Interplay of cell dynamics and epithelial tension during morphogenesis of the Drosophila pupal wing

Junctional force patterning drives both positional order and planar polarity in the auditory epithelia

Tissue function depends on the precise organisation of the constituent cells. In the cochlea, the fidelity of hearing depends on mechanosensory hair cells being consistently surrounded by supporting cells. In addition to this positional order, auditory sensitivity depends crucially on planar cell polarity. This is characterised by the alignment of the orientation of eccentrically placed hair bundles on each hair cell. These two levels of order emerge simultaneously despite the cellular fluxes that occur during cochlear development. However, the link between tissue-scale cellular rearrangements and intrinsic cellular mechanisms remains unknown. By combining experimental and theoretical approaches, we find a precise force patterning underpinning positional order and planar cell polarity. This occurs through the modulation of the levels and phospho-type of the regulatory light chain of non-muscle myosin II at specific cell-cell junctions of the auditory epithelium. We propose that the control of junctional mechanics is vital for the organisation of multi-cell-type epithelia.

Substrate stress relaxation regulates monolayer fluidity and leader cell formation for collectively migrating epithelia

Collective migration of epithelial tissues is a critical feature of developmental morphogenesis and tissue homeostasis. Coherent motion of cell collectives requires large-scale coordination of motion and force generation and is influenced by mechanical properties of the underlying substrate. While tissue viscoelasticity is a ubiquitous feature of biological tissues, its role in mediating collective cell migration is unclear. Here, we have investigated the impact of substrate stress relaxation on the migration of micropatterned epithelial monolayers. Epithelial monolayers exhibit faster collective migration on viscoelastic alginate substrates with slower relaxation timescales, which are more elastic, relative to substrates with faster stress relaxation, which exhibit more viscous loss. Faster migration on slow-relaxing substrates is associated with reduced substrate deformation, greater monolayer fluidity, and enhanced leader cell formation. In contrast, monolayers on fast-relaxing substrates generate substantial substrate deformations and are more jammed within the bulk, with reduced formation of transient lamellipodial protrusions past the monolayer edge leading to slower overall expansion. This work reveals features of collective epithelial dynamics on soft, viscoelastic materials and adds to our understanding of cell-substrate interactions at the tissue scale.

Origin of yield stress and mechanical plasticity in model biological tissues

During development and under normal physiological conditions, biological tissues are continuously subjected to substantial mechanical stresses. In response to large deformations, cells in a tissue must undergo multicellular rearrangements to maintain integrity and robustness. However, how these events are connected in time and space remains unknown. Here, using theoretical modeling, we study the mechanical plasticity of cell monolayers under large deformations. Our results suggest that the jamming-unjamming (solid-fluid) transition can vary significantly depending on the degree of deformation, implying that tissues are highly unconventional materials. We elucidate the origins of this behavior. We also demonstrate how large deformations are accommodated through a series of cellular rearrangements, similar to avalanches in non-living materials. We find that these 'tissue avalanches' are governed by stress redistribution and the spatial distribution of "soft" or vulnerable spots, which are more prone to undergo rearrangements. Finally, we propose a simple and experimentally accessible framework to infer tissue-level stress and predict avalanches based on static images.

The role of extracellular matrix viscoelasticity in development and disease

For several decades, research has studied the influence of the extracellular matrix (ECM) mechanical properties in cell response, primarily emphasising its elasticity as the main determinant of cell and tissue behaviour. However, the ECM is not purely elastic; it is viscoelastic. ECM viscoelasticity has now emerged as a major regulator of collective cell dynamics. This review highlights recent findings on the role of ECM viscoelasticity in development and pathology.

How does the extracellular matrix affect the rigidity of an embedded spheroid?

Cellularized tissue and polymer networks can both transition from floppy to rigid as a function of their control parameters, and, yet, the two systems often mechanically interact, which could affect their respective rigidities. To study this interaction, we consider a vertex model with surface tension embedded in a spring network in two dimensions. We identify two regimes with different global spheroid shapes and governed by the pressure resulting from competition between surface tension and tension in the network. In the first regime, the tissue remains compact, while in the second, a cavitation-like instability leads to the emergence of gaps at the tissue-network interface. Intriguingly, compression of the tissue promotes fluidization, while tension promotes cellular alignment and rigidification with the mechanisms driving rigidification differing on either side of the instability.

Tissue Active Matter: Integrating Mechanics and Signaling into Dynamical Models

The importance of physical forces in the morphogenesis, homeostatic function, and pathological dysfunction of multicellular tissues is being increasingly characterized, both theoretically and experimentally. Analogies between biological systems and inert materials such as foams, gels, and liquid crystals have provided striking insights into the core design principles underlying multicellular organization. However, these connections can seem surprising given that a key feature of multicellular systems is their ability to constantly consume energy, providing an active origin for the forces that they produce. Key emerging questions are, therefore, to understand whether and how this activity grants tissues novel properties that do not have counterparts in classical materials, as well as their consequences for biological function. Here, we review recent discoveries at the intersection of active matter and tissue biology, with an emphasis on how modeling and experiments can be combined to understand the dynamics of multicellular systems. These approaches suggest that a number of key biological tissue-scale phenomena, such as morphogenetic shape changes, collective migration, or fate decisions, share unifying design principles that can be described by physical models of tissue active matter.

Mechanical strain focusing at topological defect sites in regenerating Hydra

The formation of a new head during Hydra regeneration involves the establishment of a head organizer that functions as a signaling center and contains an aster-shaped topological defect in the organization of the supracellular actomyosin fibers. Here, we show that the future head region in regenerating tissue fragments undergoes multiple instances of extensive stretching and rupture events from the onset of regeneration. These recurring localized tissue deformations arise due to transient contractions of the supracellular ectodermal actomyosin fibers that focus mechanical strain at defect sites. We further show that stabilization of aster-shaped defects is disrupted by perturbations of the Wnt signaling pathway. We propose a closed-loop feedback mechanism promoting head organizer formation, and develop a biophysical model of regenerating Hydra tissues that incorporates a morphogen source activated by mechanical strain and an alignment interaction directing fibers along morphogen gradients. We suggest that this positive-feedback loop leads to mechanical strain focusing at defect sites, enhancing local morphogen production and promoting robust organizer formation.

