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.

Cell position-based evaluation of mechanical features of cells in multicellular systems

Measurement of mechanical forces of cell-cell interactions is important for studying the emergence of diverse three-dimensional morphologies of multicellular organisms. We previously reported an image-based statistical method for inferring effective pairwise forces of cell-cell interactions (i.e., attractive/repulsive forces), where a cell particle model was fitted to cell tracking data acquired by live imaging. However, because the particle model is a coarse-grained model, it remains unclear how the pairwise forces relates to sub-cellular mechanical components including cell-cell adhesive forces. Here we applied our inference method to cell tracking data generated by vertex models that assumed sub-cellular components. Through this approach, we investigated the relationship between the effective pairwise forces and various sub-cellular components: cell-cell adhesion forces, cell surface tensions, cell-extracellular matrix (ECM) adhesion, traction forces between cells and ECM, cell growth, etc. We found that the cell-cell adhesion forces were attractive, and both the cell surface tensions and cell-ECM adhesive forces were repulsive, etc. These results indicate that sub-cellular mechanical components can contribute to the effective attractive/repulsive forces of cell-cell interactions. This comprehensive analysis provides theoretical bases for linking the pairwise forces to the sub-cellular mechanical components: this showcase is useful for speculating the sub-cellular mechanical components from the information of cell positions, and for interpreting simulation results based on particle models.

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.

Effects of blastocyst elongation and implantation chamber formation on the alignment of the embryonic axis and uterine axis in mice

Embryo implantation involves a series of events that bring the embryo and maternal tissues into contact to support post-implantation development in mammals. During implantation, alignment of the embryonic-abembryonic (E-Ab) axis of the blastocyst with the mesometrial-antimesometrial (M-AM) axis of the uterus precedes post-implantation embryonic development and placentation. In the present study, we observed the morphological changes in blastocysts and the endometrial luminal epithelium (LE) that occur during the alignment of the embryonic and the uterine axes. We found that at the time that the blastocysts attached to the LE at the mural trophectoderm, the embryonic axis was not aligned with the uterine axis. Alignment of the embryonic E-Ab axis with the uterine M-AM axis occurred after E4.0, and the embryo was significantly elongated during the process. The depth of the implantation chamber (IC) correlated with the degree of alignment, suggesting that elongated embryos are oriented along the M-AM axis during IC formation. Transplantation of the Concanavalin A (Con A)-coated beads induced IC formation, and the alignment of two Con A-coated beads present in the same IC in the M-AM direction suggested that elongated materials can align along the M-AM axis. These data suggest that an elongated shape of the embryo and IC formation coordinate the alignment of the embryonic and uterine axes.

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.

Shaping epithelial lumina under pressure

The formation of fluid- or gas-filled lumina surrounded by epithelial cells pervades development and disease. We review the balance between lumen pressure and mechanical forces from the surrounding cells that governs lumen formation. We illustrate the mechanical side of this balance in several examples of increasing complexity, and discuss how recent work is beginning to elucidate how nonlinear and active mechanics and anisotropic biomechanical structures must conspire to overcome the isotropy of pressure to form complex, non-spherical lumina.

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.

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

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

Biophysical models of early mammalian embryogenesis

Embryo development is a critical and fascinating stage in the life cycle of many organisms. Despite decades of research, the earliest stages of mammalian embryogenesis are still poorly understood, caused by a scarcity of high-resolution spatial and temporal data, the use of only a few model organisms, and a paucity of truly multidisciplinary approaches that combine biological research with biophysical modeling and computational simulation. Here, we explain the theoretical frameworks and biophysical processes that are best suited to modeling the early mammalian embryo, review a comprehensive list of previous models, and discuss the most promising avenues for future work.

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.

