Vertex models of epithelial morphogenesis

Minimal cellular automaton model with heterogeneous cell sizes predicts epithelial colony growth

Regulation of cell proliferation is a crucial aspect of tissue development and homeostasis and plays a major role in morphogenesis, wound healing, and tumor invasion. A phenomenon of such regulation is contact inhibition, which describes the dramatic slowing of proliferation, cell migration and individual cell growth when multiple cells are in contact with each other. While many physiological, molecular and genetic factors are known, the mechanism of contact inhibition is still not fully understood. In particular, the relevance of cellular signaling due to interfacial contact for contact inhibition is still debated. Cellular automata (CA) have been employed in the past as numerically efficient mathematical models to study the dynamics of cell ensembles, but they are not suitable to explore the origins of contact inhibition as such agent-based models assume fixed cell sizes. We develop a minimal, data-driven model to simulate the dynamics of planar cell cultures by extending a probabilistic CA to incorporate size changes of individual cells during growth and cell division. We successfully apply this model to previous in-vitro experiments on contact inhibition in epithelial tissue: After a systematic calibration of the model parameters to measurements of single-cell dynamics, our CA model quantitatively reproduces independent measurements of emergent, culture-wide features, like colony size, cell density and collective cell migration. In particular, the dynamics of the CA model also exhibit the transition from a low-density confluent regime to a stationary postconfluent regime with a rapid decrease in cell size and motion. This implies that the volume exclusion principle, a mechanical constraint which is the only inter-cellular interaction incorporated in the model, paired with a size-dependent proliferation rate is sufficient to generate the observed contact inhibition. We discuss how our approach enables the introduction of effective bio-mechanical interactions in a CA framework for future studies.

Dynamics of multicellular swirling on micropatterned substrates

Chirality plays a crucial role in biology, as it is highly conserved and fundamentally important in the developmental process. To better understand the relationship between the chirality of individual cells and that of tissues and organisms, we develop a generalized mechanics model of chiral polarized particles to investigate the swirling dynamics of cell populations on substrates. Our analysis reveals that cells with the same chirality can form distinct chiral patterns on ring-shaped or rectangular substrates. Interestingly, our studies indicate that an excessively strong or weak individual cellular chirality hinders the formation of such chiral patterns. Our studies also indicate that there exists the influence distance of substrate boundaries in chiral patterns. Smaller influence distances are observed when cell-cell interactions are weaker. Conversely, when cell-cell interactions are too strong, multiple cells tend to be stacked together, preventing the formation of chiral patterns on substrates in our analysis. Additionally, we demonstrate that the interaction between cells and substrate boundaries effectively controls the chiral distribution of cellular orientations on ring-shaped substrates. This research highlights the significance of coordinating boundary features, individual cellular chirality, and cell-cell interactions in governing the chiral movement of cell populations and provides valuable mechanics insights into comprehending the intricate connection between the chirality of single cells and that of tissues and organisms.

Cell-size-dependent regulation of Ezrin dictates epithelial resilience to stretch by countering myosin-II-mediated contractility

The epithelial adaptations to mechanical stress are facilitated by molecular and tissue-scale changes that include the strengthening of junctions, cytoskeletal reorganization, and cell-proliferation-mediated changes in tissue rheology. However, the role of cell size in controlling these properties remains underexplored. Our experiments in the zebrafish embryonic epidermis, guided by theoretical estimations, reveal a link between epithelial mechanics and cell size, demonstrating that an increase in cell size compromises the tissue fracture strength and compliance. We show that an increase in E-cadherin levels in the proliferation-deficient epidermis restores epidermal compliance but not the fracture strength, which is largely regulated by Ezrin-an apical membrane-cytoskeleton crosslinker. We show that Ezrin fortifies the epithelium in a cell-size-dependent manner by countering non-muscle myosin-II-mediated contractility. This work uncovers the importance of cell size maintenance in regulating the mechanical properties of the epithelium and fostering protection against future mechanical stresses.

Contact area and tissue growth dynamics shape synthetic juxtacrine signaling patterns

Cell-cell communication through direct contact, or juxtacrine signaling, is important in development, disease, and many areas of physiology. Synthetic forms of juxtacrine signaling can be precisely controlled and operate orthogonally to native processes, making them a powerful reductionist tool with which to address fundamental questions in cell-cell communication in vivo. Here we investigate how cell-cell contact length and tissue growth dynamics affect juxtacrine signal responses through implementing a custom synthetic gene circuit in Drosophila wing imaginal discs alongside mathematical modeling to determine synthetic Notch (synNotch) activation patterns. We find that the area of contact between cells largely determines the extent of synNotch activation, leading to the prediction that the shape of the interface between signal-sending and signal-receiving cells will impact the magnitude of the synNotch response. Notably, synNotch outputs form a graded spatial profile that extends several cell diameters from the signal source, providing evidence that the response to juxtacrine signals can persist in cells as they proliferate away from source cells, or that cells remain able to communicate directly over several cell diameters. Our model suggests the former mechanism may be sufficient, since it predicts graded outputs without diffusion or long-range cell-cell communication. Overall, we identify that cell-cell contact area together with output synthesis and decay rates likely govern the pattern of synNotch outputs in both space and time during tissue growth, insights that may have broader implications for juxtacrine signaling in general.

Mechanochemical dynamics of collective cells and hierarchical topological defects in multicellular lumens

Collective cell dynamics is essential for tissue morphogenesis and various biological functions. However, it remains incompletely understood how mechanical forces and chemical signaling are integrated to direct collective cell behaviors underlying tissue morphogenesis. Here, we propose a three-dimensional (3D) mechanochemical theory accounting for biochemical reaction-diffusion and cellular mechanotransduction to investigate the dynamics of multicellular lumens. We show that the interplay between biochemical signaling and mechanics can trigger either pitchfork or Hopf bifurcation to induce diverse static mechanochemical patterns or generate oscillations with multiple modes both involving marked mechanical deformations in lumens. We uncover the crucial role of mechanochemical feedback in emerging morphodynamics and identify the evolution and morphogenetic functions of hierarchical topological defects including cell-level hexatic defects and tissue-level orientational defects. Our theory captures the common mechanochemical traits of collective dynamics observed in experiments and could provide a mechanistic context for understanding morphological symmetry breaking in 3D lumen-like tissues.

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.

