Department of Systems Biology, Harvard Medical School, Boston, MA 02115.
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138.
Proc Natl Acad Sci U S A. 2018 Sep 25;115(39):9738-9743. doi: 10.1073/pnas.1808053115. Epub 2018 Sep 7.
The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.
米氏方程（Michaelis-Menten，MM）基本公式描述了酶催化反应的速率如何依赖于底物浓度。通过对三个反应网络的时间尺度分离，得出了熟悉的双曲线关系。此后，该公式被发现可以描述许多生物学背景下的稳态输入-输出响应，包括单分子酶动力学、基因调控、转录、翻译和力产生。先前解释其普遍性的尝试仅限于具有规则结构或简化参数假设的网络。在这里，我们利用基于图的时间尺度分离的线性框架来推导出在一般情况下出现 MM 公式的结构条件。这些条件要求将图划分为两部分，类似于对原始 MM 图进行“粗粒化”，并且对输入变量出现的位置和方式施加约束。图的其他特征，包括参数的数值，可以是任意的，从而解释了该公式的普遍性。对于处于热力学平衡的系统，我们推导出一个必要且充分的条件。对于远离热力学平衡的系统，特别是那些具有不可逆反应的系统，会出现不同的结构条件，并且一般特征仍然是开放的。然而，我们的结果在更大的通用性上包含了文献中我们已知的所有示例。
