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

中文翻译：

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单细胞与固定环境交互的Notch-Delta信号系统的稳态和灵敏度分析

Notch-Delta 信号通路是一个高度保守的信号系统，参与了多种生长、模式和分化过程。实验表明，来自不同细胞的 Delta 会激活该途径（反式激活），而来自同一细胞的 Delta 会抑制该途径（顺式抑制）。Notch-Delta 相互作用可以将细胞切换到两种相反的命运之一：发送者（高 Delta/低 Notch）或接收者（低 Delta/高 Notch）。我们研究了 Sprnzak 等人 (2010) 的 Notch-Delta 信号模型，通过稳态分析研究细胞命运。重点放在一个具有固定外部 Delta 和 Notch 电源的单电池的基本案例上。首先，我们证明了存在一个唯一的渐近稳定的稳定状态。第二，我们推导出稳态相对于所有参数的递增/递减和渐近特性。第三，我们研究了敏感性，发现细胞命运仅对强顺式抑制下的 Notch 和 Delta 的产生率敏感。最后，我们将该模型应用于多细胞病例，发现侧抑制模式可以通过空间变化的 Delta 产生率来创建。在当前模型中没有观察到 Hopf 分岔。我们将此模型应用于多细胞病例，发现侧抑制模式可以通过空间变化的 Delta 产生率来创建。在当前模型中没有观察到 Hopf 分岔。我们将此模型应用于多细胞病例，发现侧抑制模式可以通过空间变化的 Delta 产生率来创建。在当前模型中没有观察到 Hopf 分岔。