Embryonic development is driven by highly regulated spatial and temporal patterns of gene expression. These patterns emerge in a sequential process, with simple patterns providing inputs to molecular networks, which, in turn, transform these inputs into more complex outputs. Due to the recent advances in the molecular studies of development, these networks can now be modeled at mechanistic level. Exploring complex mechanistic models is critically dependent on the use of computational techniques; at the same time, the behavior of large-scale models can often be understood through analysis of simplified models, providing a "bridge" between computational and real experiments. Rigorous mathematical techniques for analyzing the dynamics of signal processing in developmental patterning are yet to be developed. To this end, the PIs propose to pursue a number of analytical approaches for studying the spatiotemporal dynamics in patterning networks. These approaches, which rely heavily on the techniques from the calculus of variations, will be applied in the context of epithelial pattern formation in the Drosophila egg, an established experimental model for studying how simple inputs establish complex spatial patterns in development. The proposed analytical and computational work will build on the modeling and experimental results obtained by the PIs during the previous funding period. Within the framework of this application, the PIs will focus on signal processing by two-dimensional patterning networks with positive feedback loops.
The proposed research is closely linked to the experimental work on pattern formation mediated by the Epidermal Growth Factor Receptor (EGFR), a key regulator of epithelial tissues across species. Given the highly conserved nature of EGFR signaling and the generality of inductive patterning events, the work of the PIs will provide insights into a large class of development problems. In addition, the mathematical techniques developed in this proposal will be applicable to a wide range of problems modeled by nonlinear parabolic partial differential equations. This research is strongly aligned with the educational effort of the PIs that involves further development of undergraduate and graduate courses on mathematical biology of cell communication systems and interdisciplinary training of mathematics and engineering graduate and undergraduate students at NJIT and Princeton.
