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Morphogens, such as Decapentaplegic (Dpp) in Drosophila wing disc, are locally produced and spread to other regions in organs, forming gradients that control the interrelated pattern and growth of developing organs. During development, morphogen gradients must be robust to changes in intracellular and extracellular environments so as to provide target genes precise information to determine cell fates and the resulting spatial pattern on the proper position and at the proper time. However, how tissue growth affects the robust formation of morphogen gradient remains to be fully explored. Here, we model, using a reaction–advection–diffusion equation, a morphogen transport mechanism including tissue growth. By introducing and analyzing a robustness index that measures the effect of tissue growth on the formation of the morphogen gradient, we demonstrate that tissue growth can enlarge the range of morphogen gradient. In particular, tissue growth in the region near the source can efficiently filter the shift resulting from the changes in the production rate in each of steady-state and pre-steady-state situations, but the efficiency depends on growth rate and spatial position in either situation as well as on the time of development in the pre-steady-state situation. These results indicate that tissue growth is an nonnegligible factor for the robust formation of morphogen gradient.

Morphogen gradient possesses stem cells’ spatial information to control these cells’ differentiation toward distinct directions. In embryos, the formation of this gradient is a dynamic process with time. However, how the information during the process of forming morphogen gradient is measured remains unsolved. Here, according to the theory of information geometry, the information length based on Fisher information is introduced to measure the change in the information from morphogen gradient’s initial state to its steady state. We find that morphogen’s diffusion rate, degradation rate and production rate have important effects on the information length. Specially, morphogen gradation’s degradation rate and diffusion rate have opposite effects on the information length. Our results enrich the current ones about morphogen gradients.

Two models of receptor-mediated morphogen transport in a biological tissue are proposed to investigate morphogen gradient formation. The first model concerns intracellular transport of morphogen molecules, while the second describes transport along the cell surface. Both models couple via diffusivity a quasilinear degenerate parabolic equation describing the transport of the morphogens with an ordinary differential equation describing reversible binding kinetics of receptors. A detailed study of the steady states is provided, together with numerical tests which compare the models with experimental data.