Bubbly vertex dynamics: A dynamical and geometrical model for epithelial tissues with curved cell shapes

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.

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

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

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).

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

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.

Adapting a Plant Tissue Model to Animal Development: Introducing Cell Sliding into VirtualLeaf

Cell-based, mathematical modeling of collective cell behavior has become a prominent tool in developmental biology. Cell-based models represent individual cells as single particles or as sets of interconnected particles and predict the collective cell behavior that follows from a set of interaction rules. In particular, vertex-based models are a popular tool for studying the mechanics of confluent, epithelial cell layers. They represent the junctions between three (or sometimes more) cells in confluent tissues as point particles, connected using structural elements that represent the cell boundaries. A disadvantage of these models is that cell-cell interfaces are represented as straight lines. This is a suitable simplification for epithelial tissues, where the interfaces are typically under tension, but this simplification may not be appropriate for mesenchymal tissues or tissues that are under compression, such that the cell-cell boundaries can buckle. In this paper, we introduce a variant of VMs in which this and two other limitations of VMs have been resolved. The new model can also be seen as on off-the-lattice generalization of the Cellular Potts Model. It is an extension of the open-source package VirtualLeaf, which was initially developed to simulate plant tissue morphogenesis where cells do not move relative to one another. The present extension of VirtualLeaf introduces a new rule for cell-cell shear or sliding, from which cell rearrangement (T1) and cell extrusion (T2) transitions emerge naturally, allowing the application of VirtualLeaf to problems of animal development. We show that the updated VirtualLeaf yields different results than the traditional vertex-based models for differential adhesion-driven cell sorting and for the neighborhood topology of soft cellular networks.

Relating cell shape and mechanical stress in a spatially disordered epithelium using a vertex-based model

Using a popular vertex-based model to describe a spatially disordered planar epithelial monolayer, we examine the relationship between cell shape and mechanical stress at the cell and tissue level. Deriving expressions for stress tensors starting from an energetic formulation of the model, we show that the principal axes of stress for an individual cell align with the principal axes of shape, and we determine the bulk effective tissue pressure when the monolayer is isotropic at the tissue level. Using simulations for a monolayer that is not under peripheral stress, we fit parameters of the model to experimental data for Xenopus embryonic tissue. The model predicts that mechanical interactions can generate mesoscopic patterns within the monolayer that exhibit long-range correlations in cell shape. The model also suggests that the orientation of mechanical and geometric cues for processes such as cell division are likely to be strongly correlated in real epithelia. Some limitations of the model in capturing geometric features of Xenopus epithelial cells are highlighted.

Mechanical characterization of disordered and anisotropic cellular monolayers

We consider a cellular monolayer, described using a vertex-based model, for which cells form a spatially disordered array of convex polygons that tile the plane. Equilibrium cell configurations are assumed to minimize a global energy defined in terms of cell areas and perimeters; energy is dissipated via dynamic area and length changes, as well as cell neighbor exchanges. The model captures our observations of an epithelium from a Xenopus embryo showing that uniaxial stretching induces spatial ordering, with cells under net tension (compression) tending to align with (against) the direction of stretch, but with the stress remaining heterogeneous at the single-cell level. We use the vertex model to derive the linearized relation between tissue-level stress, strain, and strain rate about a deformed base state, which can be used to characterize the tissue’s anisotropic mechanical properties; expressions for viscoelastic tissue moduli are given as direct sums over cells. When the base state is isotropic, the model predicts that tissue properties can be tuned to a regime with high elastic shear resistance but low resistance to area changes, or vice versa.

Mechanocellular models of epithelial morphogenesis

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

Epithelial Folding Driven by Apical or Basal-Lateral Modulation: Geometric Features, Mechanical Inference, and Boundary Effects

During embryonic development, epithelial sheets fold into complex structures required for tissue and organ functions. Although substantial efforts have been devoted to identifying molecular mechanisms underlying epithelial folding, far less is understood about how forces deform individual cells to sculpt the overall sheet morphology. Here we describe a simple and general theoretical model for the autonomous folding of monolayered epithelial sheets. We show that active modulation of intracellular mechanics along the basal-lateral as well as the apical surfaces is capable of inducing fold formation in the absence of buckling instability. Apical modulation sculpts epithelia into shallow and V-shaped folds, whereas basal-lateral modulation generates deep and U-shaped folds. These characteristic tissue shapes remain unchanged when subject to mechanical perturbations from the surroundings, illustrating that the autonomous folding is robust against environmental variabilities. At the cellular scale, how cells change shape depends on their initial aspect ratios and the modulation mechanisms. Such cell deformation characteristics are verified via experimental measurements for a canonical folding process driven by apical modulation, indicating that our theory could be used to infer the underlying folding mechanisms based on experimental data. The mechanical principles revealed in our model could potentially guide future studies on epithelial folding in diverse systems.

