Three-dimensional Vertex Model Research Articles

Mechanochemical patterning and wave propagation in multicellular tubes

Multicellular tubes are fundamental tissues for transporting and distributing liquids and gases in living organisms. Although the molecular, cellular and mechanical aspects in tube formation have been addressed experimentally, how these factors are coupled to control tube patterning and dynamics at the tissue level remains incompletely understood. Here, we propose a three-dimensional (3D) vertex model that incorporates a mechanochemical feedback loop correlating cell deformation and actomyosin signaling pathway to probe the morphodynamics of multicellular tubes. We show that diverse patterns, including ring, helix, double helix, and labyrinth, are generated in tubes through pitchfork bifurcation, where spatial fluctuations of both biochemical signaling and 3D cell deformation are remarkably involved. The mechanochemical feedback loop enables cell oscillations via Hopf bifurcation, which induces the mechanical and chemical patterns to propagate successively as either traveling or pulse waves while their spatial configurations are maintained, strikingly distinct from the classical Turing instability. Our simulations, together with stability analysis of a minimal model, uncover the essential role of mechanochemical principles in sculpting biological tubes.

Local Mechanical Modulation-Driven Evagination in Invaginated Epithelia.

Local cells can actively create reverse bending (evagination) in invaginated epithelia, which plays a crucial role in the formation of elaborate organisms. However, the precise physical mechanism driving the evagination remains elusive. Here, we present a three-dimensional vertex model, incorporating the intrinsic cell polarity, to explore the complex morphogenesis induced by local mechanical modulations. We find that invaginated tissues can spontaneously generate local reverse bending due to the shift of the apicobasal polarity. Their exact shapes can be analytically determined by the local apicobasal differential tension and the internal stress. Our continuum theory exhibits three regions in a phase diagram controlled by these two parameters, showing curvature transitions from ordered to disordered states. Additionally, we delve into epithelial curvature transition induced by the nucleus repositioning, revealing its active contribution to the apicobasal force generation. The uncovered mechanical principles could potentially guide more studies on epithelial folding in diverse systems.

Morphological instability at topological defects in a three-dimensional vertex model for spherical epithelia

Epithelial monolayers are a central building block of complex organisms. Topological defects have emerged as important elements for single cell behavior in flat epithelia. Here we theoretically study such defects in a three-dimensional vertex model for spherical epithelia like cysts or intestinal organoids. We find that they lead to the same generic morphological instability to an icosahedral shape as it is known from spherical elastic shells like virus capsids, polymerized vesicles, or buckyballs. We derive analytical expressions for the effective stretching and bending moduli as a function of the parameters of the vertex model, in excellent agreement with computer simulations. These equations accurately predict both the buckling of a flat epithelial monolayer under uniaxial compression and the faceting transition around the topological defects in spherical epithelia. We further show that localized apico-basal tension asymmetries allow them to reduce the transition threshold to small system sizes. Published by the American Physical Society 2024

Mean-field elastic moduli of a three-dimensional, cell-based vertex model

The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible–incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes.

Single and two-cells shape analysis from energy functionals for three-dimensional vertex models.

Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetrical and asymmetrical, we deduce the evolution of the aspect ratio and the relative radius. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.

Topologically-protected interior for three-dimensional confluent cellular collectives

