Shoffner S K, Schnell Santiago
Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, MI 48109, USA.
Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, MI 48109, USA; Department of Computational Medicine & Bioinformatics, University of Michigan Medical School, Ann Arbor, MI 48109, USA; Brehm Center for Diabetes Research, University of Michigan Medical School, Ann Arbor, MI 48105, USA.
Math Biosci. 2017 May;287:122-129. doi: 10.1016/j.mbs.2016.09.001. Epub 2016 Sep 6.
The derivation of timescales is frequently introduced as an art form in papers and textbooks. The best scaling techniques require the application of physical intuition to identify dimensionless variables that are one unit order of magnitude and small parameters, which can simplify nonlinear differential equations. However, physical intuition requires prior knowledge of the solution to the dynamical systems under investigation. There are problems where the application of physical intuition is not straightforward. Therefore, it is necessary to apply mathematical techniques to estimate scales for the separation of timescales and simplification. In this review, we present three mathematical techniques - determination of pairwise balances, principle of minimum simplification and scaling by inverse rates - to scale dynamical systems with limited prior knowledge of model behavior. We illustrate the application of these techniques with the Michaelis-Menten reaction, which is widely studied to introduce scaling and simplification techniques in textbooks. We show that the pairwise balance approach, though commonly introduced as a method for nondimensionalization, can fail to derive a separation between timescales. The other techniques we review here can be applied to a number of dynamical systems, where the separation of timescales can lead to the simplification of a complex nonlinear problem.
在论文和教科书中，时间尺度的推导常常被当作一种艺术形式来介绍。最佳的标度技术需要运用物理直觉来识别无量纲变量，这些变量是一个单位数量级且为小参数，它们能够简化非线性微分方程。然而，物理直觉需要对所研究的动力系统的解有先验知识。存在一些问题，在这些问题中物理直觉的应用并非直接明了。因此，有必要应用数学技术来估计时间尺度分离和简化的标度。在这篇综述中，我们介绍三种数学技术——成对平衡的确定、最小简化原理和逆速率标度法——用于在对模型行为仅有有限先验知识的情况下对标度动力系统。我们用米氏反应来说明这些技术的应用，米氏反应在教科书中被广泛研究以引入标度和简化技术。我们表明，成对平衡方法尽管通常被作为一种无量纲化方法来介绍，但可能无法得出时间尺度之间的分离。我们在此综述的其他技术可应用于许多动力系统，在这些系统中时间尺度的分离能够导致复杂非线性问题的简化。