Tissue shear as a cue for aligning planar polarity in the developing Drosophila wing

Planar polarity establishment in epithelia requires interpretation of directional tissue-level information at cellular and molecular levels. Mechanical forces exerted during tissue morphogenesis are emerging as crucial tissue-level directional cues, yet the mechanisms by which they regulate planar polarity are poorly understood. Using the Drosophila pupal wing, we confirm that tissue stress promotes proximal-distal (PD) planar polarity alignment. Moreover, high tissue stress anisotropy can reduce the rate of accumulation and lower the stability on cell junctions of the core planar polarity protein Frizzled (Fz). Notably, under high tissue stress anisotropy, we see an increased gradient of cell flow, characterised by differential velocities across adjacent cell rows. This promotes core protein turnover at cell-cell contacts parallel to the flow direction, possibly via dissociation of transmembrane complexes by shear forces. We propose that gradients of cell flow play a critical role in establishing and maintaining PD-oriented polarity alignment in the developing pupal wing.

Origin of yield stress and mechanical plasticity in model biological tissues

During development and under normal physiological conditions, biological tissues are continuously subjected to substantial mechanical stresses. In response to large deformations cells in a tissue must undergo multicellular rearrangements in order to maintain integrity and robustness. However, how these events are connected in time and space remains unknown. Here, using computational and theoretical modeling, we studied the mechanical plasticity of epithelial monolayers under large deformations. Our results demonstrate that the jamming-unjamming (solid-fluid) transition in tissues can vary significantly depending on the degree of deformation, implying that tissues are highly unconventional materials. Using analytical modeling, we elucidate the origins of this behavior. We also demonstrate how a tissue accommodates large deformations through a collective series of rearrangements, which behave similarly to avalanches in non-living materials. We find that these 'tissue avalanches' are governed by stress redistribution and the spatial distribution of vulnerable spots. Finally, we propose a simple and experimentally accessible framework to predict avalanches and infer tissue mechanical stress based on static images.

Stored elastic bending tension as a mediator of embryonic body folding

During development, amniote vertebrate embryos transform from a flat sheet into a three-dimensional cylindrical form through ventral folding of the lateral sides of the sheet (the lateral plate [LP]) and their fusion in the ventral midline. Using a chick embryo slice system, we find that the flat stage is actually a poised balance of opposing dorsal and ventral elastic bending tensions. An intact extracellular matrix (ECM) is required for generating tension, as localized digestion of ECM dissipates tension, while removal of endoderm or ectoderm layers has no significant effect. As development proceeds, dorsal bending tension dissipates coincident with epithelial-mesenchymal transition in the dorsal LP while ventral tension is maintained, changing the balance of forces to promote ventral folding. Interference with the elastic ECM component fibrillin reduces ventral bending tension and perturbs body folding in vivo. A model is presented for the accumulation and harnessing of LP bending tension to drive body folding.

The Fat-Dachsous planar polarity pathway competes with hinge contraction to orient polarized cell behaviors during Drosophila wing morphogenesis

During tissue morphogenesis, an interplay of biochemical pathways and mechanical cues regulates polarized cell behaviors, the balance of which leads to tissues reaching their correct shape and size.1,2,3,4 A well-studied example of a biochemical regulator is the highly conserved Fat-Dachsous (Ft-Ds) pathway that coordinates planar polarized cell behaviors and growth in epithelial tissues.5,6 For instance, in the Drosophila larval wing disc, the Ft-Ds pathway acts via the atypical myosin Dachs to control tissue shape by promoting the orientation of cell divisions primarily in a proximodistal (PD) direction.7,8 Here, we investigate interactions between Ft-Ds planar polarity and mechanical forces in the developing Drosophila pupal wing. We show that in the early stages of pupal wing development (16-18 h after puparium formation), anteroposterior (AP)-oriented cell divisions and T1 transitions are controlled by the Ft-Ds pathway acting via Dachs. Shortly thereafter, PD-oriented tissue tension is induced across the wing blade by the process of hinge contraction. This opposes the control of Dachs over polarized cell behaviors in a tug-of-war fashion, resulting in more PD-oriented cell divisions and T1s. Furthermore, increased PD tissue tension stabilizes Ft along PD-oriented junctions, suggesting that biomechanical feedback on the Ft-Ds pathway resists the effects of hinge contraction on cell shape. We also show that loss of Dachs results in increased myosin-II stability at cell junctions, revealing compensatory interactions between these two myosins. Overall, we propose that Ft-Ds pathway function constitutes a mechanism whereby tissues are buffered against mechanical perturbations.

Initiation of epithelial wound closure by an active instability at the purse string

The ability of biological systems to withstand and recover from various disruptions, such as spontaneous genetic mutations and environmental damage, largely relies on intricate feedback mechanisms. We theoretically study the mechanical response of an epithelial tissue facing damage in the form of a circular wound. Our model describes a feedback loop between the generation of active forces in the actomyosin and tissue mechanics, described by the vertex model. While the exact dynamics of wound closure may be influenced by several biophysical mechanisms that interplay in a nontrivial way, our findings suggest that the closure may initiate as an active instability, triggered by a reduced myosin turnover rate at the wound's perimeter. We explore the interplay between myosin dynamics and the elastic properties of the tissue, elucidating their collective role in determining a wound's loss of stability, leading to the initiation of the closure process.

Generic theory of interacting, spinning, active polar particles: A model for cell aggregates

We present a generic framework for describing interacting, spinning, active polar particles, aimed at modeling dense cell aggregates, where cells are treated as polar, rotating objects that interact mechanically with one another and their surrounding environment. Using principles from nonequilibrium thermodynamics, we derive constitutive equations for interaction forces, torques, and polarity dynamics. We subsequently use this framework to analyze the spontaneous motion of cell doublets, uncovering a rich phase diagram of collective behaviors, including steady rotation driven by flow-polarity coupling or interactions between polarity and cell position.

