Quantifying statistics of gene product copy-number fluctuations: A stochastic hybrid systems approach

Across diverse cell types, genes are often expressed at low levels resulting in significant stochastic fluctuations (noise) in intracellular copy number of mRNAs and proteins over time. Motivated by single-cell experiments, we capture this stochasticity by considering a given gene that toggles between transcriptionally active and inactive states with the time spent in each state being an arbitrary random variable. mRNAs are synthesized from the active state as per a Poisson process and each molecule degrades after a random lifespan (i.e., random time from birth to death). Modeling this process using the Stochastic Hybrid System (SHS) formalism we derive exact analytical results for the statistical moments of molecular counts. Our results show that for fixed mean mRNA numbers, fluctuations in mRNA levels are amplified with decreasing noise in mRNA lifespan. Relaxing the Poisson process assumption, we next consider a scenario where transcription events occur such that the time between successive events is an arbitrarily distributed random variable. In this case, we show that decreasing noise in mRNA lifespan can both increase/decrease mRNA count fluctuations depending on the underlying transcriptional process. Finally, we extend these results to the protein level, where increasing noise in mRNA counts is sometimes associated with decreasing noise in protein copy numbers.