A model of morphogen transport in the presence of glypicans I

TL;DR

This paper analyzes a one-dimensional reaction-diffusion model of morphogen transport, proving well-posedness and uniqueness of steady states without artificial parameter restrictions, advancing understanding of biological gradient formation.

Contribution

It provides a rigorous mathematical analysis of a biologically relevant morphogen transport model, establishing well-posedness and steady state uniqueness using semigroup and $L_1$ techniques.

Findings

The system is well-posed with a unique steady state.

No artificial restrictions on parameters are needed.

Mathematical techniques used include semigroup theory and $L_1$ methods.

Abstract

We analyze a one dimensional version of a model of morphogen transport, a biological process governing cell differentiation. The model was proposed by Hufnagel et al. to describe the forming of morphogen gradient in the wing imaginal disc of the fruit fly. In mathematical terms the model is a system of reaction-diffusion equations which consists of two parabolic PDE's and three ODE's. The source of ligands is modelled by a Dirac Delta. Using semigroup approach and $L_1$ techniques we prove that the system is well-posed and possesses a unique steady state. All results are proved without imposing any artificial restrictions on the range of parameters.