Two different mechanisms of planar cell intercalation leading to tissue elongation

Strain softening and hysteresis arising from 3D multicellular dynamics during long-term large deformation

Living tissues exhibit complex mechanical properties, including viscoelastic and elastoplastic responses, that are crucial for regulating cell behaviors and tissue deformations. Despite their significance, the intricate properties of three-dimensional (3D) cell constructs are not well understood and are inadequately implemented in biomaterial engineering. To address this gap, we developed a numerical method to analyze the dynamic properties of cell constructs using a 3D vertex model framework. By focusing on 3D tissues composed of confluent homogeneous cells, we characterized their properties in response to various deformation magnitudes and time scales. Stress relaxation tests revealed that large deformations initially induced relaxation in the shapes of individual cells. This process is amplified by subsequent transient cell rearrangements, homogenizing cell shapes and leading to tissue fluidization. Additionally, dynamic viscoelastic analyses showed that tissues exhibited strain softening and hysteresis during large deformations. Interestingly, this strain softening originates from multicellular structures independent of cell rearrangement, while hysteresis arises from cell rearrangement. Moreover, tissues exhibit elastoplastic responses over the long term, which are well represented by the Ramberg-Osgood model. These findings highlight the characteristic properties of cell constructs emerging from their structures and rearrangements, especially during long-term large deformations. The developed method offers a new approach to uncover the dynamic nature of 3D tissue mechanics and could serve as a technical foundation for exploring tissue mechanics and advancing biomaterial engineering.

Vertex models capturing subcellular scales in epithelial tissues

Vertex models provide a robust theoretical framework for studying epithelial tissues as a network of cell boundaries. They have been pivotal in exploring properties such as cell packing geometry and rigidity transitions. Recently, extended vertex models have become instrumental in bridging the subcellular scales to the tissue scale. Here, we review extensions of the model aiming to capture experimentally observed subcellular features of epithelial tissues including heterogeneity in myosin activity across the tissue, non-uniform contractility structures, and mechanosensitive feedback loops. We discuss how these extensions change and challenge current perspectives on observables of macroscopic tissue properties. First, we find that extensions to the vertex model can change model properties significantly, impacting the critical threshold and in some cases even the existence of a rigidity transition. Second, we find that packing disorder can be explained by models employing different subcellular mechanisms, indicating a source of stochasticity and gradual local size changes as common mesoscopic motifs in the mechanics of tissue organization. We address complementary models and statistical inference, putting vertex models in a broader methodological context and we give a brief overview of software packages utilized in increasingly complex vertex model studies. Our review emphasizes the need for more comparative, systematic studies that identify specific classes of vertex models which share a set of well-defined properties, as well as a more in-depth discussion of modeling choices and their biological motivations.

Assessing mechanical agency during apical apoptotic cell extrusion

Homeostasis is necessary for epithelia to maintain barrier function and prevent the accumulation of defective cells. Unfit, excess, and dying cells in the larval zebrafish tail fin epidermis are removed via controlled cell death and extrusion. Extrusion coincides with oscillations of cell area, both in the extruding cell and its neighbors. Here, we develop a biophysical model of this process to explore the role of autonomous and non-autonomous mechanics. We vary biophysical properties and oscillatory behaviors of extruding cells and their neighbors along with tissue-wide cell density and viscosity. We find that cell autonomous processes are major contributors to the dynamics of extrusion, with the mechanical microenvironment providing a less pronounced contribution. We also find that some cells initially resist extrusion, influencing the duration of the expulsion process. Our model provides insights into the cellular dynamics and mechanics that promote elimination of unwanted cells from epithelia during homeostatic tissue maintenance.

Local and global changes in cell density induce reorganisation of 3D packing in a proliferating epithelium

Tissue morphogenesis is intimately linked to the changes in shape and organisation of individual cells. In curved epithelia, cells can intercalate along their own apicobasal axes, adopting a shape named 'scutoid' that allows energy minimization in the tissue. Although several geometric and biophysical factors have been associated with this 3D reorganisation, the dynamic changes underlying scutoid formation in 3D epithelial packing remain poorly understood. Here, we use live imaging of the sea star embryo coupled with deep learning-based segmentation to dissect the relative contributions of cell density, tissue compaction and cell proliferation on epithelial architecture. We find that tissue compaction, which naturally occurs in the embryo, is necessary for the appearance of scutoids. Physical compression experiments identify cell density as the factor promoting scutoid formation at a global level. Finally, the comparison of the developing embryo with computational models indicates that the increase in the proportion of scutoids is directly associated with cell divisions. Our results suggest that apico-basal intercalations appearing immediately after mitosis may help accommodate the new cells within the tissue. We propose that proliferation in a compact epithelium induces 3D cell rearrangements during development.

Effective mechanical potential of cell-cell interaction in tissues harboring cavity and in cell sheet toward morphogenesis

Measuring mechanical forces of cell-cell interactions is important for studying morphogenesis in multicellular organisms. We previously reported an image-based statistical method for inferring effective mechanical potentials of pairwise cell-cell interactions by fitting cell tracking data with a theoretical model. However, whether this method is applicable to tissues with non-cellular components such as cavities remains elusive. Here we evaluated the applicability of the method to cavity-harboring tissues. Using synthetic data generated by simulations, we found that the effect of expanding cavities was added to the pregiven potentials used in the simulations, resulting in the inferred effective potentials having an additional repulsive component derived from the expanding cavities. Interestingly, simulations by using the effective potentials reproduced the cavity-harboring structures. Then, we applied our method to the mouse blastocysts, and found that the inferred effective potentials can reproduce the cavity-harboring structures. Pairwise potentials with additional repulsive components were also detected in two-dimensional cell sheets, by which curved sheets including tubes and cups were simulated. We conclude that our inference method is applicable to tissues harboring cavities and cell sheets, and the resultant effective potentials are useful to simulate the morphologies.

