Solving the chemical master equation for monomolecular reaction systems analytically
Solving the chemical master equation for monomolecular reaction systems analytically
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DOI：
10.1007/s00285-006-0034-x
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发表时间：
2007-01-01
影响因子：
1.9
通讯作者：
Huisinga, Wilhelm
中科院分区：
文献类型：
--
作者：
Jahnke, Tobias;Huisinga, Wilhelm
通过不同生化反应相互作用的充分搅拌的分子种类混合物的随机动力学可以通过化学主方程（CME）精确建模。生物学和科学计算界的研究主要集中在数值技术的开发上，以通过相关马尔可夫跳跃过程的许多实现来近似 CME 的解决方案。直接求解 CME 的精确和/或有效方法领域仍然是广泛开放的，这是因为它的维数随着涉及的分子种类的数量呈指数增长。在本文中，我们提出了在基础系统受单分子反应控制的情况下，任意初始条件下 CME 的精确解公式。该解可以用多项式和乘积泊松分布的卷积来表示，其中随时间变化的参数根据传统的反应速率方程演化。这种非常结构化的表示可以轻松推断出解决方案的许多属性。模型类包含许多有趣的示例。对于更复杂的反应系统，我们的结果可以被视为构建新数值积分器的第一步，因为单分子情况的解决方案为伽辽金型方法提供了有前途的 ansatz 函数。
The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods.