Department of Physics, Humboldt University of Berlin, Newtonstr. 15, D-12489 Berlin, Germany.
J Chem Phys. 2010 Nov 21;133(19):195101. doi: 10.1063/1.3505552.
Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. 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 subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-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 subcellular 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.
细胞内的化学反应发生在毫微微升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拷贝数较低，因此浓度会有相当大的波动。相比之下，速率方程模型基于相互作用分子数量无穷大的隐含假设，或者等效地，反应在恒定的宏观浓度下在无限大的体积中发生。在本文中，我们计算了由任意数量的酶催化反应组成的化学反应网络的速率方程解的有限体积校正（或等效地，有限拷贝数校正），这些反应被限制在一个小的亚细胞隔室内部。这是通过将准稳态假设的介观版本应用于与泊松表示的化学主方程相关的精确福克-普朗克方程来实现的。该过程为有限体积校正提供了令人印象深刻的简单和紧凑的表达式。我们证明，对于限制在小体积中的酶反应网络，速率方程的预测总是会低估实际的稳态底物浓度。特别是，我们表明有限体积校正随亚细胞体积、米氏常数和酶饱和度的降低而增加。校正的大小对网络的拓扑结构敏感。该理论的预测与通常与蛋白质甲基化和代谢相关的两种类型的网络的随机模拟非常吻合。
