Analyzing the effect of cell rearrangement on Delta-Notch pattern formation

The Delta-Notch system plays a vital role in a number of areas in biology and typically forms a salt and pepper pattern in which cells strongly expressing Delta and cells strongly expressing Notch are alternately aligned via lateral inhibition. Although the spatial arrangement of the cells is important to the Delta-Notch pattern, the effect of cell rearrangement is not often considered. In this study, we provide a framework to analytically evaluate the effect of cell mixing and proliferation on Delta-Notch pattern formation in one spatial dimension. We model cell rearrangement events by a Poisson process and analyze the model while preserving the discrete properties of the spatial structure. We find that the homogeneous expression pattern is stabilized if the frequency of cell rearrangement events is sufficiently large. We analytically obtain the critical frequencies of the cell rearrangement events where the decrease of the pattern amplitude as a result of cell rearrangement is balanced by the increase in amplitude due to the Delta-Notch interaction dynamics. Our theoretical results are qualitatively consistent with experimental results, supporting the notion that the heterogeneity of expression patterns is inversely correlated with cell rearrangement \textit{in vivo}. Our framework, while applied here to the specific case of the Delta-Notch system, is applicable more widely to other pattern formation mechanisms.