大规模酶反应网络的随机理论：速率方程模型的有限拷贝数修正
Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models
摘要：
细胞内的化学反应发生在阿托至飞升的隔室体积中。在如此小的体积中实现的生理浓度意味着相互作用分子的低拷贝数，其结果是浓度的相当大的波动。相比之下，速率方程模型是基于无限大数量的相互作用分子的隐含假设，或者等价地，反应在恒定的宏观浓度下在无限体积中发生。在这篇文章中，我们计算有限体积校正（或等效的有限拷贝数校正）的化学反应网络的速率方程的解决方案，由任意大量的酶催化的反应，被限制在一个小的亚细胞室。这是通过将准稳态假设的介观版本应用于与化学主方程的泊松表示相关联的精确福克-普朗克方程来实现的。该程序产生令人印象深刻的简单和紧凑的表达式的有限体积校正。我们证明，速率方程的预测总是低估了实际的稳态底物浓度的酶反应网络限制在一个小的体积。特别是，我们表明，有限体积校正增加，减少亚细胞体积，减少米氏常数和增加酶饱和度。校正的幅度敏感地依赖于网络的拓扑结构。该理论的预测结果与两种典型的与蛋白质甲基化和代谢相关的网络的随机模拟结果非常一致。
英文摘要：
Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtolitres. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small sub-cellular compartment. This is achieved by applying a mesoscopic version of the quasi-steady state assumption to the exact Fokker-Planck equation associated with the Poisson Representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing sub-cellular volume, decreasing Michaelis-Menten constants and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.
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