In this paper, it is presented a methodology for implementing arbitrarily constructed time-homogenous Markov chains with biochemical systems. Not only discrete but also continuous-time Markov chains are allowed to be computed. By employing chemical reaction networks as a programmable language, molecular concentrations serve to denote both input and output values. One reaction network is elaborately designed for each chain. The evolution of species’ concentrations over time well matches the transient solutions of the target continuous-time Markov chain, while equilibrium concentrations can indicate the steady state probabilities. Additionally, second-order Markov chains are considered for implementation, with bimolecular reactions rather than unary ones. An original scheme is put forward to compile unimolecular systems to DNA strand displacement reactions for the sake of future physical implementations. Deterministic, stochastic and DNA simulations are provided to enhance correctness, validity and feasibility.

中文翻译：

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马尔可夫链的分子计算

在本文中，提出了一种利用生化系统实施任意构造的时间均质马尔可夫链的方法。不仅可以计算离散时间，而且可以计算连续时间的马尔可夫链。通过使用化学反应网络作为可编程语言，分子浓度既可以表示输入值，也可以表示输出值。为每个链精心设计了一个反应网络。物种浓度随时间的演变与目标连续时间马尔可夫链的瞬态解非常匹配，而平衡浓度可表明稳态概率。另外，考虑用二分子反应而不是一元反应来实现二阶马尔可夫链。为了将来的物理实现，提出了一种原始方案，将单分子系统编译成DNA链置换反应。提供确定性，随机和DNA模拟以增强正确性，有效性和可行性。