Hydrodynamic modeling of Bicoid morphogen gradient formation in Drosophila embryo

Bicoid is a maternal polarity determinant that mediates the anterior-posterior (AP) patterning in early Drosophila embryo. During oogenesis, its mRNA deposits at the anterior pole of the embryo and then translates to establish the Bicoid morphogen gradient soon after fertilization. Previous investigations indicated that the patterning is induced by the spatial gradient of Bicoid morphogen concentration, where the cytoplasmic convection plays a crucial role. The present study examines the effect of advection transport on the formation of the Bicoid morphogen gradient using direct simulation of the cytoplasmic streaming described by Navier-Stokes equations, in which the cytoplasm behaves as an incompressible Newtonian fluid. To simulate the cytoplasmic streaming originated from membrane contractions, the flow is driven by slip velocities along the cortex and the anterior-posterior axis of the cell. Results show that the Bicoid concentration distribution we obtained provides a quantitatively consistent picture with the experiment measurements, as well as the diffusive length scale. The competition among the diffusion, advection and degradation is analyzed when the cytoplasmic streaming is considered. It is found that the advection yields wavy phenomenon in the profiles of the Bicoid concentration at small diffusion coefficients, which might have important effects on the embryonic development. After the driven velocities is switched off, the interior flow evanesces gradually due to the viscous drag, the Bicoid degradation will overwhelm the advection effect.