Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-five years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the field that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume (i.e., mesoscopic) in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation (CME). Similar to the wavefunction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved: multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic differential equations that offer alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand “complex behavior” and complexity theory, and from which important biological insight can be gained.

中文翻译：

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基于化学主方程的计算细胞动力学：理解复杂性的挑战

现代分子生物学一直是计算科学的重要灵感来源。半个世纪前，理解大分子动力学的挑战已经引领计算成为研究分子生物学的工具集的一部分。25 年前，基因组科学的需求激发了整整一代对离散数学感兴趣的计算机科学家加入现在称为生物信息学的领域。在本文中，我们将根据称为化学主方程 (CME) 的随机离散状态连续时间公式，为小体积（即介观）中的生化反应系统的动力学提出新的数学理论。类似于量子力学中的波函数，与状态空间相关的动态变化的概率景观提供了生化反应系统的基本特征。动力学的随机轨迹通过使用 Gillespie 算法的模拟最为人所知。与 Metropolis 算法相比，这种蒙特卡罗采样技术不遵循具有详细平衡的过程。我们将展示几个示例，说明如何使用 CME 来模拟细胞生化系统。我们还将说明所涉及的计算挑战：多尺度现象、随机性和非线性之间的相互作用，以及宏观决定论如何从细观动力学中产生。我们指出了 CME 计算解决方案的最新进展，包括稳态景观的精确解和提供 Gilespie 算法替代方案的随机微分方程。我们认为 CME 是一个理想的系统，从中可以学习理解“复杂行为”和复杂性理论，从中可以获得重要的生物学见解。