Microdifferential Pressure Measurement Device for Cellular Microenvironments

Mechanical forces influence cellular proliferation, differentiation, tissue morphogenesis, and functional expression within the body. To comprehend the impact of these forces on living organisms, their quantification is essential. This study introduces a novel microdifferential pressure measurement device tailored for cellular-scale pressure assessments. The device comprises a glass substrate and a microchannel constructed of polydimethylsiloxane, polytetrafluoroethylene tubes, a glass capillary, and a microsyringe pump. This device obviates the need for electrical measurements, relying solely on the displacement of ultrapure water within the microchannel to assess the micropressure in embryos. First, the device was subjected to arbitrary pressures, and the relationship between the pressure and the displacement of ultrapure water in the microchannel was determined. Calibration results showed that the displacement dx [μm] could be calculated from the pressure P [Pa] using the equation dx = 0.36 P. The coefficient of determination was shown to be 0.87, indicating a linear response. When utilized to measure brain ventricular pressure in mouse embryos, the fabricated device yielded an average pressure reading of 1313 ± 640 Pa. This device can facilitate the measurement of pressure within microcavities in living tissues and other areas requiring precise and localized pressure evaluations.

Peeking into the future: inferring mechanics in dynamical tissues

Cells exert forces on each other and their environment, shaping the tissue. The resulting mechanical stresses can be determined experimentally or estimated computationally using stress inference methods. Over the years, mechanical stress inference has become a non-invasive, low-cost computational method for estimating the relative intercellular stresses and intracellular pressures of tissues. This mini-review introduces and compares the static and dynamic modalities of stress inference, considering their advantages and limitations. To date, most software has focused on static inference, which requires only a single microscopy image as input. Although applicable in quasi-equilibrium states, this approach neglects the influence that cell rearrangements might have on the inference. In contrast, dynamic stress inference relies on a time series of microscopy images to estimate stresses and pressures. Here, we discuss both static and dynamic mechanical stress inference in terms of their physical, mathematical, and computational foundations and then outline what we believe are promising avenues for in silico inference of the mechanical states of tissues.

The geometric basis of epithelial convergent extension

Shape changes of epithelia during animal development, such as convergent extension, are achieved through the concerted mechanical activity of individual cells. While much is known about the corresponding large-scale tissue flow and its genetic drivers, fundamental questions regarding local control of contractile activity on the cellular scale and its embryo-scale coordination remain open. To address these questions, we develop a quantitative, model-based analysis framework to relate cell geometry to local tension in recently obtained time-lapse imaging data of gastrulating Drosophila embryos. This analysis systematically decomposes cell shape changes and T1 rearrangements into internally driven, active, and externally driven, passive, contributions. Our analysis provides evidence that germ band extension is driven by active T1 processes that self-organize through positive feedback acting on tensions. More generally, our findings suggest that epithelial convergent extension results from the controlled transformation of internal force balance geometry which combines the effects of bottom-up local self-organization with the top-down, embryo-scale regulation by gene expression.

Mechanics of Drosophila wing deployment

During their final transformation, insects emerge from the pupal case and deploy their wings within minutes. The wings deploy from a compact origami structure, to form a planar and rigid blade that allows the insect to fly. Deployment is powered by a rapid increase in internal pressure, and by the subsequent flow of hemolymph into the deployable wing structure. Using a combination of imaging techniques, we characterize the internal and external structure of the wing in Drosophila melanogaster, the unfolding kinematics at the organ scale, and the hemolymph flow during deployment. We find that, beyond the mere unfolding of the macroscopic folds, wing deployment also involves wing expansion, with the stretching of epithelial cells and the unwrinkling of the cuticle enveloping the wing. A quantitative computational model, incorporating mechanical measurements of the viscoelastic properties and microstructure of the wing, predicts the existence of an operating point for deployment and captures the dynamics of the process. This model shows that insects exploit material and geometric nonlinearities to achieve rapid and efficient reconfiguration of soft deployable structures.

Bayesian parameter inference for epithelial mechanics

Cell-based mechanical models, such as the Cell Vertex Model (CVM), have proven useful for studying the mechanical control of epithelial tissue dynamics. We recently developed a statistical method called image-based parameter inference for formulating CVM model functions and estimating their parameters from image data of epithelial tissues. In this study, we employed Bayesian statistics to improve the utility and flexibility of image-based parameter inference. Tests on synthetic data confirmed that both our non-hierarchical and hierarchical Bayesian models provide accurate estimates of model parameters. By applying this method to Drosophila wings, we demonstrated that the reliability of parameter estimation is closely linked to the mechanical anisotropies present in the tissue. Moreover, we revealed that the cortical elasticity term is dispensable for explaining force-shape correlations in vivo. We anticipate that the flexibility of the Bayesian statistical framework will facilitate the integration of various types of information, thereby contributing to the quantitative dissection of the mechanical control of tissue dynamics.

Deep learning for rapid analysis of cell divisions in vivo during epithelial morphogenesis and repair

Cell division is fundamental to all healthy tissue growth, as well as being rate-limiting in the tissue repair response to wounding and during cancer progression. However, the role that cell divisions play in tissue growth is a collective one, requiring the integration of many individual cell division events. It is particularly difficult to accurately detect and quantify multiple features of large numbers of cell divisions (including their spatio-temporal synchronicity and orientation) over extended periods of time. It would thus be advantageous to perform such analyses in an automated fashion, which can naturally be enabled using deep learning. Hence, we develop a pipeline of deep learning models that accurately identify dividing cells in time-lapse movies of epithelial tissues in vivo. Our pipeline also determines their axis of division orientation, as well as their shape changes before and after division. This strategy enables us to analyse the dynamic profile of cell divisions within the Drosophila pupal wing epithelium, both as it undergoes developmental morphogenesis and as it repairs following laser wounding. We show that the division axis is biased according to lines of tissue tension and that wounding triggers a synchronised (but not oriented) burst of cell divisions back from the leading edge.

Deep learning reveals a damage signalling hierarchy that coordinates different cell behaviours driving wound re-epithelialisation

One of the key tissue movements driving closure of a wound is re-epithelialisation. Earlier wound healing studies describe the dynamic cell behaviours that contribute to wound re-epithelialisation, including cell division, cell shape changes and cell migration, as well as the signals that might regulate these cell behaviours. Here, we have used a series of deep learning tools to quantify the contributions of each of these cell behaviours from movies of repairing wounds in the Drosophila pupal wing epithelium. We test how each is altered after knockdown of the conserved wound repair signals Ca2+ and JNK, as well as after ablation of macrophages that supply growth factor signals believed to orchestrate aspects of the repair process. Our genetic perturbation experiments provide quantifiable insights regarding how these wound signals impact cell behaviours. We find that Ca2+ signalling is a master regulator required for all contributing cell behaviours; JNK signalling primarily drives cell shape changes and divisions, whereas signals from macrophages largely regulate cell migration and proliferation. Our studies show deep learning to be a valuable tool for unravelling complex signalling hierarchies underlying tissue repair.

