Intercellular signaling is critical in development of multicellular organisms: by regulating cell differentiation, migration, growth, and death, cell communication guides the development of tissues and organs. In adult organisms the same mechanisms are responsible for tissue repair and maintenance; defects of cell communication systems lead to a number of life-threatening pathologies. Signaling through the Epidermal Growth Factor Receptor (EGFR) is essential in a number of developmental processes across species, from fruitflies to humans, and is extensively studied at the molecular level.

Drosophila melanogaster is a major model organism for the in-vivo analysis of development at the molecular level. One is interested in the mechanisms by which cell communication by diffusing signals patterns epithelial layers. The eggshell of a mature egg of Drosophila is characterized by the presence of two dorsal appendages, a pair organ that supplies the developing embryo with oxygen. Their formation, induced in mid-oogenesis, relies on extensive communication between the oocyte and the cells of the follicular epithelium. The appendages are produced by the two groups of cells that differentiate from the epithelium under the action of the oocyte-derived signal.

In a joint work with S. Y. Shvartsman (Princeton) and D. A. Lauffenburger (MIT), Cyrill Muratov developed a mathematical model that describes the patterning events specifying the formation of dorsal appendages in Drosophila oogenesis. The model reduces to a system of coupled reaction-diffusion equations driven by a localized input and characterizes the eggshell phenotype by the number of peaks in the signaling pattern. The model is mechanistic: it is based on a biomolecular mechanism and, in the spirit of the quasi-steady state approximation, identifies the slowest relevant processes and variables responsible for the signaling patterns guiding the formation of a pair of dorsal appendages. Furthermore, the choice of the parameters of the model is guided by the available biochemical information about the relevant time and length scales, etc., of the involved processes.

The figure below presents a summary of the analysis of the signaling patterns.

Figure: (a) Steady state bifurcation diagram showing the hysteretic transitions between branches with zero, one, and two peaks. Only the stable solutions are shown. (b) This sequence of hysteretic transitions can be used to account for a number of observed phenotypic transitions. (c) Two-parameter bifurcation diagram showing the regions of existence of zero- to four-peaked solutions as a function of input amplitude and width. Transitions between qualitatively different patterns is given by lines of saddle-node bifurcations. Complex phenotypes are predicted for wide and strong inputs. (d) Space-time plot showing a transient induced by a monotonically increasing single-peaked input.