Constraining the complexity of promoter dynamics using fluctuations in gene expression

Gene expression is an inherently stochastic process with transcription of mRNAs often occurring in bursts: short periods of activity followed by typically longer periods of inactivity. While a simple model involving switching between two promoter states has been widely used to analyze transcription dynamics, recent experimental observations have provided evidence for more complex kinetic schemes underlying bursting. Specifically, experiments provide evidence for complexity in promoter dynamics during the switch from the transcriptionally inactive to the transcriptionally active state. An open question in the field is: what is the minimal complexity needed to model promoter dynamics and how can we determine this? Here, we show that measurements of mRNA fluctuations can be used to set fundamental bounds on the complexity of promoter dynamics. We study models wherein the switching time distribution from transcriptionally inactive to active states is described by a general waiting-time distribution. Using approaches from renewal theory and queueing theory, we derive analytical expressions which connect the Fano factor of mRNA distributions to the waiting-time distribution for promoter switching between inactive and active states. The results derived lead to bounds on the minimal number of promoter states and thus allow us to derive bounds on the minimal complexity of promoter dynamics based on single-cell measurements of mRNA levels.