Organoids are in vitro cellular collectives from which, for example, brain-like, or gut-like, or kidney-like structures emerge. To make quantitative predictions regarding the morphology and rheology of a cellular collective in its initial stages of development, we construct and study a three-dimensional vertex model. In such a model, the cells are represented as deformable polyhedrons with cells sharing faces such that there are no gaps between them, otherwise known as confluent. In a bulk model with periodic boundary conditions, we find a rigidity transition as a function of the target cell shape index ${s}_{0}$ with a critical value ${s}_{0}^{*}=5.39\ifmmode\pm\else\textpm\fi{}0.01$. For a confluent cellular collective with a finite boundary, and in the presence of lateral extensile and in-plane, radial extensile deformations, we find a significant boundary-bulk effect that is one-cell layer thick. More specifically, for lateral extensile deformations, the cells in the bulk are much less aligned with the direction of the lateral deformation than the cells at the boundary. For in-plane, radial deformations, the cells in the bulk exhibit much less reorientation perpendicular to the radial direction than the cells at the boundary. In other words, for both deformations, the bulk, interior cells are topologically protected from the deformations, at least over time scales much slower than the timescale for cellular rearrangements and up to reasonable amounts of strain. Our results provide an underlying mechanism for some observed cell shape patterning in organoids and in in vivo settings. Finally, we discuss the use of a cellular-based approach to designing organoids with new types of morphologies to study the intricate relationship between structure and function at the multicellular scale.

A three-dimensional vertex model for Drosophila salivary gland invagination

During epithelial morphogenesis, force generation at the cellular level not only causes cell deformation, but may also produce coordinated cell movement and rearrangement on the tissue level. In this paper, we use a novel three-dimensional vertex model to explore the roles of cellular forces during the formation of the salivary gland in the Drosophila embryo. Representing the placode as an epithelial sheet of initially columnar cells, we focus on the spatial and temporal patterning of contractile forces due to three actomyosin pools: the apicomedial actomyosin in the pit of the placode, junctional actomyosin arcs outside the pit, and a supracellular actomyosin cable along the circumference of the placode. In an in silico ‘wild type’ model, these pools are activated at different times according to experimental data. To identify the role of each myosin pool, we have also simulated various in silico ‘mutants’ in which only one or two of the myosin pools are activated. We find that the apicomedial myosin initiates a small dimple in the pit, but this is not essential for the overall invagination of the placode. The myosin arcs are the main driver of invagination and are responsible for the internalization of the apical surface. The circumferential actomyosin cable acts to constrict the opening of the developing tube, and is responsible for forming a properly shaped lumen. Cell intercalation tends to facilitate the invagination, but the geometric constraints of our model only allow a small number of intercalations, and their effect is minor. The placode invagination predicted by the model is in general agreement with experimental observations. It confirms some features of the current ‘belt-and-braces’ model for the process, and provides new insights on the separate roles of the various myosin pools and their spatio-temporal coordination.

A Mechanical Instability in Planar Epithelial Monolayers Leads to Cell Extrusion

In cell extrusion, a cell embedded in an epithelial monolayer loses its apical or basal surface and is subsequently squeezed out of the monolayer by neighboring cells. Cell extrusions occur during apoptosis, epithelial-mesenchymal transition, or precancerous cell invasion. They play important roles in embryogenesis, homeostasis, carcinogenesis, and many other biological processes. Although many of the molecular factors involved in cell extrusion are known, little is known about the mechanical basis of cell extrusion. We used a three-dimensional (3D) vertex model to investigate the mechanical stability of cells arranged in a monolayer with 3D foam geometry. We found that when the cells composing the monolayer have homogeneous mechanical properties, cells are extruded from the monolayer when the symmetry of the 3D geometry is broken because of an increase in cell density or a decrease in the number of topological neighbors around single cells. Those results suggest that mechanical instability inherent in the 3D foam geometry of epithelial monolayers is sufficient to drive epithelial cell extrusion. In the situation in which cells in the monolayer actively generate contractile or adhesive forces under the control of intrinsic genetic programs, the forces act to break the symmetry of the monolayer, leading to cell extrusion that is directed to the apical or basal side of the monolayer by the balance of contractile and adhesive forces on the apical and basal sides. Although our analyses are based on a simple mechanical model, our results are in accordance with observations of epithelial monolayers in vivo and consistently explain cell extrusions under a wide range of physiological and pathophysiological conditions. Our results illustrate the importance of a mechanical understanding of cell extrusion and provide a basis by which to link molecular regulation to physical processes.