Virtual spherical-shaped multicellular platform for simulating the morphogenetic processes of spider-like body axis formation

Remodeling of multicellular architecture is a critical developmental process for shaping the axis of a bilaterally symmetric animal body and involves coordinated cell–cell interactions and cell rearrangement. In arthropods, the early embryonic process that leads to the segmented body axis varies at the cellular and molecular levels depending on the species. Developmental studies using insect and spider model species have provided specific examples of these diversified mechanisms that regulate axis formation and segmentation in arthropod embryos. However, there are few theoretical models for how diversity in the early embryonic process occurred during evolution, in part because of a limited computational infrastructure. We developed a virtual spherical-shaped multicellular platform to reproduce body axis-forming processes. Each virtual cell behaves according to the cell vertex model, with the computational program organized in a hierarchical order from cells and tissues to whole embryos. Using an initial set of two different mechanical states for cell differentiation and global directional signals that are linked to the planar polarity of each cell, the virtual cell assembly exhibited morphogenetic processes similar to those observed in spider embryos. We found that the development of an elongating body axis is achieved through implementation of an interactive cell polarity parameter associated with edge tension at the cell–cell adhesion interface, with no local control of the cell division rate and direction. We also showed that modifying the settings can cause variation in morphogenetic processes. This platform also can embed a gene network that generates waves of gene expression in a virtual dynamic multicellular field. This study provides a computational platform for testing the development and evolution of animal body patterns.

A theoretical understanding of mammalian preimplantation development

The blastocyst has long been a hallmark system of study in developmental biology due to its importance in mammalian development and clinical relevance for assisted reproductive technologies. In recent years, the blastocyst is emerging as a system of study for mathematical modelling. In this review, we compile, to our knowledge, all models describing preimplantation development. Coupled with experiments, these models have provided insight regarding the morphogenesis and cell-fate specification throughout preimplantation development. In the case of cell-fate specification, theoretical models have provided mechanisms explaining how proportion of cell types are established and maintained when confronted to perturbations. For cell-shape based models, they have described quantitatively how mechanical forces sculpt the blastocyst and even predicted how morphogenesis could be manipulated. As theoretical biology develops, we believe the next critical stage in modelling involves an integration of cell fate and mechanics to provide integrative models of development at distinct spatiotemporal scales. We discuss how, building on a balanced base of mechanical and chemical models, the preimplantation embryo will play a key role in integrating these two faces of the same coin.

Morphogenesis of the human preimplantation embryo: bringing mechanics to the clinics

During preimplantation development, the human embryo forms the blastocyst, the structure enabling uterine implantation. The blastocyst consists of an epithelial envelope, the trophectoderm, encompassing a fluid-filled lumen, the blastocoel, and a cluster of pluripotent stem cells, the inner cell mass. This specific architecture is crucial for the implantation and further development of the human embryo. Furthermore, the morphology of the human embryo is a prime determinant for clinicians to assess the implantation potential of in vitro fertilized human embryos, which constitutes a key aspect of assisted reproduction technology. Therefore, it is crucial to understand how the human embryo builds the blastocyst. As any material, the human embryo changes shape under the action of forces. Here, we review recent advances in our understanding of the mechanical forces shaping the blastocyst. We discuss the cellular processes responsible for generating morphogenetic forces that were studied mostly in the mouse and review the literature on human embryos to see which of them may be conserved. Based on the specific morphological defects commonly observed in clinics during human preimplantation development, we discuss how mechanical forces and their underlying cellular processes may be affected. Together, we propose that bringing tissue mechanics to the clinics will advance our understanding of human preimplantation development, as well as our ability to help infertile couples to have babies.

A multiscale model via single-cell transcriptomics reveals robust patterning mechanisms during early mammalian embryo development

During early mammalian embryo development, a small number of cells make robust fate decisions at particular spatial locations in a tight time window to form inner cell mass (ICM), and later epiblast (Epi) and primitive endoderm (PE). While recent single-cell transcriptomics data allows scrutinization of heterogeneity of individual cells, consistent spatial and temporal mechanisms the early embryo utilize to robustly form the Epi/PE layers from ICM remain elusive. Here we build a multiscale three-dimensional model for mammalian embryo to recapitulate the observed patterning process from zygote to late blastocyst. By integrating the spatiotemporal information reconstructed from multiple single-cell transcriptomic datasets, the data-informed modeling analysis suggests two major processes critical to the formation of Epi/PE layers: a selective cell-cell adhesion mechanism (via EphA4/EphrinB2) for fate-location coordination and a temporal attenuation mechanism of cell signaling (via Fgf). Spatial imaging data and distinct subsets of single-cell gene expression data are then used to validate the predictions. Together, our study provides a multiscale framework that incorporates single-cell gene expression datasets to analyze gene regulations, cell-cell communications, and physical interactions among cells in complex geometries at single-cell resolution, with direct application to late-stage development of embryogenesis.

