Applied Mathematics Graduate Program, University of Colorado, Boulder, CO, USA,
Bull Math Biol. 2013 Nov;75(11):2118-49. doi: 10.1007/s11538-013-9884-8. Epub 2013 Sep 10.
In previous work, we have introduced a "linear framework" for time-scale separation in biochemical systems, which is based on a labelled, directed graph, G, and an associated linear differential equation, dx/dt = L(G) ∙ x, where L(G) is the Laplacian matrix of G. Biochemical nonlinearity is encoded in the graph labels. Many central results in molecular biology can be systematically derived within this framework, including those for enzyme kinetics, allosteric proteins, G-protein coupled receptors, ion channels, gene regulation at thermodynamic equilibrium, and protein post-translational modification. In the present paper, in response to new applications, which accommodate nonequilibrium mechanisms in eukaryotic gene regulation, we lay out the mathematical foundations of the framework. We show that, for any graph and any initial condition, the dynamics always reaches a steady state, which can be algorithmically calculated. If the graph is not strongly connected, which may occur in gene regulation, we show that the dynamics can exhibit flexible behavior that resembles multistability. We further reveal an unexpected equivalence between deterministic Laplacian dynamics and the master equations of continuous-time Markov processes, which allows rigorous treatment within the framework of stochastic, single-molecule mechanisms.
在之前的工作中，我们介绍了一种用于生化系统时标分离的“线性框架”，它基于一个标记有向图 G 和一个相关的线性微分方程 dx/dt = L(G) ∙ x，其中 L(G) 是 G 的拉普拉斯矩阵。生化非线性性被编码在图标签中。许多分子生物学中的重要结果都可以在这个框架中系统地推导出来，包括酶动力学、别构蛋白、G 蛋白偶联受体、离子通道、热力学平衡下的基因调控以及蛋白质翻译后修饰。在本文中，为了应对新的应用，即适应真核基因调控中的非平衡机制，我们阐述了该框架的数学基础。我们表明，对于任何图和任何初始条件，动力学总是会达到一个稳定状态，可以通过算法计算。如果图不是强连通的，这可能会在基因调控中发生，我们表明动力学可以表现出类似于多稳定性的灵活行为。我们进一步揭示了确定性拉普拉斯动力学和连续时间马尔可夫过程的主方程之间的意外等价性，这允许在随机单分子机制的框架内进行严格处理。
