Generalized travelling wave solutions for a microscopic chemotaxis model.

Abstract

In biology cell migration is one of the most critical processes, for it is decisive in the mechanisms leading to the beginning of life. The collective migration of cells via wave motion plays a key role in understanding many essential steps in developmental processes. It is often modelled as a system of partial differential equations (PDEs). We investigate in a one-dimensional microscopic model, the formation of travelling bands (via wave motion) of bacteria E coli, caused by the chemotactic response of cells to a signal moving with constant speed. We also look at the impact of cell growth and unbiased turning rate on the behaviour of our system. The model derives from the experimental observation reported in Budrene and Berg (1991, 1995). In the first problem we tackle, we overlook the proliferation of the cells and we search for travelling wave solutions in the case where the cells do not starve. We show that, using a group theoretical approach, a larger class of travelling wave solutions than that obtained from the standard ansatz is possible. By applying realistic initial and boundary conditions, we restrict the general solutions appropriately. This is the first time that explicit travelling wave solutions have been obtained for this system of equations. In particular, we treat the full system, including non-zero diffusivity terms, unlike previous approaches. Importantly, we provide biologically relevant solutions. The second problem focuses on the metabolism effect in the case of starvation. It was observed experimentally that a low concentration of nutrients may not cause the band to break up, but rather impel the cells to consume the excreted signal. Here cell growth is allowed with constant rate. We use asymptotic methods to prove the existence of travelling wave solutions in both the case of diffusivity and non diffusivity. Significant results have been obtained. In the last problem we incorporate the proliferation of the cells in the case of non-limiting resources. Constant cell growth and a nutrient dependent proliferation rate are considered. We combine a dynamical systems analysis with other analytic methods to investigate the behaviour of the solutions. Travelling wave solutions have been obtained both for high chemotactic sensitivity, and also in the case of no chemotaxis. Explicit, biologically and pertinent solutions have been provided, confirming the validation of the model.