Local accumulation time for the formation of morphogen gradients from a Lévy diffusion process

Morphogen gradients provide very precise spatial information on the control of cell fate specification in many developing tissues. In previous studies, morphogen gradient formation was commonly modelled as Fickian diffusion. However, the complexity of morphogen transport and anisotropy of intracellular and extracellular environments in vivo can lead to Lévy flights of morphogens. In this case, a natural question is whether morphogen gradients reach steady states on timescales relevant to developmental patterning. Here, we build and analyse a Lévy diffusion model of morphogen transport, which is based on a continuous time random walk with a long-tailed jump length distribution. Importantly, we derive the analytical expression of local accumulation time that provides a time scale that characterizes relaxation to a steady state at an arbitrary position within the patterned field, and shows that this time depends on cell positions in a nonlinear and asymmetric manner. Our analytical result provides an explicit connection between the key parameters of the problem and the time needed to reach a steady state value at an arbitrarily given position, which is important for a better understanding of tissue patterning by morphogen gradients in a more real case.