Collective cell dynamics and luminal fluid flow in the epididymis: A mechanobiological perspective

BACKGROUND

The mammalian epididymis is a specialized duct system that serves a critical role in sperm maturation and storage. Its distinctive, highly coiled tissue morphology provides a unique opportunity to investigate the link between form and function in reproductive biology. Although recent genetic studies have identified key genes and signaling pathways involved in the development and physiological functions of the epididymis, there has been limited discussion about the underlying dynamic and mechanical processes that govern these phenomena.

AIMS

In this review, we aim to address this gap by examining two key aspects of the epididymis across its developmental and physiological phases.

RESULTS AND DISCUSSION

First, we discuss how the complex morphology of the Wolffian/epididymal duct emerges through collective cell dynamics, including duct elongation, cell proliferation, and arrangement during embryonic development. Second, we highlight dynamic aspects of luminal fluid flow in the epididymis, essential for regulating the microenvironment for sperm maturation and motility, and discuss how this phenomenon emerges and interplays with epididymal epithelial cells.

CONCLUSION

This review not only aims to summarize current knowledge but also to provide a starting point for further exploration of mechanobiological aspects related to the cellular and extracellular fluid dynamics in the epididymis.

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.

Local and global changes in cell density induce reorganisation of 3D packing in a proliferating epithelium

Tissue morphogenesis is intimately linked to the changes in shape and organisation of individual cells. In curved epithelia, cells can intercalate along their own apicobasal axes adopting a shape named "scutoid" that allows energy minimization in the tissue. Although several geometric and biophysical factors have been associated with this 3D reorganisation, the dynamic changes underlying scutoid formation in 3D epithelial packing remain poorly understood. Here we use live-imaging of the sea star embryo coupled with deep learning-based segmentation, to dissect the relative contributions of cell density, tissue compaction, and cell proliferation on epithelial architecture. We find that tissue compaction, which naturally occurs in the embryo, is necessary for the appearance of scutoids. Physical compression experiments identify cell density as the factor promoting scutoid formation at a global level. Finally, the comparison of the developing embryo with computational models indicates that the increase in the proportion of scutoids is directly associated with cell divisions. Our results suggest that apico-basal intercalations appearing just after mitosis may help accommodate the new cells within the tissue. We propose that proliferation in a compact epithelium induces 3D cell rearrangements during development.

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.

Curvature-Induced Cell Rearrangements in Biological Tissues

On a curved surface, epithelial cells can adapt to geometric constraints by tilting and by exchanging their neighbors from apical to basal sides, known as an apico-basal topological transition 1 (AB-T1). The relationship between cell tilt, AB-T1s, and tissue curvature still lacks a unified understanding. Here, we propose a general framework for cell packing in curved environments and explain the formation of AB-T1s from the perspective of strain anisotropy. We find that steep curvature gradients can lead to cell tilting and induce AB-T1s. Alternatively, pressure differences across the epithelial tissue can drive AB-T1s in regions of large curvature anisotropy. The two mechanisms compete to determine the impact of tissue geometry and mechanics on optimized cell rearrangements in three dimensions.

Computational approaches for simulating luminogenesis

Lumens, liquid-filled cavities surrounded by polarized tissue cells, are elementary units involved in the morphogenesis of organs. Theoretical modeling and computations, which can integrate various factors involved in biophysics of morphogenesis of cell assembly and lumens, may play significant roles to elucidate the mechanisms in formation of such complex tissue with lumens. However, up to present, it has not been documented well what computational approaches or frameworks can be applied for this purpose and how we can choose the appropriate approach for each problem. In this review, we report some typical lumen morphologies and basic mechanisms for the development of lumens, focusing on three keywords - mechanics, hydraulics and geometry - while outlining pros and cons of the current main computational strategies. We also describe brief guidance of readouts, i.e., what we should measure in experiments to make the comparison with the model's assumptions and predictions.

Differential Cellular Stiffness Contributes to Tissue Elongation on an Expanding Surface

Pattern formation and morphogenesis of cell populations is essential for successful embryogenesis. Steinberg proposed the differential adhesion hypothesis, and differences in cell–cell adhesion and interfacial tension have proven to be critical for cell sorting. Standard theoretical models such as the vertex model consider not only cell–cell adhesion/tension but also area elasticity of apical cell surfaces and viscous friction forces. However, the potential contributions of the latter two parameters to pattern formation and morphogenesis remain to be determined. In this theoretical study, we analyzed the effect of both area elasticity and the coefficient of friction on pattern formation and morphogenesis. We assumed the presence of two cell populations, one population of which is surrounded by the other. Both populations were placed on the surface of a uniformly expanding environment analogous to growing embryos, in which friction forces are exerted between cell populations and their expanding environment. When the area elasticity or friction coefficient in the cell cluster was increased relative to that of the surrounding cell population, the cell cluster was elongated. In comparison with experimental observations, elongation of the notochord in mice is consistent with the hypothesis based on the difference in area elasticity but not the difference in friction coefficient. Because area elasticity is an index of cellular stiffness, we propose that differential cellular stiffness may contribute to tissue elongation within an expanding environment.

Programming cell growth into different cluster shapes using diffusible signals

Advances in genetic engineering technologies have allowed the construction of artificial genetic circuits, which have been used to generate spatial patterns of differential gene expression. However, the question of how cells can be programmed, and how complex the rules need to be, to achieve a desired tissue morphology has received less attention. Here, we address these questions by developing a mathematical model to study how cells can collectively grow into clusters with different structural morphologies by secreting diffusible signals that can influence cellular growth rates. We formulate how growth regulators can be used to control the formation of cellular protrusions and how the range of achievable structures scales with the number of distinct signals. We show that a single growth inhibitor is insufficient for the formation of multiple protrusions but may be achieved with multiple growth inhibitors, and that other types of signals can regulate the shape of protrusion tips. These examples illustrate how our approach could potentially be used to guide the design of regulatory circuits for achieving a desired target structure.

