Title: Solving the chemical master equation for monomolecular reaction systems analytically: a Doi-Peliti path integral view Show Chinese title

Title: 解析求解单分子反应系统的化学主方程：Doi-Peliti路径积分视角

Abstract: The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME -- which typically consists of infinitely many coupled differential equations -- are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find an explicit and exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. We also find a formal exact time-dependent solution for any CME whose list of reactions involves only zero and first order reactions, which may be the most general result currently known. Show Chinese abstract

Abstract: 化学主方程（CME）是对相互作用分子的基本描述，常用于建模化学动力学和噪声基因调控网络。 CME的精确时变解——通常由无限多个耦合微分方程组成——很少见，对于数值基准测试和理解更复杂系统的行为很有价值。 Jahnke和Huisinga对单分子反应系统的CME精确时变解的开创性计算是最具普遍性的解析结果之一；然而，它很难推广，因为它关键依赖于单分子反应的性质。 在本文中，我们使用Doi-Peliti路径积分方法重新推导了Jahnke和Huisinga关于单分子反应系统时变概率分布和矩的结果，这种方法将求解CME转化为评估许多积分。 虽然Doi-Peliti方法不太直观，但它更加机械，因此更容易推广。 为了说明Doi-Peliti方法如何超越Jahnke和Huisinga的方法，我们还找到了一个涉及自催化反应的问题的显式且精确的时变解，Jahnke和Huisinga认为他们的方法无法解决这个问题。 我们还找到了任何只包含零级和一级反应的CME的正式精确时变解，这可能是目前最普遍的结果。