Mathematical model of the formation of morphogen gradients through membrane-associated non-receptors

The importance of morphogens is a central concept in developmental biology. Multiple-fate patterning and the robustness of the morphogen gradient are essential for embryo development. The ways by which morphogens diffuse from a local source to form long distance gradients can differ from one morphogen to the other, and for the same morphogen in different organs. This paper will study the mechanism by which morphogens diffuse through the aid of membrane-associated non-receptors and will investigate how the membrane-associated non-receptors help the morphogen to form long distance gradients and to achieve good robustness. Such a mechanism has been reported for some morphogens that are rapidly turned over. We will establish a set of reaction-diffusion equations to model the dynamical process of morphogen gradient formation. Under the assumption of rapid morphogen degradation, we discuss the existence, uniqueness, local stability, approximation solution, and the robustness of the steady-state gradient. The results in this paper show that when the morphogen is rapidly turned over, diffusion of the morphogen through membrane-associated non-receptors is a possible strategy to form a long distance multiple-fate gradient that is locally stable and is robust against the changes in morphogen synthesis rate.