Elisabeth G. Rens

The small GTPases Rac and Rho are known to regulate eukaryotic cell shape, promoting front protrusion (Rac) or rear retraction (Rho) of the cell edge. Such cell deformation changes the contact and adhesion of cell to the extracellular matrix (ECM), while ECM signaling through integrin receptors also affects GTPase activity. We develop and investigate a model for this three-way feedback loop in 1D and 2D spatial domains, as well as in a fully deforming 2D cell shapes with detailed adhesion-bond biophysics. The model consists of reaction–diffusion equations solved numerically with open-source software, Morpheus, and with custom-built cellular Potts model simulations. We find a variety of patterns and cell behaviors, including persistent polarity, flipped front-back cell polarity oscillations, spiral waves, and random protrusion-retraction. We show that the observed spatial patterns depend on the cell shape, and vice versa.

The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.

Epithelial branching morphogenesis drives the development of organs such as the lung, salivary gland, kidney and the mammary gland. It involves cell proliferation, cell differentiation and cell migration. An elaborate network of chemical and mechanical signals between the epithelium and the surrounding mesenchymal tissues regulates the formation and growth of branching organs. Surprisingly, when cultured in isolation from mesenchymal tissues, many epithelial tissues retain the ability to exhibit branching morphogenesis even in the absence of proliferation. In this work, we propose a simple, experimentally plausible mechanism that can drive branching morphogenesis in the absence of proliferation and cross-talk with the surrounding mesenchymal tissue. The assumptions of our mathematical model derive from in vitro observations of the behaviour of mammary epithelial cells. These data show that autocrine secretion of the growth factor TGF β 1 inhibits the formation of cell protrusions, leading to curvature-dependent inhibition of sprouting. Our hybrid cellular Potts and partial-differential equation model correctly reproduces the experimentally observed tissue-geometry-dependent determination of the sites of branching, and it suffices for the formation of self-avoiding branching structures in the absence and also in the presence of cell proliferation. This article is part of the theme issue ‘Multi-scale analysis and modelling of collective migration in biological systems’.

Epithelial branching morphogenesis drives the development of organs such as the lung, salivary gland, kidney and the mammary gland. It involves cell proliferation, cell differentiation and cell migration. An elaborate network of chemical and mechanical signals between the epithelium and the surrounding mesenchymal tissues regulates the formation and growth of branching organs. Surprisingly, when cultured in isolation from mesenchymal tissues, many epithelial tissues retain the ability to exhibit branching morphogenesis even in absence of proliferation. In this work, we propose a simple, experimentally-plausible mechanism that can drive branching morphogenesis in absence of proliferation and cross-talk with the surrounding mesenchymal tissue. The assumptions of our mathematical model derive from in vitro observations of the behavior of mammary epithelial cells. These data show that autocrine secretion of the growth factor TGF β 1 inhibits the formation of cell protrusions, leading to curvature dependent inhibition of sprouting. Our hybrid cellular Potts and partial-differential equation model correctly reproduces the experimentally observed tissue-geometry dependent determination of the sites of branching, and it suffices for the formation of self-avoiding branching structures in absence and also in presence of cell proliferation.

Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian. Using the fact that a force field is easily derived from a scalar energy (F = −∇H), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We predict the traction forces exerted by single cells of various shapes and sizes on a 2D substrate. While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we approximate the force field inside the cell domain using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces exerted by cells on one another in classic cell-sorting experiments. Cells exert forces on their surroundings and on one another. In simulations of cell shape using the Cellular Potts Model (CPM), the dynamics of deforming cell shapes is traditionally represented by an energy-minimization method. We use this CPM energy, the Hamiltonian, to derive and visualize the corresponding forces exerted by the cells. We use the fact that force is the negative gradient of energy to assign forces to the CPM cell edges, and then extend the results to approximate interior forces by interpolation. We show that this method works for single as well as multiple interacting model cells, both static and motile. Finally, we show favorable comparison between predicted forces and real forces measured experimentally.

The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.

Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian while permitting stochastic fluctuations. Using the fact that a force field is easily derived from a scalar energy ( F = −∇ H ), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We display the predicted forces for single cells of various shapes and sizes (relative to cell rest-area and cell rest-perimeter). While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we infer internal forces using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces associated with the dynamics of classic cell-sorting experiments in larger clusters of model cells. Author summary: Cells exert forces on their surroundings and on one another. In simulations of cell shape using the Cellular Potts Model (CPM), the dynamics of deforming cell shapes is traditionally represented by an energy-minimization method. We use this CPM energy, the Hamiltonian, to derive and visualize the corresponding forces exerted by the cells. We use the fact that force is the negative gradient of energy to assign forces to the CPM cell edges, and then extend the results to interior forces by interpolation. We show that this method works for single as well as multiple interacting model cells, both static and motile. Finally, we show favorable comparison between predicted forces and real forces measured experimentally.

Mathematical modeling is an essential approach for the understanding of complex multicellular behaviors in tissue morphogenesis. Here, we review the cellular Potts model (CPM; also known as the Glazier‐Graner‐Hogeweg model), an effective computational modeling framework. We discuss its usability for modeling complex developmental phenomena by examining four fundamental examples of tissue morphogenesis: (i) cell sorting, (ii) cyst formation, (iii) tube morphogenesis in kidney development, and (iv) blood vessel formation. The review provides an introduction for biologists for starting simulation analysis using the CPM framework.