After wounding, a G-protein coupled receptor promotes the restoration of tension in epithelial cells

The maintenance of epithelial barrier function involves cellular tension, with cells pulling on their neighbors to maintain epithelial integrity. Wounding interrupts cellular tension, which may serve as an early signal to initiate epithelial repair. To characterize how wounds alter cellular tension we used a laser-recoil assay to map cortical tension around wounds in the epithelial monolayer of the Drosophila pupal notum. Within a minute of wounding, there was widespread loss of cortical tension along both radial and tangential directions. This tension loss was similar to levels observed with Rok inactivation. Tension was subsequently restored around the wound, first in distal cells and then in proximal cells, reaching the wound margin ∼10 min after wounding. Restoring tension required the GPCR Mthl10 and the IP3 receptor, indicating the importance of this calcium signaling pathway known to be activated by cellular damage. Tension restoration correlated with an inward-moving contractile wave that has been previously reported; however, the contractile wave itself was not affected by Mthl10 knockdown. These results indicate that cells may transiently increase tension and contract in the absence of Mthl10 signaling, but that pathway is critical for fully resetting baseline epithelial tension after it is disrupted by wounding.

Regulation of Chromatin Modifications through Coordination of Nucleus Size and Epithelial Cell Morphology Heterogeneity

Cell morphology heterogeneity within epithelial collectives is a pervasive phenomenon intertwined with tissue mechanical properties. Despite its widespread occurrence, the underlying mechanisms driving cell morphology heterogeneity and its consequential biological ramifications remain elusive. Here, we investigate the dynamic evolution of epithelial cell morphology and nucleus morphology during crowding, unveiling a consistent correlation between the two. Our investigation reveals a persistent log-normal probability distribution characterizing both cell and nucleus areas across diverse crowding stages and epithelial model systems. We showed that this morphological diversity arises from asymmetric partitioning during cell division and is perpetuated through actomyosin-mediated regulation of cell-nucleus size coordination. Moreover, we provide insights into the impact of nucleus morphology on chromatin dynamics, demonstrating that constraining nucleus area leads to downregulation of the euchromatic mark H3K9ac and upregulation of the heterochromatic mark H3K27me3 through modulation of histone demethylase UTX expression. These findings under-score the significance of cell morphology heterogeneity as a driver of chromatin state diversity, shaping functional variability within epithelial tissues.

Choice of friction coefficient deeply affects tissue behaviour in stochastic epithelial vertex models

To understand the mechanisms that coordinate the formation of biological tissues, the use of numerical implementations is necessary. The complexity of such models involves many assumptions and parameter choices that result in unpredictable consequences, obstructing the comparison with experimental data. Here, we focus on vertex models, a family of spatial models used extensively to simulate the dynamics of epithelial tissues. Usually, in the literature, the choice of the friction coefficient is not addressed using quasi-static deformation arguments that generally do not apply to realistic scenarios. In this manuscript, we discuss the role that the choice of friction coefficient has on the relaxation times and consequently in the conditions of cell cycle progression and division. We explore the effects that these changes have on the morphology, growth rate and topological transitions of the tissue dynamics. These results provide a deeper understanding of the role that an accurate mechanical description plays in the use of vertex models as inference tools. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'.

Matrix confinement modulates 3D spheroid sorting and burst-like collective migration

While it is known that cells with differential adhesion tend to segregate and preferentially sort, the physical forces governing sorting and invasion in heterogeneous tumors remain poorly understood. To investigate this, we tune matrix confinement, mimicking changes in the stiffness and confinement of the tumor microenvironment, to explore how physical confinement influences individual and collective cell migration in 3D spheroids. High levels of confinement lead to cell sorting while reducing matrix confinement triggers the collective fluidization of cell motion. Cell sorting, which depends on cell-cell adhesion, is crucial to this phenomenon. Burst-like migration does not occur for spheroids that have not undergone sorting, regardless of the degree of matrix confinement. Using computational Self-Propelled Voronoi modeling, we show that spheroid sorting and invasion into the matrix depend on the balance between cell-generated forces and matrix resistance. The findings support a model where matrix confinement modulates 3D spheroid sorting and unjamming in an adhesion-dependent manner, providing insights into the mechanisms of cell sorting and migration in the primary tumor and toward distant metastatic sites. STATEMENT OF SIGNIFICANCE: The mechanical properties of the tumor microenvironment significantly influence cancer cell migration within the primary tumor, yet how these properties affect intercellular interactions in heterogeneous tumors is not well understood. By utilizing calcium and calcium chelators, we dynamically alter collagen-alginate hydrogel stiffness and investigate tumor cell behavior within co-culture spheroids in response to varying degrees of matrix confinement. High confinement is found to trigger cell sorting while reducing confinement for sorted spheroids facilitates collective cell invasion. Notably, without prior sorting, spheroids do not exhibit burst-like migration, regardless of confinement levels. This work establishes that matrix confinement and intercellular adhesion regulate 3D spheroid dynamics, offering insights into cellular organization and migration within the primary tumor.

Topological balance of cell distributions in plane monolayers

Most of normal proliferative epithelia of plants and metazoans are topologically invariant and characterized by similar cell distributions according to the number of cell neighbors (DCNs). Here we study peculiarities of these distributions and explain why the DCN obtained from the location of intercellular boundaries and that based on the Voronoi tessellation with nodes located on cell nuclei may differ from each other. As we demonstrate, special microdomains where four or more intercellular boundaries converge are topologically charged. Using this fact, we deduce a new equation describing the topological balance of the DCNs. The developed theory is applied for a series of microphotographs of non-tumoral epithelial cells of the human cervix (HCerEpiC) to improve the image processing near the edges of microphotographs and reveal the topological invariance of the examined monolayers. Special contact microdomains may be present in epithelia of various natures, however, considering the well-known vertex model of epithelium, we show that such contacts are absent in the usual solid-like state of the model and appear only in the liquid-like cancer state. Also, we discuss a possible biological role of special contacts in context of proliferative epithelium dynamics and tissue morphogenesis.

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.

SimuCell3D: three-dimensional simulation of tissue mechanics with cell polarization

The three-dimensional (3D) organization of cells determines tissue function and integrity, and changes markedly in development and disease. Cell-based simulations have long been used to define the underlying mechanical principles. However, high computational costs have so far limited simulations to either simplified cell geometries or small tissue patches. Here, we present SimuCell3D, an efficient open-source program to simulate large tissues in three dimensions with subcellular resolution, growth, proliferation, extracellular matrix, fluid cavities, nuclei and non-uniform mechanical properties, as found in polarized epithelia. Spheroids, vesicles, sheets, tubes and other tissue geometries can readily be imported from microscopy images and simulated to infer biomechanical parameters. Doing so, we show that 3D cell shapes in layered and pseudostratified epithelia are largely governed by a competition between surface tension and intercellular adhesion. SimuCell3D enables the large-scale in silico study of 3D tissue organization in development and disease at a great level of detail.

