Receptor-Based Models with Diffusion-Driven Instability for Pattern Formation in Hydra

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Abstract
The aim of this paper 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 analyzed 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. Numerical simulations are done to illustrate the analysis. The four-variable model is able to capture some results from cutting experiments and reflects de novo pattern formation from dissociated cells.