Cell signaling is at the basis of many biological processes such as development, tissue repair, and homeostasis. It can be carried out by different mechanisms. Here we are focusing on ligand mediated cell-to-cell signaling in which a molecule (ligand) is free to move into the extra-cellular medium. On the cell layer surface, it can bind to its molecule-specific receptors located on the cell plasma membrane. This mechanism is the subject of many experimental and theoretical studies on many model biological systems, such as the follicular epithelium of the Drosophila egg, which motivates this work.
Here, we present a general mathematical model that incorporates the processes that characterize ligand mediated intercellular signaling in epithelial layers, and we study this model under various assumptions. This model is characterized by nonlinear reaction-diffusion type dynamics. The nonlinearities mainly arise at cell layer where feedbacks can be generated through e.g. ligand mediated ligand release. A direct consequence of this nonlinear behavior is a possibility of existence of multiple steady states and traveling wave solutions. In this thesis, we investigate these types of solutions numerically and analytically.
Looking for steady solutions of parabolic reaction-diffusion equations leads to nonlinear elliptic boundary-value problems. We take advantage of this property and develop an extension of the method of Optimal Grids for elliptic problems. We call this method Compensated Optimal Grids. We present its application and study its convergence properties. Ultimately, we show that this method can be spectrally accurate when applied to two dimensional problems, and fourth order accurate in the three dimensional case, despite the use of simple nearest-neighbor stencils.
We conclude our study by investigating signal transmission in the presence of bulk degradation. In particular, we study the multiplicity of steady states and construct traveling waves solutions for a mathematical model that accounts for degradation in the bulk, also obtaining biophysical conditions in which signal transmission is possible.