Using Continuous Limit Techniques and Stochastic Computational Modeling to Predict the Biological Behavior of Aggregating Cells

Mathematical modeling and computational simulations can be used to predict the average behavior of many cell-cell interactions. In this thesis, results on modeling the predatory bacteria myxococcus xanthus and the slug forming amoeba dictyostelium discoideum are presented as well as results on random placement of non-overlapping cells in an individual based model, and continuous limits of discrete stochastic systems. Periodic reversals in systems of self-propelled rod shaped bacteria can lead to ordering of cells, thus enabling them to effectively resolve traffic jams formed during swarming and maximize their swarming rate. A connection is described between a one dimensional cell-based stochastic model of reversing non-overlapping bacteria and a non-linear diffusion equation. Boltzmann- Matano analysis is used to determine the nonlinear diffusion equation corresponding to the specific reversal frequency. It is shown that cell populations with high reversal frequencies are able to spread out effectively at high densities. If the cells rarely reverse then they are able to spread out at lower densities but are not able to spread out at higher densities. The amoeba Dictyostelium is able to sense the location of other members of its species and aggregate, using chemotaxis. During aggregation, the amoeba creates and follows the gradient of a diffusive chemical. Through local behavioral chemotactic rules, the cells are able to cluster together and create slugs. PDE equations describing this behavior have been derived in (Lushnikov, 2008). Existence, uniqueness and boundedness of solutions of this equation are proven in (Alber, 2009) and this thesis. The 1D steady state solutions and their stability are also analyzed. It is predicted that stable solutions correspond to biologically realizable aggregates, and conditions for determining whether or not certain amoeba aggregation patterns will occur in nature are presented.