A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating cell behavior in a cell aggregate

Distinct cellular and junctional dynamics independently regulate the rotation and elongation of the embryonic gut in Drosophila

Complex organ structures are formed with high reproducibility. To achieve such intricate morphologies, the responsible epithelium undergoes multiple simultaneous shape changes, such as elongation and folding. However, these changes have typically been assessed separately. In this study, we revealed how distinct shape changes are controlled during internal organ morphogenesis. The Drosophila embryonic hindgut undergoes left-right asymmetric rotation and anteroposterior elongation in a tissue-autonomous manner driven by cell sliding and convergent extension, respectively, in the hindgut epithelia. However, the regulation of these processes remains unclear. Through genetic analysis and live imaging, we demonstrated that cell sliding and convergent extension are independently regulated by Myosin1D and E-cadherin, and Par-3, respectively, whereas both require MyosinII activity. Using a mathematical model, we demonstrated that independently regulated cellular dynamics can simultaneously cause shape changes in a single mechanical system using anisotropic edge contraction. Our findings indicate that distinct cellular dynamics sharing a common apparatus can be independently and simultaneously controlled to form complex organ shapes. This suggests that such a mechanism may be a general strategy during complex tissue morphogenesis.

A two-dimensional vertex model for curvy cell-cell interfaces at the subcellular scale

Cross-sections of cell shapes in a tissue monolayer typically resemble a tiling of convex polygons. Yet, examples exist where the polygons are not convex with curved cell-cell interfaces, as seen in the adaxial epidermis. To date, two-dimensional vertex models predicting the structure and mechanics of cell monolayers have been mostly limited to convex polygons. To overcome this limitation, we introduce a framework to study curvy cell-cell interfaces at the subcellular scale within vertex models by using a parametrized curve between vertices that is expanded in a Fourier series and whose coefficients represent additional degrees of freedom. This extension to non-convex polygons allows for cells with the same shape index, or dimensionless perimeter, to be, for example, either elongated or globular with lobes. In the presence of applied, anisotropic stresses, we find that local, subcellular curvature or buckling can be energetically more favourable than larger scale deformations involving groups of cells. Inspired by recent experiments, we also find that local, subcellular curvature at cell-cell interfaces emerges in a group of cells in response to the swelling of additional cells surrounding the group. Our framework, therefore, can account for a wider array of multicellular responses to constraints in the tissue environment.

Characterizing Cellular Physiological States with Three-Dimensional Shape Descriptors for Cell Membranes

The shape of a cell as defined by its membrane can be closely associated with its physiological state. For example, the irregular shapes of cancerous cells and elongated shapes of neuron cells often reflect specific functions, such as cell motility and cell communication. However, it remains unclear whether and which cell shape descriptors can characterize different cellular physiological states. In this study, 12 geometric shape descriptors for a three-dimensional (3D) object were collected from the previous literature and tested with a public dataset of ~400,000 independent 3D cell regions segmented based on fluorescent labeling of the cell membranes in Caenorhabditis elegans embryos. It is revealed that those shape descriptors can faithfully characterize cellular physiological states, including (1) cell division (cytokinesis), along with an abrupt increase in the elongation ratio; (2) a negative correlation of cell migration speed with cell sphericity; (3) cell lineage specification with symmetrically patterned cell shape changes; and (4) cell fate specification with differential gene expression and differential cell shapes. The descriptors established may be used to identify and predict the diverse physiological states in numerous cells, which could be used for not only studying developmental morphogenesis but also diagnosing human disease (e.g., the rapid detection of abnormal cells).

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

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

Graph topological transformations in space-filling cell aggregates

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

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.

