Transient power-law behaviour following induction distinguishes between competing models of stochastic gene expression

Andrew G Nicoll, Juraj Szavits-Nossan, Martin R Evans, Ramon Grima

What features of transcription can be learnt by fitting mathematical models of gene expression to mRNA count data? Given a suite of models, fitting to data selects an optimal one, thus identifying a probable transcriptional mechanism. Whilst attractive, the utility of this methodology remains unclear. Here, we sample steady-state, single-cell mRNA count distributions from parameters in the physiological range, and show they cannot be used to confidently estimate the number of inactive gene states, i.e. the number of rate-limiting steps in transcriptional initiation. Distributions from over 99% of the parameter space generated using models with 2, 3, or 4 inactive states can be well fit by one with a single inactive state. However, we show that if the mRNA lifetime is hours long, then for many minutes following induction, the increase in the mean mRNA count obeys a power law whose exponent equals the sum of the number of states visited from the initial inactive state to the active state and the number of rate-limiting post-transcriptional processing steps. Our study shows that non-linear regression estimation of the exponent from eukaryotic data is sufficient to estimate the total number of regulatory steps in transcription initiation, splicing, and nuclear export.