  Chemotaxis is a directed cell movement in response to external chemical
stimuli. In this paper, we propose a simple model for the origin of chemotaxis
- namely how a directed movement in response to an external chemical signal may
occur based on purely reaction-diffusion equations reflecting inner working of
the cells. The model is inspired by the well-studied role of the rho-GTPase
Cdc42 regulator of cell polarity, in particular in yeast cells. We analyze
several versions of the model in order to better understand its analytic
properties, and prove global regularity in one and two dimensions. Using
computer simulations, we demonstrate that in the framework of this model, at
least in certain parameter regimes, the speed of the directed movement appears
to be proportional to the size of the gradient of signalling chemical. This
coincides with the form of the chemical drift in the most studied mean field
model of chemotaxis, the Keller-Segel equation.