Epithelial tissue folding pattern in confined geometry

The primordium of the exoskeleton of an insect is epithelial tissue with characteristic patterns of folds. As the insect develops from larva to pupa, the spreading of these folds produces the three-dimensional shape of the exoskeleton of the insect. It is known that the three-dimensional exoskeleton shape has already been encoded in characteristic patterns of folds in the primordium; however, a description of how the epithelial tissue forms with the characteristic patterns of folds remains elusive. The present paper suggests a possible mechanism for the formation of the folding pattern. During the primordium development, because of the epithelial tissue is surrounded by other tissues, cell proliferation proceeds within a confined geometry. To elucidate the mechanics of the folding of the epithelial tissue in the confined geometry, we employ a three-dimensional vertex model that expresses tissue deformations based on cell mechanical behaviors and apply the model to examine the effects of cell divisions and the confined geometry on epithelial folding. Our simulation results suggest that the orientation of the axis of cell division is sufficient to cause different folding patterns in silico and that the restraint of out-of-plane deformation due to the confined geometry determines the interspacing of the folds.

Feedback from Tissue Mechanics Self-Organizes Efficient Outgrowth of Plant Organ

Plant organ outgrowth superficially appears like the continuous mechanical deformation of a sheet of cells. Yet, how precisely cells as individual mechanical entities can act to morph a tissue reliably and efficiently into three dimensions during outgrowth is still puzzling, especially when cells are tightly connected as in plant tissue. In plants, the mechanics of cells within a tissue is particularly well-defined because individual cell growth is essentially the mechanical yielding of the cell wall in response to internal turgor pressure. Cell-wall stiffness is controlled by biological signaling, which is impacted by stresses, and hence, cell growth is observed to respond to mechanical stresses building up within a tissue. What is the role of the mechanical feedback during morphing of tissue in three dimensions? Here, we develop a three-dimensional vertex model to investigate tissue mechanics at the onset of organ outgrowth at the tip of a plant shoot. We find that organ height is primarily governed by the ratio of growth rates of faster-growing cells initiating the organ versus slower-growing cells surrounding them. Remarkably, the outgrowth rate is higher when cell growth responds to the tissue-wide mechanical stresses. Our quantitative analysis of simulation data shows that tissue mechanical feedback on cell growth can act via a twofold mechanism. First, the feedback guides patterns of cellular growth. Second, the feedback modifies the stress patterns on the cells, consequently amplifying and propagating growth anisotropies. This mechanism may allow plants to grow organs efficiently out of the meristem by reorganizing the cellular growth rather than inflating growth rates.

Apical Junctional Fluctuations Lead to Cell Flow while Maintaining Epithelial Integrity

Epithelial sheet integrity is robustly maintained during morphogenesis, which is essential to shape organs and embryos. While maintaining the planar monolayer in three-dimensional space, cells dynamically flow via rearranging their connections between each other. However, little is known about how cells maintain the plane sheet integrity in three-dimensional space and provide cell flow in the in-plane sheet. In this study, using a three-dimensional vertex model, we demonstrate that apical junctional fluctuations allow stable cell rearrangements while ensuring monolayer integrity. In addition to the fluctuations, direction-dependent contraction on the apical cell boundaries, which corresponds to forces from adherens junctions, induces cell flow in a definite direction. We compared the kinematic behaviors of this apical-force-driven cell flow with those of typical cell flow that is driven by forces generated on basal regions and revealed the characteristic differences between them. These differences can be used to distinguish the mechanism of epithelial cell flow observed in experiments, i.e., whether it is apical- or basal-force-driven. Our numerical simulations suggest that cells actively generate fluctuations and use them to regulate both epithelial integrity and plasticity during morphogenesis.

Tissue curvature and apicobasal mechanical tension imbalance instruct cancer morphogenesis.