Bubble packing, eccentricity, and notochord development

This paper develops a theoretical basis for the observed relationship between cell arrangements in notochords and analog physical models, and the eccentricity of their cross sections. Three models are developed and analyzed, of the mechanics of cell packing in sheaths. The key ratios governing the packing patterns and eccentricity are cells per unit length λ, tension ratio Γ, and eccentricity e. For flexible and semi-flexible sheaths, the optimal packing pattern shifts from "bamboo", with a symmetric cross section, to "staircase", with an eccentric cross section, at a critical value λ = 1.13. In rigid tubes, this threshold is lowered as imposed eccentricity is increased. Patterns can be observed which are not optimal; pattern transitions may occur below or above the critical λ values. The eccentricity of staircase patterns in flexible and semi-flexible tubes is found to be dependent on the tension ratio Γ, increasing as sheath tension decreases relative to interior cell tension. A novel "serpentine" packing pattern appears for low Γ near the critical λ. The developmental utility of enforcing notochord eccentricity is discussed, as well as potential mechanisms for such control.

Left-handed cardiac looping by cell chirality is mediated by position-specific convergent extensions

In the embryonic heart development of mammals and birds, a straight initial heart tube undergoes left-handed helical looping, which is a remarkable and puzzling event. We are interested in the mechanism of this chiral helical looping. Recently, observations were reported that myocardial cells in the embryonic chick heart show intrinsic chirality of rotation. The chirality of myocardial cells, via anisotropic polarization of Golgi inside the cells, leads to a left-right (LR) asymmetry of cell shape. On cell boundaries of LR asymmetric cells, phosphorylated myosin and N-cadherin are enriched. Such LR asymmetric cellular circumstances lead to a large-scale three-dimensional chiral structure, the left-handed helical loop. However, the physical mechanism of this looping is unclear. Computer simulations were performed using a cell-based three-dimensional mathematical model assuming an anterior-rightward-biased contractile force of the cell boundaries on the ventral surface of the heart (orientation of a clock hand pointing to 10 to 11 o'clock). An initially straight heart tube was successfully remodeled to the left-handed helical tube via frequent convergent extension (CE) of collective cells, which corresponds to the previously reported observations of chick heart development. Although we assumed that the biased boundary contractile force was uniform all over the ventral side, orientations of the CEs became position specific on the anterior, posterior, right, and left regions on the ventral tube. Such position-specific CEs produced the left-handed helical loop. In addition, our results suggest the loop formation process consists of two distinct phases of preparation and explicit looping. Intrinsic cell properties of chirality in this investigation were discussed relating to extrinsic factors investigated by other researches. Finally, because CE is generally exerted in the axial developmental process across different animal species, we discussed the contribution of CE to the chiral heart structure across species of chick, mouse, Xenopus, and zebrafish.

Dissecting Organismal Morphogenesis by Bridging Genetics and Biophysics

Multicellular organisms develop complex shapes from much simpler, single-celled zygotes through a process commonly called morphogenesis. Morphogenesis involves an interplay between several factors, ranging from the gene regulatory networks determining cell fate and differentiation to the mechanical processes underlying cell and tissue shape changes. Thus, the study of morphogenesis has historically been based on multidisciplinary approaches at the interface of biology with physics and mathematics. Recent technological advances have further improved our ability to study morphogenesis by bridging the gap between the genetic and biophysical factors through the development of new tools for visualizing, analyzing, and perturbing these factors and their biochemical intermediaries. Here, we review how a combination of genetic, microscopic, biophysical, and biochemical approaches has aided our attempts to understand morphogenesis and discuss potential approaches that may be beneficial to such an inquiry in the future. Expected final online publication date for the Annual Review of Genetics, Volume 55 is November 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

The complex three-dimensional organization of epithelial tissues

Understanding the cellular organization of tissues is key to developmental biology. In order to deal with this complex problem, researchers have taken advantage of reductionist approaches to reveal fundamental morphogenetic mechanisms and quantitative laws. For epithelia, their two-dimensional representation as polygonal tessellations has proved successful for understanding tissue organization. Yet, epithelial tissues bend and fold to shape organs in three dimensions. In this context, epithelial cells are too often simplified as prismatic blocks with a limited plasticity. However, there is increasing evidence that a realistic approach, even from a reductionist perspective, must include apico-basal intercalations (i.e. scutoidal cell shapes) for explaining epithelial organization convincingly. Here, we present an historical perspective about the tissue organization problem. Specifically, we analyze past and recent breakthroughs, and discuss how and why simplified, but realistic, in silico models require scutoidal features to address key morphogenetic events.

Multicellular contractility contributes to the emergence of mesothelioma nodules

Malignant pleural mesothelioma (MPM) has an overall poor prognosis and unsatisfactory treatment options. MPM nodules, protruding into the pleural cavity may have growth and spreading dynamics distinct that of other solid tumors. We demonstrate that multicellular aggregates can develop spontaneously in the majority of tested MPM cell lines when cultured at high cell density. Surprisingly, the nodule-like aggregates do not arise by excessive local cell proliferation, but by myosin II-driven cell contractility. Prominent actin cables, spanning several cells, are abundant both in cultured aggregates and in MPM surgical specimens. We propose a computational model for in vitro MPM nodule development. Such a self-tensioned Maxwell fluid exhibits a pattern-forming instability that was studied by analytical tools and computer simulations. Altogether, our findings may underline a rational for targeting the actomyosin system in MPM.

Isotropic expansion of external environment induces tissue elongation and collective cell alignment

Cell movement is crucial for morphogenesis in multicellular organisms. Growing embryos or tissues often expand isotropically, i.e., uniformly, in all dimensions. On the surfaces of these expanding environments, which we call "fields," cells are subjected to frictional forces and move passively in response. However, the potential roles of isotropically expanding fields in morphogenetic events have not been investigated well. Our previous mathematical simulations showed that a tissue was elongated on an isotropically expanding field (Imuta et al., 2014). However, the underlying mechanism remains unclarified, and how cells behave during tissue elongation was not investigated. In this study, we mathematically analyzed the effect of isotropically expanding fields using a vertex model, a standard type of multi-cellular model. We found that cells located on fields were elongated along a similar direction each other and exhibited a columnar configuration with nearly single-cell width. Simultaneously, it was confirmed that the cell clusters were also elongated, even though field expansion was absolutely isotropic. We then investigated the mechanism underlying these counterintuitive phenomena. In particular, we asked whether the dynamics of elongation was predominantly determined by the properties of the field, the cell cluster, or both. Theoretical analyses involving simplification of the model revealed that cell clusters have an intrinsic ability to asymmetrically deform, leading to their elongation. Importantly, this ability is effective only under the non-equilibrium conditions provided by field expansion. This may explain the elongation of the notochord, located on the surface of the growing mouse embryo. We established the mechanism underlying tissue elongation induced by isotropically expanding external environments, and its involvement in collective cell alignment with cell elongation, providing key insight into morphogenesis involving multiple adjacent tissues.

