Mathematical Models of Self-Organized Patterning of human Embryonic Stem Cells (hESCs)

Development of an organism from a single celled zygote is a highly complicated process which consists of division of cells and specialized differentiation. Due to the difficulties posed by direct observations in vivo, mathematical modeling is an important component of studies concerning embryonic development. We describe a reaction-diffusion model for the activator-inhibitor cascade including BMP4, Wnt, Nodal and their inhibitors. We use Robin boundary conditions to effectively model the physical realities of the system. Numerical simulations are used to study these reaction diffusion systems using finite element approximations. We also present a detailed theoretical proof of the existence and stability of steady state solutions for a specialized activator-inhibitor system, subject to certain sufficient conditions on the data of the problem.