Measuring and Modeling Single-Cell Heterogeneity and Fate Decision in Mouse Embryos

Cellular heterogeneity is a property of any living system; however, its relationship with cellular fate decision remains an open question. Recent technological advances have enabled valuable insights, especially in complex systems such as the mouse embryo. In this review, we discuss recent studies that characterize cellular heterogeneity at different levels during mouse development, from the two-cell stage up to gastrulation. In addition to key experimental findings, we review mathematical modeling approaches that help researchers interpret these findings. Disentangling the role of heterogeneity in cell fate decision will likely rely on the refined integration of experiments, large-scale omics data, and mathematical modeling, complemented by the use of synthetic embryos and gastruloids as promising in vitro models.

Predicting pattern formation in embryonic stem cells using a minimalist, agent-based probabilistic model

The mechanisms of pattern formation during embryonic development remain poorly understood. Embryonic stem cells in culture self-organise to form spatial patterns of gene expression upon geometrical confinement indicating that patterning is an emergent phenomenon that results from the many interactions between the cells. Here, we applied an agent-based modelling approach in order to identify plausible biological rules acting at the meso-scale within stem cell collectives that may explain spontaneous patterning. We tested different models involving differential motile behaviours with or without biases due to neighbour interactions. We introduced a new metric, termed stem cell aggregate pattern distance (SCAPD) to probabilistically assess the fitness of our models with empirical data. The best of our models improves fitness by 70% and 77% over the random models for a discoidal or an ellipsoidal stem cell confinement respectively. Collectively, our findings show that a parsimonious mechanism that involves differential motility is sufficient to explain the spontaneous patterning of the cells upon confinement. Our work also defines a region of the parameter space that is compatible with patterning. We hope that our approach will be applicable to many biological systems and will contribute towards facilitating progress by reducing the need for extensive and costly experiments.

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.

Biophysics in oviduct: Planar cell polarity, cilia, epithelial fold and tube morphogenesis, egg dynamics

Organs and tissues in multi-cellular organisms exhibit various morphologies. Tubular organs have multi-scale morphological features which are closely related to their functions. Here we discuss morphogenesis and the mechanical functions of the vertebrate oviduct in the female reproductive tract, also known as the fallopian tube. The oviduct functions to convey eggs from the ovary to the uterus. In the luminal side of the oviduct, the epithelium forms multiple folds (or ridges) well-aligned along the longitudinal direction of the tube. In the epithelial cells, cilia are formed orienting toward the downstream of the oviduct. The cilia and the folds are supposed to be involved in egg transportation. Planar cell polarity (PCP) is developed in the epithelium, and the disruption of the Celsr1 gene, a PCP related-gene, causes randomization of both cilia and fold orientations, discontinuity of the tube, inefficient egg transportation, and infertility. In this review article, we briefly introduce various biophysical and biomechanical issues in the oviduct, including physical mechanisms of formation of PCP and organized cilia orientation, epithelial cell shape regulation, fold pattern formation generated by mechanical buckling, tubulogenesis, and egg transportation regulated by fluid flow. We also mention about possible roles of the oviducts in egg shape formation and embryogenesis, sinuous patterns of tubes, and fold and tube patterns observed in other tubular organs such as the gut, airways, etc.

Hybrid cell-centred/vertex model for multicellular systems with equilibrium-preserving remodelling

We present a hybrid cell-centred/vertex model for mechanically simulating planar cellular monolayers undergoing cell reorganisation. Cell centres are represented by a triangular nodal network, while the cell boundaries are formed by an associated vertex network. The two networks are coupled through a kinematic constraint which we allow to relax progressively. Special attention is paid to the change of cell-cell connectivity due to cell reorganisation or remodelling events. We handle these situations by using a variable resting length and applying an Equilibrium-Preserving Mapping on the new connectivity, which computes a new set of resting lengths that preserve nodal and vertex equilibrium. We illustrate the properties of the model by simulating monolayers subjected to imposed extension and during a wound healing process. The evolution of forces and the Equilibrium-Preserving Mapping are analysed during the remodelling events. As a by-product, the proposed technique enables to recover fully vertex or fully cell-centred models in a seamless manner by modifying a numerical parameter of the model.