Substrate stress relaxation regulates monolayer fluidity and leader cell formation for collectively migrating epithelia

Collective migration of epithelial tissues is a critical feature of developmental morphogenesis and tissue homeostasis. Coherent motion of cell collectives requires large scale coordination of motion and force generation and is influenced by mechanical properties of the underlying substrate. While tissue viscoelasticity is a ubiquitous feature of biological tissues, its role in mediating collective cell migration is unclear. Here, we have investigated the impact of substrate stress relaxation on the migration of micropatterned epithelial monolayers. Epithelial monolayers exhibit faster collective migration on viscoelastic alginate substrates with slower relaxation timescales, which are more elastic, relative to substrates with faster stress relaxation, which exhibit more viscous loss. Faster migration on slow-relaxing substrates is associated with reduced substrate deformation, greater monolayer fluidity, and enhanced leader cell formation. In contrast, monolayers on fast-relaxing substrates generate substantial substrate deformations and are more jammed within the bulk, with reduced formation of transient lamellipodial protrusions past the monolayer edge leading to slower overall expansion. This work reveals features of collective epithelial dynamics on soft, viscoelastic materials and adds to our understanding of cell-substrate interactions at the tissue scale.

Significance Statement

Groups of cells must coordinate their movements in order to sculpt organs during development and maintain tissues. The mechanical properties of the underlying substrate on which cells reside are known to influence key aspects of single and collective cell migration. Despite being a nearly universal feature of biological tissues, the role of viscoelasticity (i.e., fluid-like and solid-like behavior) in collective cell migration is unclear. Using tunable engineered biomaterials, we demonstrate that sheets of epithelial cells display enhanced migration on slower-relaxing (more elastic) substrates relative to faster-relaxing (more viscous) substrates. Building our understanding of tissue-substrate interactions and collective cell dynamics provides insights into approaches for tissue engineering and regenerative medicine, and therapeutic interventions to promote health and treat disease.

Multicell-Fold: geometric learning in folding multicellular life

During developmental processes such as embryogenesis, how a group of cells fold into specific structures, is a central question in biology. However, it remains a major challenge to understand and predict the behavior of every cell within the living tissue over time during such intricate processes. Here we present a geometric deep-learning model that can accurately capture the highly convoluted interactions among cells. We demonstrate that multicellular data can be represented with both granular and foam-like physical pictures through a unified graph data structure, considering both cellular interactions and cell junction networks. Using this model, we achieve interpretable 4-D morphological sequence alignment, and predicting cell rearrangements before they occur at single-cell resolution. Furthermore, using neural activation map and ablation studies, we demonstrate cell geometries and cell junction networks together regulate morphogenesis at single-cell precision. This approach offers a pathway toward a unified dynamic atlas for a variety of developmental processes.

Control of Modular Tissue Flows Shaping the Embryo in Avian Gastrulation

Avian gastrulation requires coordinated flows of thousands of cells to form the body plan. We quantified these flows using their fundamental kinematic units: one attractor and two repellers constituting its Dynamic Morphoskeleton (DM). We have also elucidated the mechanistic origin of the attractor, marking the primitive streak (PS), and controlled its shape, inducing gastrulation flows in the chick embryo that are typical of other vertebrates. However, the origins of repellers and dynamic embryo shape remain unclear. Here, we address these questions using active matter physics and experiments. Repeller 1, separating the embryo proper (EP) from extraembryonic (EE) tissues, arises from the tug-of-war between EE epiboly and EP isotropic myosin-induced active stress. Repeller 2, bisecting the anterior and posterior PS and associated with embryo shape change, arises from anisotropic myosin-induced active intercalation in the mesendoderm. Combining mechanical confinement with inhibition of mesendoderm induction, we eliminated either one or both repellers, as predicted by our model. Our results reveal a remarkable modularity of avian gastrulation flows delineated by the DM, uncovering the mechanistic roles of EE epiboly, EP active constriction, mesendoderm intercalation and ingression. These findings offer a new perspective for deconstructing morphogenetic flows, uncovering their modular origin, and aiding synthetic morphogenesis.

The Geometric Basis of Epithelial Convergent Extension

Shape changes of epithelia during animal development, such as convergent extension, are achieved through concerted mechanical activity of individual cells. While much is known about the corresponding large scale tissue flow and its genetic drivers, fundamental questions regarding local control of contractile activity on cellular scale and its embryo-scale coordination remain open. To address these questions, we develop a quantitative, model-based analysis framework to relate cell geometry to local tension in recently obtained timelapse imaging data of gastrulating Drosophila embryos. This analysis provides a systematic decomposition of cell shape changes and T1-rearrangements into internally driven, active, and externally driven, passive, contributions. Our analysis provides evidence that germ band extension is driven by active T1 processes that self-organize through positive feedback acting on tensions. More generally, our findings suggest that epithelial convergent extension results from controlled transformation of internal force balance geometry which combines the effects of bottom-up local self-organization with the top-down, embryo-scale regulation by gene expression.

Graph topological transformations in space-filling cell aggregates

Cell rearrangements are fundamental mechanisms driving large-scale deformations of living tissues. In three-dimensional (3D) space-filling cell aggregates, cells rearrange through local topological transitions of the network of cell-cell interfaces, which is most conveniently described by the vertex model. Since these transitions are not yet mathematically properly formulated, the 3D vertex model is generally difficult to implement. The few existing implementations rely on highly customized and complex software-engineering solutions, which cannot be transparently delineated and are thus mostly non-reproducible. To solve this outstanding problem, we propose a reformulation of the vertex model. Our approach, called Graph Vertex Model (GVM), is based on storing the topology of the cell network into a knowledge graph with a particular data structure that allows performing cell-rearrangement events by simple graph transformations. Importantly, when these same transformations are applied to a two-dimensional (2D) polygonal cell aggregate, they reduce to a well-known T1 transition, thereby generalizing cell-rearrangements in 2D and 3D space-filling packings. This result suggests that the GVM's graph data structure may be the most natural representation of cell aggregates and tissues. We also develop a Python package that implements GVM, relying on a graph-database-management framework Neo4j. We use this package to characterize an order-disorder transition in 3D cell aggregates, driven by active noise and we find aggregates undergoing efficient ordering close to the transition point. In all, our work showcases knowledge graphs as particularly suitable data models for structured storage, analysis, and manipulation of tissue data.

