The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modeling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis, and this process has since been mathematically analyzed. In this paper, we provide an exact expression for the expected number of self-sustaining autocatalytic networks that will form in a general chemical reaction system, and the expected number of these networks that will also be uninhibited (by some molecule produced by the system). Using these equations, we are able to describe the patterns of catalysis and inhibition that maximize or minimize the expected number of such networks. We apply our results to derive a general theorem concerning the trade-off between catalysis and inhibition, and to provide some insight into the extent to which the expected number of self-sustaining autocatalytic networks coincides with the probability that at least one such system is present.

化学反应系统中自我维持的自催化网络的出现已被研究为模拟生命系统如何首次出现的可能机制。几十年来，人们已经知道，这种网络将在简单的随机催化过程下在聚合物系统内（在裂解和连接反应下）形成，并且这个过程已经被数学分析。在本文中，我们为在一般化学反应系统中形成的自持自催化网络的预期数量提供了一个精确的表达式，以及这些网络的预期数量也将不受抑制（由系统产生的某些分子） . 使用这些方程，我们能够描述最大化或最小化此类网络的预期数量的催化和抑制模式。