Eirikur Palsson

A three-dimensional model of cell movement in multicellular systems

A mathematical model for cell movement in multicellular systems has been developed that allows us to simulate and visualize, in three dimensions, individual cell movements in a number of multicellular systems. These include cell movement during aggregation and slug stage of Dictyostelium discoideum, embryogenesis, limb formation and wound healing. The model is quite adaptable to a number of systems, due to the way it is designed. The building blocks of the model are individual cells, where each cell has certain given properties that are not necessarily the same for all cells. The basic properties are that a cell can deform under force (either stretch or compress), while conserving its volume, it adheres to other cells and it can generate an active motive force. The response of a cell depends on its internal parameter state, and on the information it receives from its external environment, which includes neighbor cells, the extracellular matrix and chemical signals. The net force on a cell is calculated by summing up all the forces that a cell experiences at its surroundings. Each cell is then moved and deformed according to the equations of motion and deformation. Finally, the net movement of all the cells gives the collective movement of the entire tissue. Here we introduce this model and show examples of its applications and compare the results with experimental data. In the first simulations, we show how different cell types can be sorted out based solely on differences in adhesion. We compare our results to cell sorting experiments done by Steinberg and co-workers [R.A. Foty, C.M. Pfleger, G. Forgacs, M.S. Steinberg, Development 122 (1996) 1611–1620; M.S. Steinberg, Reconstruction of tissues by dissociated cells, Science 141 (1963) 3579] using values for adhesion within the range of the experimental values, and show that the model reproduces the experiments very well. We also present results from simulations of Dictyostelium movements. We first modeled the aggregation stage, where cells are aggregating chemotactically, towards a signaling center, in response to cAMP waves. In these simulations one can observe stream formation and how the mound arises due to the inward motion of the cells towards the signaling center. Later we studied simulations of 2D slugs, and compared them to observations of 2D slugs done by Bonner [J.T. Bonner, Proc. Natl. Acad. Sci. USA 95 (1998) 9355–9359].

A 3-D model used to explore how cell adhesion and stiffness affect cell sorting and movement in multicellular systems

A three-dimensional mathematical model is used to determine the effects of adhesion and cell signalling on cell movements during the aggregation and slug stages of Dictyostelium discoideum (Dd) and to visualize cell sorting. The building blocks of the model are individual deformable ellipsoidal cells, where movement depends on internal parameter state (cell size and stiffness) and on external cues from the neighboring cells, extracellular matrix, and chemical signals. Cell movement and deformation are calculated from equations of motion using the total force acting on each cell, ensuring that forces are balanced. The simulations show that the sorting patterns of prestalk and prespore cells, emerging during the slug stage, depend critically on the type of cell adhesion and not just on chemotactic differences between cells. This occurs because cell size and stiffness can prevent the otherwise faster cells from passing the slower cells. The patterns are distinctively different when the prestalk cells are more or less adhesive than the prespore cells. These simulations suggest that sorting is not solely due to differential chemotaxis, and that differences in both adhesion strength and type between different cell types play a very significant role, both in Dictyostelium and other systems.

Mathematical model of macrophage-facilitated breast cancer cells invasion

Mortality from breast cancer stems from its tendency to invade into surrounding tissues and organs. Experiments have shown that this metastatic process is facilitated by macrophages in a short-ranged chemical signalling loop. Macrophages secrete epidermal growth factor, EGF, and respond to the colony stimulating factor 1, CSF-1. Tumor cells secrete CSF-1 and respond to EGF. In this way, the cells coordinate aggregation and cooperative migration. Here we investigate this process in a model for in vitro interactions using two distinct but related mathematical approaches. In the first, we analyze and simulate a set of partial differential equations to determine conditions for aggregation. In the second, we use a cell-based discrete 3D simulation to follow the fates and motion of individual cells during aggregation. Linear stability analysis of the PDE model reveals that decreasing the chemical secretion, chemotaxis coefficients or density of cells or increasing the chemical degradation in the model could eliminate the spontaneous aggregation of cells. Simulations with the discrete model show that the ratio between tumor cells and macrophages in aggregates increases when the EGF secretion parameter is increased. The results also show how CSF-1/CSF-1R autocrine signalling in tumor cells affects the ratio between the two cell types. Comparing the continuum results with simulations of a discrete cell-based model, we find good qualitative agreement.

On the evolution of omnivory in a community context

Omnivory is extremely common in animals, yet theory predicts that when given a choice of resources specialization should be favored over being generalist. The evolution of a feeding phenotype involves complex interactions with many factors other than resource choice alone, including environmental heterogeneity, resource quality, availability, and interactions with other organisms. We applied an evolutionary simulation model to examine how ecological conditions shape evolution of feeding phenotypes (e.g., omnivory), by varying the quality and availability (absolute and relative) of plant and animal (prey) resources. Resulting feeding phenotypes were defined by the relative contribution of plants and prey to diets of individuals. We characterized organisms using seven traits that were allowed to evolve freely in different simulated environments, and we asked which traits are important for different feeding phenotypes to evolve among interacting organisms. Carnivores, herbivores, and omnivores all coexisted without any requirement in the model for a synergistic effect of eating plant and animal prey. Omnivores were most prevalent when ratio of plants and animal prey was low, and to a lesser degree, when habitat productivity was high. A key result of the model is that omnivores evolved through many different combinations of trait values and environmental contexts. Specific combinations of traits tended to form emergent trait complexes, and under certain environmental conditions, are expressed as omnivorous feeding phenotypes. The results indicate that relative availabilities of plants and prey (over the quality of resources) determine an individuals feeding class and that feeding phenotypes are often the product of convergent evolution of emergent trait complexes under specific environmental conditions. Foraging outcomes appear to be consequences of degree and type of phenotypic specialization for plant and animal prey, navigation and exploitation of the habitat, reproduction, and interactions with other individuals in a heterogeneous environment. Omnivory should not be treated as a fixed strategy, but instead a pattern of phenotypic expression, emerging from diverse genetic sources and coevolving across a range of ecological contexts.

A 3-D Deformable Ellipsoidal Cell Model with Cell Adhesion and Signaling

In this chapter a three-dimensional model of ellipsoidal cells is presented and used to study how cell-cell signaling, cell adhesion, chemotaxis and differentiation all work together in a coordinated fashion to give rise to the developed organism. The Dictyostelium discoideum is used as a model system, since it is simple, yet has all the basic cell-cell interactions. Another goal of introducing this model is to achieve visualization of cell movements and signal propagation in 3-D space.

Polarization and migration in the zebrafish posterior lateral line system

Collective cell migration plays an important role in development. Here, we study the posterior lateral line primordium (PLLP) a group of about 100 cells, destined to form sensory structures, that migrates from head to tail in the zebrafish embryo. We model mutually inhibitory FGF-Wnt signalling network in the PLLP and link tissue subdivision (Wnt receptor and FGF receptor activity domains) to receptor-ligand parameters. We then use a 3D cell-based simulation with realistic cell-cell adhesion, interaction forces, and chemotaxis. Our model is able to reproduce experimentally observed motility with leading cells migrating up a gradient of CXCL12a, and trailing (FGF receptor active) cells moving actively by chemotaxis towards FGF ligand secreted by the leading cells. The 3D simulation framework, combined with experiments, allows an investigation of the role of cell division, chemotaxis, adhesion, and other parameters on the shape and speed of the PLLP. The 3D model demonstrates reasonable behaviour of control as well as mutant phenotypes.