Morphogenetic action through flux-limited spreading

• We introduce a nonlinear model with flux-limited spreading describing Shh-GLI signaling, with restricted velocity for morphogen propagation.
• Our flux-limited model states that cells are exposed to different morphogen concentrations at different times, contrary to Fickʼs law predictions.
• Our flux-limited spreading model corroborates the desensitization of the morphogen response with respect to time, describing true biological Hedgehog-Gli patterns.
• We suggest that spatially restricted information transfer through cytonemes is the biological counterpart of flux-limiters in our flux-limited spreading model.
• We detect and measure Hh-morphogen particles velocities in cell extensions (cytonemes) for different biological systems: Drosophila, vertebrates or cell cultures.

A central question in biology is how secreted morphogens act to induce different cellular responses within a group of cells in a concentration-dependent manner. Modeling morphogenetic output in multicellular systems has so far employed linear diffusion, which is the normal type of diffusion associated with Brownian processes. However, there is evidence that at least some morphogens, such as Hedgehog (Hh) molecules, may not freely diffuse. Moreover, the mathematical analysis of such models necessarily implies unrealistic instantaneous spreading of morphogen molecules, which are derived from the assumptions of Brownian motion in its continuous formulation. A strict mathematical model considering Fickʼs diffusion law predicts morphogen exposure of the whole tissue at the same time. Such a strict model thus does not describe true biological patterns, even if similar and attractive patterns appear as results of applying such simple model. To eliminate non-biological behaviors from diffusion models we introduce flux-limited spreading (FLS), which implies a restricted velocity for morphogen propagation and a nonlinear mechanism of transport. Using FLS and focusing on intercellular Hh-Gli signaling, we model a morphogen gradient and highlight the propagation velocity of morphogen particles as a new key biological parameter. This model is then applied to the formation and action of the Sonic Hh (Shh) gradient in the vertebrate embryonic neural tube using our experimental data on Hh spreading in heterologous systems together with published data. Unlike linear diffusion models, FLS modeling predicts concentration fronts and the evolution of gradient dynamics and responses over time. In addition to spreading restrictions by extracellular binding partners, we suggest that the constraints imposed by direct bridges of information transfer such as nanotubes or cytonemes underlie FLS. Indeed, we detect and measure morphogen particle velocity in such cell extensions in different systems.