Mechanical forces across compartments coordinate cell shape and fate transitions to generate tissue architecture

Morphogenesis and cell state transitions must be coordinated in time and space to produce a functional tissue. An excellent paradigm to understand the coupling of these processes is mammalian hair follicle development, which is initiated by the formation of an epithelial invagination-termed placode-that coincides with the emergence of a designated hair follicle stem cell population. The mechanisms directing the deformation of the epithelium, cell state transitions and physical compartmentalization of the placode are unknown. Here we identify a key role for coordinated mechanical forces stemming from contractile, proliferative and proteolytic activities across the epithelial and mesenchymal compartments in generating the placode structure. A ring of fibroblast cells gradually wraps around the placode cells to generate centripetal contractile forces, which, in collaboration with polarized epithelial myosin activity, promote elongation and local tissue thickening. These mechanical stresses further enhance compartmentalization of Sox9 expression to promote stem cell positioning. Subsequently, proteolytic remodelling locally softens the basement membrane to facilitate a release of pressure on the placode, enabling localized cell divisions, tissue fluidification and epithelial invagination into the underlying mesenchyme. Together, our experiments and modelling identify dynamic cell shape transformations and tissue-scale mechanical cooperation as key factors for orchestrating organ formation.

Differences in boundary behavior in the 3D vertex and Voronoi models

An important open question in the modeling of biological tissues is how to identify the right scale for coarse-graining, or equivalently, the right number of degrees of freedom. For confluent biological tissues, both vertex and Voronoi models, which differ only in their representation of the degrees of freedom, have effectively been used to predict behavior, including fluid-solid transitions and cell tissue compartmentalization, which are important for biological function. However, recent work in 2D has hinted that there may be differences between the two models in systems with heterotypic interfaces between two tissue types, and there is a burgeoning interest in 3D tissue models. Therefore, we compare the geometric structure and dynamic sorting behavior in mixtures of two cell types in both 3D vertex and Voronoi models. We find that while the cell shape indices exhibit similar trends in both models, the registration between cell centers and cell orientation at the boundary are significantly different between the two models. We demonstrate that these macroscopic differences are caused by changes to the cusp-like restoring forces introduced by the different representations of the degrees of freedom at the boundary, and that the Voronoi model is more strongly constrained by forces that are an artifact of the way the degrees of freedom are represented. This suggests that vertex models may be more appropriate for 3D simulations of tissues with heterotypic contacts.

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.

Differential cell motion: A mathematical model of anterior posterior sorting

Here, we investigate how a subpopulation of cells can move through an aggregate of cells. Using a stochastic force-based model of Dictyostelium discoideum when the population is forming a slug, we simulate different strategies for prestalk cells to reliably move to the front of the slug while omitting interaction with the substrate thus ignoring the overall motion of the slug. Of the mechanisms that we simulated, prestalk cells being more directed is the best strategy followed by increased asymmetric motive forces for prestalk cells. The lifetime of the cell adhesion molecules, while not enough to produce differential motion, did modulate the results of the strategies employed. Finally, understanding and simulating the appropriate boundary conditions are essential to correctly predict the motion.

Coarse-graining the vertex model and its response to shear

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

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

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

On the reaction-diffusion type modelling of the self-propelled object motion

In this study, we propose a mathematical model of self-propelled objects based on the Allen-Cahn type phase-field equation. We combine it with the equation for the concentration of surfactant used in previous studies to construct a model that can handle self-propelled object motion with shape change. A distinctive feature of our mathematical model is that it can represent both deformable self-propelled objects, such as droplets, and solid objects, such as camphor disks, by controlling a single parameter. Furthermore, we demonstrate that, by taking the singular limit, this phase-field based model can be reduced to a free boundary model, which is equivalent to the [Formula: see text]-gradient flow model of self-propelled objects derived by the variational principle from the interfacial energy, which gives a physical interpretation to the phase-field model.