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.

Combining Turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching

This study demonstrates computational simulations of multicellular deformation coupled with chemical patterning in the three-dimensional (3D) space. To address these aspects, we proposes a novel mathematical model, where a reaction–diffusion system is discretely expressed at a single cell level and combined with a 3D vertex model. To investigate complex phenomena emerging from the coupling of patterning and deformation, as an example, we employed an activator–inhibitor system and converted the activator concentration of individual cells into their growth rate. Despite the simplicity of the model, by growing a monolayer cell vesicle, the coupling system provided rich morphological dynamics such as undulation, tubulation, and branching. Interestingly, the morphological variety depends on the difference in time scales between patterning and deformation, and can be partially understood by the intrinsic hysteresis in the activator-inhibitor system with domain growth. Importantly, the model can be applied to 3D multicellular dynamics that couple the reaction–diffusion patterning with various cell behaviors, such as deformation, rearrangement, division, apoptosis, differentiation, and proliferation. Thus, the results demonstrate the significant advantage of the proposed model as well as the biophysical importance of exploring spatiotemporal dynamics of the coupling phenomena of patterning and deformation in 3D space.

Mechanocellular models of epithelial morphogenesis

Embryonic epithelia achieve complex morphogenetic movements, including in-plane reshaping, bending and folding, through the coordinated action and rearrangement of individual cells. Technical advances in molecular and live-imaging studies of epithelial dynamics provide a very real opportunity to understand how cell-level processes facilitate these large-scale tissue rearrangements. However, the large datasets that we are now able to generate require careful interpretation. In combination with experimental approaches, computational modelling allows us to challenge and refine our current understanding of epithelial morphogenesis and to explore experimentally intractable questions. To this end, a variety of cell-based modelling approaches have been developed to describe cell-cell mechanical interactions, ranging from vertex and 'finite-element' models that approximate each cell geometrically by a polygon representing the cell's membrane, to immersed boundary and subcellular element models that allow for more arbitrary cell shapes. Here, we review how these models have been used to provide insights into epithelial morphogenesis and describe how such models could help future efforts to decipher the forces and mechanical and biochemical feedbacks that guide cell and tissue-level behaviour. In addition, we discuss current challenges associated with using computational models of morphogenetic processes in a quantitative and predictive way.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Four simple rules that are sufficient to generate the mammalian blastocyst

Early mammalian development is both highly regulative and self-organizing. It involves the interplay of cell position, predetermined gene regulatory networks, and environmental interactions to generate the physical arrangement of the blastocyst with precise timing. However, this process occurs in the absence of maternal information and in the presence of transcriptional stochasticity. How does the preimplantation embryo ensure robust, reproducible development in this context? It utilizes a versatile toolbox that includes complex intracellular networks coupled to cell-cell communication, segregation by differential adhesion, and apoptosis. Here, we ask whether a minimal set of developmental rules based on this toolbox is sufficient for successful blastocyst development, and to what extent these rules can explain mutant and experimental phenotypes. We implemented experimentally reported mechanisms for polarity, cell-cell signaling, adhesion, and apoptosis as a set of developmental rules in an agent-based in silico model of physically interacting cells. We find that this model quantitatively reproduces specific mutant phenotypes and provides an explanation for the emergence of heterogeneity without requiring any initial transcriptional variation. It also suggests that a fixed time point for the cells' competence of fibroblast growth factor (FGF)/extracellular signal-regulated kinase (ERK) sets an embryonic clock that enables certain scaling phenomena, a concept that we evaluate quantitatively by manipulating embryos in vitro. Based on these observations, we conclude that the minimal set of rules enables the embryo to experiment with stochastic gene expression and could provide the robustness necessary for the evolutionary diversification of the preimplantation gene regulatory network.

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.

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.

D'Arcy Thompson's 'on Growth and form': From soap bubbles to tissue self-organization

Tissues are thought to behave like fluids with a given surface tension. Differences in tissue surface tension (TST) have been proposed to trigger cell sorting and tissue envelopment. D'Arcy Thompson in his seminal book 'On Growth and Form' has introduced this concept of differential TST as a key physical mechanism dictating tissue formation and organization within the developing organism. Over the past century, many studies have picked up the concept of differential TST and analyzed the role and cell biological basis of TST in development, underlining the importance and influence of this concept in developmental biology.

