Stochastic modeling of the mRNA kinetic process: A delay chemical master-equation model based on a survival probability

Gene expression is a random and complex process involving multistep biochemical reactions. This process is commonly modeled by using a delay chemical master equation (DCME), which introduces delays to simplify the intermediate reaction steps. However, such a model is difficult to achieve the stationary analytical solution because it contains one or more joint probabilities. In this paper, we investigate a two-stage gene expression, consisting of an active or inactive promoter, mRNA degradation, RNA polymerase (RNAP) transcription with a delay due to the time interval from RNAP's initialization to the production of a mature RNA. Unlike previous DCME models of gene expression, we introduce a survival probability derived from the undelayed reactions to establish a new DCME model without the joint probabilities. Through our research, the exact stationary solution of this model is obtained, and numerical simulations for it demonstrate that the delay has a remarkable impact on the probability distribution of mRNA number, as well as on the mean and Fano factor of this distribution. Our results provide valuable insights into the study of stochastic delay gene expression.