NEGATIVE FEEDBACK IN MORPHOGEN GRADIENTS

In this paper, the effects of a conventional form of negative feedback on receptor synthesis rate as a mechanism to induce robustness of biological development of the Drosophila wing imaginal disc is analyzed through the initial-boundary value problem in a basic partial differential equation model for the extracellular biological development activities. The existence, uniqueness, linear stability and monotonicity of the steady state signaling gradient are established rigorously. Solutions are then obtained for a special case of the steady state problem to show explicitly the effects of the chosen form of negative feedback. It is evident from the results that the principal effect of such a feedback mechanism is to render the signaling bound morphogen gradient more uniformly distributed except for a narrow layer adjacent to the edge of the posterior compartment of the wing disc. While the change in the magnitude of the maximum signaling morphogen concentration near the ligand source may be kept at an acceptable level, the leveling and flattening of the gradient render it less differential in space (except in the boundary layer) and hence more deviating from the desired gradient for the target biological pattern. The conclusion is then shown to apply to the general case with the help of the theoretical results on monotonicity. These results suggest that negative feedback on the receptor synthesis rate of the chosen form is not effective for promoting robustness. In fact, it drives the system somewhat in the opposite direction, a conclusion supported by the simulation results. The findings in turn suggest more useful forms of the negative feedback for mediating receptor synthesis to offset a higher ligand synthesis rate for our types of systems.