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

Discrete network models of endothelial cells and their interactions with the substrate

Endothelial cell monolayers line the inner surfaces of blood and lymphatic vessels. They are continuously exposed to different mechanical loads, which may trigger mechanobiological signals and hence play a role in both physiological and pathological processes. Computer-based mechanical models of cells contribute to a better understanding of the relation between cell-scale loads and cues and the mechanical state of the hosting tissue. However, the confluency of the endothelial monolayer complicates these approaches since the intercellular cross-talk needs to be accounted for in addition to the cytoskeletal mechanics of the individual cells themselves. As a consequence, the computational approach must be able to efficiently model a large number of cells and their interaction. Here, we simulate cytoskeletal mechanics by means of molecular dynamics software, generally suitable to deal with large, locally interacting systems. Methods were developed to generate models of single cells and large monolayers with hundreds of cells. The single-cell model was considered for a comparison with experimental data. To this end, we simulated cell interactions with a continuous, deformable substrate, and computationally replicated multistep traction force microscopy experiments on endothelial cells. The results indicate that cell discrete network models are able to capture relevant features of the mechanical behaviour and are thus well-suited to investigate the mechanics of the large cytoskeletal network of individual cells and cell monolayers.

Active viscoelastic models for cell and tissue mechanics

Living cells are out of equilibrium active materials. Cell-generated forces are transmitted across the cytoskeleton network and to the extracellular environment. These active force interactions shape cellular mechanical behaviour, trigger mechano-sensing, regulate cell adaptation to the microenvironment and can affect disease outcomes. In recent years, the mechanobiology community has witnessed the emergence of many experimental and theoretical approaches to study cells as mechanically active materials. In this review, we highlight recent advancements in incorporating active characteristics of cellular behaviour at different length scales into classic viscoelastic models by either adding an active tension-generating element or adjusting the resting length of an elastic element in the model. Summarizing the two groups of approaches, we will review the formulation and application of these models to understand cellular adaptation mechanisms in response to various types of mechanical stimuli, such as the effect of extracellular matrix properties and external loadings or deformations.

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.

Junctional and cytoplasmic contributions in wound healing

Wound healing is characterized by the re-epitheliation of a tissue through the activation of contractile forces concentrated mainly at the wound edge. While the formation of an actin purse string has been identified as one of the main mechanisms, far less is known about the effects of the viscoelastic properties of the surrounding cells, and the different contribution of the junctional and cytoplasmic contractilities. In this paper, we simulate the wound healing process, resorting to a hybrid vertex model that includes cell boundary and cytoplasmic contractilities explicitly, together with a differentiated viscoelastic rheology based on an adaptive rest-length. From experimental measurements of the recoil and closure phases of wounds in the Drosophila wing disc epithelium, we fit tissue viscoelastic properties. We then analyse in terms of closure rate and energy requirements the contributions of junctional and cytoplasmic contractilities. Our results suggest that reduction of junctional stiffness rather than cytoplasmic stiffness has a more pronounced effect on shortening closure times, and that intercalation rate has a minor effect on the stored energy, but contributes significantly to shortening the healing duration, mostly in the later stages.

Development of a New 3D Hybrid Model for Epithelia Morphogenesis

Many epithelial developmental processes like cell migration and spreading, cell sorting, or T1 transitions can be described as planar deformations. As such, they can be studied using two-dimensional tools and vertex models that can properly predict collective dynamics. However, many other epithelial shape changes are characterized by out-of-plane mechanics and three-dimensional effects, such as bending, cell extrusion, delamination, or invagination. Furthermore, during planar cell dynamics or tissue repair in monolayers, spatial intercalation between the apical and basal sides has even been detected. Motivated by this lack of symmetry with respect to the midsurface, we here present a 3D hybrid model that allows us to model differential contractility at the apical, basal or lateral sides. We use the model to study the effects on wound closure of solely apical or lateral contractile contributions and show that an apical purse-string can be sufficient for full closure when it is accompanied by volume preservation.

Epithelial Migration and Non-adhesive Periderm Are Required for Digit Separation during Mammalian Development

The fusion of digits or toes, syndactyly, can be part of complex syndromes, including van der Woude syndrome. A subset of van der Woude cases is caused by dominant-negative mutations in the epithelial transcription factor Grainyhead like-3 (GRHL3), and Grhl3^-/-mice have soft-tissue syndactyly. Although impaired interdigital cell death of mesenchymal cells causes syndactyly in multiple genetic mutants, Grhl3^-/- embryos had normal interdigital cell death, suggesting alternative mechanisms for syndactyly. We found that in digit separation, the overlying epidermis forms a migrating interdigital epithelial tongue (IET) when the epithelium invaginates to separate the digits. Normally, the non-adhesive surface periderm allows the IET to bifurcate as the digits separate. In contrast, in Grhl3^-/- embryos, the IET moves normally between the digits but fails to bifurcate because of abnormal adhesion of the periderm. Our study identifies epidermal developmental processes required for digit separation.

A Diffuse Interface Framework for Modeling the Evolution of Multi-cell Aggregates as a Soft Packing Problem Driven by the Growth and Division of Cells

We present a model for cell growth, division and packing under soft constraints that arise from the deformability of the cells as well as of a membrane that encloses them. Our treatment falls within the framework of diffuse interface methods, under which each cell is represented by a scalar phase field and the zero level set of the phase field represents the cell membrane. One crucial element in the treatment is the definition of a free energy density function that penalizes cell overlap, thus giving rise to a simple model of cell-cell contact. In order to properly represent cell packing and the associated free energy, we include a simplified representation of the anisotropic mechanical response of the underlying cytoskeleton and cell membrane through penalization of the cell shape change. Numerical examples demonstrate the evolution of multi-cell clusters and of the total free energy of the clusters as a consequence of growth, division and packing.