Engineering tools for quantifying and manipulating forces in epithelia

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

Long-term adherent cell dynamics emerging from energetic and frictional interactions at the interface

Cell adhesion plays an important role in a wide range of biological situations, including embryonic development, cancer invasion, and wound healing. Although several computational models describing adhesion dynamics have been proposed, models applicable to long-term, large-length-scale cell dynamics are lacking. In this study we investigated possible states of long-term adherent cell dynamics in three-dimensional space by constructing a continuum model of interfacial interactions between adhesive surfaces. In this model a pseudointerface is supposed between each pair of triangular elements that discretize cell surfaces. By introducing a distance between each pair of elements, the physical properties of the interface are given by interfacial energy and friction. The proposed model was implemented into the model of a nonconservative fluid cell membrane where the cell membrane dynamically flows with turnover. Using the implemented model, numerical simulations of adherent cell dynamics on a substrate under flow were performed. The simulations not only reproduced the previously reported dynamics of adherent cells, such as detachment, rolling, and fixation on the substrate, but also discovered other dynamic states, including cell slipping and membrane flow patterns, corresponding to behaviors that occur on much longer timescales than the dissociation of adhesion molecules. These results illustrate the variety of long-term adherent cell dynamics, which are more diverse than the short-term ones. The proposed model can be extended to arbitrarily shaped membranes, thus being useful for the mechanical analysis of a wide range of long-term cell dynamics where adhesion is essential.

Precise alternating cellular pattern in the inner ear by coordinated hopping intercalations and delaminations

The mammalian hearing organ, the organ of Corti, is one of the most organized tissues in mammals. It contains a precisely positioned array of alternating sensory hair cells (HCs) and nonsensory supporting cells. How such precise alternating patterns emerge during embryonic development is not well understood. Here, we combine live imaging of mouse inner ear explants with hybrid mechano-regulatory models to identify the processes that underlie the formation of a single row of inner hair cells (IHCs). First, we identify a previously unobserved morphological transition, termed "hopping intercalation," that allows cells differentiating toward IHC fate to "hop" under the apical plane into their final position. Second, we show that out-of-row cells with low levels of the HC marker Atoh1 delaminate. Last, we show that differential adhesion between cell types contributes to straightening of the IHC row. Our results support a mechanism for precise patterning based on coordination between signaling and mechanical forces that is likely relevant for many developmental processes.

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.

On modeling the multiscale mechanobiology of soft tissues: Challenges and progress

Tissues grow and remodel in response to mechanical cues, extracellular and intracellular signals experienced through various biological events, from the developing embryo to disease and aging. The macroscale response of soft tissues is typically nonlinear, viscoelastic anisotropic, and often emerges from the hierarchical structure of tissues, primarily their biopolymer fiber networks at the microscale. The adaptation to mechanical cues is likewise a multiscale phenomenon. Cell mechanobiology, the ability of cells to transform mechanical inputs into chemical signaling inside the cell, and subsequent regulation of cellular behavior through intra- and inter-cellular signaling networks, is the key coupling at the microscale between the mechanical cues and the mechanical adaptation seen macroscopically. To fully understand mechanics of tissues in growth and remodeling as observed at the tissue level, multiscale models of tissue mechanobiology are essential. In this review, we summarize the state-of-the art modeling tools of soft tissues at both scales, the tissue level response, and the cell scale mechanobiology models. To help the interested reader become more familiar with these modeling frameworks, we also show representative examples. Our aim here is to bring together scientists from different disciplines and enable the future leap in multiscale modeling of tissue mechanobiology.

Impact of environmental asymmetry on epithelial morphogenesis

Epithelial folding is a universal biological phenomenon in morphogenesis, typical examples being brain gyri, villi of the intestinal tract, and imaginal discs in invertebrates. During epithelial morphogenesis, the physical constraints imposed by the surrounding microenvironment on epithelial tissue play critical roles in folding morphology. In this study, we focused on the asymmetry of the environmental constraints sandwiching the epithelial sheet and introduced the degree of asymmetry, which indicates whether the basal or apical side of the epithelium is closer to the constraint wall. Then, we investigated the relationship between the degree of asymmetry and epithelial folding morphology using three-dimensional vertex simulations. The results show that the folding patterns of the epithelial sheets change from spot patterns to labyrinth patterns and then to hole patterns as the degree of asymmetry changes. Furthermore, we examined the pattern formation in terms of the equation of out-of-plane displacement of the sheet derived from the mechanical energy functional.

