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Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain, which is theoretically well understood. Here and in the companion paper, we extend this theory to finite temperatures for a very broad class of under-constrained systems. In the vicinity of the athermal transition point, we derive from first principles expressions for elastic properties such as isotropic tension $t$ and shear modulus $G$ on temperature $T$, isotropic strain $\varepsilon$, and shear strain $\gamma$, which we confirm numerically. These expressions contain only three phenomenological parameters, entropic rigidity $\kappa_S$, energetic rigidity $\kappa_E$, and a parameter $b_\varepsilon$ describing the interaction between isotropic and shear strain. Our results imply that in under-constrained systems, entropic and energetic rigidity interact like two springs in series. This also allows for a simple explanation of the previously numerically observed scaling relation $t\sim G\sim T^{1/2}$ at $\varepsilon=\gamma=0$. Our work unifies the physics of systems as diverse as polymer fibers & networks, membranes, and vertex models for biological tissues., Comment: 6 pages, 3 figures

Athermal (i.e. zero-temperature) under-constrained systems are typically floppy, but they can be rigidified by the application of external strain. Following our recently developed analytical theory for the athermal limit, here and in the companion paper, we extend this theory to under-constrained systems at finite temperatures. Close to the athermal transition point, we derive from first principles the partition sum for a broad class of under-constrained systems, from which we obtain analytic expressions for elastic material properties such as isotropic tension $t$ and shear modulus $G$ in terms of isotropic strain $\varepsilon$, shear strain $\gamma$, and temperature $T$. Our work unifies the physics of systems as diverse as polymer fibers & networks, membranes, and vertex models for biological tissues., Comment: 19 pages, 1 figure

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 show that 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: (i) 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. (ii) 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., Comment: 20 pages, 7 figures

Biological cells can actively tune their intracellular architecture according to their overall shape. Here we explore the rheological implication of such coupling in a minimal model of a dense cellular material where each cell exerts an active mechanical stress along its axis of elongation. Increasing the active stress amplitude leads to several transitions. An initially hexagonal crystal motif is first destabilized into a solid with anisotropic cells. Increasing activity further, we find a re-entrant transition to a regime with finite hexatic order and finite shear modulus, in which cells arrange according to a rhombile pattern with periodically arranged rosette structures. The shear modulus vanishes again at a third threshold beyond which spontaneous tissue flows arise. In this last regime, we observe the emergence of cell shape patterns called topological defects, with flow and stress fields around defects agreeing with those observed in epithelial tissue experiments. We further provide a testable prediction of cell-cell rearrangement hotspots near topological defects. Overall, our work connects seemingly distinct features - e.g. rosettes and topological defects - observed across various types of epithelial tissues., Comment: 27 pages; 21 figures

Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e.\ networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the mechanical network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus $G$ scales linearly with isotropic stress $T$ close to the rigidity transition; however, this seems to be at odds with recent numerical studies suggesting an exponent between $G$ and $T$ that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition linear scaling, $G\sim T$, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class., Comment: 17 pages, 10 figures

Phase separation can drive spatial organization of multicomponent mixtures. For instance in developing animal embryos, effective phase separation descriptions have been used to account for the spatial organization of different tissue types. Similarly, separation of different tissue types and the emergence of a polar organization is also observed in cell aggregates mimicking early embryonic axis formation. Here, we describe such aggregates as deformable two-phase fluid droplets, which are suspended in a fluid environment (third phase). Using hybrid finite-volume Lattice-Boltzmann simulations, we numerically explore the out-of-equilibrium routes that can lead to the polar equilibrium state of such a droplet (Janus droplet). We focus on the interplay between spinodal decomposition and advection with hydrodynamic flows driven by interface tensions, which we characterize by a Peclet number $Pe$. Consistent with previous work, for large $Pe$ the coarsening process is generally accelerated. However, for intermediate $Pe$ we observe long-lived, strongly elongated droplets, where both phases form an alternating stripe pattern. We show that these ``croissant'' states are close to mechanical equilibrium and coarsen only slowly through diffusive fluxes in an Ostwald-ripening-like process. Finally, we show that a surface tension asymmetry between both droplet phases leads to transient, rotationally symmetric states whose resolution leads to flows reminiscent of Marangoni flows. Our work highlights the importance of advection for the phase separation process in finite, deformable systems., Comment: 11 pages, 6 figures

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 cell-generated 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, and 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.

