We consider emergent collective behavior of a multicellular biological system. Specifically, we investigate the role of hypoxia (lack of oxygen) in migration of brain tumor cells. We performed two series of cell migration experiments. In the first set of experiments, cell migration away from a tumor spheroid was investigated. The second set of experiments was performed in a typical wound-healing geometry: Cells were placed on a substrate, a scratch was made, and cell migration into the gap was investigated. Experiments show a surprising result: Cells under normal and hypoxic conditions have migrated the same distance in the "spheroid" experiment, while in the "scratch" experiment cells under normal conditions migrated much faster than under hypoxic conditions. To explain this paradox, we formulate a discrete stochastic model for cell dynamics. The theoretical model explains our experimental observations and suggests that hypoxia decreases both the motility of cells and the strength of cell-cell adhesion. The theoretical predictions were further verified in independent experiments.

We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front speed as a power series expansion in a parameter, p, that gives the relative size of proliferation and diffusion processes for the invading cells. In two dimensions the model becomes the Eden model for p ≈ 1. In both one and two dimensions for small p, front propagation for this model should approach that of the Fisher-Kolmogorov equation. However, as in other cases, this discrete model approaches Fisher-Kolmogorov behavior slowly.

We study the in vitro dynamics of the malignant brain tumor glioblastoma multiforme. The growing tumor consists of a dense proliferating zone and an outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior: one cell line invades in a spherically symmetric manner, but another gives rise to branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a qualitative agreement with experimental findings for the two types of cells.

We present a stochastic model which describes fronts of cells invading a wound. In the model cells can move, proliferate, and experience cell-cell adhesion. We find several qualitatively different regimes of front motion and analyze the transitions between them. Above a critical value of adhesion and for small proliferation large isolated clusters are formed ahead of the front. This is mapped onto the well-known ferromagnetic phase transition in the Ising model. For large adhesion, and larger proliferation the clusters become connected (at some fixed time). For adhesion below the critical value the results are similar to our previous work which neglected adhesion. The results are compared with experiments, and possible directions of future work are proposed.