Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA.
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA.
Bull Math Biol. 2023 Mar 31;85(5):39. doi: 10.1007/s11538-023-01136-5.
Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications.
连续时间马尔可夫链经常被用作化学反应网络的随机模型，特别是在系统生物学这个不断发展的领域。对于这些随机化学反应网络（SCRN）来说，一个基本问题是理解这些系统的随机行为对化学反应速率参数的依赖关系。为了解决这个问题，在本文中，我们开发了一些理论工具，称为比较定理，这些定理为 SCRN 提供了随机排序结果。这些定理为这些网络模型中的参数提供了单调依赖的充分条件，使我们能够在适当的条件下获得关于瞬态和稳态行为的信息。这些定理利用了 SCRN 的结构性质，而不仅仅是一般连续时间马尔可夫链的结构性质。此外，我们还推导出了两个定理，用于比较具有不同参数值或相同参数和不同初始条件的 SCRN 的固定分布和平均首次通过时间。这些工具是针对在一般（有限或可数无限）状态空间中取值的 SCRN 开发的，也可用于非质量作用动力学模型。当倾向函数有界时，我们的证明方法给出了一种显式方法来耦合两个可比较的 SCRN，可以用来以可比较的方式同时模拟它们的样本路径。我们通过应用于酶动力学模型和染色质修饰的表观遗传调控模型来说明我们的结果。