Dynamic interplay of microtubule and actomyosin forces drive tissue extension

In order to shape a tissue, individual cell-based mechanical forces have to be integrated into a global force pattern. Over the last decades, the importance of actomyosin contractile arrays, which are the key constituents of various morphogenetic processes, has been established for many tissues. Recent studies have demonstrated that the microtubule cytoskeleton mediates folding and elongation of the epithelial sheet during Drosophila morphogenesis, placing microtubule mechanics on par with actin-based processes. While these studies establish the importance of both cytoskeletal systems during cell and tissue rearrangements, a mechanistic understanding of their functional hierarchy is currently missing. Here, we dissect the individual roles of these two key generators of mechanical forces during epithelium elongation in the developing Drosophila wing. We show that wing extension, which entails columnar-to-cuboidal cell shape remodeling in a cell-autonomous manner, is driven by anisotropic cell expansion caused by the remodeling of the microtubule cytoskeleton from apico-basal to planarly polarized. Importantly, cell and tissue elongation is not associated with Myosin activity. Instead, Myosin II exhibits a homeostatic role, as actomyosin contraction balances polarized microtubule-based forces to determine the final cell shape. Using a reductionist model, we confirm that pairing microtubule and actomyosin-based forces is sufficient to recapitulate cell elongation and the final cell shape. These results support a hierarchical mechanism whereby microtubule-based forces in some epithelial systems prime actomyosin-generated forces.

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Balancing competing effects of tissue growth and cytoskeletal regulation during Drosophila wing disc development

How a developing organ robustly coordinates the cellular mechanics and growth to reach a final size and shape remains poorly understood. Through iterations between experiments and model simulations that include a mechanistic description of interkinetic nuclear migration, we show that the local curvature, height, and nuclear positioning of cells in the Drosophila wing imaginal disc are defined by the concurrent patterning of actomyosin contractility, cell-ECM adhesion, ECM stiffness, and interfacial membrane tension. We show that increasing cell proliferation via different growth-promoting pathways results in two distinct phenotypes. Triggering proliferation through insulin signaling increases basal curvature, but an increase in growth through Dpp signaling and Myc causes tissue flattening. These distinct phenotypic outcomes arise from differences in how each growth pathway regulates the cellular cytoskeleton, including contractility and cell-ECM adhesion. The coupled regulation of proliferation and cytoskeletal regulators is a general strategy to meet the multiple context-dependent criteria defining tissue morphogenesis.

Inter-plane feedback coordinates cell morphogenesis and maintains 3D tissue organization in the Drosophila pupal retina

How complex organs coordinate cellular morphogenetic events to achieve three-dimensional (3D) form is a central question in development. The question is uniquely tractable in the late Drosophila pupal retina, where cells maintain stereotyped contacts as they elaborate the specialized cytoskeletal structures that pattern the apical, basal and longitudinal planes of the epithelium. In this study, we combined cell type-specific genetic manipulation of the cytoskeletal regulator Abelson (Abl) with 3D imaging to explore how the distinct cellular morphogenetic programs of photoreceptors and interommatidial pigment cells (IOPCs) organize tissue pattern to support retinal integrity. Our experiments show that photoreceptor and IOPC terminal differentiation is unexpectedly interdependent, connected by an intercellular feedback mechanism that coordinates and promotes morphogenetic change across orthogonal tissue planes to ensure correct 3D retinal pattern. We propose that genetic regulation of specialized cellular differentiation programs combined with inter-plane mechanical feedback confers spatial coordination to achieve robust 3D tissue morphogenesis.

The extracellular matrix in tissue morphogenesis: No longer a backseat driver

The forces driving tissue morphogenesis are thought to originate from cellular activities. While it is appreciated that extracellular matrix (ECM) may also be involved, ECM function is assumed to be simply instructive in modulating the cellular behaviors that drive changes to tissue shape. However, there is increasing evidence that the ECM may not be the passive player portrayed in developmental biology textbooks. In this review we highlight examples of embryonic ECM dynamics that suggest cell-independent activity, along with developmental processes during which localized ECM alterations and ECM-autonomous forces are directing changes to tissue shape. Additionally, we discuss experimental approaches to unveil active ECM roles during tissue morphogenesis. We propose that it may be time to rethink our general definition of morphogenesis as a cellular-driven phenomenon and incorporate an underappreciated, and surprisingly dynamic ECM.

Programmed disassembly of a microtubule-based membrane protrusion network coordinates 3D epithelial morphogenesis in Drosophila

Comprehensive analysis of cellular dynamics during the process of morphogenesis is fundamental to understanding the principles of animal development. Despite recent advancements in light microscopy, how successive cell shape changes lead to complex three-dimensional tissue morphogenesis is still largely unresolved. Using in vivo live imaging of Drosophila wing development, we have studied unique cellular structures comprising a microtubule-based membrane protrusion network. This network, which we name here the Interplanar Amida Network (IPAN), links the two wing epithelium leaflets. Initially, the IPAN sustains cell-cell contacts between the two layers of the wing epithelium through basal protrusions. Subsequent disassembly of the IPAN involves loss of these contacts, with concomitant degeneration of aligned microtubules. These processes are both autonomously and non-autonomously required for mitosis, leading to coordinated tissue proliferation between two wing epithelia. Our findings further reveal that a microtubule organization switch from non-centrosomal to centrosomal microtubule-organizing centers (MTOCs) at the G2/M transition leads to disassembly of non-centrosomal microtubule-derived IPAN protrusions. These findings exemplify how cell shape change-mediated loss of inter-tissue contacts results in 3D tissue morphogenesis.