Tubular epithelia are a basic building block of organs and a common site of cancer occurrence1-4. During tumorigenesis, transformed cells overproliferate and epithelial architecture is disrupted. However, the biophysical parameters that underlie the adoption of abnormal tumour tissue shapes are unknown. Here we show in the pancreas of mice that the morphology of epithelial tumours is determined by the interplay of cytoskeletal changes in transformed cells and the existing tubular geometry. To analyse the morphological changes in tissue architecture during the initiation of cancer, we developed a three-dimensional whole-organ imaging technique that enables tissue analysis at single-cell resolution. Oncogenic transformation of pancreatic ducts led to two types of neoplastic growth: exophytic lesions that expanded outwards from the duct and endophytic lesions that grew inwards to the ductal lumen. Myosin activity was higher apically than basally in wild-type cells, but upon transformation this gradient was lost in both lesion types. Three-dimensional vertex model simulations and a continuum theory of epithelial mechanics, which incorporate the cytoskeletal changes observed in transformed cells, indicated that the diameter of the source epithelium instructs the morphology of growing tumours. Three-dimensional imaging revealed that-consistent with theory predictions-small pancreatic ducts produced exophytic growth, whereas large ducts deformed endophytically. Similar patterns of lesion growth were observed in tubular epithelia of the liver and lung; this finding identifies tension imbalance andtissue curvature as fundamental determinants of epithelial tumorigenesis.

Versatile three-dimensional vertex model: mechanical feedback system in morphogenesis

Versatile three-dimensional vertex model: mechanical feedback system in morphogenesis

Versatile three-dimensional vertex model

Versatile three-dimensional vertex model

Chiral cell sliding drives left-right asymmetric organ twisting.

Polarized epithelial morphogenesis is an essential process in animal development. While this process is mostly attributed to directional cell intercalation, it can also be induced by other mechanisms. Using live-imaging analysis and a three-dimensional vertex model, we identified 'cell sliding,' a novel mechanism driving epithelial morphogenesis, in which cells directionally change their position relative to their subjacent (posterior) neighbors by sliding in one direction. In Drosophila embryonic hindgut, an initial left-right (LR) asymmetry of the cell shape (cell chirality in three dimensions), which occurs intrinsically before tissue deformation, is converted through LR asymmetric cell sliding into a directional axial twisting of the epithelial tube. In a Drosophila inversion mutant showing inverted cell chirality and hindgut rotation, cell sliding occurs in the opposite direction to that in wild-type. Unlike directional cell intercalation, cell sliding does not require junctional remodeling. Cell sliding may also be involved in other cases of LR-polarized epithelial morphogenesis.

Topological graph description of multicellular dynamics based on vertex model

Vertex models are generally powerful tools for exploring biological insights into multicellular dynamics. In these models, a multicellular structure is represented by a network, which is dynamically rearranged using topological operations. Remarkably, the topological dynamics of the network are important in guaranteeing the results from the models and their biological implications. However, it remains unclear whether the entire pattern of multicellular topological dynamics can be accurately expressed by a set of operators in the models. Surprisingly, vertex models have been empirically used for several decades without any mathematical verification. In this study, we propose a rigorous two-/three-dimensional (2D/3D) vertex model to describe multicellular topological dynamics. To do this, we classify several types of vertex models from a graph-theoretic perspective. Based on the classification, mathematical analyses reveal several conditions that enable us to apply the operators accurately without topological errors. Under these conditions, the operators can completely express the entire pattern of multicellular topological dynamics. From these results, we newly propose rigorous 2D/3D vertex models that can be applied to general multicellular dynamics, and we clarify several points to verify the results obtained from previous models.

実器官の多細胞ダイナミクスを解析する汎用三次元バーテックスモデルの開発

実器官の多細胞ダイナミクスを解析する汎用三次元バーテックスモデルの開発

Mechanical roles of apical constriction, cell elongation, and cell migration during neural tube formation in Xenopus.