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.

Comparing individual-based approaches to modelling the self-organization of multicellular tissues

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

Gene Expression Noise Enhances Robust Organization of the Early Mammalian Blastocyst

A critical event in mammalian embryo development is construction of an inner cell mass surrounded by a trophoectoderm (a shell of cells that later form extraembryonic structures). We utilize multi-scale, stochastic modeling to investigate the design principles responsible for robust establishment of these structures. This investigation makes three predictions, each supported by our quantitative imaging. First, stochasticity in the expression of critical genes promotes cell plasticity and has a critical role in accurately organizing the developing mouse blastocyst. Second, asymmetry in the levels of noise variation (expression fluctuation) of Cdx2 and Oct4 provides a means to gain the benefits of noise-mediated plasticity while ameliorating the potentially detrimental effects of stochasticity. Finally, by controlling the timing and pace of cell fate specification, the embryo temporally modulates plasticity and creates a time window during which each cell can continually read its environment and adjusts its fate. These results suggest noise has a crucial role in maintaining cellular plasticity and organizing the blastocyst.

A critical event in mammalian embryo development is construction of a mass of embryonic stem cells surrounded by a distinct shell that later forms the placenta along with other structures. Despite sustained investigation, multiple hypotheses for what is responsible for this organization persist and it remains unclear what is responsible for the robust organization (remarkable ability for embryos to pattern correctly) of these structures. Here, we utilize multi-scale, stochastic modeling along with fluorescence imaging to investigate the factors that contribute to the incredible robustness of this organizational process. Results point to two factors that contribute to this robustness: 1) the timing and pace of cell fate specification and 2) stochastic gene regulatory effects. The former creates a window of time during which each cell can continually read their environment and adjust their gene expressions (and consequently fate) in response to dynamic rearrangements of cells arising from cell divisions and motions. The latter improves cell plasticity, providing the capability for cells to adjust to changes in their local environment. Fluorescence imaging results demonstrate that the magnitude and structure of gene expression variations match those predicted to promote organizational robustness.

A set of simple cell processes is sufficient to model spiral cleavage

During cleavage, different cellular processes cause the zygote to become partitioned into a set of cells with a specific spatial arrangement. These processes include the orientation of cell division according to: an animal-vegetal gradient; the main axis (Hertwig's rule) of the cell; and the contact areas between cells or the perpendicularity between consecutive cell divisions (Sachs' rule). Cell adhesion and cortical rotation have also been proposed to be involved in spiral cleavage. We use a computational model of cell and tissue biomechanics to account for the different existing hypotheses about how the specific spatial arrangement of cells in spiral cleavage arises during development. Cell polarization by an animal-vegetal gradient, a bias to perpendicularity between consecutive cell divisions (Sachs' rule), cortical rotation and cell adhesion, when combined, reproduce the spiral cleavage, whereas other combinations of processes cannot. Specifically, cortical rotation is necessary at the 8-cell stage to direct all micromeres in the same direction. By varying the relative strength of these processes, we reproduce the spatial arrangement of cells in the blastulae of seven different invertebrate species.

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.

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.

Towards understanding the roles of position and geometry on cell fate decisions during preimplantation development

The first lineage segregation event in mouse embryos produces two separate cell populations: inner cell mass and trophectoderm. This is understood to be brought about by cells sensing their position within the embryo and differentiating accordingly. The cellular and molecular underpinnings of this process remain under investigation and have variously been considered to be completely stochastic or alternately, subject to some predisposition set up at fertilisation or before. Here, we consider these views in light of recent publications, discuss the possible role of cell geometry and mechanical forces in this process and describe how modelling could contribute in addressing this issue.