Patterned apoptosis has an instructive role for local growth and tissue shape regulation in a fast-growing epithelium

What regulates organ size and shape remains one fundamental mystery of modern biology. Research in this area has primarily focused on deciphering the regulation in time and space of growth and cell division, while the contribution of cell death has been overall neglected. This includes studies of the Drosophila wing, one of the best-characterized systems for the study of growth and patterning, undergoing massive growth during larval stage and important morphogenetic remodeling during pupal stage. So far, it has been assumed that cell death was relatively neglectable in this tissue both during larval stage and pupal stage, and as a result, the pattern of growth was usually attributed to the distribution of cell division. Here, using systematic mapping and registration combined with quantitative assessment of clone size and disappearance as well as live imaging, we outline a persistent pattern of cell death and clone elimination emerging in the larval wing disc and persisting during pupal wing morphogenesis. Local variation of cell death is associated with local variation of clone size, pointing to an impact of cell death on local growth that is not fully compensated by proliferation. Using morphometric analyses of adult wing shape and genetic perturbations, we provide evidence that patterned death locally and globally affects adult wing shape and size. This study describes a roadmap for precise assessment of the contribution of cell death to tissue shape and outlines an important instructive role of cell death in modulating quantitatively local growth and morphogenesis of a fast-growing tissue.

Mechanical regulation of substrate adhesion and de-adhesion drives a cell-contractile wave during Drosophila tissue morphogenesis

During morphogenesis, mechanical forces induce large-scale deformations; yet, how forces emerge from cellular contractility and adhesion is unclear. In Drosophila embryos, a tissue-scale wave of actomyosin contractility coupled with adhesion to the surrounding vitelline membrane drives polarized tissue invagination. We show that this process emerges subcellularly from the mechanical coupling between myosin II activation and sequential adhesion/de-adhesion to the vitelline membrane. At the wavefront, integrin clusters anchor the actin cortex to the vitelline membrane and promote activation of myosin II, which in turn enhances adhesion in a positive feedback. Following cell detachment, cortex contraction and advective flow amplify myosin II. Prolonged contact with the vitelline membrane prolongs the integrin-myosin II feedback, increases integrin adhesion, and thus slows down cell detachment and wave propagation. The angle of cell detachment depends on adhesion strength and sets the tensile forces required for detachment. Thus, we document how the interplay between subcellular mechanochemical feedback and geometry drives tissue morphogenesis.

Assembly, dynamics and remodeling of epithelial cell junctions throughout development

Cell junctions play key roles in epithelial integrity. During development, when epithelia undergo extensive morphogenesis, these junctions must be remodeled in order to maintain mechanochemical barriers and ensure the cohesion of the tissue. In this Review, we present a comprehensive and integrated description of junctional remodeling mechanisms in epithelial cells during development, from embryonic to adult epithelia. We largely focus on Drosophila, as quantitative analyses in this organism have provided a detailed characterization of the molecular mechanisms governing cell topologies, and discuss the conservation of these mechanisms across metazoans. We consider how changes at the molecular level translate to tissue-scale irreversible deformations, exploring the composition and assembly of cellular interfaces to unveil how junctions are remodeled to preserve tissue homeostasis during cell division, intercalation, invagination, ingression and extrusion.

Core PCP mutations affect short-time mechanical properties but not tissue morphogenesis in the Drosophila pupal wing

How morphogenetic movements are robustly coordinated in space and time is a fundamental open question in biology. We study this question using the wing of Drosophila melanogaster, an epithelial tissue that undergoes large-scale tissue flows during pupal stages. Previously, we showed that pupal wing morphogenesis involves both cellular behaviors that allow relaxation of mechanical tissue stress, as well as cellular behaviors that appear to be actively patterned (Etournay et al., 2015). Here, we show that these active cellular behaviors are not guided by the core planar cell polarity (PCP) pathway, a conserved signaling system that guides tissue development in many other contexts. We find no significant phenotype on the cellular dynamics underlying pupal morphogenesis in mutants of core PCP. Furthermore, using laser ablation experiments, coupled with a rheological model to describe the dynamics of the response to laser ablation, we conclude that while core PCP mutations affect the fast timescale response to laser ablation they do not significantly affect overall tissue mechanics. In conclusion, our work shows that cellular dynamics and tissue shape changes during Drosophila pupal wing morphogenesis do not require core PCP as an orientational guiding cue.

A mechanochemical model recapitulates distinct vertebrate gastrulation modes

During vertebrate gastrulation, an embryo transforms from a layer of epithelial cells into a multilayered gastrula. This process requires the coordinated movements of hundreds to tens of thousands of cells, depending on the organism. In the chick embryo, patterns of actomyosin cables spanning several cells drive coordinated tissue flows. Here, we derive a minimal theoretical framework that couples actomyosin activity to global tissue flows. Our model predicts the onset and development of gastrulation flows in normal and experimentally perturbed chick embryos, mimicking different gastrulation modes as an active stress instability. Varying initial conditions and a parameter associated with active cell ingression, our model recapitulates distinct vertebrate gastrulation morphologies, consistent with recently published experiments in the chick embryo. Altogether, our results show how changes in the patterning of critical cell behaviors associated with different force-generating mechanisms contribute to distinct vertebrate gastrulation modes via a self-organizing mechanochemical process.

Mechanochemical Active Feedback Generates Convergence Extension in Epithelial Tissue

Convergence extension, the simultaneous elongation of tissue along one axis while narrowing along a perpendicular axis, occurs during embryonic development. A fundamental process that contributes to shaping the organism, it happens in many different species and tissue types. Here, we present a minimal continuum model, that can be directly linked to the controlling microscopic biochemistry, which shows spontaneous convergence extension. It is comprised of a 2D viscoelastic active material with a mechanochemical active feedback mechanism coupled to a substrate via friction. Robust convergent extension behavior emerges beyond a critical value of the activity parameter and is controlled by the boundary conditions and the coupling to the substrate. Oscillations and spatial patterns emerge in this model when internal dissipation dominates over friction, as well as in the active elastic limit.

