Mean-field theory accurately captures the variation of copy number distributions across the mRNA's life cycle

We consider a stochastic model where a gene switches between two states, an mRNA transcript is released in the active state and subsequently it undergoes an arbitrary number of sequential unimolecular steps before being degraded. The reactions effectively describe various stages of the mRNA life cycle such as initiation, elongation, termination, splicing, export and degradation. We construct a novel mean-field approach that leads to closed-form steady-state distributions for the number of transcript molecules at each stage of the mRNA life cycle. By comparison with stochastic simulations, we show that the approximation is highly accurate over all of parameter space, independent of the type of expression (constitutive or bursty) and of the shape of the distribution (unimodal, bimodal and nearly bimodal). The theory predicts that in a population of identical cells, any bimodality is gradually washed away as the mRNA progresses through its life cycle.