A hybrid system model of planar cell polarity signaling in Drosophila melanogaster wing epithelium

A hybrid system model of a 4-protein signaling mechanism within an array of cells in the Drosophila melanogaster (fruit fly) wing is presented and analyzed. While the protein interaction model between cells is biologically fairly well understood, the interaction resulting in spatial distribution of proteins within each cell is not. We propose a weak interaction mechanism between proteins within each cell, and we incorporate this with a model of the protein interaction between cells to form a hybrid model, in which the continuous growth and decay of proteins is modeled using continuous state linear differential equations, and the discrete switches which activate and deactivate proteins are modeled using finite state machines. We then present a preliminary model validation: equilibrium analysis of the hybrid model provides allowable parameter ranges for growth and decay rates, as well as conditions on the cell network boundary conditions, for the biologically feasible steady state protein concentrations to occur. We conclude with a comparison of our analytical results against experimental image data obtained in collaboration with the Stanford University School of Medicine: we present the design of our image analysis algorithm and results for time stamped images of protein localization within the Drosophila wing.