Although tissues are usually studied in isolation, this situation rarely occurs in biology, as cells, tissues and organs coexist and interact across scales to determine both shape and function. Here, we take a quantitative approach combining data from recent experiments, mathematical modelling and Bayesian parameter inference, to describe the self-assembly of multiple epithelial sheets by growth and collision. We use two simple and well-studied continuum models, where cells move either randomly or following population pressure gradients. After suitable calibration, both models prove to be practically identifiable, and can reproduce the main features of single tissue expansions. However, our findings reveal that whenever tissue–tissue interactions become relevant, the random motion assumption can lead to unrealistic behaviour. Under this setting, a model accounting for population pressure from different cell populations is more appropriate and shows a better agreement with experimental measurements. Finally, we discuss how tissue shape and pressure affect multi-tissue collisions. Our work thus provides a systematic approach to quantify and predict complex tissue configurations with applications in the design of tissue composites and more generally in tissue engineering.

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Cell-cell adhesion is one the most fundamental mechanisms regulating collective cell migration during tissue development, homeostasis and repair, allowing cell populations to selforganize and eventually form and maintain complex tissue shapes. Cells interact with each other via the formation of protrusions or filopodia and they adhere to other cells through binding of cell surface proteins. The resulting adhesive forces are then related to cell size and shape and, often, continuum models represent them by nonlocal attractive interactions. In this paper, we present a new continuum model of cell-cell adhesion which can be derived from a general nonlocal model in the limit of short-range interactions. This new model is local, resembling a system of thin-film type equations, with the various model parameters playing the role of surface tensions between different cell populations. Numerical simulations in one and two dimensions reveal that the local model maintains the diversity of cell sorting patterns observed both in experiments and in previously used nonlocal models. In addition, it also has the advantage of having explicit stationary solutions, which provides a direct link between the model parameters and the differential adhesion hypothesis.

Marine fisheries are an important source of food supply and play an important economic role in many regions worldwide. However, due to aggressive fishing practices they are increasingly overexploited. Marine reserves have the potential to alleviate this problem and moreover, they also provide a physical area where an alternative economic activity can exist without being in conflict with fishing gear. Here we explore the idea of combining multiple economic activities in a marine ecosystem, namely: fishing and tourism. We use a model in which the fish population evolves according to a reaction-diffusion partial differential equation, and we consider the interactions between fishing and tourism. We use optimal control theory to find, depending on the model parameters, the optimal management strategy. The results show that, subject to certain conditions, it is possible to have two different revenue streams in the same habitat in contrast with the classical view of competing uses. We also corroborate that marine reserves emerge as the optimal strategy and that the presence of visitors in these areas generates larger profits than if only fishing was considered.

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Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equation is in general density-dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium type equation on networks, which can be coarse-grained to obtain a known nonlinear equation. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.

We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.