Self-sustained planar intercalations due to mechanosignaling feedbacks lead to robust axis extension during morphogenesis

Tissue elongation is a necessary process in metazoans to implement their body plans that is not fully understood. Here we propose a mechanism based on the interplay between cellular mechanics and primordia patterning that results in self-sustained planar intercalations. Thus, we show that a location-dependent modulation of the mechanical properties of cells leads to robust axis extension. To illustrate the plausibility of this mechanism, we test it against different patterning models by means of computer simulations of tissues where we implemented mechano-signaling feedbacks. Our results suggest that robust elongation relies on a trade-off between cellular and tissue strains that is orchestrated through the cleavage orientation. In the particular context of axis extension in Turing-patterned tissues, we report that different directional cell activities cooperate synergetically to achieve elongation. Altogether, our findings help to understand how the axis extension phenomenon emerges from the dynamics of individual cells.

The Chiral Looping of the Embryonic Heart Is Formed by the Combination of Three Axial Asymmetries

In mammals and birds, embryonic development of the heart involves conversion of a straight tubular structure into a three-dimensional helical loop, which is a chiral structure. We investigated theoretically the mechanism of helical loop formation of the mouse embryonic heart, especially focusing on determination of left-/right-handedness of the helical loop. In geometrical terms, chirality is the result of the combination of three axial asymmetries in three-dimensional space. We hypothesized the following correspondences between axial asymmetries and morphogenesis (bending and displacement): the dorsal-ventral asymmetry by ventral bending of a straight tube of the initial heart and the left-right and anterior-posterior asymmetries, the left-right asymmetry by rightward displacement of the heart tube, which is confined to the anterior region of the tube. Morphogenesis of chiral looping of the embryonic heart is a large-scaled event of the multicellular system in which substantial physical force operates dynamically. Using computer simulations with a cell-based physico-mechanical model and experiments with mouse embryos, we confirmed the hypothesis. We conclude that rightward displacement of the tube determines the left-handed screw of the loop. The process of helix loop formation consists of three steps: 1) the left-right biasing system involving Nodal-related signals that leads to left-right asymmetry in the embryonic body; 2) the rightward displacement of the tube; and finally 3) the left-handed helical looping. Step 1 is already established. Step 3 is elucidated by our study, which highlights the need for step 2 to be clarified; namely, we explore how the left-right asymmetry in the embryonic body leads to the rightward displacement of the heart tube.

Viscoelastic voyages - Biophysical perspectives on cell intercalation during Drosophila gastrulation

Developmental processes are driven by a combination of cytoplasmic, cortical, and surface-associated forces. However, teasing apart the contributions of these forces and how a viscoelastic cell responds has long been a key question in developmental biology. Recent advances in applying biophysical approaches to these questions is leading to a fundamentally new understanding of morphogenesis. In this review, we discuss how computational analysis of experimental findings and in silico modeling of Drosophila gastrulation processes has led to a deeper comprehension of the physical principles at work in the early embryo. We also summarize many of the emerging methodologies that permit biophysical analysis as well as those that provide direct and indirect measurements of force directions and magnitudes. Finally, we examine the multiple frameworks that have been used to model tissue and cellular behaviors.

Biomechanical modeling of invasive breast carcinoma under a dynamic change in cell phenotype: collective migration of large groups of cells

According to recent studies, cancer is an evolving complex ecosystem. It means that tumor cells are well differentiated and involved in heterotypic interactions with their microenvironment competing for available resources to proliferate and survive. In this paper, we propose a chemo-mechanical model for the growth of specific subtypes of an invasive breast carcinoma. The model suggests that a carcinoma is a heterogeneous entity comprising cells of different phenotypes, which perform different functions in a tumor. Every cell is represented by an elastic polygon changing its form and size under pressure from the tissue. The mechanical model is based on the elastic potential energy of the tissue including the effects of contractile forces within the cell perimeter and the elastic resistance to stretching or compressing the cell with respect to the reference area. A tissue can evolve via mechanisms of cell division and intercalation. The phenotype of each cell is determined by its environment and can dynamically change via an epithelial-mesenchymal transition and vice versa. The phenotype defines the cell adhesion to the adjacent tissue and the ability to divide. In this part, we focus on the forms of collective migration of large groups of cells. Numerical simulations show the different architectural subtypes of invasive carcinoma. For each communication, we examine the dynamics of the cell population and evaluate the complexity of the pattern in terms of the synergistic paradigm. The patterns are compared with the morphological structures previously identified in clinical studies.

The same but different: cell intercalation as a driver of tissue deformation and fluidity

The ability of cells to exchange neighbours, termed intercalation, is a key feature of epithelial tissues. Intercalation is predominantly associated with tissue deformations that drive morphogenesis. More recently, however, intercalation that is not associated with large-scale tissue deformations has been described both during animal development and in mature epithelial tissues. This latter form of intercalation appears to contribute to an emerging phenomenon that we refer to as tissue fluidity-the ability of cells to exchange neighbours without changing the overall dimensions of the tissue. Here, we discuss the contribution of junctional dynamics to intercalation governing both morphogenesis and tissue fluidity. In particular, we focus on the relative roles of junctional contractility and cell-cell adhesion as the driving forces behind intercalation. These two contributors to junctional mechanics can be used to simulate cellular intercalation in mechanical computational models, to test how junctional cell behaviours might regulate tissue fluidity and contribute to the maintenance of tissue integrity and the onset of disease.This article is part of the Theo Murphy meeting issue 'Mechanics of development'.

