Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA.
J Chem Phys. 2018 Nov 14;149(18):185101. doi: 10.1063/1.5050808.
Stochasticity plays important roles in reaction systems. Vector fields of probability flux and velocity characterize time-varying and steady-state properties of these systems, including high probability paths, barriers, checkpoints among different stable regions, as well as mechanisms of dynamic switching among them. However, conventional fluxes on continuous space are ill-defined and are problematic when at the boundaries of the state space or when copy numbers are small. By re-defining the derivative and divergence operators based on the discrete nature of reactions, we introduce new formulations of discrete fluxes. Our flux model fully accounts for the discreetness of both the state space and the jump processes of reactions. The reactional discrete flux satisfies the continuity equation and describes the behavior of the system evolving along directions of reactions. The species discrete flux directly describes the dynamic behavior in the state space of the reactants such as the transfer of probability mass. With the relationship between these two fluxes specified, we show how to construct time-evolving and steady-state global flow-maps of probability flux and velocity in the directions of every species at every microstate and how they are related to the outflow and inflow of probability fluxes when tracing out reaction trajectories. We also describe how to impose proper conditions enabling exact quantification of flux and velocity in the boundary regions, without the difficulty of enforcing artificial reflecting conditions. We illustrate the computation of probability flux and velocity using three model systems, namely, the birth-death process, the bistable Schlögl model, and the oscillating Schnakenberg model.
随机性在反应系统中起着重要作用。概率通量和速度的向量场描述了这些系统的时变和稳态特性，包括高概率路径、不同稳定区域之间的障碍和检查点，以及它们之间动态切换的机制。然而，连续空间中的常规通量定义不明确，并且在状态空间的边界处或拷贝数较小时会出现问题。通过基于反应的离散性质重新定义导数和散度算子，我们引入了离散通量的新公式。我们的通量模型充分考虑了状态空间和反应跳跃过程的离散性。反应性离散通量满足连续性方程，并描述了沿着反应方向演化的系统行为。物种离散通量直接描述了反应物在状态空间中的动态行为，例如概率质量的转移。指定了这两个通量之间的关系后，我们展示了如何在每个微状态下的每个物种的方向上构建概率通量和速度的时变和稳态全局流图，以及它们如何与追踪反应轨迹时的概率通量的流出和流入相关。我们还描述了如何在边界区域施加适当的条件，以便在不执行人工反射条件的情况下准确量化通量和速度。我们使用三个模型系统（即生灭过程、双稳态 Schlögl 模型和振荡 Schnakenberg 模型）来说明概率通量和速度的计算。
