Effects of Sog on Dpp-Receptor Binding

Concentration gradients of morphogens are known to be instrumental in cell signaling and tissue patterning. Of interest here is how the presence of a competitor of BMP ligands affects cell signaling. The effects of Sog on the binding of Dpp with cell receptors are analyzed for dorsal-ventral morphogen gradient formation in vertebrate and Drosophila embryos. This prototype system includes diffusing ligands, degradation of morphogens, and cleavage of Dpp-Sog complexes by Tolloid to free up Dpp. Simple and biologically meaningful necessary and sufficient conditions for the existence of a steady state gradient configuration are established, and existence theorems are proved. For high Sog production rates (relative to the Dpp production rate), it is found that the steady state configuration exhibits a more intense Dpp-receptor concentration near the dorsal midline. Numerical simulations of the evolution of the system show that, beyond some threshold Sog production rate, the transient Dpp-receptor concentration at the dorsal midline would become more intense than that of the steady state, before subsiding and approaching a nonuniform steady state of lower magnitude. The magnitude of the transient concentration has been found to increase by several fold with increasing Sog production rate. The highly intense Dpp activity at and around the dorsal midline is consistent with available experimental observations and other analytical studies.