Modeling stochastic gene expression: From Markov to non-Markov models

Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.