Department of Biomedical Engineering, Boston University, 44 Cummington Mall, Boston, MA 02215, USA.
Biol Direct. 2013 Mar 22;8:7. doi: 10.1186/1745-6150-8-7.
The Michaelis-Menten equation, proposed a century ago, describes the kinetics of enzyme-catalyzed biochemical reactions. Since then, this equation has been used in countless, increasingly complex models of cellular metabolism, often including time-dependent enzyme levels. However, even for a single reaction, there remains a fundamental disconnect between our understanding of the reaction kinetics, and the regulation of that reaction through changes in the abundance of active enzyme.
We revisit the Michaelis-Menten equation under the assumption of a time-dependent enzyme concentration. We show that all temporal enzyme profiles with the same average enzyme level yield identical substrate degradation- a simple analytical conclusion that can be thought of as an invariance principle, and which we validate experimentally using a β-galactosidase assay. The ensemble of all time-dependent enzyme trajectories with the same average concentration constitutes a space of functions. We develop a simple model of biological fitness which assigns a cost to each of these trajectories (in the form of a function of functions, i.e. a functional). We then show how one can use variational calculus to analytically infer temporal enzyme profiles that minimize the overall enzyme cost. In particular, by separately treating the static costs of amino acid sequestration and the dynamic costs of protein production, we identify a fundamental cellular tradeoff.
The overall metabolic outcome of a reaction described by Michaelis-Menten kinetics is ultimately determined by the average concentration of the enzyme during a given time interval. This invariance in analogy to path-independent phenomena in physics, suggests a new way in which variational calculus can be employed to address biological questions. Together, our results point to possible avenues for a unified approach to studying metabolism and its regulation.
This article was reviewed by Sergei Maslov, William Hlavacek and Daniel Kahn.
米氏方程，一个世纪前提出的，描述了酶促生化反应的动力学。从那时起，这个方程已经被用于无数越来越复杂的细胞代谢模型中，通常包括时间依赖性的酶水平。然而，即使对于单一反应，我们对反应动力学的理解与通过改变活性酶丰度来调节该反应之间仍然存在根本的脱节。
我们在假设时间依赖性酶浓度的情况下重新考察了米氏方程。我们表明，所有具有相同平均酶水平的时间相关酶谱都产生相同的底物降解-这是一个简单的分析结论，可以被认为是不变原理，我们使用β-半乳糖苷酶测定实验验证了这一点。所有具有相同平均浓度的时间相关酶轨迹的集合构成了一个函数空间。我们开发了一种简单的生物适应性模型，该模型为每个轨迹分配一个成本（以函数的形式，即函数）。然后，我们展示了如何使用变分微积分来分析推断出最小化整体酶成本的时间相关酶谱。特别是，通过分别处理氨基酸隔离的静态成本和蛋白质生产的动态成本，我们确定了一个基本的细胞权衡。
米氏动力学描述的反应的整体代谢结果最终由给定时间间隔内酶的平均浓度决定。这种不变性与物理中无路径相关现象的类比，为变分微积分在解决生物学问题方面提供了一种新的方法。总的来说，我们的结果为研究代谢及其调节的统一方法提供了可能的途径。
这篇文章由 Sergei Maslov、William Hlavacek 和 Daniel Kahn 进行了评审。
