Aggregation of a distributed source in morphogen gradient formation

In the development of a biological entity, ligands (such as Decapentaplegic (Dpp) along the anterior-posterior axis of the Drosophila wing imaginal disc) are synthesized at a localized source and transported away from the source for binding with cell surface receptors to form concentration gradients of ligand-receptor complexes for cell signaling. Generally speaking, activities such as diffusion and reversible binding with degradable receptors also take place in the region of ligand production. The effects of such morphogen activities in the region of localized distributed ligand source on the ligand-receptor concentration gradient in the entire biological entity have been modeled and analyzed as System F in [1]. In this paper, we deduce from System F, a related end source model (System A) in which the effects of the distributed ligand source is replaced by an idealized point stimulus at the border between the (posterior) chamber and the ligand production region that simulates the average effects of the ligand activities in the production zone. This aggregated end source model is shown to adequately reproduce the significant implications of System F and to contain the corresponding ad hoc point source model, System R of [2], as a special case. Because of its simpler mathematical structure and the absence of any limitation on the ligand synthesis rate for the existence of steady-state gradients, System A type models are expected to be used widely. An example of such application is the recent study of the inhibiting effects of the formation of nonsignaling ligand-nonreceptor complexes [3].