稀释化学反应网络的化学计量和动力学自催化
Stoechiometric and dynamical autocatalysis for diluted chemical reaction networks
摘要：
自催化作用是化学和生物化学系统复制能力的基础。最近，Blokhuis等人给出了反应网络的自催化的化学计量学定义，指出存在反应的组合，使得所有自催化物种的平衡严格为正，并研究了最小的自催化网络，称为{ \em autocatalytic cores}。相比之下，自发自催化-即反应网络内部所有物质从稀释状态即低浓度开始的指数放大-是一种动力学性质。我们在这里介绍一个拓扑条件（顶部）的自催化，即：限制反应网络的描述高度稀释的物种，我们假设存在一个强连接的组件拥有至少一个反应与多个产品（包括多个副本的单一物种）。我们发现这个条件是化学计量自催化的充分必要条件。当降解反应具有足够小的速率时，拓扑条件进一步确保动态自催化，其特征在于严格正的李雅普诺夫指数给出系统的瞬时指数增长率。证明一般是基于辅助马尔可夫链的研究。我们提供的例子一般的自催化核心的类型学Blokhuis等人的I型和III型。在配套文章中，李雅普诺夫指数和行为的增长制度进行了定量研究超出了目前的稀释制度。
英文摘要：
Autocatalysis underlies the ability of chemical and biochemical systems to replicate. Recently, Blokhuis et al. gave a stoechiometric definition of autocatalysis for reaction networks, stating the existence of a combination of reactions such that the balance for all autocatalytic species is strictly positive, and investigated minimal autocatalytic networks, called {\em autocatalytic cores}. By contrast, spontaneous autocatalysis -- namely, exponential amplification of all species internal to a reaction network, starting from a diluted regime, i.e. low concentrations -- is a dynamical property. We introduce here a topological condition (Top) for autocatalysis, namely: restricting the reaction network description to highly diluted species, we assume existence of a strongly connected component possessing at least one reaction with multiple products (including multiple copies of a single species). We find this condition to be necessary and sufficient for stoechiometric autocatalysis. When degradation reactions have small enough rates, the topological condition further ensures dynamical autocatalysis, characterized by a strictly positive Lyapunov exponent giving the instantaneous exponential growth rate of the system. The proof is generally based on the study of auxiliary Markov chains. We provide as examples general autocatalytic cores of Type I and Type III in the typology of Blokhuis et al. In a companion article, Lyapunov exponents and the behavior in the growth regime are studied quantitatively beyond the present diluted regime .
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