Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable [Formula: see text]. Homeostasis plays an important role in the regulation of biological systems, cf. Ferrell (Cell Syst 2:62-67, 2016), Tang and McMillen (J Theor Biol 408:274-289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to [Formula: see text] is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387-407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input-output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1-24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input-output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input-output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760-773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305-364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264-275, 2014. https://doi.org/10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104-110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma et al. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities.

中文翻译：

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三节点输入输出网络中的动态平衡极小。

稳态发生在一个系统中，在该系统中，输出变量在输入变量变化的间隔上近似恒定[公式：请参见文本]。稳态在生物系统的调节中起着重要的作用。Ferrell（Cell Syst 2：62-67，2016），Tang和McMillen（J Theor Biol 408：274-289，2016），Nijhout等。（BMC Biol 13:79，2015）和Nijhout等人。（Wiley Interdiscip Rev Syst Biol Med 11：e1440，2018）。在控制理论文献中给出了一种在数学模型中寻找稳态的方法，该点是输出变量相对于[公式：参见文本]的导数等于零的点。这些点称为完美稳态或完美适应。另外，Golubitsky和Stewart（J Math Biol 74：387-407，2017）使用了无穷小的动态平衡概念（即，输入输出函数的导数在零点处为零），将奇异性理论引入稳态研究。里德等。（Bull Math Biol 79（9）：1-24，2017）给出了三节点化学反应系统中极小的稳态的两个例子：前馈激发和底物抑制。在本文中，我们假设有一个输入节点，一个输出节点并且它们是截然不同的，因此有13个不同的三节点网络导致78个三节点输入-输出网络配置。不同的配置基于哪个节点是输入节点，哪个节点是输出节点。尽管如此，我们仍然表明，三节点输入输出网络只有三种基本机制可以导致极小的动态平衡，我们称其为结构动态平衡，Haldane动态平衡，和零降解稳态。Ma等人给出了这种分类的实质部分。（Cell 138：760-773，2009）和Ferrell（2016）等。我们的贡献包括使用通用的允许系统（Golubitsky和Stewart in Bull Am Math Soc 43：305-364，2006）而不是特定的生化模型进行完整分类，将无限小稳态的类型与简单路径的图论存在联系起来，以及为使用奇异性理论研究更高维度的稳态奇异性提供基础，例如Nijhout和Reed引入的椅子奇异性（Integr Comp Biol 54（2）：264-275，2014.https：//doi.org/10.1093/icb/ icu010）和Nijhout等人。（Math Biosci 257：104-110，2014）。参见Golubitsky和Stewart（2017）。这些机制中的前两个通过前馈激发和底物抑制来说明。只有当网络具有作为子网络的前馈回路时，才会发生结构稳态。也就是说，当有两个不同的简单路径将输入节点连接到输出节点时。而且，当网络仅仅是前馈环基序本身时，其中的一条路径必须是兴奋性的，而另一条路径必须是抑制性的，以支持无限小的动态平衡。当从输入节点到输出节点只有一条简单的路径，然后仅当沿着该路径的耦合之一具有强度0时，才会发生Haldane稳态。Ma等人的生物化学实例说明了零降解稳态。（2009）；仅当第三个节点的退化常数为0时，才会发生这种动态平衡。