Scutoids are a geometrical solution to three-dimensional packing of epithelia

As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modelling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbours in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term "scutoid". The detailed analysis of diverse tissues confirms that generation of apico-basal intercalations between cells is a common feature during morphogenesis. Using biophysical arguments, we propose that scutoids make possible the minimization of the tissue energy and stabilize three-dimensional packing. Hence, we conclude that scutoids are one of nature's solutions to achieve epithelial bending. Our findings pave the way to understand the three-dimensional organization of epithelial organs.

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.

Many organs arise from simple sheets and tubes of cells. During development these sheets bend and deform into the more complex shape of the final organ. This can be seen, for example, in the hindgut of fruit flies, which is an organ that is equivalent to our intestines. Initially, the hindgut is a simple tube of cells. Later the hindgut develops a twist to the left that renders its right and left sides non-symmetrical. During twisting, the cells in the hindgut also change shape. It was not known how this shape change and other behaviors of the cells cause the hindgut to twist.

Inaki et al. have now filmed how the hindgut develops in live fruit flies and produced computer simulations of the development process. The results suggest that a previously unidentified type of cell behavior called ‘cell sliding’ is responsible for twisting the hindgut. During sliding, the cells stay in contact with their neighbors as they move in a single direction. Sliding is triggered by the cells in the hindgut taking on a more symmetrical shape.

Cell sliding may prove to be a common way to shape organs, many of which feature non-symmetrical twisted tubes of cells. In the future, learning how to control cell sliding could help researchers to create organs and biological structures in the laboratory that could be used in organ transplants and regenerative medicine.

Simulation of Cell Patterning Triggered by Cell Death and Differential Adhesion in Drosophila Wing

The Drosophila wing exhibits a well-ordered cell pattern, especially along the posterior margin, where hair cells are arranged in a zigzag pattern in the lateral view. Based on an experimental result observed during metamorphosis of Drosophila, we considered that a pattern of initial cells autonomously develops to the zigzag pattern through cell differentiation, intercellular communication, and cell death (apoptosis) and performed computer simulations of a cell-based model of vertex dynamics for tissues. The model describes the epithelial tissue as a monolayer cell sheet of polyhedral cells. Their vertices move according to equations of motion, minimizing the sum total of the interfacial and elastic energies of cells. The interfacial energy densities between cells are introduced consistently with an ideal zigzag cell pattern, extracted from the experimental result. The apoptosis of cells is modeled by gradually reducing their equilibrium volume to zero and by assuming that the hair cells prohibit neighboring cells from undergoing apoptosis. Based on experimental observations, we also assumed wing elongation along the proximal-distal axis. Starting with an initial cell pattern similar to the micrograph experimentally obtained just before apoptosis, we carried out the simulations according to the model mentioned above and successfully reproduced the ideal zigzag cell pattern. This elucidates a physical mechanism of patterning triggered by cell apoptosis theoretically and exemplifies, to our knowledge, a new framework to study apoptosis-induced patterning. We conclude that the zigzag cell pattern is formed by an autonomous communicative process among the participant cells.

Regulation of gastrulation movements by emergent cell and tissue interactions

It is during gastrulation that the primordial germ layers are specified, embryonic axes become morphologically manifest, and the embryonic body plan begins to take shape. As morphogenetic movements push and pull nascent tissues into position within the gastrula, new interactions are established between neighboring cells and tissues. These interactions represent an emergent property within gastrulating embryos, and serve to regulate and promote ensuing morphogenesis that establishes the next set of cell/tissue contacts, and so on. Several recent studies demonstrate the critical roles of such interactions during gastrulation, including those between germ layers, along embryonic axes, and at tissue boundaries. Emergent tissue interactions result from - and result in - morphogen signaling, cell contacts, and mechanical forces within the gastrula. Together, these comprise a dynamic and complex regulatory cascade that drives gastrulation morphogenesis.

Mechanical role of the spatial patterns of contractile cells in invagination of growing epithelial tissue

Epithelial invagination is one of the fundamental deformation modes during morphogenesis, and is essential for deriving the three-dimensional shapes of organs from a flat epithelial sheet. Invagination occurs in an orderly manner according to the spatial pattern of the contractile cells; however, it remains elusive how tissue deformation can be caused by cellular activity in the patterned region. In this study, we investigated the mechanical role of the spatial patterns of the contractile cells in invagination of growing tissue using multicellular dynamics simulations. We found that cell proliferation and apical constriction were responsible for expanding the degree of tissue deformation and determining the location of the deformation, respectively. The direction of invagination depended on the spatial pattern of the contractile cells. Further, comparing the simulation results of surface and line contractions as possible modes of apical constriction, we found that the direction of invagination differed between these two modes even if the spatial pattern was the same. These results indicate that the buckling of the epithelial cell sheet caused by cell proliferation causes the invagination, with the direction and location determined by the configuration of the wedge-shaped cells given by the spatial pattern of the contractile cells.