Using Histologic Image Analysis to Understand Biophysical Regulations of Epithelial Cell Morphology

Epithelial mechanics and mechanobiology have become 2 important research fields in life sciences and bioengineering. These fields investigate how physical factors induced by cell adhesion and collective behaviors can directly regulate biologic processes, such as organ development and disease progression. Cell mechanics and mechanobiology thus make exciting biophysics education topics to illustrate how fundamental physics principles play a role in regulating cell biology. However, the field currently lacks hands-on activities that engage students in learning science and outreach programs in these topics. One such area is the development of robust hands-on modules that allow students to observe features of cell shape and mechanics and connect them to fundamental physics principles. Here, we demonstrate a workflow that engages students in studying epithelial cell mechanics by using commercial histology slides of frog skin. We show that by using recently developed artificial intelligence-based image-segmentation tools, students can easily quantify different cell morphologic features in a high-throughput manner. Using our workflow, students can reproduce 2 essential findings in cell mechanics: the common gamma distribution of normalized cell aspect ratio in jammed epithelia and the constant ratio between the nuclear and cellular area. Importantly, because the only required instrument for this active learning module is a readily available light microscope and a computer, our module is relatively low cost, as well as portable. These features make the module scalable for students at various education levels and outreach programs. This highly accessible education module provides a fun and engaging way to introduce students to the world of epithelial tissue mechanics.

TubULAR: tracking in toto deformations of dynamic tissues via constrained maps

A common motif in biology is the arrangement of cells into tubes, which further transform into complex shapes. Traditionally, analysis of dynamic tissues has relied on inspecting static snapshots, live imaging of cross-sections or tracking isolated cells in three dimensions. However, capturing the interplay between in-plane and out-of-plane behaviors requires following the full surface as it deforms and integrating cell-scale motions into collective, tissue-scale deformations. Here, we present an analysis framework that builds in toto maps of tissue deformations by following tissue parcels in a static material frame of reference. Our approach then relates in-plane and out-of-plane behaviors and decomposes complex deformation maps into elementary contributions. The tube-like surface Lagrangian analysis resource (TubULAR) provides an open-source implementation accessible either as a standalone toolkit or as an extension of the ImSAnE package used in the developmental biology community. We demonstrate our approach by analyzing shape change in the embryonic Drosophila midgut and beating zebrafish heart. The method naturally generalizes to in vitro and synthetic systems and provides ready access to the mechanical mechanisms relating genetic patterning to organ shape change.

Constitutive model for the rheology of biological tissue

The rheology of biological tissue is key to processes such as embryo development, wound healing, and cancer metastasis. Vertex models of confluent tissue monolayers have uncovered a spontaneous liquid-solid transition tuned by cell shape; and a shear-induced solidification transition of an initially liquidlike tissue. Alongside this jamming/unjamming behavior, biological tissue also displays an inherent viscoelasticity, with a slow time and rate-dependent mechanics. With this motivation, we combine simulations and continuum theory to examine the rheology of the vertex model in nonlinear shear across a full range of shear rates from quastistatic to fast, elucidating its nonlinear stress-strain curves after the inception of shear of finite rate, and its steady state flow curves of stress as a function of strain rate. We formulate a rheological constitutive model that couples cell shape to flow and captures both the tissue solid-liquid transition and its rich linear and nonlinear rheology.

Spatiotemporal remodeling of extracellular matrix orients epithelial sheet folding

Biological systems are inherently noisy; however, they produce highly stereotyped tissue morphology. Drosophila pupal wings show a highly stereotypic folding through uniform expansion and subsequent buckling of wing epithelium within a surrounding cuticle sac. The folding pattern produced by buckling is generally stochastic; it is thus unclear how buckling leads to stereotypic tissue folding of the wings. We found that the extracellular matrix (ECM) protein, Dumpy, guides the position and direction of buckling-induced folds. Dumpy anchors the wing epithelium to the overlying cuticle at specific tissue positions. Tissue-wide alterations of Dumpy deposition and degradation yielded different buckling patterns. In summary, we propose that spatiotemporal ECM remodeling shapes stereotyped tissue folding through dynamic interactions between the epithelium and its external structures.

Active cell divisions generate fourfold orientationally ordered phase in living tissue

Morphogenesis, the process through which genes generate form, establishes tissue-scale order as a template for constructing the complex shapes of the body plan. The extensive growth required to build these ordered substrates is fuelled by cell proliferation, which, naively, should destroy order. Understanding how active morphogenetic mechanisms couple cellular and mechanical processes to generate order-rather than annihilate it-remains an outstanding question in animal development. We show that cell divisions are the primary drivers of tissue flow, leading to a fourfold orientationally ordered phase. Waves of anisotropic cell proliferation propagate across the embryo with precise patterning. Defects introduced into the nascent lattice by cell divisions are moved out of the tissue bulk towards the boundary by subsequent divisions. Specific cell proliferation rates and orientations enable cell divisions to organize rather than fluidize the tissue. We observe this using live imaging and tissue cartography to analyse the dynamics of fourfold tissue ordering in the trunk segmental ectoderm of the crustacean Parhyale hawaiensis beginning 72 h after egg lay. The result is a robust, active mechanism for generating global orientational order in a non-equilibrium system that sets the stage for the subsequent development of shape and form.

Interspersed expression of CUP-SHAPED COTYLEDON2 and REDUCED COMPLEXITY shapes Cardamine hirsuta complex leaf form

How genetically regulated growth shapes organ form is a key problem in developmental biology. Here, we investigate this problem using the leaflet-bearing complex leaves of Cardamine hirsuta as a model. Leaflet development requires the action of two growth-repressing transcription factors: REDUCED COMPLEXITY (RCO), a homeodomain protein, and CUP-SHAPED COTYLEDON2 (CUC2), a NAC-domain protein. However, how their respective growth-repressive actions are integrated in space and time to generate complex leaf forms remains unknown. By using live imaging, we show that CUC2 and RCO are expressed in an interspersed fashion along the leaf margin, creating a distinctive striped pattern. We find that this pattern is functionally important because forcing RCO expression in the CUC2 domain disrupts auxin-based marginal patterning and can abolish leaflet formation. By combining genetic perturbations with time-lapse imaging and cellular growth quantifications, we provide evidence that RCO-mediated growth repression occurs after auxin-based leaflet patterning and in association with the repression of cell proliferation. Additionally, through the use of genetic mosaics, we show that RCO is sufficient to repress both cellular growth and proliferation in a cell-autonomous manner. This mechanism of growth repression is different to that of CUC2, which occurs in proliferating cells. Our findings clarify how the two growth repressors RCO and CUC2 coordinate to subdivide developing leaf primordia into distinct leaflets and generate the complex leaf form. They also indicate different relationships between growth repression and cell proliferation in the patterning and post-patterning stages of organogenesis.

