Steady state and sensitivity analysis of a Notch-Delta signaling system of one single cell interacting with fixed environment

The Notch–Delta signaling pathway is a highly conserved signaling system that par-<br>takes in a diverse process of growth, patterns and differentiation. Experiments have<br>shown that Delta from different cells activates this pathway (trans-activation) while<br>Delta from the same cell inhibits this pathway (cis-inhibition). The Notch–Delta inter-<br>actions could switch a cell to one of the two opposite fates: either Sender (high Delta/low<br>Notch) or Receiver (low Delta/high Notch). We studied a Notch–Delta signaling model<br>from Sprinzak et al., (2010), to investigate the cell fate through steady state analysis.<br>The focus was placed on a fundamental case of one single cell with fixed external Delta<br>and Notch supplies. First, we proved there exists a unique steady state which is asymp-<br>totically stable. Second, we derived the increasing/decreasing and asymptotic properties<br>of the steady state with respect to all the parameters. Third, we studied the sensitivity<br>and discovered the cell fate is only sensitive to the production rates of Notch and Delta<br>under strong cis-inhibition. Finally, we applied this model to multi-cellular cases and<br>found that the lateral inhibition pattern could be created with the spatially varied Delta<br>production rate. The Hopf bifurcation is not observed in the current model.