Active viscoelastic models for cell and tissue mechanics

Living cells are out of equilibrium active materials. Cell-generated forces are transmitted across the cytoskeleton network and to the extracellular environment. These active force interactions shape cellular mechanical behaviour, trigger mechano-sensing, regulate cell adaptation to the microenvironment and can affect disease outcomes. In recent years, the mechanobiology community has witnessed the emergence of many experimental and theoretical approaches to study cells as mechanically active materials. In this review, we highlight recent advancements in incorporating active characteristics of cellular behaviour at different length scales into classic viscoelastic models by either adding an active tension-generating element or adjusting the resting length of an elastic element in the model. Summarizing the two groups of approaches, we will review the formulation and application of these models to understand cellular adaptation mechanisms in response to various types of mechanical stimuli, such as the effect of extracellular matrix properties and external loadings or deformations.

The mechanical influence of densification on epithelial architecture

Epithelial tissues are the most abundant tissue type in animals, lining body cavities and generating compartment barriers. The function of a monolayered epithelial tissue-whether protective, secretory, absorptive, or filtrative-relies on the side-by-side arrangement of its component cells. The mechanical parameters that determine the shape of epithelial cells in the apical-basal plane are not well-understood. Epithelial tissue architecture in culture is intimately connected to cell density, and cultured layers transition between architectures as they proliferate. This prompted us to ask to what extent epithelial architecture emerges from two mechanical considerations: A) the constraints of densification and B) cell-cell adhesion, a hallmark feature of epithelial cells. To address these questions, we developed a novel polyline cell-based computational model and used it to make theoretical predictions about epithelial architecture upon changes to density and cell-cell adhesion. We tested these predictions using cultured cell experiments. Our results show that the appearance of extended lateral cell-cell borders in culture arises as a consequence of crowding-independent of cell-cell adhesion. However, cadherin-mediated cell-cell adhesion is associated with a novel architectural transition. Our results suggest that this transition represents the initial appearance of a distinctive epithelial architecture. Together our work reveals the distinct mechanical roles of densification and adhesion to epithelial layer formation and provides a novel theoretical framework to understand the less well-studied apical-basal plane of epithelial tissues.

Modelling cell shape in 3D structured environments: A quantitative comparison with experiments

Cell shape plays a fundamental role in many biological processes, including adhesion, migration, division and development, but it is not clear which shape model best predicts three-dimensional cell shape in structured environments. Here, we compare different modelling approaches with experimental data. The shapes of single mesenchymal cells cultured in custom-made 3D scaffolds were compared by a Fourier method with surfaces that minimize area under the given adhesion and volume constraints. For the minimized surface model, we found marked differences to the experimentally observed cell shapes, which necessitated the use of more advanced shape models. We used different variants of the cellular Potts model, which effectively includes both surface and bulk contributions. The simulations revealed that the Hamiltonian with linear area energy outperformed the elastic area constraint in accurately modelling the 3D shapes of cells in structured environments. Explicit modelling the nucleus did not improve the accuracy of the simulated cell shapes. Overall, our work identifies effective methods for accurately modelling cellular shapes in complex environments.

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.

Toward a predictive understanding of epithelial cell death

Epithelial cell death is highly prevalent during development and tissue homeostasis. While we have a rather good understanding of the molecular regulators of programmed cell death, especially for apoptosis, we still fail to predict when, where, how many and which specific cells will die in a tissue. This likely relies on the much more complex picture of apoptosis regulation in a tissular and epithelial context, which entails cell autonomous but also non-cell autonomous factors, diverse feedback and multiple layers of regulation of the commitment to apoptosis. In this review, we illustrate this complexity of epithelial apoptosis regulation by describing these different layers of control, all demonstrating that local cell death probability is a complex emerging feature. We first focus on non-cell autonomous factors that can locally modulate the rate of cell death, including cell competition, mechanical input and geometry as well as systemic effects. We then describe the multiple feedback mechanisms generated by cell death itself. We also outline the multiple layers of regulation of epithelial cell death, including the coordination of extrusion and regulation occurring downstream of effector caspases. Eventually, we propose a roadmap to reach a more predictive understanding of cell death regulation in an epithelial context.

"Patchiness" in mechanical stiffness across a tumor as an early-stage marker for malignancy

Mechanical phenotyping of tumors, either at an individual cell level or tumor cell population level is gaining traction as a diagnostic tool. However, the extent of diagnostic and prognostic information that can be gained through these measurements is still unclear. In this work, we focus on the heterogeneity in mechanical properties of cells obtained from a single source such as a tissue or tumor as a potential novel biomarker. We believe that this heterogeneity is a conventionally overlooked source of information in mechanical phenotyping data. We use mechanics-based in-silico models of cell-cell interactions and cell population dynamics within 3D environments to probe how heterogeneity in cell mechanics drives tissue and tumor dynamics. Our simulations show that the initial heterogeneity in the mechanical properties of individual cells and the arrangement of these heterogenous sub-populations within the environment can dictate overall cell population dynamics and cause a shift towards the growth of malignant cell phenotypes within healthy tissue environments. The overall heterogeneity in the cellular mechanotype and their spatial distributions is quantified by a "patchiness" index, which is the ratio of the global to local heterogeneity in cell populations. We observe that there exists a threshold value of the patchiness index beyond which an overall healthy population of cells will show a steady shift towards a more malignant phenotype. Based on these results, we propose that the "patchiness" of a tumor or tissue sample, can be an early indicator for malignant transformation and cancer occurrence in benign tumors or healthy tissues. Additionally, we suggest that tissue patchiness, measured either by biochemical or biophysical markers, can become an important metric in predicting tissue health and disease likelihood just as landscape patchiness is an important metric in ecology.

