A free boundary model of epithelial dynamics

A mathematical model of wound healing in bovine corneal endothelium

We developed a mathematical model to describe healing processes in bovine corneal endothelial (BCE) cells in culture, triggered by mechanical wounds with parallel edges. Previous findings from our laboratory show that, in these cases, BCE monolayers exhibit an approximately constant healing velocity. Also, that caspase-dependent apoptosis occurs, with the fraction of apoptotic cells increasing with the distance traveled by the healing edge. In addition, in this study we report the novel findings that, for wound scratch assays performed preserving the basal extracellular matrix: i) the healing cells increase their en face surface area in a characteristic fashion, and ii) the average length of the segments of the cell columns actively participating in the healing process increases linearly with time. These latter observations preclude the utilization of standard traveling wave formalisms to model wound healing in BCE cells. Instead, we developed and studied a simple phenomenological model based on a plausible formula for the spreading dynamics of the individual healing cells, that incorporates original evidence about the process in BCE cells. The model can be simulated to: i) obtain an approximately constant healing velocity; ii) reproduce the profile of the healing cell areas, and iii) obtain approximately linear time dependences of the mean cell area and average length of the front active segments per column. In view of its accuracy to account for the experimental observations, the model can also be acceptably employed to quantify the appearance of apoptotic cells during BCE wound healing. The strategy utilized here could offer a novel formal framework to represent modifications undergone by some epithelial cell lines during wound healing.

The role of mechanical interactions in EMT

The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF-β, produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.

A free boundary mechanobiological model of epithelial tissues

In this study, we couple intracellular signalling and cell-based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction-diffusion equations are derived to describe tissue-level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac-Rho pathway where cell- and tissue-level mechanics are directly related to intracellular signalling. Second, we study an activator-inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell-level processes included in the discrete model.

Mechanical Cell Competition in Heterogeneous Epithelial Tissues

Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than an intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.

From a discrete model of chemotaxis with volume-filling to a generalized Patlak-Keller-Segel model

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.

Spatial structure arising from chase-escape interactions with crowding

Movement of individuals, mediated by localised interactions, plays a key role in numerous processes including cell biology and ecology. In this work, we investigate an individual-based model accounting for various intraspecies and interspecies interactions in a community consisting of two distinct species. In this framework we consider one species to be chasers and the other species to be escapees, and we focus on chase-escape dynamics where the chasers are biased to move towards the escapees, and the escapees are biased to move away from the chasers. This framework allows us to explore how individual-level directional interactions scale up to influence spatial structure at the macroscale. To focus exclusively on the role of motility and directional bias in determining spatial structure, we consider conservative communities where the number of individuals in each species remains constant. To provide additional information about the individual-based model, we also present a mathematically tractable deterministic approximation based on describing the evolution of the spatial moments. We explore how different features of interactions including interaction strength, spatial extent of interaction, and relative density of species influence the formation of the macroscale spatial patterns.

Polarization wave at the onset of collective cell migration

Collective cell migration underlies morphogenesis, tissue regeneration, and cancer progression. How the biomechanical coupling between epithelial cells triggers and coordinates the collective migration is an open question. Here, we develop a one-dimensional model for an epithelial monolayer which predicts that after the onset of migration at an open boundary, cells in the bulk of the epithelium are gradually recruited into outward-directed motility, exhibiting traveling-wave-like behavior. We find an exact formula for the speed of this motility wave proportional to the square root of the cells' contractility, which accounts for cortex tension and adhesion between adjacent cells.

Reversible signal transmission in an active mechanical metamaterial

Mechanical metamaterials are designed to enable unique functionalities, but are typically limited by an initial energy state and require an independent energy input to function repeatedly. Our study introduces a theoretical active mechanical metamaterial that incorporates a biological reaction mechanism to overcome this key limitation of passive metamaterials. Our material allows for reversible mechanical signal transmission, where energy is reintroduced by the biologically motivated reaction mechanism. By analysing a coarse-grained continuous analogue of the discrete model, we find that signals can be propagated through the material by a travelling wave. Analysis of the continuum model provides the region of the parameter space that allows signal transmission, and reveals similarities with the well-known FitzHugh-Nagumo system. We also find explicit formulae that approximate the effect of the time scale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.

A one-dimensional individual-based mechanical model of cell movement in heterogeneous tissues and its coarse-grained approximation

Mechanical heterogeneity in biological tissues, in particular stiffness, can be used to distinguish between healthy and diseased states. However, it is often difficult to explore relationships between cellular-level properties and tissue-level outcomes when biological experiments are performed at a single scale only. To overcome this difficulty, we develop a multi-scale mathematical model which provides a clear framework to explore these connections across biological scales. Starting with an individual-based mechanical model of cell movement, we subsequently derive a novel coarse-grained system of partial differential equations governing the evolution of the cell density due to heterogeneous cellular properties. We demonstrate that solutions of the individual-based model converge to numerical solutions of the coarse-grained model, for both slowly-varying-in-space and rapidly-varying-in-space cellular properties. We discuss applications of the model, such as determining relative cellular-level properties and an interpretation of data from a breast cancer detection experiment.