Morphogen protein gradients play a vital role in regulating spatial pattern formation during development. The most commonly accepted mechanism of protein gradient formation involves the diffusion and degradation of morphogens from a localized source. However, there is growing experimental evidence for a direct cell-to-cell signaling mechanism via thin actin-rich cellular extensions known as cytonemes. Recent modeling studies of cytoneme-based morphogenesis in invertebrates ignore the discrete nature of vesicular transport along cytonemes, focusing on deterministic continuum models. In this paper, we develop an impulsive signaling model of morphogen gradient formation in invertebrates, which takes into account the discrete and stochastic nature of vesicular transport along cytonemes. We begin by solving a first passage time problem with sticky boundaries to determine the expected time to deliver a vesicle to a target cell, assuming that there is a 'nucleation' time for injecting the vesicle into the cytoneme. We then use queuing theory to analyze the impulsive model of morphogen gradient formation in the case of multiple cytonemes and multiple targets. In particular, we determine the steady-state mean and variance of the morphogen distribution across a one-dimensional array of target cells. The mean distribution recovers the spatially decaying morphogen gradient of previous deterministic models. However, the burst-like nature of morphogen transport can lead to Fano factors greater than unity across the array of cells, resulting in significant fluctuations at more distant target sites.

中文翻译：

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基于细胞因子的形态发生子梯度形成的脉冲信号模型。

形态发生蛋白梯度在调节发育过程中空间格局的形成中起着至关重要的作用。蛋白质梯度形成的最普遍接受的机制涉及来自局部来源的吗啡原的扩散和降解。但是，越来越多的实验证据表明，通过富含肌动蛋白的薄细胞延伸（称为细胞因子），可以直接在细胞间传递信号。在无脊椎动物中，基于细胞因子的形态发生的最新模型研究忽略了沿细胞因子的囊泡转运的离散性质，重点是确定性连续体模型。在本文中，我们开发了无脊椎动物中形态发生剂梯度形成的脉冲信号模型，该模型考虑了沿细胞因子的水泡运输的离散性和随机性。我们首先解决具有粘性边界的第一次通过时间问题，以确定将囊泡注射到细胞因子中的“成核”时间，从而确定将囊泡递送至靶细胞的预期时间。然后，我们使用排队论来分析在多个细胞因子和多个靶点的情况下形态发生剂梯度形成的冲动模型。特别是，我们确定了目标细胞的一维阵列上形态发生子分布的稳态均值和方差。平均分布恢复了先前确定性模型的空间衰减的形态发生子梯度。然而，形态发生子运输的突如其来的性质会导致Fano因子在整个细胞阵列中大于单位，从而在更远的靶位点产生明显的波动。