Morphogen gradient formation in a complex environment: an anomalous diffusion model.

Current models of morphogen-induced patterning assume that morphogens undergo normal, or Fickian, diffusion, although the validity of this assumption has never been examined. Here we argue that the interaction of morphogens with the complex extracellular surrounding may lead to anomalous diffusion. We present a phenomenological model that captures this interaction, and derive the properties of the morphogen profile under conditions of anomalous (non-Fickian) diffusion. In this context we consider the continuous time random walk formalism and extend its application to account for degradation of morphogen particles. We show that within the anomalous diffusion model, morphogen profiles are fundamentally distinct from the corresponding Fickian profiles. Differences were found in several key aspects, including the role of degradation in determining the profile, the rate by which it spreads in time and its long-term behavior. We analyze the effect of an abrupt change in the extracellular environment on the concentration profiles. Furthermore, we discuss the robustness of the morphogen distribution to fluctuations in morphogen production rate, and describe a feedback mechanism that can buffer such fluctuations. Our study also provides rigorous criteria to distinguish experimentally between Fickian and anomalous modes of morphogen transport.