Topological graph description of multicellular dynamics based on vertex model

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

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

Couple stresses and discrete potentials in the vertex model of cellular monolayers

The vertex model is widely used to simulate the mechanical properties of confluent epithelia and other multicellular tissues. This inherently discrete framework allows a Cauchy stress to be attributed to each cell, and its symmetric component has been widely reported, at least for planar monolayers. Here, we consider the stress attributed to the neighbourhood of each tricellular junction, evaluating in particular its leading-order antisymmetric component and the associated couple stresses, which characterise the degree to which individual cells experience (and resist) in-plane bending deformations. We develop discrete potential theory for localised monolayers having disordered internal structure and use this to derive the analogues of Airy and Mindlin stress functions. These scalar potentials typically have broad-banded spectra, highlighting the contributions of small-scale defects and boundary layers to global stress patterns. An affine approximation attributes couple stresses to pressure differences between cells sharing a trijunction, but simulations indicate an additional role for non-affine deformations.

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.

The Random Plots Graph Generation Model for Studying Systems with Unknown Connection Structures

We consider the problem of modeling complex systems where little or nothing is known about the structure of the connections between the elements. In particular, when such systems are to be modeled by graphs, it is unclear what vertex degree distributions these graphs should have. We propose that, instead of attempting to guess the appropriate degree distribution for a poorly understood system, one should model the system via a set of sample graphs whose degree distributions cover a representative range of possibilities and account for a variety of possible connection structures. To construct such a representative set of graphs, we propose a new random graph generator, Random Plots, in which we (1) generate a diversified set of vertex degree distributions and (2) target a graph generator at each of the constructed distributions, one-by-one, to obtain the ensemble of graphs. To assess the diversity of the resulting ensembles, we (1) substantialize the vague notion of diversity in a graph ensemble as the diversity of the numeral characteristics of the graphs within this ensemble and (2) compare such formalized diversity for the proposed model with that of three other common models (Erdős–Rényi–Gilbert (ERG), scale-free, and small-world). Computational experiments show that, in most cases, our approach produces more diverse sets of graphs compared with the three other models, including the entropy-maximizing ERG. The corresponding Python code is available at GitHub.

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.

A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation

We discuss several continuum cell-cell adhesion models based on the underlying microscopic assumptions. We propose an improvement on these models leading to sharp fronts and intermingling invasion fronts between different cell type populations. The model is based on basic principles of localized repulsion and nonlocal attraction due to adhesion forces at the microscopic level. The new model is able to capture both qualitatively and quantitatively experiments by Katsunuma et al. (2016). We also review some of the applications of these models in other areas of tissue growth in developmental biology. We finally explore the resulting qualitative behavior due to cell-cell repulsion.

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.

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.