Thickness of epithelia on wavy substrates: measurements and continuous models

We measured the thickness of MDCK epithelia grown on substrates with a sinusoidal profile. We show that while at long wavelength the profile of the epithelium follows that of the substrate, at short wavelengths cells are thicker in valleys than on ridges. This is reminiscent of the so-called «healing length in the case of a thin liquid film wetting a rough solid substrate. We explore the ability of continuum mechanics models to account for these observations. Modeling the epithelium as a thin liquid film, with surface tension, does not fully account for the measurements. Neither does modeling the epithelium as a thin incompressible elastic film. On the contrary, the addition of an apical active stress gives satisfactory agreement with measurements, with one fitting parameter, the ratio between the active stress and the elastic modulus.

Polarized interfacial tension induces collective migration of cells, as a cluster, in a 3D tissue

In embryogenesis and cancer invasion, cells collectively migrate as a cluster in 3D tissues. Many studies have elucidated mechanisms of either individual or collective cell migration on 2D substrates; however, it remains unclear how cells collectively migrate as a cluster through 3D tissues. To address this issue, we considered the interfacial tension at cell-cell boundaries expressing cortical actomyosin contractions and cell-cell adhesive interactions. The strength of this tension is polarized; i.e., spatially biased within each cell according to a chemoattractant gradient. Using a 3D vertex model, we performed numerical simulations of multicellular dynamics in 3D space. The simulations revealed that the polarized interfacial tension enables cells to migrate collectively as a cluster through a 3D tissue. In this mechanism, interfacial tension induces unidirectional flow of each cell surface from the front to the rear along the cluster surface. Importantly, this mechanism does not necessarily require convection of cells, i.e., cell rearrangement, within the cluster. Moreover, several migratory modes were induced, depending on the strengths of polarity, adhesion, and noise; i.e., cells migrate either as single cells, as a cluster, or aligned like beads on a string, as occurs in embryogenesis and cancer invasion. These results indicate that the simple expansion and contraction of cell-cell boundaries enables cells to move directionally forward and to produce the variety of collective migratory movements observed in living systems.

Tiling mechanisms of the Drosophila compound eye through geometrical tessellation

Tiling patterns are observed in many biological structures. The compound eye is an interesting example of tiling and is often constructed by hexagonal arrays of ommatidia, the optical unit of the compound eye. Hexagonal tiling may be common due to mechanical restrictions such as structural robustness, minimal boundary length, and space-filling efficiency. However, some insects exhibit tetragonal facets.^1-4 Some aquatic crustaceans, such as shrimp and lobsters, have evolved with tetragonal facets.^5-8 Mantis shrimp is an insightful example as its compound eye has a tetragonal midband region sandwiched between hexagonal hemispheres.^9,10 This casts doubt on the naive explanation that hexagonal tiles recur in nature because of their mechanical stability. Similarly, tetragonal tiling patterns are also observed in some Drosophila small-eye mutants, whereas the wild-type eyes are hexagonal, suggesting that the ommatidial tiling is not simply explained by such mechanical restrictions. If so, how are the hexagonal and tetragonal patterns controlled during development? Here, we demonstrate that geometrical tessellation determines the ommatidial tiling patterns. In small-eye mutants, the hexagonal pattern is transformed into a tetragonal pattern as the relative positions of neighboring ommatidia are stretched along the dorsal-ventral axis. We propose that the regular distribution of ommatidia and their uniform growth collectively play an essential role in the establishment of tetragonal and hexagonal tiling patterns in compound eyes.

