Well-mixed stochastic chemical kinetics are properly modeled by the chemical master equation (CME) and associated Markov jump processes in molecule number space. If the reactants are present in large amounts, however, corresponding simulations of the stochastic dynamics become computationally expensive and model reductions are demanded. The classical model reduction approach uniformly rescales the overall dynamics to obtain deterministic systems characterized by ordinary differential equations, the well-known mass action reaction rate equations. For systems with multiple scales, there exist hybrid approaches that keep parts of the system discrete while another part is approximated either using Langevin dynamics or deterministically. This paper aims at giving a coherent overview of the different hybrid approaches, focusing on their basic concepts and the relation between them. We derive a novel general description of such hybrid models that allows expressing various forms by one type of equation. We also check in how far the approaches apply to model extensions of the CME for dynamics which do not comply with the central well-mixed condition and require some spatial resolution. A simple but meaningful gene expression system with negative self-regulation is analysed to illustrate the different approximation qualities of some of the hybrid approaches discussed. Especially, we reveal the cause of error in the case of small volume approximations.

中文翻译：

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化学反应网络的混合模型：多尺度理论及其在基因调控系统中的应用

良好混合的随机化学动力学可以通过化学主方程（CME）和分子数空间中相关的马尔可夫跃迁过程进行正确建模。但是，如果反应物大量存在，则随机动力学的相应模拟在计算上变得昂贵，并且需要简化模型。经典的模型约简方法统一地重新缩放整体动力学，以获得由常微分方程，众所周知的质量作用反应速率方程表征的确定性系统。对于具有多个比例的系统，存在混合方法，该方法可以使系统的某些部分保持离散，而另一部分则使用Langevin动力学或确定性地进行近似。本文旨在对不同的混合方法进行一致的概述，专注于它们的基本概念以及它们之间的关系。我们得到了这种混合模型的新颖概括描述，该模型允许通过一种类型的方程式表达各种形式。我们还检查了该方法在多大程度上适用于CME的模型扩展，以实现不符合中央充分混合条件且需要一定空间分辨率的动力学。分析了一个具有负自我调节功能的简单但有意义的基因表达系统，以说明所讨论的某些混合方法的不同近似质量。尤其是，我们揭示了在小体积逼近情况下的错误原因。我们还检查了该方法在多大程度上适用于CME的模型扩展，以实现不符合中央充分混合条件且需要一定空间分辨率的动力学。分析了一个具有负自我调节作用的简单但有意义的基因表达系统，以说明所讨论的某些混合方法的不同近似质量。尤其是，我们揭示了在小体积逼近情况下的错误原因。我们还检查了该方法在多大程度上适用于CME的模型扩展，以实现不符合中央充分混合条件且需要一定空间分辨率的动力学。分析了一个具有负自我调节功能的简单但有意义的基因表达系统，以说明所讨论的某些混合方法的不同近似质量。尤其是，我们揭示了在小体积逼近情况下的错误原因。