Analysis of non-Markovian effects in generalized birth-death models

Abstract

Birth-death processes are a fundamental reaction module for which we can find its prototypes in many scientific fields. For such a kind of module, if all the reaction events are Markovian, the reaction kinetics is simple. However, experimentally observable quantities are in general consequences of a series of reactions, implying that the synthesis of a macromolecule in general involve multiple middle reaction steps with some reactions that would not be specified by experiments. This multistep process can create molecular memory between reaction events, leading to non-Markovian behavior. Based on the theoretical framework established in a recent paper published in [ 39 ], we find that the effect of non-Markovianity is equivalent to the introduction of a feedback, non-Markovianity always amplifies the mean level of the product if the death reaction is non-Markovian but always reduces the mean level if the birth reaction is non-Markovian, and in contrast to Markovianity, non-Markovianity can reduce or amplify the product noise, depending on the details of waiting-time distributions characterizing reaction events. Examples analysis indicates that non-Markovianity, whose effects were neglected in previous studies, can significantly impact gene expression.