Retina organoids: Window into the biophysics of neuronal systems

With a kind of magnetism, the human retina draws the eye of neuroscientist and physicist alike. It is attractive as a self-organizing system, which forms as a part of the central nervous system via biochemical and mechanical cues. The retina is also intriguing as an electro-optical device, converting photons into voltages to perform on-the-fly filtering before the signals are sent to our brain. Here, we consider how the advent of stem cell derived in vitro analogs of the retina, termed retina organoids, opens up an exploration of the interplay between optics, electrics, and mechanics in a complex neuronal network, all in a Petri dish. This review presents state-of-the-art retina organoid protocols by emphasizing links to the biochemical and mechanical signals of in vivo retinogenesis. Electrophysiological recording of active signal processing becomes possible as retina organoids generate light sensitive and synaptically connected photoreceptors. Experimental biophysical tools provide data to steer the development of mathematical models operating at different levels of coarse-graining. In concert, they provide a means to study how mechanical factors guide retina self-assembly. In turn, this understanding informs the engineering of mechanical signals required to tailor the growth of neuronal network morphology. Tackling the complex developmental and computational processes in the retina requires an interdisciplinary endeavor combining experiment and theory, physics, and biology. The reward is enticing: in the next few years, retina organoids could offer a glimpse inside the machinery of simultaneous cellular self-assembly and signal processing, all in an in vitro setting.

Modeling Epiblast Shape in Implanting Mammalian Embryos

An indispensable prerequisite of mammalian development is successful morphogenesis in the epiblast, the embryonic tissue that gives rise to all differentiated cells of the adult mammal. The right control of both epiblast morphogenesis and the events that regulate its shape in particular during implantation is henceforth of tremendous importance. However, monitoring the process of development in implanting human embryos is ethically and technically challenging, making it difficult to troubleshoot when things go wrong, as it is unfortunately the case with over 30% of pregnancy failures. Although modern in vitro techniques have proven very insightful lately, more tools are needed in the quest to elucidate mammalian and human development. Mathematical and computational modeling position themselves as helpful complementary tools in the biologist's toolbox, enabling the exploration of the living in silico, beyond the boundaries set by ethical concerns and the potential limitations of wet lab techniques. Here, we show how mathematical modeling and computer simulations can be used to emulate and investigate mechanisms driving epiblast shape changes in mouse and human embryos during implantation.

Derivation of continuum models from discrete models of mechanical forces in cell populations

In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.

Quantifying the link between local structure and cellular rearrangements using information in models of biological tissues

Machine learning techniques have been used to quantify the relationship between local structural features and variations in local dynamical activity in disordered glass-forming materials. To date these methods have been applied to an array of standard (Arrhenius and super-Arrhenius) glass formers, where work on "soft spots" indicates a connection between the linear vibrational response of a configuration and the energy barriers to non-linear deformations. Here we study the Voronoi model, which takes its inspiration from dense epithelial monolayers and which displays anomalous, sub-Arrhenius scaling of its dynamical relaxation time with decreasing temperature. Despite these differences, we find that the likelihood of rearrangements can nevertheless vary by several orders of magnitude within the model tissue and extract a local structural quantity, "softness," that accurately predicts the temperature dependence of the relaxation time. We use an information-theoretic measure to quantify the extent to which softness determines impending topological rearrangements; we find that softness captures nearly all of the information about rearrangements that is obtainable from structure, and that this information is large in the solid phase of the model and decreases rapidly as state variables are varied into the fluid phase.

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.

Computational modelling unveils how epiblast remodelling and positioning rely on trophectoderm morphogenesis during mouse implantation

Understanding the processes by which the mammalian embryo implants in the maternal uterus is a long-standing challenge in embryology. New insights into this morphogenetic event could be of great importance in helping, for example, to reduce human infertility. During implantation the blastocyst, composed of epiblast, trophectoderm and primitive endoderm, undergoes significant remodelling from an oval ball to an egg cylinder. A main feature of this transformation is symmetry breaking and reshaping of the epiblast into a “cup”. Based on previous studies, we hypothesise that this event is the result of mechanical constraints originating from the trophectoderm, which is also significantly transformed during this process. In order to investigate this hypothesis we propose MG# (MechanoGenetic Sharp), an original computational model of biomechanics able to reproduce key cell shape changes and tissue level behaviours in silico. With this model, we simulate epiblast and trophectoderm morphogenesis during implantation. First, our results uphold experimental findings that repulsion at the apical surface of the epiblast is essential to drive lumenogenesis. Then, we provide new theoretical evidence that trophectoderm morphogenesis indeed can dictate the cup shape of the epiblast and fosters its movement towards the uterine tissue. Our results offer novel mechanical insights into mouse peri-implantation and highlight the usefulness of agent-based modelling methods in the study of embryogenesis.

