Elaborate regulatory feedback processes are thought to make biological development robust, that is, resistant to changes induced by genetic or environmental perturbations. How this might be done is still not completely understood. Previous numerical simulations on reaction-diffusion models of Dpp gradients in Drosophila wing imaginal disc have showed that feedback (of the Hill function type) on (signaling) receptors and/or non-(signaling) receptors are of limited effectiveness in promoting robustness. Spatial nonuniformity of the feedback processes has also been shown theoretically to lead to serious shape distortion and a principal cause for ineffectiveness. Through mathematical modeling and analysis, the present article shows that spatially uniform nonlocal feedback mechanisms typically modify gradient shape through a shape parameter (that does not change with location). This in turn enables us to uncover new multi-feedback instrument for effective promotion of robust signaling gradients.

中文翻译：

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有关受体和非受体合成的监管反馈，以实现可靠的信号传递。

精心制定的监管反馈过程被认为可以使生物学发展变得健壮，即抵抗遗传或环境扰动引起的变化。如何做到这一点仍未完全理解。以前在果蝇翅膀假想盘中Dpp梯度的反应扩散模型的数值模拟表明，对（信号）受体和/或非（信号）受体的（希尔函数类型的）反馈在增强鲁棒性方面效果有限。反馈过程的空间不均匀性在理论上也已显示出会导致严重的形状失真和无效的主要原因。通过数学建模和分析，本文显示，空间均匀的非局部反馈机制通常会通过形状参数（不会随位置变化）来修改梯度形状。反过来，这使我们能够发现新的多反馈仪器，以有效地增强鲁棒的信号梯度。