Complexities of recapitulating polygenic effects in natural populations: replication of genetic effects on wing shape in artificially selected and wild-caught populations of Drosophila melanogaster

Identifying the genetic architecture of complex traits is important to many geneticists, including those interested in human disease, plant and animal breeding, and evolutionary genetics. Advances in sequencing technology and statistical methods for genome-wide association studies have allowed for the identification of more variants with smaller effect sizes, however, many of these identified polymorphisms fail to be replicated in subsequent studies. In addition to sampling variation, this failure to replicate reflects the complexities introduced by factors including environmental variation, genetic background, and differences in allele frequencies among populations. Using Drosophila melanogaster wing shape, we ask if we can replicate allelic effects of polymorphisms first identified in a genome-wide association studies in three genes: dachsous, extra-macrochaete, and neuralized, using artificial selection in the lab, and bulk segregant mapping in natural populations. We demonstrate that multivariate wing shape changes associated with these genes are aligned with major axes of phenotypic and genetic variation in natural populations. Following seven generations of artificial selection along the dachsous shape change vector, we observe genetic differentiation of variants in dachsous and genomic regions containing other genes in the hippo signaling pathway. This suggests a shared direction of effects within a developmental network. We also performed artificial selection with the extra-macrochaete shape change vector, which is not a part of the hippo signaling network, but showed a largely shared direction of effects. The response to selection along the emc vector was similar to that of dachsous, suggesting that the available genetic diversity of a population, summarized by the genetic (co)variance matrix (G), influenced alleles captured by selection. Despite the success with artificial selection, bulk segregant analysis using natural populations did not detect these same variants, likely due to the contribution of environmental variation and low minor allele frequencies, coupled with small effect sizes of the contributing variants.

DeXtrusion: automatic recognition of epithelial cell extrusion through machine learning in vivo

Accurately counting and localising cellular events from movies is an important bottleneck of high-content tissue/embryo live imaging. Here, we propose a new methodology based on deep learning that allows automatic detection of cellular events and their precise xyt localisation on live fluorescent imaging movies without segmentation. We focused on the detection of cell extrusion, the expulsion of dying cells from the epithelial layer, and devised DeXtrusion: a pipeline based on recurrent neural networks for automatic detection of cell extrusion/cell death events in large movies of epithelia marked with cell contour. The pipeline, initially trained on movies of the Drosophila pupal notum marked with fluorescent E-cadherin, is easily trainable, provides fast and accurate extrusion predictions in a large range of imaging conditions, and can also detect other cellular events, such as cell division or cell differentiation. It also performs well on other epithelial tissues with reasonable re-training. Our methodology could easily be applied for other cellular events detected by live fluorescent microscopy and could help to democratise the use of deep learning for automatic event detections in developing tissues.

Sculpting with stiffness: rigidity as a regulator of morphogenesis

From a physical perspective, morphogenesis of tissues results from interplay between their material properties and the mechanical forces exerted on them. The importance of mechanical forces in influencing cell behaviour is widely recognised, whereas the importance of tissue material properties in vivo, like stiffness, has only begun to receive attention in recent years. In this mini-review, we highlight key themes and concepts that have emerged related to how tissue stiffness, a fundamental material property, guides various morphogenetic processes in living organisms.

Robust statistical properties of T1 transitions in a multi-phase field model of cell monolayers

Large-scale tissue deformation which is fundamental to tissue development hinges on local cellular rearrangements, such as T1 transitions. In the realm of the multi-phase field model, we analyse the statistical and dynamical properties of T1 transitions in a confluent monolayer. We identify an energy profile that is robust to changes in several model parameters. It is characterized by an asymmetric profile with a fast increase in energy before the T1 transition and a sudden drop after the T1 transition, followed by a slow relaxation. The latter being a signature of the fluidity of the cell monolayer. We show that T1 transitions are sources of localised large deformation of the cells undergoing the neighbour exchange, and they induce other T1 transitions in the nearby cells leading to a chaining of events that propagate local cell deformation to large scale tissue flows.

Engineering tools for quantifying and manipulating forces in epithelia

The integrity of epithelia is maintained within dynamic mechanical environments during tissue development and homeostasis. Understanding how epithelial cells mechanosignal and respond collectively or individually is critical to providing insight into developmental and (patho)physiological processes. Yet, inferring or mimicking mechanical forces and downstream mechanical signaling as they occur in epithelia presents unique challenges. A variety of in vitro approaches have been used to dissect the role of mechanics in regulating epithelia organization. Here, we review approaches and results from research into how epithelial cells communicate through mechanical cues to maintain tissue organization and integrity. We summarize the unique advantages and disadvantages of various reduced-order model systems to guide researchers in choosing appropriate experimental systems. These model systems include 3D, 2D, and 1D micromanipulation methods, single cell studies, and noninvasive force inference and measurement techniques. We also highlight a number of in silico biophysical models that are informed by in vitro and in vivo observations. Together, a combination of theoretical and experimental models will aid future experiment designs and provide predictive insight into mechanically driven behaviors of epithelial dynamics.

The role of non-affine deformations in the elastic behavior of the cellular vertex model

The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A₀, and perimeter, P₀. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p_*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p₀ < p_*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.

Tuneable defect-curvature coupling and topological transitions in active shells

Recent experimental observations have suggested that topological defects can facilitate the creation of sharp features in developing embryos. Whereas these observations echo established knowledge about the interplay between geometry and topology in two-dimensional passive liquid crystals, the role of activity has mostly remained unexplored. In this article we focus on deformable shells consisting of either polar or nematic active liquid crystals and demonstrate that activity renders the mechanical coupling between defects and curvature much more involved and versatile than previously thought. Using a combination of linear stability analysis and three-dimensional computational fluid dynamics, we demonstrate that such a coupling can in fact be tuned, depending on the type of liquid crystal order, the specific structure of the defect (i.e. asters or vortices) and the nature of the active forces. In polar systems, this can drive a spectacular transition from spherical to toroidal topology, in the presence of large extensile activity. Our analysis strengthens the idea that defects could serve as topological morphogens and provides a number of predictions that could be tested in in vitro studies, for instance in the context of organoids.