Manipulating Mesenchymal Stem Cell Differentiation on Nanopattern Constructed through Cell-Mediated Mineralization

The extracellular matrix microenvironment, including chemical constituents and topological structure, plays key role in regulating the cell behavior, such as adhesion, proliferation, differentiation, apoptosis, etc. Until now, to investigate the relationship between surface texture and cell response, various ordered patterns have been prepared on the surface of different matrixes, whereas almost all these strategies depend on advanced instruments or severe synthesis conditions. Herein, cell-mediated mineralization method has been applied to construct nanopattern on the surface of β-TCP scaffold. The formation process, morphology, and composition of the final pattern were characterized, and a possible mineralization mechanism has been proposed. Moreover, the cell behavior on the nanopattern has been investigated, and the results showed that the mouse bone marrow mesenchyme stem cells (mBMSCs) display good affinity with the nanopattern, which was manifested by the good proliferation and osteogenic differentiation status of cells. The synthetic strategy may shed light to construct advanced topological structures on other matrixes for bone repair.

A three-dimensional vertex model forDrosophilasalivary gland invagination

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

Poisson-bracket formulation of the dynamics of fluids of deformable particles

Using the Poisson-bracket method, we derive continuum equations for a fluid of deformable particles in two dimensions. Particle shape is quantified in terms of two continuum fields: an anisotropy density field that captures the deformations of individual particles from regular shapes and a shape tensor density field that quantifies both particle elongation and nematic alignment of elongated shapes. We explicitly consider the example of a dense biological tissue as described by the Vertex model energy, where cell shape has been proposed as a structural order parameter for a liquid-solid transition. The hydrodynamic model of biological tissue proposed here captures the coupling of cell shape to flow and provides a starting point for modeling the rheology of dense tissue.

From genes to shape during metamorphosis: a history

Metamorphosis (Greek for a state of transcending-form or change-in-shape) refers to a dramatic transformation of an animal's body structure that occurs after development of the embryo or larva in many species. The development of a fly (or butterfly) from a crawling larva (or caterpillar) that forms a pupa (or chrysalis) before eclosing as a flying adult is a classic example of metamorphosis that captures the imagination and has been immortalized in children's books. Powerful genetic experiments in the fruit fly Drosophila melanogaster have revealed how genes can instruct the behaviour of individual cells to control patterns of tissue growth, mechanical force, cell-cell adhesion and cell-matrix adhesion drive morphogenetic change in epithelial tissues. Together, the distribution of mass, force and resistance determines cell shape changes, cell-cell rearrangements, and/or the orientation of cell divisions to generate the final form of the tissue. In organising tissue shape, genes harness the power of self-organisation to determine the collective behaviour of molecules and cells, which can often be reproduced in computer simulations of cell polarity and/or tissue mechanics. This review highlights fundamental discoveries in epithelial morphogenesis made by pioneers who were fascinated by metamorphosis, including D'Arcy Thompson, Conrad Waddington, Dianne Fristrom and Antonio Garcia-Bellido.

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.

