Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

随机反应网络是描述随机波动相关的物种之间相互作用的基本模型。主方程提供了离散状态空间中概率分布的演变，该空间由每个物种的种群计数向量组成。然而，由于其精确解通常难以捉摸，因此已经提出了几种解析近似值。确定性速率方程 (DRE) 作为一个紧凑的微分方程系统给出了宏观近似，用于估计每个物种的平均种群，但在非线性相互作用动力学的情况下它可能不准确。在这里，我们提出了有限状态扩展（FSE），通过将离散状态空间的选定子集的主方程动力学与 DRE 的平均种群动力学耦合，在随机反应网络的微观和宏观解释之间进行中介的分析方法。一种算法将网络转换为扩展的网络，其中每个离散状态表示为进一步不同的物种。这种转换准确地保留了随机动力学，但扩展网络的 DRE 可以解释为对原始网络的修正。由于固有噪声、多尺度种群和多稳定性，在挑战最先进技术的模型中证明了 FSE 的有效性。一种算法将网络转换为扩展的网络，其中每个离散状态表示为进一步不同的物种。这种转换准确地保留了随机动力学，但扩展网络的 DRE 可以解释为对原始网络的修正。由于固有噪声、多尺度种群和多稳定性，在挑战最先进技术的模型中证明了 FSE 的有效性。一种算法将网络转换为扩展的网络，其中每个离散状态表示为进一步不同的物种。这种转换准确地保留了随机动力学，但扩展网络的 DRE 可以解释为对原始网络的修正。由于固有噪声、多尺度种群和多稳定性，在挑战最先进技术的模型中证明了 FSE 的有效性。