Nagahata Yutaka, Kobayashi Masato, Toda Mikito, Maeda Satoshi, Taketsugu Tetsuya, Komatsuzaki Tamiki
The Institute for Chemical Reaction Design and Discovery, Hokkaido University, Sapporo 001-0021, Japan.
Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0020, Japan.
Proc Natl Acad Sci U S A. 2024 May 21;121(21):e2317781121. doi: 10.1073/pnas.2317781121. Epub 2024 May 17.
Complex networks are pervasive in various fields such as chemistry, biology, and sociology. In chemistry, first-order reaction networks are represented by a set of first-order differential equations, which can be constructed from the underlying energy landscape. However, as the number of nodes increases, it becomes more challenging to understand complex kinetics across different timescales. Hence, how to construct an interpretable, coarse-graining scheme that preserves the underlying timescales of overall reactions is of crucial importance. Here, we develop a scheme to capture the underlying hierarchical subsets of nodes, and a series of coarse-grained (reduced-dimensional) rate equations between the subsets as a function of time resolution from the original reaction network. Each of the coarse-grained representations guarantees to preserve the underlying slow characteristic timescales in the original network. The crux is the construction of a lumping scheme incorporating a similarity measure in deciphering the underlying timescale hierarchy, which does not rely on the assumption of equilibrium. As an illustrative example, we apply the scheme to four-state Markovian models and Claisen rearrangement of allyl vinyl ether (AVE), and demonstrate that the reduced-dimensional representation accurately reproduces not only the slowest but also the faster timescales of overall reactions although other reduction schemes based on equilibrium assumption well reproduce the slowest timescale but fail to reproduce the second-to-fourth slowest timescales with the same accuracy. Our scheme can be applied not only to the reaction networks but also to networks in other fields, which helps us encompass their hierarchical structures of the complex kinetics over timescales.
复杂网络在化学、生物学和社会学等各个领域普遍存在。在化学中，一级反应网络由一组一阶微分方程表示，这些方程可以从潜在的能量景观构建。然而，随着节点数量的增加，理解不同时间尺度上的复杂动力学变得更具挑战性。因此，如何构建一种可解释的粗粒化方案，以保留整体反应的潜在时间尺度至关重要。在这里，我们开发了一种方案来捕获节点的潜在层次子集，以及一系列粗粒化（降维）速率方程，这些方程描述了子集中原始反应网络随时间分辨率的函数关系。每个粗粒化表示都保证保留原始网络中潜在的慢特征时间尺度。关键在于构建一种集总方案，该方案在解读潜在时间尺度层次结构时纳入了相似性度量，且不依赖于平衡假设。作为一个示例，我们将该方案应用于四态马尔可夫模型和烯丙基乙烯基醚（AVE）的克莱森重排，并证明降维表示不仅能准确再现整体反应的最慢时间尺度，还能再现较快的时间尺度，而其他基于平衡假设的简化方案虽然能很好地再现最慢时间尺度，但无法以相同的精度再现第二至第四慢的时间尺度。我们的方案不仅可以应用于反应网络，还可以应用于其他领域的网络，这有助于我们理解它们在时间尺度上复杂动力学的层次结构。
