Generalized master equation for stochastic gene expression with non-Markovian degradation

mRNAs and proteins are believed to degrade in non-Markovian fashions, challenging the common assumption of exponential mRNA and protein lifetimes. Here, we use the queue theory to derive a generalized master equation for a three-stage model of gene expression with nonexponential mRNA and protein degradations. Analytical and numerical results show that the protein degradation memory enlarges the bimodal region and increases the protein noise, whereas the mRNA degradation memory enhances bimodality but reduces the protein noise by lowering the translational burst variance. While these results provide insight into the roles of degradation memories in shaping gene-expression patterns, the generalized master equation is a useful tool for studying gene-expression dynamics in complex cellular environments.