ERK-mediated curvature feedback regulates branching morphogenesis in lung epithelial tissue

Intricate branching patterns emerge in internal organs due to the recurrent occurrence of simple deformations in epithelial tissues. During murine lung development, epithelial cells in distal tips of the single tube require fibroblast growth factor (FGF) signals emanating from their surrounding mesenchyme to form repetitive tip bifurcations. However, it remains unknown how the cells employ FGF signaling to convert their behaviors to achieve the recursive branching processes. Here, we show a mechano-chemical regulatory system underlying lung branching morphogenesis, orchestrated by extracellular signal-regulated kinase (ERK) as a downstream driver of FGF signaling. We found that tissue-scale curvature regulated ERK activity in the lung epithelium using two-photon live cell imaging and mechanical perturbations. ERK activation occurs specifically in epithelial tissues exhibiting positive curvature, regardless of whether the change in curvature was attributable to morphogenesis or perturbations. Moreover, ERK activation accelerates actin polymerization preferentially at the apical side of cells, mechanically contributing to the extension of the apical membrane, culminating in a reduction of epithelial tissue curvature. These results indicate the existence of a negative feedback loop between tissue curvature and ERK activity that transcends spatial scales. Our mathematical model confirms that this regulatory mechanism is sufficient to generate the recursive branching processes. Taken together, we propose that ERK orchestrates a curvature feedback loop pivotal to the self-organized patterning of tissues.

Matrix confinement modulates 3D spheroid sorting and burst-like collective migration

While it is known that cells with differential adhesion tend to segregate and preferentially sort, the physical forces governing sorting and invasion in heterogeneous tumors remain poorly understood. To investigate this, we tune matrix confinement, mimicking changes in the stiffness and confinement of the tumor microenvironment, to explore how physical confinement influences individual and collective cell migration in 3D spheroids. High levels of confinement lead to cell sorting while reducing matrix confinement triggers the collective fluidization of cell motion. Cell sorting, which depends on cell-cell adhesion, is crucial to this phenomenon. Burst-like migration does not occur for spheroids that have not undergone sorting, regardless of the degree of matrix confinement. Using computational Self-Propelled Voronoi modeling, we show that spheroid sorting and invasion into the matrix depend on the balance between cell-generated forces and matrix resistance. The findings support a model where matrix confinement modulates 3D spheroid sorting and unjamming in an adhesion-dependent manner, providing insights into the mechanisms of cell sorting and migration in the primary tumor and toward distant metastatic sites.

A mathematical framework for the emergence of winners and losers in cell competition

Cell competition is a process in multicellular organisms where cells interact with their neighbours to determine a "winner" or "loser" status. The loser cells are eliminated through programmed cell death, leaving only the winner cells to populate the tissue. Cell competition is context-dependent; the same cell type can win or lose depending on the cell type it is competing against. Hence, winner/loser status is an emergent property. A key question in cell competition is: how do cells acquire their winner/loser status? In this paper, we propose a mathematical framework for studying the emergence of winner/loser status based on a set of quantitative criteria that distinguishes competitive from non-competitive outcomes. We apply this framework in a cell-based modelling context, to both highlight the crucial role of active cell death in cell competition and identify the factors that drive cell competition.

Biomimetic growth in polymer gels

By modeling gels growing in confined environments, we uncover a biomimetic feedback mechanism between the evolving gel and confining walls that enables significant control over the properties of the grown gel. Our new model describes the monomer adsorption, polymerization and cross-linking involved in forming new networks and the resultant morphology and mechanical behavior of the grown gel. Confined between two hard walls, a thin, flat "parent" gel undergoes buckling; removal of the walls returns the gel to the flat structure. Polymerization and cross-linking in the confined parent generates the next stage of growth, forming a random copolymer network (RCN). When the walls are removed, the RCN remains in the buckled state, simultaneously "locking in" these patterns and increasing the Young's modulus by two orders of magnitude. Confinement of thicker gels between harder or softer 3D walls leads to controllable mechanical heterogeneities, where the Young's modulus between specific domains can differ by three orders of magnitude. These systems effectively replicate the feedback between mechanics and morphology in biological growth, where mechanical forces guide the structure formation throughout stages of growth. The findings provide new guidelines for shaping "growing materials" and introducing new approaches to matching form and function in synthetic systems.

Minimal vertex model explains how the amnioserosa avoids fluidization during Drosophila dorsal closure

Dorsal closure is a process that occurs during embryogenesis of Drosophila melanogaster . During dorsal closure, the amnioserosa (AS), a one-cell thick epithelial tissue that fills the dorsal opening, shrinks as the lateral epidermis sheets converge and eventually merge. During this process, the aspect ratio of amnioserosa cells increases markedly. The standard 2-dimensional vertex model, which successfully describes tissue sheet mechanics in multiple contexts, would in this case predict that the tissue should fluidize via cell neighbor changes. Surprisingly, however, the amnioserosa remains an elastic solid with no such events. We here present a minimal extension to the vertex model that explains how the amnioserosa can achieve this unexpected behavior. We show that continuous shrinkage of the preferred cell perimeter and cell perimeter polydispersity lead to the retention of the solid state of the amnioserosa. Our model accurately captures measured cell shape and orientation changes and predicts non-monotonic junction tension that we confirm with laser ablation experiments.

Significance Statement

During embryogenesis, cells in tissues can undergo significant shape changes. Many epithelial tissues fluidize, i.e. cells exchange neighbors, when the average cell aspect ratio increases above a threshold value, consistent with the standard vertex model. During dorsal closure in Drosophila melanogaster , however, the amnioserosa tissue remains solid even as the average cell aspect ratio increases well above threshold. We introduce perimeter polydispersity and allow the preferred cell perimeters, usually held fixed in vertex models, to decrease linearly with time as seen experimentally. With these extensions to the standard vertex model, we capture experimental observations quantitatively. Our results demonstrate that vertex models can describe the behavior of the amnioserosa in dorsal closure by allowing normally fixed parameters to vary with time.

Mechanical control of neural plate folding by apical domain alteration

Vertebrate neural tube closure is associated with complex changes in cell shape and behavior, however, the relative contribution of these processes to tissue folding is not well understood. At the onset of Xenopus neural tube folding, we observed alternation of apically constricted and apically expanded cells. This apical domain heterogeneity was accompanied by biased cell orientation along the anteroposterior axis, especially at neural plate hinges, and required planar cell polarity signaling. Vertex models suggested that dispersed isotropically constricting cells can cause the elongation of adjacent cells. Consistently, in ectoderm, cell-autonomous apical constriction was accompanied by neighbor expansion. Thus, a subset of isotropically constricting cells may initiate neural plate bending, whereas a 'tug-of-war' contest between the force-generating and responding cells reduces its shrinking along the body axis. This mechanism is an alternative to anisotropic shrinking of cell junctions that are perpendicular to the body axis. We propose that apical domain changes reflect planar polarity-dependent mechanical forces operating during neural folding.