Vertex models: from cell mechanics to tissue morphogenesis

Tissue morphogenesis requires the collective, coordinated motion and deformation of a large number of cells. Vertex model simulations for tissue mechanics have been developed to bridge the scales between force generation at the cellular level and tissue deformation and flows. We review here various formulations of vertex models that have been proposed for describing tissues in two and three dimensions. We discuss a generic formulation using a virtual work differential, and we review applications of vertex models to biological morphogenetic processes. We also highlight recent efforts to obtain continuum theories of tissue mechanics, which are effective, coarse-grained descriptions of vertex models.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

The world of epithelial sheets

An epithelium is a layer of closely connected cells covering the body or lining a body cavity. In this review, several fundamental questions are addressed regarding the epithelium. (i) While an epithelium functions as barrier against the external environment, how is barrier function maintained during its construction? (ii) What determines the apical and basal sides of epithelial layer? (iii) Is there any relationship between the apical side of the epithelium and the apical membrane of an epithelial cell? (iv) Why are hepatocytes (liver cells) called epithelial, even though they differ completely from column-like shape of typical epithelial cells? Keeping these questions in mind, multiple shapes of epithelia were considered, extracting a few of their elemental processes, and constructing a virtual world of epithelia by combining them. Epithelial cells were also classified into several types based on the number of apical domains of each cell. In addition, an intracellular organelle was introduced within epithelial cells, the vacuolar apical compartment (VAC), which is produced within epithelial cells surrounded by external cell matrix (ECM). The VAC interacts with areas of cell-cell contact of the cell surface membrane and is converted to apical membrane. The properties of VACs enable us to answer the initial questions posed above. Finally, the genetic and molecular mechanisms of epithelial morphogenesis are discussed.

Modelling apical columnar epithelium mechanics from circumferential contractile fibres

Simple columnar epithelia are formed by individual epithelial cells connecting together to form single cell high sheets. They are a main component of many important body tissues and are heavily involved in both normal and cancerous cell activities. Prior experimental observations have identified a series of contractile fibres around the circumference of a cross section located in the upper (apical) region of each cell. While other potential mechanisms have been identified in both the experimental and theoretical literature, these circumferential fibres are considered to be the most likely mechanism controlling movement of this cross section. Here, we investigated the impact of circumferential contractile fibres on movement of the cross section by creating an alternate model where movement is driven from circumferential contractile fibres, without any other potential mechanisms. In this model, we utilised a circumferential contractile fibre representation based on investigations into the movement of contractile fibres as an individual system, treated circumferential fibres as a series of units, and matched our model simulation to experimental geometries. By testing against laser ablation datasets sourced from existing literature, we found that circumferential fibres can reproduce the majority of cross-sectional movements. We also investigated model predictions related to various aspects of cross-sectional movement, providing insights into epithelium mechanics and demonstrating the usefulness of our modelling approach.

A node-based version of the cellular Potts model

The cellular Potts model (CPM) is a lattice-based Monte Carlo method that uses an energetic formalism to describe the phenomenological mechanisms underlying the biophysical problem of interest. We here propose a CPM-derived framework that relies on a node-based representation of cell-scale elements. This feature has relevant consequences on the overall simulation environment. First, our model can be implemented on any given domain, provided a proper discretization (which can be regular or irregular, fixed or time evolving). Then, it allowed an explicit representation of cell membranes, whose displacements realistically result in cell movement. Finally, our node-based approach can be easily interfaced with continuous mechanics or fluid dynamics models. The proposed computational environment is here applied to some simple biological phenomena, such as cell sorting and chemotactic migration, also in order to achieve an analysis of the performance of the underlying algorithm. This work is finally equipped with a critical comparison between the advantages and disadvantages of our model with respect to the traditional CPM and to some similar vertex-based approaches.

Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium

Cellular rearrangements between olfactory cells and supporting cells, driven by the different expression and distribution of nectins and cadherins, are required for mosaic cellular patterning in the olfactory epithelium.

In the olfactory epithelium (OE), olfactory cells (OCs) and supporting cells (SCs), which express different cadherins, are arranged in a characteristic mosaic pattern in which OCs are enclosed by SCs. However, the mechanism underlying this cellular patterning is unclear. Here, we show that the cellular pattern of the OE is established by cellular rearrangements during development. In the OE, OCs express nectin-2 and N-cadherin, and SCs express nectin-2, nectin-3, E-cadherin, and N-cadherin. Heterophilic trans-interaction between nectin-2 on OCs and nectin-3 on SCs preferentially recruits cadherin via α-catenin to heterotypic junctions, and the differential distributions of cadherins between junctions promote cellular intercalations, resulting in the formation of the mosaic pattern. These observations are confirmed by model cell systems, and various cellular patterns are generated by the combinatorial expression of nectins and cadherins. Collectively, the synergistic action of nectins and cadherins generates mosaic pattern, which cannot be achieved by a single mechanism.

Filopodial-Tension Model of Convergent-Extension of Tissues

In convergent-extension (CE), a planar-polarized epithelial tissue elongates (extends) in-plane in one direction while shortening (converging) in the perpendicular in-plane direction, with the cells both elongating and intercalating along the converging axis. CE occurs during the development of most multicellular organisms. Current CE models assume cell or tissue asymmetry, but neglect the preferential filopodial activity along the convergent axis observed in many tissues. We propose a cell-based CE model based on asymmetric filopodial tension forces between cells and investigate how cell-level filopodial interactions drive tissue-level CE. The final tissue geometry depends on the balance between external rounding forces and cell-intercalation traction. Filopodial-tension CE is robust to relatively high levels of planar cell polarity misalignment and to the presence of non-active cells. Addition of a simple mechanical feedback between cells fully rescues and even improves CE of tissues with high levels of polarity misalignments. Our model extends easily to three dimensions, with either one converging and two extending axes, or two converging and one extending axes, producing distinct tissue morphologies, as observed in vivo.

The development of an embryo from a fertilized egg to an adult organism requires not only cell proliferation and differentiation, but also numerous types of tissue restructuring. The development of a relatively round initial embryo into one elongated along its rostral-caudal axis involves coordinated tissue elongation and cell reorganization in one or more groups of cells or tissues. Counterintuitively, in many organisms, cells in elongating tissues elongate and increase their protrusive activity in the direction perpendicular to the axis of elongation (convergent extension). Experimental and theoretical studies have not determined how this cell-level oriented protrusive activity leads to observed tissue-level changes in morphology. We propose a filopodial-tension model that shows how tension from oriented cell protrusions leads to observed patterns of tissue CE.

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.