Neural tube closure is an important and necessary process during the development of the central nervous system. The formation of the neural tube structure from a flat sheet of neural epithelium requires several cell morphogenetic events and tissue dynamics to account for the mechanics of tissue deformation. Cell elongation changes cuboidal cells into columnar cells, and apical constriction then causes them to adopt apically narrow, wedge-like shapes. In addition, the neural plate in Xenopus is stratified, and the non-neural cells in the deep layer (deep cells) pull the overlying superficial cells, eventually bringing the two layers of cells to the midline. Thus, neural tube closure appears to be a complex event in which these three physical events are considered to play key mechanical roles. To test whether these three physical events are mechanically sufficient to drive neural tube formation, we employed a three-dimensional vertex model and used it to simulate the process of neural tube closure. The results suggest that apical constriction cued the bending of the neural plate by pursing the circumference of the apical surface of the neural cells. Neural cell elongation in concert with apical constriction further narrowed the apical surface of the cells and drove the rapid folding of the neural plate, but was insufficient for complete neural tube closure. Migration of the deep cells provided the additional tissue deformation necessary for closure. To validate the model, apical constriction and cell elongation were inhibited in Xenopus laevis embryos. The resulting cell and tissue shapes resembled the corresponding simulation results.Electronic supplementary materialThe online version of this article (doi:10.1007/s10237-016-0794-1) contains supplementary material, which is available to authorized users.

Three-Dimensional Vertex Simulation on Smooth Surface Maintenance of Growing Epithelial Tissue Based on Intercellular Mechano-Feedback

Shape of epithelial tissue is appropriately accomplished by a series of tissue deformation during morphogenesis. Therefore, driving forces of tissue deformation can be controlled not only by gene expressions but also by tissue shape at any instant. In this study we focused on the mechanosensitivity of alpha-catenin, an intercellular adhesion component, and suggested a mechano-feedback relationship between apical constriction and cell shape in which apical constriction induces a cell shape change, while the shape change in surrounding cells simultaneously modifies the apical constriction of a cell. Using a three-dimensional vertex model framework, we constructed a mathematical model expressing the feedback relationship in which the modification of the apical constriction force at the cell is achieved based on the difference in the circumferential length of its surrounding cells. We performed simulations on a growing spheroid-like tissue with and without mechano-feedback, in which cell proliferation occurs in a monolayer sheet of a spherical shell. Simulation results revealed that the apical surface of the growing tissue with mechano-feedback was a smooth spherical shape, whereas the apical surface without mechano-feedback was not smooth spherical. These results resembles the experimental results of spheroid formation in which a spherical tissue shape was formed in spheroids with wild-type alpha-catenin and not in spheroids with mutant alpha-catenin that lacked mechano-sensitivity. In the simulations, we found that the apical constriction force approached a certain range over time by mechano-feedback. Further analysis suggested that continuous modifications through mechano-feedback are necessary for maintaining a smooth apical surface by suppressing cells from becoming apically very narrow wedge-like shapes that contribute to disturbing the smoothness of the tissue surface. Consequently, we suggest that mechano-feedback plays a role in maintaining a smooth apical surface of the growing tissue.

Three-dimensional vertex model for simulating multicellular morphogenesis.

During morphogenesis, various cellular activities are spatiotemporally coordinated on the protein regulatory background to construct the complicated, three-dimensional (3D) structures of organs. Computational simulations using 3D vertex models have been the focus of efforts to approach the mechanisms underlying 3D multicellular constructions, such as dynamics of the 3D monolayer or multilayer cell sheet like epithelia as well as the 3D compacted cell aggregate, including dynamic changes in layer structures. 3D vertex models enable the quantitative simulation of multicellular morphogenesis on the basis of single-cell mechanics, with complete control of various cellular activities such as cell contraction, growth, rearrangement, division, and death. This review describes the general use of the 3D vertex model, along with its applications to several simplified problems of developmental phenomena.