Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan.
Phys Rev E. 2019 Jan;99(1-1):012416. doi: 10.1103/PhysRevE.99.012416.
Many biological activities are induced by cellular chemical reactions of diffusing reactants. The dynamics of such systems can be captured by stochastic reaction networks. A recent numerical study has shown that diffusion can significantly enhance the fluctuations in gene regulatory networks. However, the universal relation between diffusion and stochastic system dynamics remains veiled. Within the approximation of reaction-diffusion master equation (RDME), we find general relation that the steady-state distribution in complex balanced networks is diffusion-independent. Here, complex balance is the nonequilibrium generalization of detailed balance. We also find that for a diffusion-included network with a Poisson-like steady-state distribution, the diffusion can be ignored at steady state. We then derive a necessary and sufficient condition for networks holding such steady-state distributions. Moreover, we show that for linear reaction networks the RDME reduces to the chemical master equation, which implies that the stochastic dynamics of networks is unaffected by diffusion at any arbitrary time. Our findings shed light on the fundamental question of when diffusion can be neglected, or (if nonnegligible) its effects on the stochastic dynamics of the reaction network.
许多生物活性是由扩散反应物的细胞化学反应诱导的。这种系统的动力学可以通过随机反应网络来捕捉。最近的一项数值研究表明，扩散可以显著增强基因调控网络中的波动。然而，扩散和随机系统动力学之间的普遍关系仍然不清楚。在反应-扩散主方程（RDME）的近似下，我们发现复杂平衡网络中的稳态分布与扩散无关。这里，复杂平衡是平衡态详细平衡的非平衡推广。我们还发现，对于具有泊松型稳态分布的包含扩散的网络，在稳态时可以忽略扩散。然后，我们推导出具有这种稳态分布的网络的必要和充分条件。此外，我们表明，对于线性反应网络，RDME 简化为化学主方程，这意味着在任何任意时间，网络的随机动力学不受扩散的影响。我们的发现揭示了扩散何时可以忽略，或者（如果不可忽略）其对反应网络的随机动力学的影响的基本问题。
