Kinetics of receptor occupancy during morphogen gradient formation

During embryogenesis, sheets of cells are patterned by concentration profiles of morphogens, molecules that act as dose-dependent regulators of gene expression and cell differentiation. Concentration profiles of morphogens can be formed by a source-sink mechanism, whereby an extracellular protein is secreted from a localized source, diffuses through the tissue and binds to cell surface receptors. A morphogen molecule bound to its receptor can either dissociate or be internalized by the cell. The effects of morphogens on cells depend on the occupancy of surface receptors, which in turn depends on morphogen concentration. In the simplest case, the local concentrations of the morphogen and morphogen-receptor complexes monotonically increase with time from zero to their steady-state values. Here, we derive analytical expressions for the time scales which characterize the formation of the steady-state concentrations of both the diffusible morphogen molecules and morphogen-receptor complexes at a given point in the patterned tissue.