenesis. Here we ask under which conditions protein concentration gradients (e.g. morphogen gradients), which are known to control large-scale coordination among cells, can stabilize such deformations. To this end, we study a hydrodynamic model of an active polar material. To account for the effect of the protein gradient, the polar field is coupled to the boundary-provided gradient of a scalar field that also advects with material flows. Focusing on the large system size limit, we show in particular: (a) the system can be stable for an effectively extensile coupling between scalar field gradient and active stresses, i.e. gradient-extensile coupling, while it is always unstable for a gradient-contractile coupling. Intriguingly, there are many systems in the biological literature that are gradient-extensile, while we could not find any that are clearly gradient-contractile. (b) Stability is strongly affected by the way polarity magnitude is controlled. Taken together, our findings, if experimentally confirmed, suggest new developmental principles that are directly rooted in active matter physics.
We present an approach to understand geometric-incompatibility–induced rigidity in underconstrained materials, including subisostatic 2D spring networks and 2D and 3D vertex models for dense ...biological tissues. We show that in all these models a geometric criterion, represented by a minimal length ℓ̄min, determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but also the precise magnitudes for bulk modulus and shear modulus discontinuities at the rigidity transition as well as the magnitude of the Poynting effect. We also predict from first principles that the ratio of the excess shear modulus to the shear stress should be inversely proportional to the critical strain with a prefactor of 3. We propose that this factor of 3 is a general hallmark of geometrically induced rigidity in underconstrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Finally, our results may lay important foundations for ways to estimate ℓ̄min from measurements of local geometric structure and thus help develop methods to characterize large-scale mechanical properties from imaging data.
•The mechanical properties of biological tissues are important for their function.•Models predict constitutive laws explaining how tissues deform in response to forces.•Cellular contributions to ...overall tissue deformation can be precisely quantified.•Cell shape may indicate whether a tissue behaves as a solid or a fluid.•Cell motility can induce unexpected collective behavior on the tissue scale.
In multi-cellular organisms, morphogenesis translates processes at the cellular scale into tissue deformation at the scale of organs and organisms. To understand how biochemical signaling regulates tissue form and function, we must understand the mechanical forces that shape cells and tissues. Recent progress in developing mechanical models for tissues has led to quantitative predictions for how cell shape changes and polarized cell motility generate forces and collective behavior on the tissue scale. In particular, much insight has been gained by thinking about biological tissues as physical materials composed of cells. Here we review these advances and discuss how they might help shape future experiments in developmental biology.
Collective cell migration is a highly regulated process involved in wound healing, cancer metastasis, and morphogenesis. Mechanical interactions among cells provide an important regulatory mechanism ...to coordinate such collective motion. Using a self-propelled Voronoi (SPV) model that links cell mechanics to cell shape and cell motility, we formulate a generalized mechanical inference method to obtain the spatiotemporal distribution of cellular stresses from measured traction forces in motile tissues and show that such traction-based stresses match those calculated from instantaneous cell shapes. We additionally use stress information to characterize the rheological properties of the tissue. We identify a motility-induced swim stress that adds to the interaction stress to determine the global contractility or extensibility of epithelia. We further show that the temporal correlation of the interaction shear stress determines an effective viscosity of the tissue that diverges at the liquid–solid transition, suggesting the possibility of extracting rheological information directly from traction data.
Within developing embryos, tissues flow and reorganize dramatically on timescales as short as minutes. This includes epithelial tissues, which often narrow and elongate in convergent extension ...movements due to anisotropies in external forces or in internal cellgenerated forces. However, the mechanisms that allow or prevent tissue reorganization, especially in the presence of strongly anisotropic forces, remain unclear. We study this question in the converging and extending Drosophila germband epithelium, which displays planar-polarized myosin II and experiences anisotropic forces from neighboring tissues. We show that, in contrast to isotropic tissues, cell shape alone is not sufficient to predict the onset of rapid cell rearrangement. From theoretical considerations and vertex model simulations, we predict that in anisotropic tissues, two experimentally accessible metrics of cell patterns—the cell shape index and a cell alignment index—are required to determine whether an anisotropic tissue is in a solid-like or fluid-like state.We show that changes in cell shape and alignment over time in the Drosophila germband predict the onset of rapid cell rearrangement in both wild-type and snail twist mutant embryos, where our theoretical prediction is further improved when we also account for cell packing disorder. These findings suggest that convergent extension is associated with a transition to more fluid-like tissue behavior, which may help accommodate tissue-shape changes during rapid developmental events.
Segmentation and tracking of cells in long-term time-lapse experiments has emerged as a powerful method to understand how tissue shape changes emerge from the complex choreography of constituent ...cells. However, methods to store and interrogate the large datasets produced by these experiments are not widely available. Furthermore, recently developed methods for relating tissue shape changes to cell dynamics have not yet been widely applied by biologists because of their technical complexity. We therefore developed a database format that stores cellular connectivity and geometry information of deforming epithelial tissues, and computational tools to interrogate it and perform multi-scale analysis of morphogenesis. We provide tutorials for this computational framework, called TissueMiner, and demonstrate its capabilities by comparing cell and tissue dynamics in vein and inter-vein subregions of the Drosophila pupal wing. These analyses reveal an unexpected role for convergent extension in shaping wing veins.
How tissue shape emerges from the collective mechanical properties and behavior of individual cells is not understood. We combine experiment and theory to study this problem in the developing wing ...epithelium of Drosophila. At pupal stages, the wing-hinge contraction contributes to anisotropic tissue flows that reshape the wing blade. Here, we quantitatively account for this wing-blade shape change on the basis of cell divisions, cell rearrangements and cell shape changes. We show that cells both generate and respond to epithelial stresses during this process, and that the nature of this interplay specifies the pattern of junctional network remodeling that changes wing shape. We show that patterned constraints exerted on the tissue by the extracellular matrix are key to force the tissue into the right shape. We present a continuum mechanical model that quantitatively describes the relationship between epithelial stresses and cell dynamics, and how their interplay reshapes the wing.
Tissue, cell, and nucleus morphology change during tumor progression. In 2D confluent cell cultures, different tissue states, such as fluid (unjammed) and solid (jammed), are correlated with cell ...shapes. These results do not have to apply a priori to three dimensions. Cancer cell motility requires and corresponds to a fluidization of the tumor tissue on the bulk level. Here, we investigate bulk tissue fluidity in 3D and determine how it correlates with cell and nucleus shape. In patient samples of mamma and cervix carcinoma, we find areas where cells can move or are immobile. We compare 3D cell spheroids composed of cells from a cancerous and a noncancerous cell line. Through bulk mechanical spheroid-fusion experiments and single live-cell tracking, we show that the cancerous sample is fluidized by active cells moving through the tissue. The healthy, epithelial sample with immobile cells behaves more solidlike. 3D segmentations of the samples show that the degree of tissue fluidity correlates with elongated cell and nucleus shapes. This correlation links cell shapes to cell motility and bulk mechanical behavior. We find two active states of matter in solid tumors: an amorphous glasslike state with characteristics of 3D cell jamming and a disordered fluid state. Individual cell and nucleus shape may serve as a marker for metastatic potential to foster personalized cancer treatment.
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