Receptor based models for pattern formation in hydra

The aim of this work is to show under which conditions a receptor-based model can produce and regulate patterns. Such model is applied to the pattern formation and regulation in a fresh water polyp, hydra.

The model is based on the idea that both head and foot formation could be controlled by receptor-ligand binding. Positional value is determined by the density of bound receptors. The model is defined in the form of reaction-diffusion equations coupled with ordinary differential equations.

The objective is to check what minimal processes are sufficient to produce patterns in the framework of a diffusion-driven (Turing-type) instability. Three-variable (describing the dynamics of ligands, free and bound receptors) and four-variable models (including also an enzyme cleaving the ligand) are analysed and compared. The minimal three-variable model takes into consideration the density of free receptors, bound receptors and ligands. In such model patterns can evolve only if self-enhancement of free receptors, i.e. a positive feedback loop between the production of new free receptors and their present density, is assumed.

The final pattern strongly depends on initial conditions. In the four-variable model a diffusion-driven instability occurs without the assumption that free receptors stimulate their own synthesis. It is shown that gradient in the density of bound receptors occurs if there is also a second diffusible substance, an enzyme, which degrades ligands. The four-variable model is able to capture some results from cutting experiments and reflects {\it{de novo}} pattern formation from dissociated cells.

The results of grafting experiments suggest that model should involve a memory-based relation. It is shown that the model is able to capture results from experiments if the dynamics of production of ligands and enzyme are described by the system of ordinary differential equations showing hysteresis.