Robustness of force and stress inference in an epithelial tissue

During morphogenesis, the shape of a tissue emerges from collective cellular behaviors, which are in part regulated by mechanical and biochemical interactions between cells. Quantification of force and stress is therefore necessary to analyze the mechanisms controlling tissue morphogenesis. Recently, a mechanical measurement method based on force inference from cell shapes and connectivity has been developed. It is non-invasive, and can provide space-time maps of force and stress within an epithelial tissue, up to prefactors. We previously performed a comparative study of three force-inference methods, which differ in their approach of treating indefiniteness in an inverse problem between cell shapes and forces. In the present study, to further validate and compare the three force inference methods, we tested their robustness by measuring temporal fluctuation of estimated forces. Quantitative data of cell-level dynamics in a developing tissue suggests that variation of forces and stress will remain small within a short period of time (~minutes). Here, we showed that cell-junction tensions and global stress inferred by the Bayesian force inference method varied less with time than those inferred by the method that estimates only tension. In contrast, the amplitude of temporal fluctuations of estimated cell pressures differs less between different methods. Altogether, the present study strengthens the validity and robustness of the Bayesian force-inference method.

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.

Single cells get together: High-resolution approaches to study the dynamics of early mouse development

Embryonic development is a complex and highly dynamic process during which individual cells interact with one another, adopt different identities and organize themselves in three-dimensional space to generate an entire organism. Recent technical developments in genomics and high-resolution quantitative imaging are making it possible to study cellular populations at single-cell resolution and begin to integrate different inputs, for example genetic, physical and chemical factors, that affect cell differentiation over spatial and temporal scales. The preimplantation mouse embryo allows the analysis of cell fate decisions in vivo with high spatiotemporal resolution. In this review we highlight how the application of live imaging and single-cell resolution analysis pipelines is providing an unprecedented level of insight on the processes that shape the earliest stages of mammalian development.

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.

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.

Mechanobiology and developmental control

Morphogenesis is the remarkable process by which cells self-assemble into complex tissues and organs that exhibit specialized form and function during embryological development. Many of the genes and chemical cues that mediate tissue and organ formation have been identified; however, these signals alone are not sufficient to explain how tissues and organs are constructed that exhibit their unique material properties and three-dimensional forms. Here, we review work that has revealed the central role that physical forces and extracellular matrix mechanics play in the control of cell fate switching, pattern formation, and tissue development in the embryo and how these same mechanical signals contribute to tissue homeostasis and developmental control throughout adult life.

The mechanical anisotropy in a tissue promotes ordering in hexagonal cell packing

Many epithelial tissues pack cells into a honeycomb pattern to support their structural and functional integrity. Developmental changes in cell packing geometry have been shown to be regulated by both mechanical and biochemical interactions between cells; however, it is largely unknown how molecular and cellular dynamics and tissue mechanics are orchestrated to realize the correct and robust development of hexagonal cell packing. Here, by combining mechanical and genetic perturbations along with live imaging and Bayesian force inference, we investigate how mechanical forces regulate cellular dynamics to attain a hexagonal cell configuration in the Drosophila pupal wing. We show that tissue stress is oriented towards the proximal-distal axis by extrinsic forces acting on the wing. Cells respond to tissue stretching and orient cell contact surfaces with the stretching direction of the tissue, thereby stabilizing the balance between the intrinsic cell junction tension and the extrinsic force at the cell-population level. Consequently, under topological constraints of the two-dimensional epithelial sheet, mismatches in the orientation of hexagonal arrays are suppressed, allowing more rapid relaxation to the hexagonal cell pattern. Thus, our results identify the mechanism through which the mechanical anisotropy in a tissue promotes ordering in cell packing geometry.

A molecular basis for developmental plasticity in early mammalian embryos

Early mammalian embryos exhibit remarkable plasticity, as highlighted by the ability of separated early blastomeres to produce a whole organism. Recent work in the mouse implicates a network of transcription factors in governing the establishment of the primary embryonic lineages. A combination of genetics and embryology has uncovered the organisation and function of the components of this network, revealing a gradual resolution from ubiquitous to lineage-specific expression through a combination of defined regulatory relationships, spatially organised signalling, and biases from mechanical inputs. Here, we summarise this information, link it to classical embryology and propose a molecular framework for the establishment and regulation of developmental plasticity.

A self-organization framework for symmetry breaking in the mammalian embryo

The mechanisms underlying the appearance of asymmetry between cells in the early embryo and consequently the specification of distinct cell lineages during mammalian development remain elusive. Recent experimental advances have revealed unexpected dynamics of and new complexity in this process. These findings can be integrated in a new unified framework that regards the early mammalian embryo as a self-organizing system.