具有一般动力学的化学反应网络的反应裂解与复平衡分布
Reaction cleaving and complex-balanced distributions for chemical reaction networks with general kinetics
AI总结：
研究随机反应网络的复平衡平稳分布，通过对反应图循环的条件刻画其存在性与形式，推广了亏值定理，并提出分解强连通图为不相交循环的迭代方法以保持等价动力学与复平衡性。
摘要：
反应网络已成为生物科学中的主要建模框架，从流行病学和种群生物学到遗传学和细胞生物学。近年来，在随机反应网络（SRNs）方面取得了很大进展，这些网络被建模为连续时间马尔可夫链（CTMCs）及其平稳分布。我们关注复平衡平稳分布，其中从一个复合物流出的概率流等于流入该复合物的流。我们通过对反应图（一个有向图）的循环条件，刻画了具有任意转移函数的SRNs的复平衡分布的存在性和形式。此外，我们给出了复平衡分布存在的一个充分条件，并给出了它何时也是必要条件的精确条件。对于质量作用动力学（及其某些推广）或者有向图的连通分量是循环的情况，该充分条件也是必要的。此外，我们陈述了一个亏值定理，这是随机质量作用动力学的亏值定理向任意随机动力学的推广。该定理给出了复平衡分布存在的参数空间的余维数。为了实现这一点，我们构建了一个迭代过程，将强连通反应图分解为不相交的循环，使得相应的SRN具有等价的动力学并保持复平衡性，前提是原始SRN具有此性质。这种分解可能具有独立的意义，并可能适用于一般的边标记有向图。
英文摘要：
Reaction networks have become a major modelling framework in the biological sciences from epidemiology and population biology to genetics and cellular biology. In recent years, much progress has been made on stochastic reaction networks (SRNs), modelled as continuous time Markov chains (CTMCs) and their stationary distributions. We are interested in complex-balanced stationary distributions, where the probability flow out of a complex equals the flow into the complex. We characterise the existence and the form of complex-balanced distributions of SRNs with arbitrary transition functions through conditions on the cycles of the reaction graph (a digraph). Furthermore, we give a sufficient condition for the existence of a complex-balanced distribution and give precise conditions for when it is also necessary. The sufficient condition is also necessary for mass-action kinetics (and certain generalisations of that) or if the connected components of the digraph are cycles. Moreover, we state a deficiency theorem, a generalisation of the deficiency theorem for stochastic mass-action kinetics to arbitrary stochastic kinetics. The theorem gives the co-dimension of the parameter space for which a complex-balanced distribution exists. To achieve this, we construct an iterative procedure to decompose a strongly connected reaction graph into disjoint cycles, such that the corresponding SRN has equivalent dynamics and preserves complex-balancedness, provided the original SRN had so. This decomposition might have independent interest and might be applicable to edge-labelled digraphs in general.
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