A mechanical model for diversified insect wing margin shapes

The wings in different insect species are morphologically distinct with regards to their size, outer contour (margin) shape, venation, and pigmentation. The basis of the diversity of wing margin shapes remains unknown, despite the fact that gene networks governing the Drosophila wing development have been well characterised. Among the different types of wing margin shapes, smoothly curved contour is the most frequently found and implies the existence of a highly organised, multicellular mechanical structure. Here, we developed a mechanical model for diversified insect wing margin shapes, in which non-uniform bending stiffness of the wing margin is considered. We showed that a variety of spatial distribution of the bending stiffness could reproduce diverse wing margin shapes. Moreover, the inference of the distribution of the bending stiffness from experimental images indicates a common spatial profile among insects tested. We further studied the effect of the intrinsic tension of the wing blade on the margin shape and on the inferred bending stiffness. Finally, we implemented the bending stiffness of the wing margin in the cell vertex model of the wing blade, and confirmed that the hybrid model retains the essential feature of the margin model. We propose that in addition to morphogenetic processes in the wing blade, the spatial profile of the bending stiffness in the wing margin can play a pivotal role in shaping insect wings.

Active Vertex Model for cell-resolution description of epithelial tissue mechanics

We introduce an Active Vertex Model (AVM) for cell-resolution studies of the mechanics of confluent epithelial tissues consisting of tens of thousands of cells, with a level of detail inaccessible to similar methods. The AVM combines the Vertex Model for confluent epithelial tissues with active matter dynamics. This introduces a natural description of the cell motion and accounts for motion patterns observed on multiple scales. Furthermore, cell contacts are generated dynamically from positions of cell centres. This not only enables efficient numerical implementation, but provides a natural description of the T1 transition events responsible for local tissue rearrangements. The AVM also includes cell alignment, cell-specific mechanical properties, cell growth, division and apoptosis. In addition, the AVM introduces a flexible, dynamically changing boundary of the epithelial sheet allowing for studies of phenomena such as the fingering instability or wound healing. We illustrate these capabilities with a number of case studies.

Vertex stability and topological transitions in vertex models of foams and epithelia

In computer simulations of dry foams and of epithelial tissues, vertex models are often used to describe the shape and motion of individual cells. Although these models have been widely adopted, relatively little is known about their basic theoretical properties. For example, while fourfold vertices in real foams are always unstable, it remains unclear whether a simplified vertex model description has the same behavior. Here, we study vertex stability and the dynamics of T1 topological transitions in vertex models. We show that, when all edges have the same tension, stationary fourfold vertices in these models do indeed always break up. In contrast, when tensions are allowed to depend on edge orientation, fourfold vertices can become stable, as is observed in some biological systems. More generally, our formulation of vertex stability leads to an improved treatment of T1 transitions in simulations and paves the way for studies of more biologically realistic models that couple topological transitions to the dynamics of regulatory proteins.

Lateral inhibition-induced pattern formation controlled by the size and geometry of the cell

Pattern formation in development biology is one of the fundamental processes by which cells change their functions. It is based on the communication of cells via intra- and intercellular dynamics of biochemicals. Thus, the cell is directly involved in biochemical interactions. However, many theoretical approaches describing biochemical pattern formation have usually neglected the cell's role or have simplified the subcellular process without considering cellular aspects despite the cell being the environment where biochemicals interact. On the other hand, recent experimental observations suggest that a change in the physical conditions of cell-to-cell contact can result in a change in cell fate and tissue patterning in a lateral inhibition system. Here we develop a mathematical model by which biochemical dynamics can be directly observed with explicitly expressed cell structure and geometry in higher dimensions, and reconsider pattern formation by lateral inhibition of the Notch-Delta signaling pathway. We explore how the physical characteristic of cell, such as cell geometry or size, influences the biochemical pattern formation in a multi-cellular system. Our results suggest that a property based on cell geometry can be a novel mechanism for symmetry breaking inducing cell asymmetry. We show that cell volume can critically influence cell fate determination and pattern formation at the tissue level, and the surface area of the cell-to-cell contact can directly affect the spatial range of patterning.

Modelling wound closure in an epithelial cell sheet using the cellular Potts model

We use a two-dimensional cellular Potts model to represent the behavior of an epithelial cell layer and describe its dynamics in response to a microscopic wound.