Amphiphilic cationic cyclodextrin nanovesicles: a versatile cue for guiding cell adhesion

It is well known that amphiphilic cationic β-cyclodextrins (amβCDs) form nanovesicles able to release their cargo in aqueous solution upon applying different stimuli. In addition they can be selectively positioned onto substrates by unconventional soft lithography. This makes them a powerful tool for designing environments where different cues can be externally supplied to the cells helping to achieve good control of their fate. Lithographically controlled wetting (LCW) of amβCD nanovesicles loaded with fluorescein isothiocyanate (FITC), amβCD/FITC, has been used here to fabricate geometrically functionalized surfaces, thus achieving multiscale control of the cell environment. The amβCD functionalization was strongly influenced by the surface energy of the underlying substrates that, according to their hydrophobicity, orient the amβCD in a different way, thus "offering" different portions to the cells. The structure of the pattern was characterized both over large scales exploiting the FITC fluorescence and at the nanoscale by atomic force microscopy. Cell guidance and aCD/FITC cell internalization were demonstrated in human neuroblastoma SHSY5Y cells.

Mechanics of the cell: Interaction mechanisms and mechanobiological models

The importance of cell mechanics has long been recognized for the cell development and function. Biomechanics plays an important role in cell metabolism, regulation of mechanotransduction pathways and also modulation of nuclear response. The mechanical properties of the cell are likely determined by, among many others, the cytoskeleton elasticity, membrane tension and cell-substrate adhesion. This coordinated but complex mechanical interplay is required however, for the cell to respond to and influence in a reciprocal manner the chemical and mechanical signals from the extracellular matrix (ECM). In an effort to better and more fully understand the cell mechanics, the role of nuclear mechanics has emerged as an important contributor to the overall cellular mechanics. It is not too difficult to appreciate the physical connection between the nucleus and the cytoskeleton network that may be connected to the ECM through the cell membrane. Transmission of forces from ECM through this connection is essential for a wide range of cellular behaviors and functions such as cytoskeletal reorganization, nuclear movement, cell migration and differentiation. Unlike the cellular mechanics that can be measured using a number of biophysical techniques that were developed in the past few decades, it still remains a daunting challenge to probe the nuclear mechanics directly. In this paper, we therefore aim to provide informative description of the cell membrane and cytoskeleton mechanics, followed by unique computational modeling efforts to elucidate the nucleus-cytoskeleton coupling. Advances in our knowledge of complete cellular biomechanics and mechanotransduction may lead to clinical relevance and applications in mechano-diseases such as atherosclerosis, stem cell-based therapies, and the development of tissue engineered products.

Osmotic gradients induce stable dome morphogenesis on extracellular matrix

One of the fundamental processes in morphogenesis is dome formation, but many of the mechanisms involved are unexplored. Previous in vitro studies showed that an osmotic gradient is the driving factor of dome formation. However, these investigations were performed without extracellular matrix (ECM), which provides structural support to morphogenesis. With the use of ECM, we observed that basal hypertonic stress induced stable domes in vitro that have not been seen in previous studies. These domes developed as a result of ECM swelling via aquaporin water transport activity. Based on computer simulation, uneven swelling, with a positive feedback between cell stretching and enhanced water transport, was a cause of dome formation. These results indicate that osmotic gradients induce dome morphogenesis via both enhanced water transport activity and subsequent ECM swelling.

A Mechanical Instability in Planar Epithelial Monolayers Leads to Cell Extrusion

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

Mechanisms of endothelial cell coverage by pericytes: computational modelling of cell wrapping and in vitro experiments

Pericytes (PCs) wrap around endothelial cells (ECs) and perform diverse functions in physiological and pathological processes. Although molecular interactions between ECs and PCs have been extensively studied, the morphological processes at the cellular level and their underlying mechanisms have remained elusive. In this study, using a simple cellular Potts model, we explored the mechanisms for EC wrapping by PCs. Based on the observed in vitro cell wrapping in three-dimensional PC-EC coculture, the model identified four putative contributing factors: preferential adhesion of PCs to the extracellular matrix (ECM), strong cell-cell adhesion, PC surface softness and larger PC size. While cell-cell adhesion can contribute to the prevention of cell segregation and the degree of cell wrapping, it cannot determine the orientation of cell wrapping alone. While atomic force microscopy revealed that PCs have a larger Young's modulus than ECs, the experimental analyses supported preferential ECM adhesion and size asymmetry. We also formulated the corresponding energy minimization problem and numerically solved this problem for specific cases. These results give biological insights into the role of PC-ECM adhesion in PC coverage. The modelling framework presented here should also be applicable to other cell wrapping phenomena observed in vivo.