Minimal vertex model explains how the amnioserosa avoids fluidization during Drosophila dorsal closure

Dorsal closure is a process that occurs during embryogenesis of Drosophila melanogaster. During dorsal closure, the amnioserosa (AS), a one-cell thick epithelial tissue that fills the dorsal opening, shrinks as the lateral epidermis sheets converge and eventually merge. During this process, the aspect ratio of amnioserosa cells increases markedly. The standard 2-dimensional vertex model, which successfully describes tissue sheet mechanics in multiple contexts, would in this case predict that the tissue should fluidize via cell neighbor changes. Surprisingly, however, the amnioserosa remains an elastic solid with no such events. We here present a minimal extension to the vertex model that explains how the amnioserosa can achieve this unexpected behavior. We show that continuous shrink-age of the preferred cell perimeter and cell perimeter polydispersity lead to the retention of the solid state of the amnioserosa. Our model accurately captures measured cell shape and orientation changes and predicts non-monotonic junction tension that we confirm with laser ablation experiments.

Connecting theory and experiment in cell and tissue mechanics

Understanding complex living systems, which are fundamentally constrained by physical phenomena, requires combining experimental data with theoretical physical and mathematical models. To develop such models, collaborations between experimental cell biologists and theoreticians are increasingly important but these two groups often face challenges achieving mutual understanding. To help navigate these challenges, this Perspective discusses different modelling approaches, including bottom-up hypothesis-driven and top-down data-driven models, and highlights their strengths and applications. Using cell mechanics as an example, we explore the integration of specific physical models with experimental data from the molecular, cellular and tissue level up to multiscale input. We also emphasize the importance of constraining model complexity and outline strategies for crosstalk between experimental design and model development. Furthermore, we highlight how physical models can provide conceptual insights and produce unifying and generalizable frameworks for biological phenomena. Overall, this Perspective aims to promote fruitful collaborations that advance our understanding of complex biological systems.

Bridging the gap between collective motility and epithelial-mesenchymal transitions through the active finite voronoi model

We introduce an active version of the recently proposed finite Voronoi model of epithelial tissue. The resultant Active Finite Voronoi (AFV) model enables the study of both confluent and non-confluent geometries and transitions between them, in the presence of active cells. Our study identifies six distinct phases, characterized by aggregation-segregation, dynamical jamming-unjamming, and epithelial-mesenchymal transitions (EMT), thereby extending the behavior beyond that observed in previously studied vertex-based models. The AFV model with rich phase diagram provides a cohesive framework that unifies the well-observed progression to collective motility via unjamming with the intricate dynamics enabled by EMT. This approach should prove useful for challenges in developmental biology systems as well as the complex context of cancer metastasis. The simulation code is also provided.

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.

The Effect of Substrate Stiffness on Elastic Force Transmission in the Epithelial Monolayers over Short Timescales

Purpose

The importance of mechanical forces and microenvironment in guiding cellular behavior has been widely accepted. Together with the extracellular matrix (ECM), epithelial cells form a highly connected mechanical system subjected to various mechanical cues from their environment, such as ECM stiffness, and tensile and compressive forces. ECM stiffness has been linked to many pathologies, including tumor formation. However, our understanding of the effect of ECM stiffness and its heterogeneities on rapid force transduction in multicellular systems has not been fully addressed.

Methods

We used experimental and computational methods. Epithelial cells were cultured on elastic hydrogels with fluorescent nanoparticles. Single cells were moved by a micromanipulator, and epithelium and substrate deformation were recorded. We developed a computational model to replicate our experiments and quantify the force distribution in the epithelium. Our model further enabled simulations with local stiffness gradients.

Results

We found that substrate stiffness affects the force transduction and the cellular deformation following an external force. Also, our results indicate that the heterogeneities, e.g., gradients, in the stiffness can substantially influence the strain redistribution in the cell monolayers. Furthermore, we found that the cells' apico-basal elasticity provides a level of mechanical isolation between the apical cell-cell junctions and the basal focal adhesions.

Conclusions

Our simulation results show that increased ECM stiffness, e.g., due to a tumor, can mechanically isolate cells and modulate rapid mechanical signaling between cells over distances. Furthermore, the developed model has the potential to facilitate future studies on the interactions between epithelial monolayers and elastic substrates.

Supplementary Information

The online version of this article (10.1007/s12195-023-00772-0) contains supplementary material, which is available to authorized users.

Shape-Tension Coupling Produces Nematic Order in an Epithelium Vertex Model

We study the vertex model for epithelial tissue mechanics extended to include coupling between the cell shapes and tensions in cell-cell junctions. This coupling represents an active force which drives the system out of equilibrium and leads to the formation of nematic order interspersed with prominent, long-lived +1 defects. The defects in the nematic ordering are coupled to the shape of the cell tiling, affecting cell areas and coordinations. This intricate interplay between cell shape, size, and coordination provides a possible mechanism by which tissues could spontaneously develop long-range polarity through local mechanical forces without resorting to long-range chemical patterning.

New tools reveal PCP-dependent polarized mechanics in the cortex and cytoplasm of single cells during convergent extension

Understanding biomechanics of biological systems is crucial for unraveling complex processes like tissue morphogenesis. However, current methods for studying cellular mechanics in vivo are limited by the need for specialized equipment and often provide limited spatiotemporal resolution. Here we introduce two new techniques, Tension by Transverse Fluctuation (TFlux) and in vivo microrheology, that overcome these limitations. They both offer time-resolved, subcellular biomechanical analysis using only fluorescent reporters and widely available microscopes. Employing these two techniques, we have revealed a planar cell polarity (PCP)-dependent mechanical gradient both in the cell cortex and the cytoplasm of individual cells engaged in convergent extension. Importantly, the non-invasive nature of these methods holds great promise for its application for uncovering subcellular mechanical variations across a wide array of biological contexts.

Summary

Non-invasive imaging-based techniques providing time-resolved biomechanical analysis at subcellular scales in developing vertebrate embryos.

Shape-driven confluent rigidity transition in curved biological tissues

Collective cell motions underlie structure formation during embryonic development. Tissues exhibit emergent multicellular characteristics such as jamming, rigidity transitions, and glassy dynamics, but there remain questions about how those tissue-scale dynamics derive from local cell-level properties. Specifically, there has been little consideration of the interplay between local tissue geometry and cellular properties influencing larger-scale tissue behaviors. Here, we consider a simple two-dimensional computational vertex model for confluent tissue monolayers, which exhibits a rigidity phase transition controlled by the shape index (ratio of perimeter to square root area) of cells, on surfaces of constant curvature. We show that the critical point for the rigidity transition is a function of curvature such that positively curved systems are likely to be in a less rigid, more fluid, phase. Likewise, negatively curved systems (saddles) are likely to be in a more rigid, less fluid, phase. A phase diagram we generate for the curvature and shape index constitutes a testable prediction from the model. The curvature dependence is interesting because it suggests a natural explanation for more dynamic tissue remodeling and facile growth in regions of higher surface curvature. Conversely, we would predict stability at the base of saddle-shaped budding structures without invoking the need for biochemical or other physical differences. This concept has potential ramifications for our understanding of morphogenesis of budding and branching structures.