We present a novel approach to understand geometric-incompatibility-induced rigidity in under-constrained materials, including sub-isostatic 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 $\bar\ell_\mathrm{min}$, 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 {\em 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 three, and propose that this factor of three is a general hallmark of geometrically induced rigidity in under-constrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Lastly, our results may lay important foundations for ways to estimate $\bar\ell_\mathrm{min}$ from measurements of local geometric structure, and thus help develop methods to characterize large-scale mechanical properties from imaging data., Comment: 10 pages, 5 figures

Although cell shape can reflect the mechanical and biochemical properties of the cell and its environment, quantification of 3D cell shapes within 3D tissues remains difficult, typically requiring digital reconstruction from a stack of 2D images. We investigate a simple alternative technique to extract information about the 3D shapes of cells in a tissue; this technique connects the ensemble of 3D shapes in the tissue with the distribution of 2D shapes observed in independent 2D slices. Using cell vertex model geometries, we find that the distribution of 2D shapes allows clear determination of the mean value of a 3D shape index. We analyze the errors that may arise in practice in the estimation of the mean 3D shape index from 2D imagery and find that typically only a few dozen cells in 2D imagery are required to reduce uncertainty below 2\%. This framework could be naturally extended to estimate additional 3D geometric features and quantify their uncertainty in other materials.

Vertex models are a popular approach to modeling the mechanical and dynamical properties of dense biological tissues, describing the tissue as a network of connected polygons representing the cells. Recently a class of two-dimensional vertex models was shown to exhibit a disordered rigidity transition controlled by the preferred cellular geometry, echoing experimental findings. An attractive variant of these models uses a Voronoi tessellation to describe the cells and endows them with a non-equilibrium model of cellular motility, leading to rich, glassy behavior. This glassy behavior was suggested to be inextricably linked to an underlying jamming transition. We test this conjecture, exploring the low-effective-temperature limit of the Voronoi model by studying cell trajectories from detailed dynamical simulations in combination with rigidity measurements of energy-minimized disordered cell configurations. We find that the zero-temperature limit of this model has no unjamming transition., Comment: 6 pages, 4 figures

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: the model is solid-like below a dimensionless surface area of $s_0^\ast\approx5.413$, 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., Comment: 6 pages main text + 13 pages appendix, 3 main text figures + 6 appendix figures

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 spatio-temporal 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., Comment: 12 pages, 9 figures

We present a hydrodynamic theory to describe shear flows in developing epithelial tissues. We introduce hydrodynamic fields corresponding to state properties of constituent cells as well as a contribution to overall tissue shear flow due to rearrangements in cell network topology. We then construct a generic linear constitutive equation for the shear rate due to topological rearrangements and we investigate a novel rheological behaviour resulting from memory effects in the tissue. We identify two distinct active cellular processes: generation of active stress in the tissue, and actively driven topological rearrangements. We find that these two active processes can produce distinct cellular and tissue shape changes, depending on boundary conditions applied on the tissue. Our findings have consequences for the understanding of tissue morphogenesis during development.

In this article, we propose a general framework to study the dynamics and topology of cellular networks that capture the geometry of cell packings in two-dimensional tissues. Such epithelia undergo large-scale deformation during morphogenesis of a multicellular organism. Large-scale deformations emerge from many individual cellular events such as cell shape changes, cell rearrangements, cell divisions, and cell extrusions. Using a triangle-based representation of cellular network geometry, we obtain an exact decomposition of large-scale material deformation. Interestingly, our approach reveals contributions of correlations between cellular rotations and elongation as well as cellular growth and elongation to tissue deformation. Using this Triangle Method, we discuss tissue remodeling in the developing pupal wing of the fly Drosophila melanogaster., Comment: 26 pages, 18 figures