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.

Polarized cellular mechano-response system for maintaining radial size in developing epithelial tubes

Size control in biological tissues involves multicellular communication via mechanical forces during development. Although fundamental cellular behaviours in response to mechanical stimuli underlie size maintenance during morphogenetic processes, the mechanisms underpinning the cellular mechano-response system that maintains size along an axis of a polarized tissue remain elusive. Here, we show how the diameter of an epithelial tube is maintained during murine epididymal development by combining quantitative imaging, mechanical perturbation and mathematical modelling. We found that epithelial cells counteract compressive forces caused by cell division exclusively along the circumferential axis of the tube to produce polarized contractile forces, eventually leading to an oriented cell rearrangement. Moreover, a mathematical model that includes the polarized mechano-responsive regime explains how the diameter of proliferating tubes is maintained. Our findings pave the way for an improved understanding of the cellular response to mechanical forces that involves collective multicellular behaviours for organizing diverse tissue morphologies.

Summary: Polarized cellular constriction responding to mechanical stress controls the diameter of a developing epithelial tube during murine epididymal development.

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.

Apical Junctional Fluctuations Lead to Cell Flow while Maintaining Epithelial Integrity

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

Inferring statistical properties of 3D cell geometry from 2D slices

Although cell shape can reflect the mechanical and biochemical properties of the cell and its environment, quantification of 3D cell shapes within 3D tissues remains difficult, typically requiring digital reconstruction from a stack of 2D images. We investigate a simple alternative technique to extract information about the 3D shapes of cells in a tissue; this technique connects the ensemble of 3D shapes in the tissue with the distribution of 2D shapes observed in independent 2D slices. Using cell vertex model geometries, we find that the distribution of 2D shapes allows clear determination of the mean value of a 3D shape index. We analyze the errors that may arise in practice in the estimation of the mean 3D shape index from 2D imagery and find that typically only a few dozen cells in 2D imagery are required to reduce uncertainty below 2%. Even though we developed the method for isotropic animal tissues, we demonstrate it on an anisotropic plant tissue. This framework could also be naturally extended to estimate additional 3D geometric features and quantify their uncertainty in other materials.

A Dynamic Non-Manifold Mesh Data Structure to Represent Biological Materials

Computational models of biological materials enable researchers to gain insight and make testable predictions of quantitative dynamic responses to stimuli. These models are particularly challenging to develop because biological materials are (1) highly heterogeneous containing both biological cells and complex substances such as extra-cellular medium, (2) undergo structural rearrangement (3) couple biological cells with their environment via chemical and mechanical processes. Existing numerical approaches excel at either describing biological cells or solids and fluids, but have difficulty integrating them into a single simulation approach. We present a novel dynamic non-manifold mesh data structure that naturally represents biological materials with coupled chemical and mechanical processes and structural rearrangement in a unified way.

Strain-triggered mechanical feedback in self-organizing optic-cup morphogenesis

Organogenesis is a self-organizing process of multiple cells in three-dimensional (3D) space, where macroscopic tissue deformations are robustly regulated by multicellular autonomy. It is clear that this robust regulation requires cells to sense and modulate 3D tissue formation across different scales, but its underlying mechanisms are still unclear. To address this question, we developed a versatile computational model of 3D multicellular dynamics at single-cell resolution and combined it with the 3D culture system of pluripotent stem cell-derived optic-cup organoid. The complementary approach enabled quantitative prediction of morphogenesis and its corresponding verification and elucidated that the macroscopic 3D tissue deformation is fed back to individual cellular force generations via mechanosensing. We hereby conclude that mechanical force plays a key role as a feedback regulator to establish the robustness of organogenesis.