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

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.

From soap bubbles to multicellular organisms: Unraveling the role of cell adhesion and physical constraints in tile pattern formation and tissue morphogenesis

Tile patterns, in which numerous cells are arranged in a regular pattern, are found in a variety of multicellular organisms and play important functional roles. Such regular arrangements of cells are regulated by various cell adhesion molecules. On the other hand, cell shape is also known to be regulated by physical constraints similar to those of soap bubbles. In particular, circumference minimization plays an important role, and cell adhesion negatively affects this process, thereby regulating tissue morphogenesis based on physical properties. Here, we focus on the Drosophila compound eye and the mouse auditory epithelium, and summarize the mechanisms of tile pattern formation by cell adhesion molecules such as cadherins, Irre Cell Recognition Modules (IRMs), and nectins. Phenomena that cannot be explained by physical stability based on cortical tension alone have been reported in the tile pattern formation in the compound eye, suggesting that previously unexplored forces such as cellular concentric expansion force may play an important role. We would like to summarize perspectives for future research on the mechanisms of tissue morphogenesis.

Development of Bioactive Scaffolds for Orthopedic Applications by Designing Additively Manufactured Titanium Porous Structures: A Critical Review

We overview recent findings achieved in the field of model-driven development of additively manufactured porous materials for the development of a new generation of bioactive implants for orthopedic applications. Porous structures produced from biocompatible titanium alloys using selective laser melting can present a promising material to design scaffolds with regulated mechanical properties and with the capacity to be loaded with pharmaceutical products. Adjusting pore geometry, one could control elastic modulus and strength/fatigue properties of the engineered structures to be compatible with bone tissues, thus preventing the stress shield effect when replacing a diseased bone fragment. Adsorption of medicals by internal spaces would make it possible to emit the antibiotic and anti-tumor agents into surrounding tissues. The developed internal porosity and surface roughness can provide the desired vascularization and osteointegration. We critically analyze the recent advances in the field featuring model design approaches, virtual testing of the designed structures, capabilities of additive printing of porous structures, biomedical issues of the engineered scaffolds, and so on. Special attention is paid to highlighting the actual problems in the field and the ways of their solutions.

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.

Appropriate tension sensitivity of α-catenin ensures rounding morphogenesis of epithelial spheroids

The adherens junction (AJ) is an actin filament-anchoring junction. It plays a central role in epithelial morphogenesis through cadherin-based recognition and adhesion among cells. The stability and plasticity of AJs are required for the morphogenesis. An actin-binding α-catenin is an essential component of the cadherin-catenin complex and functions as a tension transducer that changes its conformation and induces AJ development in response to tension. Despite much progress in understanding molecular mechanisms of tension sensitivity of α-catenin, its significance on epithelial morphogenesis is still unknown. Here we show that the tension sensitivity of α-catenin is essential for epithelial cells to form round spheroids through proper multicellular rearrangement. Using a novel in vitro suspension culture model, we found that epithelial cells form round spheroids even from rectangular-shaped cell masses with high aspect ratios without using high tension and that increased tension sensitivity of α-catenin affected this morphogenesis. Analyses of AJ formation and cellular tracking during rounding morphogenesis showed cellular rearrangement, probably through AJ remodeling. The rearrangement occurs at the cell mass level, but not single-cell level. Hypersensitive α-catenin mutant-expressing cells did not show cellular rearrangement at the cell mass level, suggesting that the appropriate tension sensitivity of α-catenin is crucial for the coordinated round morphogenesis.Key words: α-catenin, vinculin, adherens junction, morphogenesis, mechanotransduction.

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.

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.

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.

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.

Neuronal differentiation influences progenitor arrangement in the vertebrate neuroepithelium

Cell division, movement and differentiation contribute to pattern formation in developing tissues. This is the case in the vertebrate neural tube, in which neurons differentiate in a characteristic pattern from a highly dynamic proliferating pseudostratified epithelium. To investigate how progenitor proliferation and differentiation affect cell arrangement and growth of the neural tube, we used experimental measurements to develop a mechanical model of the apical surface of the neuroepithelium that incorporates the effect of interkinetic nuclear movement and spatially varying rates of neuronal differentiation. Simulations predict that tissue growth and the shape of lineage-related clones of cells differ with the rate of differentiation. Growth is isotropic in regions of high differentiation, but dorsoventrally biased in regions of low differentiation. This is consistent with experimental observations. The absence of directional signalling in the simulations indicates that global mechanical constraints are sufficient to explain the observed differences in anisotropy. This provides insight into how the tissue growth rate affects cell dynamics and growth anisotropy and opens up possibilities to study the coupling between mechanics, pattern formation and growth in the neural tube.

Summary: A mechanical model of the vertebrate neuroepithelium, based on experimental observations, suggests that the rate of neuronal differentiation influences tissue growth and the shape of lineage-related clones.

Emergence of robust patterns from local rules during plant development

The formation of spatial and temporal patterns is an essential component of organismal development. Patterns can be observed on every level from subcellular to organismal and may emerge from local rules that correspond to the interactions between molecules, cells, or tissues. The emergence of robust patterns may seem in contradiction with the prominent heterogeneity at subcellular and cellular scales, however it has become increasingly clear that heterogeneity can be instrumental for pattern formation. Here we review recent examples in plant development, involving genetic regulation, cell arrangement, growth and signal gradient. We discuss how patterns emerge from local rules, whether heterogeneity is stochastic or can be patterned, and whether stochastic noise is amplified or requires filtering for robust patterns to be achieved. We also stress the importance of modelling in investigating such questions.

