The Scaling and Shift of Morphogen Gene Expression Boundary in a Nonlinear Reaction Diffusion System

The scaling and shift of the gene expression boundary in a developing embryo are two key problems with regard to morphogen gradient formation in developmental biology. In this study, a bigradient model was applied to a nonlinear reaction diffusion system (NRDS) to investigate the location of morphogen gene expression boundary. In contrast to the traditional synthesis–diffusion–degradation model, the introduction of NRDS in this study contributes to the precise gene expression boundary at arbitrary location along the anterior-posterior axis other than simply midembryo even when the linear characteristic lengths of two morphogens are equal. The scaling location depends on the ratio of two morphogen influxes (\(w\)) and concentrations (\(r\)) as well as the nonlinear reaction diffusion parameters (\(\alpha , n\)). We also formulate a direct relationship between the shift in the gene expression boundary and the influx of morphogen and find that enhancing the morphogen influx is helpful to build up a robust gene expression boundary. By analyzing the robustness of the morphogen gene expression boundary and comparing with the relevant results in linear reaction diffusion system, we determine the precise range of the ratio of the two morphogen influxes with a lower shift in the morphogen gene expression boundary and increased system robustness.