| In this thesis, we consider a mathematical model of a particular step in the development of Drosophila. The characterizing feature of this step, which we will try to mimic, is the isolation of the morphogens Wg to the ventral region and Dpp of the dorsal region to the leg imaginal disc. The morphogens under consideration are wingless (Wg) and decapentaplegic (Dpp). Experimental evidence has suggested that the production rates of Wg and Dpp may be mutually inhibited and self-enhanced with respect to concentrations of bound Wg and Dpp. We will examine the role of inhibition and auto-catalysis in the formation of spatial gradients. The system will be modeled by two coupled reaction-diffusion equations governing the concentration of free morphogen. These equations will be coupled to ordinary differential equations governing the concentration of bound morphogen. Many of the parameters used in the mathematical model will come from experimental observation, so the model may be tested quantitatively as well qualitatively.; In chapter 2, we will use the Hill equation to model activation/inhibition. This choice results in the necessary condition that the Hill coefficient be greater than 2. In chapter 3, we will examine the possibility of territory formation due solely to auto-activation. We will study the stability of heteroclinic connection for this reduced model with calculation of exponentially small eigenvalue. We will show that the constructed solution for a single rnorphogen is unstable. In chapter 4, we derive the Hill type dynamics and sequential and independent ligand binding. We will use numerical simulations to demonstrate that a coupled system of two morphogens with auto-activation and cross inhibition can generate stable solutions. |
