Robust multiple-fate morphogen gradients are essential for embryo development. Here, we analyze mathematically a model of morphogen gradient (such as Dpp in Drosophila wing imaginal disc) formation in the presence of non-receptors with both diffusion of free morphogens and the movement of morphogens bound to non-receptors. Under the assumption of rapid degradation of unbound morphogen, we introduce a method of functional boundary value problem and prove the existence, uniqueness and linear stability of a biologically acceptable steady-state solution. Next, we investigate the robustness of this steady-state solution with respect to significant changes in the morphogen synthesis rate. We prove that the model is able to produce robust biological morphogen gradients when production and degradation rates of morphogens are large enough and non-receptors are abundant. Our results provide mathematical and biological insight to a mechanism of achieving stable robust long distance morphogen gradients. Key elements of this mechanism are rapid turnover of morphogen to non-receptors of neighoring cells resulting in significant degradation and transport of non-receptor-morphogen complexes, the latter moving downstream through a "bucket brigade" process.

中文翻译：

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通过膜相关非受体的“桶旅”运输的形态素梯度的稳健性。

强大的多命运形态发生梯度对于胚胎发育至关重要。在这里，我们在数学上分析了在非受体存在下形态发生素梯度（如果蝇翼成虫盘中的 Dpp）形成的模型，其中既有游离形态发生素的扩散，又有形态发生素与非受体结合的运动。在未结合的形态发生素快速降解的假设下，我们引入了函数边值问题的方法，并证明了生物学上可接受的稳态解的存在性、唯一性和线性稳定性。接下来，我们研究了这种稳态解决方案在形态发生素合成速率的显着变化方面的稳健性。我们证明，当形态素的产生和降解率足够大且非受体丰富时，该模型能够产生强大的生物形态素梯度。我们的结果为实现稳定稳健的长距离形态发生素梯度的机制提供了数学和生物学见解。这种机制的关键要素是形态发生素快速转换为邻近细胞的非受体，导致非受体-形态发生素复合物的显着降解和运输，后者通过“桶旅”过程向下游移动。