Multiscale modeling of tumor growth induced by circadian rhythm disruption in epithelial tissue

We propose a multiscale chemo-mechanical model of cancer tumor development in epithelial tissue. The model is based on the transformation of normal cells into a cancerous state triggered by a local failure of spatial synchronization of the circadian rhythm. The model includes mechanical interactions and a chemical signal exchange between neighboring cells, as well as a division of cells and intercalation that allows for modification of the respective parameters following transformation into the cancerous state. The numerical simulations reproduce different dephasing patterns--spiral waves and quasistationary clustering, with the latter being conducive to cancer formation. Modification of mechanical properties reproduces a distinct behavior of invasive and localized carcinoma.

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.

Myosin-II-mediated cell shape changes and cell intercalation contribute to primitive streak formation

Primitive streak formation in the chick embryo involves large scale highly coordinated flows of over 100.000 cells in the epiblast. These large scale tissue flows and deformations can be correlated with specific anisotropic cell behaviours in the forming mesendoderm through a combined light-sheet microscopy and computational analysis. Relevant behaviours include apical contraction, elongation along the apical-basal axis followed by ingression as well as asynchronous directional cell intercalation of small groups of mesendoderm cells. Cell intercalation is associated with sequential, directional contraction of apical junctions, the onset, localisation and direction of which correlate strongly with the appearance of active Myosin II cables in aligned apical junctions in neighbouring cells. Use of a class specific Myosin inhibitors and gene specific knockdowns show that apical contraction and intercalation are Myosin II dependent and also reveal critical roles for Myosin I and Myosin V family members in the assembly of junctional Myosin II cables.

The biophysical nature of cells: potential cell behaviours revealed by analytical and computational studies of cell surface mechanics

BACKGROUND

The biophysical characteristics of cells determine their shape in isolation and when packed within tissues. Cells can form regular or irregular epithelial structures, round up and form clusters, or deform and attach to substrates. The acquired shape of cells and tissues is a consequence of (i) internal cytoskeletal processes, such as actin polymerisation and cortical myosin contraction, (ii) adhesion molecules within the cell membrane that interact with substrates and neighbouring cells, and (iii) processes that regulate cell volume. Although these processes seem relatively simple, when combined they unleash a rich variety of cellular behaviour that is not readily understandable outside a theoretical framework.

METHODS

We perform a mathematical analysis of a commonly used class of model formalisms that describe cell surface mechanics using an energy-based approach. Predictions are then confirmed through comparison with the computational outcomes of a Vertex model and 2D and 3D simulations of the Cellular Potts model.

RESULTS

The analytical study reveals the complete possible spectrum of single cell behaviour and tissue packing in both 2D and 3D, by taking the typical core elements of cell surface mechanics into account: adhesion, cortical tension and volume conservation. We show that from an energy-based description, forces and tensions can be derived, as well as the prediction of cell behaviour and tissue packing, providing an intuitive and biologically relevant mapping between modelling parameters and experiments.

CONCLUSIONS

The quantitative cellular behaviours and biological insights agree between the analytical study and the diverse computational model formalisms, including the Cellular Potts model. This illustrates the generality of energy-based approaches for cell surface mechanics and highlights how meaningful and quantitative comparisons between models can be established. Moreover, the mathematical analysis reveals direct links between known biophysical properties and specific parameter settings within the Cellular Potts model.

Coupling intercellular molecular signalling with multicellular deformation for simulating three-dimensional tissue morphogenesis

During morphogenesis, three-dimensional (3D) multicellular structures emerge from biochemical and mechanical interplays among cells. In particular, by organizing their gradient within tissues, the diffusible signalling molecules play an essential role in producing the spatio-temporal patterns of cell status such as the differentiation states. Notably, this biochemical patterning can be dynamically coupled with multicellular deformations by signal-dependent cell activities such as contraction, adhesion, migration, proliferation and apoptosis. However, the mechanism by which these cellular activities mediate the interactions between multicellular deformations and patterning is still unknown. Herein, we propose a novel framework of a 3D vertex model to express molecular signalling among the mechanically deforming cells. By specifying a density of signalling molecules for each cell, we express their transport between neighbouring cells. By simulating signal-dependent epithelial growth, we found various types of tissue morphogenesis such as arrest, expansion, invagination and evagination. In the expansion phase, growth molecules were widely diffused with increasing tissue volume, which diluted the growth molecules in order to support the autonomous suppression of tissue growth. These results indicate that the proposed model successfully expresses 3D multicellular deformations dynamically coupled with biochemical patterning. We expect our proposed model to be a useful tool for predicting new phenomena emerging from mechanochemical coupling in multicellular morphogenesis.

Regulation of tissue morphodynamics: an important role for actomyosin contractility

Forces arising from contractile actomyosin filaments help shape tissue form during morphogenesis. Developmental events that result from actomyosin contractility include tissue elongation, bending, budding, and collective migration. Here, we highlight recent insights into these morphogenetic processes from the perspective of actomyosin contractility as a key regulator. Emphasis is placed on a range of results obtained through live imaging, culture, and computational methods. Combining these approaches in the future has the potential to generate a robust, quantitative understanding of tissue morphodynamics.

Segment-specific adhesion as a driver of convergent extension

Convergent extension, the simultaneous extension and narrowing of tissues, is a crucial event in the formation of the main body axis during embryonic development. It involves processes on multiple scales: the sub-cellular, cellular and tissue level, which interact via explicit or intrinsic feedback mechanisms. Computational modelling studies play an important role in unravelling the multiscale feedbacks underlying convergent extension. Convergent extension usually operates in tissue which has been patterned or is currently being patterned into distinct domains of gene expression. How such tissue patterns are maintained during the large scale tissue movements of convergent extension has thus far not been investigated. Intriguingly, experimental data indicate that in certain cases these tissue patterns may drive convergent extension rather than requiring safeguarding against convergent extension. Here we use a 2D Cellular Potts Model (CPM) of a tissue prepatterned into segments, to show that convergent extension tends to disrupt this pre-existing segmental pattern. However, when cells preferentially adhere to cells of the same segment type, segment integrity is maintained without any reduction in tissue extension. Strikingly, we demonstrate that this segment-specific adhesion is by itself sufficient to drive convergent extension. Convergent extension is enhanced when we endow our in silico cells with persistence of motion, which in vivo would naturally follow from cytoskeletal dynamics. Finally, we extend our model to confirm the generality of our results. We demonstrate a similar effect of differential adhesion on convergent extension in tissues that can only extend in a single direction (as often occurs due to the inertia of the head region of the embryo), and in tissues prepatterned into a sequence of domains resulting in two opposing adhesive gradients, rather than alternating segments.