Nature-inspired designs for disordered acoustic bandgap materials

We introduce an amorphous mechanical metamaterial inspired by how cells pack in biological tissues. The spatial heterogeneity in the local stiffness of these materials has been recently shown to impact the mechanics of confluent biological tissues and cancer tumor invasion. Here we use this bio-inspired structure as a design template to construct mechanical metamaterials and show that this heterogeneity can give rise to amorphous cellular solids with large, tunable acoustic bandgaps. Unlike acoustic crystals with periodic structures, the bandgaps here are directionally isotropic and robust to defects due to their complete lack of positional order. Possible ways to manipulate bandgaps are explored with a combination of the tissue-level elastic modulus and local stiffness heterogeneity of cells. To further demonstrate the existence of bandgaps, we dynamically perturb the system with an external sinusoidal wave in the perpendicular and horizontal directions. The transmission coefficients are calculated and show valleys that coincide with the location of bandgaps. Experimentally this design should lead to the engineering of self-assembled rigid acoustic structures with full bandgaps that can be controlled via mechanical tuning and promote applications in a broad area from vibration isolations to mechanical waveguides.

Contributions of cell behavior to geometric order in embryonic cartilage

During early development, cartilage provides shape and stability to the embryo while serving as a precursor for the skeleton. Correct formation of embryonic cartilage is hence essential for healthy development. In vertebrate cranial cartilage, it has been observed that a flat and laterally extended macroscopic geometry is linked to regular microscopic structure consisting of tightly packed, short, transversal clonar columns. However, it remains an ongoing challenge to identify how individual cells coordinate to successfully shape the tissue, and more precisely which mechanical interactions and cell behaviors contribute to the generation and maintenance of this columnar cartilage geometry during embryogenesis. Here, we apply a three-dimensional cell-based computational model to investigate mechanical principles contributing to column formation. The model accounts for clonal expansion, anisotropic proliferation and the geometrical arrangement of progenitor cells in space. We confirm that oriented cell divisions and repulsive mechanical interactions between cells are key drivers of column formation. In addition, the model suggests that column formation benefits from the spatial gaps created by the extracellular matrix in the initial configuration, and that column maintenance is facilitated by sequential proliferative phases. Our model thus correctly predicts the dependence of local order on division orientation and tissue thickness. The present study presents the first cell-based simulations of cell mechanics during cranial cartilage formation and we anticipate that it will be useful in future studies on the formation and growth of other cartilage geometries.

Dynamics and functions of E-cadherin complexes in epithelial cell and tissue morphogenesis

Cell-cell adhesion is at the center of structure and dynamics of epithelial tissue. E-cadherin-catenin complexes mediate Ca2+-dependent trans-homodimerization and constitute the kernel of adherens junctions. Beyond the basic function of cell-cell adhesion, recent progress sheds light the dynamics and interwind interactions of individual E-cadherin-catenin complex with E-cadherin superclusters, contractile actomyosin and mechanics of the cortex and adhesion. The nanoscale architecture of E-cadherin complexes together with cis-interactions and interactions with cortical actomyosin adjust to junctional tension and mechano-transduction by reinforcement or weakening of specific features of the interactions. Although post-translational modifications such as phosphorylation and glycosylation have been implicated, their role for specific aspects of in E-cadherin function has remained unclear. Here, we provide an overview of the E-cadherin complex in epithelial cell and tissue morphogenesis focusing on nanoscale architectures by super-resolution approaches and post-translational modifications from recent, in particular in vivo, studies. Furthermore, we review the computational modelling in E-cadherin complexes and highlight how computational modelling has contributed to a deeper understanding of the E-cadherin complexes.

Assessing mechanical agency during apical apoptotic cell extrusion

Epithelial tissues maintain homeostasis through the continual addition and removal of cells. Homeostasis is necessary for epithelia to maintain barrier function and prevent the accumulation of defective cells. Unfit, excess, and dying cells can be removed from epithelia by the process of extrusion. Controlled cell death and extrusion in the epithelium of the larval zebrafish tail fin coincides with oscillation of cell area, both in the extruding cell and its neighbors. Both cell-autonomous and non-autonomous factors have been proposed to contribute to extrusion but have been challenging to test by experimental approaches. Here we develop a dynamic cell-based biophysical model that recapitulates the process of oscillatory cell extrusion to test and compare the relative contributions of these factors. Our model incorporates the mechanical properties of individual epithelial cells in a two-dimensional simulation as repelling active particles. The area of cells destined to extrude oscillates with varying durations or amplitudes, decreasing their mechanical contribution to the epithelium and surrendering their space to surrounding cells. Quantitative variations in cell shape and size during extrusion are visualized by a hybrid weighted Voronoi tessellation technique that renders individual cell mechanical properties directly into an epithelial sheet. To explore the role of autonomous and non-autonomous mechanics, we vary the biophysical properties and behaviors of extruding cells and neighbors such as the period and amplitude of repulsive forces, cell density, and tissue viscosity. Our data suggest that cell autonomous processes are major contributors to the dynamics of extrusion, with the mechanical microenvironment providing a less pronounced contribution. Our computational model based on in vivo data serves as a tool to provide insights into the cellular dynamics and localized changes in mechanics that promote elimination of unwanted cells from epithelia during homeostatic tissue maintenance.

General, open-source vertex modeling in biological applications using Tissue Forge

Vertex models are a widespread approach for describing the biophysics and behaviors of multicellular systems, especially of epithelial tissues. Vertex models describe a wide variety of developmental scenarios and behaviors like cell rearrangement and tissue folding. Often, these models are implemented as single-use or closed-source software, which inhibits reproducibility and decreases accessibility for researchers with limited proficiency in software development and numerical methods. We developed a physics-based vertex model methodology in Tissue Forge, an open-source, particle-based modeling and simulation environment. Our methodology describes the properties and processes of vertex model objects on the basis of vertices, which allows integration of vertex modeling with the particle-based formalism of Tissue Forge, enabling an environment for developing mixed-method models of multicellular systems. Our methodology in Tissue Forge inherits all features provided by Tissue Forge, delivering open-source, extensible vertex modeling with interactive simulation, real-time simulation visualization and model sharing in the C, C++ and Python programming languages and a Jupyter Notebook. Demonstrations show a vertex model of cell sorting and a mixed-method model of cell migration combining vertex- and particle-based models. Our methodology provides accessible vertex modeling for a broad range of scientific disciplines, and we welcome community-developed contributions to our open-source software implementation.

