Local accumulation times of morphogen gradient with tissue growth

During development of multicellular organisms, morphogen is a kind of signaling molecules produced at a local region and transported into a development field, which acts as dose-dependent regulators of gene expression, directs and controls cellular differentiation. Some experiments showed that during embryo development, dynamical processes of morphogen transport always accompany the tissue growth. However, how tissue growth affects morphogen gradients remains to be explored. To answer this problem, we propose a reaction-diffusion-convection model for morphogen transport. For this model, we mainly investigate local accumulation times (LATs) of morphogen gradients, which are a measure for time of forming the steady state of morphogen gradients. In this paper, we simplify the method of calculating the LATs and use this method to obtain analytic expressions of the LATs for uniform and linear growth, respectively. Besides, for tissue nonuniform growth, we apply an approximation method of the LATs to study them. This paper shows that tissue growth can shorten the LATs of morphogen gradients.