The process of convergent extension is a major contributor to the formation of the anterior-posterior body axis in the early embryo. Convergent extension refers to the directed movement of cells that leads to the extension of tissue in one direction and narrowing of the tissue in the perpendicular direction. Often, convergent extension occurs in tissue which already contains distinct domains of gene expression such as segments, and it is unclear how these patterns are maintained despite extensive cell movement. Interestingly, experimental evidence suggests that these tissue patterns may drive rather than be compromised by convergent extension. However, existing computational models aimed at unravelling the mechanisms of convergent extension have thus far only studied the process in homogeneous tissues. With our model, we demonstrate that in a segmented tissue, preferential adhesion of cells to other cells within the same segment type is required to maintain the tissue pattern during convergent extension. Furthermore, such segment-specific adhesion is by itself capable of driving convergent extension.

Vertex models of epithelial morphogenesis

The dynamic behavior of epithelial cell sheets plays a central role during numerous developmental processes. Genetic and imaging studies of epithelial morphogenesis in a wide range of organisms have led to increasingly detailed mechanisms of cell sheet dynamics. Computational models offer a useful means by which to investigate and test these mechanisms, and have played a key role in the study of cell-cell interactions. A variety of modeling approaches can be used to simulate the balance of forces within an epithelial sheet. Vertex models are a class of such models that consider cells as individual objects, approximated by two-dimensional polygons representing cellular interfaces, in which each vertex moves in response to forces due to growth, interfacial tension, and pressure within each cell. Vertex models are used to study cellular processes within epithelia, including cell motility, adhesion, mitosis, and delamination. This review summarizes how vertex models have been used to provide insight into developmental processes and highlights current challenges in this area, including progressing these models from two to three dimensions and developing new tools for model validation.

Cell models lead to understanding of multi-cellular morphogenesis consisting of successive self-construction of cells

Morphogenesis of multi-cellular organisms occurs through cell behaviours within a cell aggregate. Cell behaviours have been described using cell models involving equations of motion for cells. Cells in cell models construct shapes of the cell aggregate by themselves. Here, a history of cell models, the cell centre model and the vertex cell model, which we have constructed, are described. Furthermore, the application of these cell models is explained in detail. These cell models have been applied to transformation of cell aggregates to become spherical, formation of mammalian blastocysts and cell intercalation in elongating tissues. These are all elemental processes of morphogenesis and take place in succession during the whole developmental process. A chain of successive elemental processes leads to morphogenesis. Finally, we highlight that cell models are indispensable to understand the process whereby genes direct biological shapes.

Computational analysis of three-dimensional epithelial morphogenesis using vertex models

The folding of epithelial sheets, accompanied by cell shape changes and rearrangements, gives rise to three-dimensional structures during development. Recently, some aspects of epithelial morphogenesis have been modeled using vertex models, in which each cell is approximated by a polygon; however, these models have been largely confined to two dimensions. Here, we describe an adaptation of these models in which the classical two-dimensional vertex model is embedded in three dimensions. This modification allows for the construction of complex three-dimensional shapes from simple sheets of cells. We describe algorithmic, computational, and biophysical aspects of our model, with the view that it may be useful for formulating and testing hypotheses regarding the mechanical forces underlying a wide range of morphogenetic processes.

Vertex dynamics simulations of viscosity-dependent deformation during tissue morphogenesis

In biological development, multiple cells cooperate to form tissue morphologies based on their mechanical interactions; namely active force generation and passive viscoelastic response. In particular, the dynamic processes of tissue deformations are governed by the viscous properties of the tissues. These properties are spatially inhomogeneous because they depend on the tissue constituents, such as cytoplasm, cytoskeleton, basement membrane and extracellular matrix. The multicellular mechanics of tissue morphogenesis have been investigated in vertex dynamics models. However, conventional models are applicable only to quasi-static deformation processes, which do not account for tissue viscosities. We propose a vertex dynamics model that simulates the viscosity-dependent dynamic deformation processes during tissue morphogenesis. By incorporating local velocity fields into the governing equation of vertex movements, the model turns Galilean invariant. In addition, the viscous properties of tissue components are newly expressed by formulating friction forces on vertices as functions of the relative velocities among the vertices. The advantages of the proposed model are examined by epithelial growth simulations under the employed condition for quasi-static processes. As a result, the epithelial vesicle simulated by the proposed model is linearly elongated with nearly free stress, while that simulated by the conventional model is undulated with compressive residual stress. Therefore, the proposed model is able to reflect the timescale of deformations by satisfying Galilean invariance. Next, the applicability of the proposed model is assessed in epithelial growth simulations of viscous extracellular materials. In this test, the epithelial vesicles are deformed into tubular shapes by oriented cell divisions, and their morphologies are extremely sensitive to extracellular viscosity. Therefore, the dynamic deformations in the proposed model depend on the viscous properties of tissue components. The proposed model will be useful for simulating dynamic deformation processes of tissue morphogenesis depending on viscous properties of various tissue components.

Interplay between intercellular signaling and cell movement in development

Cell movement and local intercellular signaling are crucial components of morphogenesis during animal development. Intercellular signaling regulates the collective movement of a cell population via direct cell-cell contact. Cell movement, conversely, can influence local intercellular signaling by rearranging neighboring cells. Here, we first discuss theoretical models that address how intercellular signaling regulates collective cell movement during development. Examples include neural crest cell migration, convergent extension, and cell movement during vertebrate axis elongation. Second, we review theoretical studies on how cell movement may affect intercellular signaling, using the segmentation clock in zebrafish as an example. We propose that interplay between cell movement and intercellular signaling must be considered when studying morphogenesis in embryonic development.