Coarse-graining the vertex model and its response to shear

Tissue dynamics and collective cell motion are crucial biological processes. Their biological machinery is mostly known, and simulation models such as the active vertex model exist and yield reasonable agreement with experimental observations such as tissue fluidization or fingering. However, a good and well-founded continuum description for tissues remains to be developed. In this work, we derive a macroscopic description for a two-dimensional cell monolayer by coarse-graining the vertex model through the Poisson bracket approach. We obtain equations for cell density, velocity, and the cellular shape tensor. We then study the homogeneous steady states, their stability (which coincides with thermodynamic stability), and especially their behavior under an externally applied shear. Our results contribute to elucidate the interplay between flow and cellular shape. The obtained macroscopic equations present a good starting point for adding cell motion, morphogenetic, and other biologically relevant processes.

Couple stresses and discrete potentials in the vertex model of cellular monolayers

The vertex model is widely used to simulate the mechanical properties of confluent epithelia and other multicellular tissues. This inherently discrete framework allows a Cauchy stress to be attributed to each cell, and its symmetric component has been widely reported, at least for planar monolayers. Here, we consider the stress attributed to the neighbourhood of each tricellular junction, evaluating in particular its leading-order antisymmetric component and the associated couple stresses, which characterise the degree to which individual cells experience (and resist) in-plane bending deformations. We develop discrete potential theory for localised monolayers having disordered internal structure and use this to derive the analogues of Airy and Mindlin stress functions. These scalar potentials typically have broad-banded spectra, highlighting the contributions of small-scale defects and boundary layers to global stress patterns. An affine approximation attributes couple stresses to pressure differences between cells sharing a trijunction, but simulations indicate an additional role for non-affine deformations.

Topological packing statistics of living and nonliving matter

Complex disordered matter is of central importance to a wide range of disciplines, from bacterial colonies and embryonic tissues in biology to foams and granular media in materials science to stellar configurations in astrophysics. Because of the vast differences in composition and scale, comparing structural features across such disparate systems remains challenging. Here, by using the statistical properties of Delaunay tessellations, we introduce a mathematical framework for measuring topological distances between general three-dimensional point clouds. The resulting system-agnostic metric reveals subtle structural differences between bacterial biofilms as well as between zebrafish brain regions, and it recovers temporal ordering of embryonic development. We apply the metric to construct a universal topological atlas encompassing bacterial biofilms, snowflake yeast, plant shoots, zebrafish brain matter, organoids, and embryonic tissues as well as foams, colloidal packings, glassy materials, and stellar configurations. Living systems localize within a bounded island-like region of the atlas, reflecting that biological growth mechanisms result in characteristic topological properties.

Continuum description of confluent tissues with spatial heterogeneous activity

A continuum description is built to characterize the stationary and transient deformations of confluent tissues subject to heterogeneous activities. By defining a coarse-grained texture matrix field to represent the shape and size of cells, we derive the coarse-grained stress tensor for the vertex model. Activity in the tissue takes the form of inhomogeneous apical contractions, which can be modeled as reductions of the vertex model reference areas or perimeters representing activity in the medial and perimeter regions of the cells, respectively. For medial activity, the extra stress is just an isotropic pressure, while for perimeter activity, it also has a deviatoric component, which is aligned with the texture matrix. The predictions of the continuum description are compared with the average spatiotemporal deformations obtained in simulations of the vertex model subject to localized apical contractions, showing an excellent agreement, even if the active patch is as small as one cell. The fluctuations around the average are more prominent when the activity is in the medial region due to the lack of negative active shape feedback, which, coupled with the confluent property, increases cellular shape and size variations.

Mechanics of cell-cell junctions

Tissue cells in epithelial or endothelial monolayers are connected through cell-cell junctions, which are stabilized by transmembrane E-cadherin bonds and intracellular actin filaments. These bonds and junctions play a crucial role in maintaining the barrier function of epithelia and endothelia and are believed to transmit forces between cells. Additionally, E-cadherin bonds can impact the shape of cell-cell junctions. In this study, we develop a continuum mechanical model of the cell-cell junction by explicitly incorporating the cell membrane, distributions of E-cadherin bonds, cytoplasmic fluid pressure, and F-actin dynamics. The static force-balanced version of the model is able to analyze the influences of cell cortical tension, actin dynamics, and cytoplasmic pressure on the junction shape and E-cadherin bonds. Furthermore, an extended model that incorporates fluid flow, across the cell boundary as well as around the cell, is also examined. This model can couple cell-shape changes with cell cortical tension and fluid flow, and predicts the additional effect of fluid motion on cell-cell junction mechanics. Taken together, our models serve as an intermediate link between molecular-scale models of cell-junction molecules and cell-scale models of tissue and epithelia.

Cell-center-based model for simulating three-dimensional monolayer tissue deformation

The shape of the epithelial monolayer can be depicted as a curved tissue in three-dimensional (3D) space, where individual cells are tightly adhered to one another. The 3D morphogenesis of these tissues is governed by cell dynamics, and a variety of mathematical modeling and simulation studies have been conducted to investigate this process. One promising approach is the cell-center model, which can account for the discreteness of cells. The cell nucleus, which is considered to correspond to the cell center, can be observed experimentally. However, there has been a shortage of cell-center models specifically tailored for simulating 3D monolayer tissue deformation. In this study, we developed a mathematical model based on the cell-center model to simulate 3D monolayer tissue deformation. Our model was confirmed by simulating the in-plane deformation, out-of-plane deformation, and invagination due to apical constriction.