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

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

Topological graph description of multicellular dynamics based on vertex model

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

Chiral cell sliding drives left-right asymmetric organ twisting

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

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

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

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

Riding the cell jamming boundary: Geometry, topology, and phase of human corneal endothelium

It is important to assess the viability of eye-banked corneas prior to transplantation due to inherent senescence and known loss of endothelial cells during surgical manipulation. Corneal endothelial cells have a complex basal and paracellular shape making them challenging to accurately measure, particularly in oedematous ex vivo tissue. This study used calibrated centroidal Voronoi Diagrams to segment cells in images of these human corneas, in order to characterize endothelial geometry, topology, and phase. Hexagonal cells dominated the endothelia, with most comprised of five different pleomorphs exhibiting self-similar topological coarsening through most of the endothelial cell density range. There was a linear relationship between cell size and shape, though cells with greater than six sides were present in larger proportions than cells with less. Hexagonal cell regularity was stable and largely independent of density. Cell and tissue phase was also examined, using the cell shape index relative to the recently discovered 'cell jamming' phase transition boundary. Images showed fluid endothelia with a range of shape indices spanning the boundary, independent of density but dependent on hexagonal regularity. The cells showed a bimodal distribution centred at the boundary, with the largest proportion of cells on the fluid side. A shoulder at the boundary suggested phase switching via shape transformation across the energy barrier, with cells either side having distinctly different size and shape characteristics. Regular hexagonal cells were closest to the boundary. This study showed the corneal endothelium acts as a glassy viscous foam characterized by well-established physical laws. Endothelial cell death transiently and locally increases cell fluidity, which is subsequently arrested by jamming of the pleomorphically diverse cell collective, via rearrangement and shape change of a small proportion of cells, which become locked in place by their neighbours thereby maintaining structural equilibrium with little energy expenditure.

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.

'The Forms of Tissues, or Cell-aggregates': D'Arcy Thompson's influence and its limits

In two chapters of his book On Growth and Form, D'Arcy Thompson used numerous biological and physical observations to show how principles from mathematics and physics - such as pressure differences, surface tension and viscosity - could explain cell shapes and packing within tissues. In this Review, we depict influences that enabled the genesis of his ideas, report examples of his visionary observations and trace his impact over the past 100 years. Recently, his ideas have been revisited as a new field of research emerged, linking cell-level physics with epithelial tissue structure and development. We critically discuss the potential and the limitations of both Thompson's and the modern approaches.

Vertex models: from cell mechanics to tissue morphogenesis

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

The mechanical properties of a cell-based numerical model of epithelium

In this work we use a computational cell-based model to study the influence of the mechanical properties of cells on the mechanics of epithelial tissues. We analyze the effect of the model parameters on the elasticity and the mechanical response of tissues subjected to stress loading application. We compare our numerical results with experimental measurements of epithelial cell monolayer mechanics. Unlike previous studies, we have been able to estimate in physical units the parameter values that match the experimental results. A key observation is that the model parameters must vary with the tissue strain. In particular, it was found that, while the perimeter contractility and the area elasticity of cells remain constant at lower strains (<20%), they must increase to respond to larger strains (>20%). However, above a threshold of 50% extension, the cells stop counteracting the tissue strain and reduce both their perimeter contractility and area elasticity.

Statistical analyses in trials for the comprehensive understanding of organogenesis and histogenesis in humans and mice

Statistical analyses based on the quantitative data from real multicellular organisms are useful as inductive-type studies to analyse complex morphogenetic events in addition to deductive-type analyses using mathematical models. Here, we introduce several of our trials for the statistical analysis of organogenesis and histogenesis of human and mouse embryos and foetuses. Multidimensional scaling has been applied to prove the existence and examine the mode of interkinetic nuclear migration, a regulatory mechanism of stem cell proliferation/differentiation in epithelial tubular tissues. Several statistical methods were used on morphometric data from human foetuses to establish the multidimensional standard growth curve and to describe the relation among the developing organs and body parts. Although the results are still limited, we show that these analyses are not only useful to understand the normal and abnormal morphogenesis in humans and mice but also to provide clues that could correlate aspects of prenatal developmental events with postnatal diseases.

Capabilities and Limitations of Tissue Size Control through Passive Mechanical Forces

Embryogenesis is an extraordinarily robust process, exhibiting the ability to control tissue size and repair patterning defects in the face of environmental and genetic perturbations. The size and shape of a developing tissue is a function of the number and size of its constituent cells as well as their geometric packing. How these cellular properties are coordinated at the tissue level to ensure developmental robustness remains a mystery; understanding this process requires studying multiple concurrent processes that make up morphogenesis, including the spatial patterning of cell fates and apoptosis, as well as cell intercalations. In this work, we develop a computational model that aims to understand aspects of the robust pattern repair mechanisms of the Drosophila embryonic epidermal tissues. Size control in this system has previously been shown to rely on the regulation of apoptosis rather than proliferation; however, to date little work has been done to understand the role of cellular mechanics in this process. We employ a vertex model of an embryonic segment to test hypotheses about the emergence of this size control. Comparing the model to previously published data across wild type and genetic perturbations, we show that passive mechanical forces suffice to explain the observed size control in the posterior (P) compartment of a segment. However, observed asymmetries in cell death frequencies across the segment are demonstrated to require patterning of cellular properties in the model. Finally, we show that distinct forms of mechanical regulation in the model may be distinguished by differences in cell shapes in the P compartment, as quantified through experimentally accessible summary statistics, as well as by the tissue recoil after laser ablation experiments.

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.