Topological properties and shape of proliferative and nonproliferative cell monolayers

During embryonic development, structures with complex geometry can emerge from planar epithelial monolayers; studying these shape transitions is of key importance for revealing the biophysical laws involved in the morphogenesis of biological systems. Here, using the example of normal proliferative monkey kidney (COS) cell monolayers, we investigate global and local topological characteristics of this model system in dependence on its shape. The obtained distributions of cells by their valence demonstrate a difference between the spherical and planar monolayers. In addition, in both spherical and planar monolayers, the probability of observing a pair of neighboring cells with certain valences depends on the topological charge of the pair. The zero topological charge of the cell pair can increase the probability for the cells to be the nearest neighbors. We then test and confirm that analogous relationships take place in a more ordered spherical system with a larger fraction of 6-valent cells, namely, in the nonproliferative epithelium (follicular system) of ascidian species oocytes. However, unlike spherical COS cell monolayers, ascidian monolayers are prone to nonrandom agglomeration of 6-valent cells and have linear topological defects called scars and pleats. The reasons for this difference in morphology are discussed. The morphological peculiarities found are compared with predictions of the widely used vertex model of epithelium.

Effective mechanical potential of cell-cell interaction explains three-dimensional morphologies during early embryogenesis

Mechanical forces are critical for the emergence of diverse three-dimensional morphologies of multicellular systems. However, it remains unclear what kind of mechanical parameters at cellular level substantially contribute to tissue morphologies. This is largely due to technical limitations of live measurements of cellular forces. Here we developed a framework for inferring and modeling mechanical forces of cell-cell interactions. First, by analogy to coarse-grained models in molecular and colloidal sciences, we approximated cells as particles, where mean forces (i.e. effective forces) of pairwise cell-cell interactions are considered. Then, the forces were statistically inferred by fitting the mathematical model to cell tracking data. This method was validated by using synthetic cell tracking data resembling various in vivo situations. Application of our method to the cells in the early embryos of mice and the nematode Caenorhabditis elegans revealed that cell-cell interaction forces can be written as a pairwise potential energy in a manner dependent on cell-cell distances. Importantly, the profiles of the pairwise potentials were quantitatively different among species and embryonic stages, and the quantitative differences correctly described the differences of their morphological features such as spherical vs. distorted cell aggregates, and tightly vs. non-tightly assembled aggregates. We conclude that the effective pairwise potential of cell-cell interactions is a live measurable parameter whose quantitative differences can be a parameter describing three-dimensional tissue morphologies.

Polychaetoid/ZO-1 strengthens cell junctions under tension while localizing differently than core adherens junction proteins

During embryonic development, dramatic cell shape changes and movements reshape the embryonic body plan. These require robust but dynamic linkage between the cell-cell adherens junctions and the force-generating actomyosin cytoskeleton. Our view of this linkage has evolved, and we now realize linkage is mediated by mechanosensitive multiprotein complexes assembled via multivalent connections. Here we combine genetic, cell biological, and modeling approaches to define the mechanism of action and functions of an important player, Drosophila polychaetoid, homologue of mammalian ZO-1. Our data reveal that Pyd reinforces cell junctions under elevated tension, and facilitates cell rearrangements. Pyd is important to maintain junctional contractility and in its absence cell rearrangements stall. We next use structured illumination microscopy to define the molecular architecture of cell-cell junctions during these events. The cadherin-catenin complex and Cno both localize to puncta along the junctional membrane, but are differentially enriched in different puncta. Pyd, in contrast, exhibits a distinct localization to strands that extend out from the region occupied by core junction proteins. We then discuss the implications for the protein network at the junction-cytoskeletal interface, suggesting different proteins localize and function in distinct ways, perhaps in distinct subcomplexes, but combine to produce robust connections.

Tissue flows are tuned by actomyosin-dependent mechanics in developing embryos

Rapid epithelial tissue flows are essential to building and shaping developing embryos. However, the mechanical properties of embryonic epithelial tissues and the factors that control these properties are not well understood. Actomyosin generates contractile tensions and contributes to the mechanical properties of cells and cytoskeletal networks in vitro, but it remains unclear how the levels and patterns of actomyosin activity contribute to embryonic epithelial tissue mechanics in vivo. To dissect the roles of cell-generated tensions in the mechanics of flowing epithelial tissues, we use optogenetic tools to manipulate actomyosin contractility with spatiotemporal precision in the Drosophila germband epithelium, which rapidly flows during body axis elongation. We find that manipulating actomyosin-dependent tensions by either optogenetic activation or deactivation of actomyosin alters the solid-fluid mechanical properties of the germband epithelium, leading to changes in cell rearrangements and tissue-level flows. Optogenetically activating actomyosin leads to increases in the overall level but decreases in the anisotropy of tension in the tissue, whereas optogenetically deactivating actomyosin leads to decreases in both the level and anisotropy of tension compared to in wild-type embryos. We find that optogenetically activating actomyosin results in more solid-like (less fluid-like) tissue properties, which is associated with reduced cell rearrangements and tissue flow compared to in wild-type embryos. Optogenetically deactivating actomyosin also results in more solid-like properties than in wild-type embryos but less solid-like properties compared to optogenetically activating actomyosin. Together, these findings indicate that increasing the overall tension level is associated with more solid-like properties in tissues that are relatively isotropic, whereas high tension anisotropy fluidizes the tissue. Our results reveal that epithelial tissue flows in developing embryos involve the coordinated actomyosin-dependent regulation of the mechanical properties of tissues and the tensions driving them to flow in order to achieve rapid tissue remodeling.

A cell membrane model that reproduces cortical flow-driven cell migration and collective movement

Many fundamental biological processes are dependent on cellular migration. Although the mechanical mechanisms of single-cell migration are relatively well understood, those underlying migration of multiple cells adhered to each other in a cluster, referred to as cluster migration, are poorly understood. A key reason for this knowledge gap is that many forces-including contraction forces from actomyosin networks, hydrostatic pressure from the cytosol, frictional forces from the substrate, and forces from adjacent cells-contribute to cell cluster movement, making it challenging to model, and ultimately elucidate, the final result of these forces. This paper describes a two-dimensional cell membrane model that represents cells on a substrate with polygons and expresses various mechanical forces on the cell surface, keeping these forces balanced at all times by neglecting cell inertia. The model is discrete but equivalent to a continuous model if appropriate replacement rules for cell surface segments are chosen. When cells are given a polarity, expressed by a direction-dependent surface tension reflecting the location dependence of contraction and adhesion on a cell boundary, the cell surface begins to flow from front to rear as a result of force balance. This flow produces unidirectional cell movement, not only for a single cell but also for multiple cells in a cluster, with migration speeds that coincide with analytical results from a continuous model. Further, if the direction of cell polarity is tilted with respect to the cluster center, surface flow induces cell cluster rotation. The reason why this model moves while keeping force balance on cell surface (i.e., under no net forces from outside) is because of the implicit inflow and outflow of cell surface components through the inside of the cell. An analytical formula connecting cell migration speed and turnover rate of cell surface components is presented.

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.