Dimensionality reduction via path integration for computing mRNA distributions

Inherent stochasticity in gene expression leads to distributions of mRNA copy numbers in a population of identical cells. These distributions are determined primarily by the multitude of states of a gene promoter, each driving transcription at a different rate. In an era where single-cell mRNA copy number data are more and more available, there is an increasing need for fast computations of mRNA distributions. In this paper, we present a method for computing separate distributions for each species of mRNA molecules, i.e. mRNAs that have been either partially or fully processed post-transcription. The method involves the integration over all possible realizations of promoter states, which we cast into a set of linear ordinary differential equations of dimension [Formula: see text], where M is the number of available promoter states and [Formula: see text] is the mRNA copy number of species j up to which one wishes to compute the probability distribution. This approach is superior to solving the Master equation (ME) directly in two ways: (a) the number of coupled differential equations in the ME approach is [Formula: see text], where [Formula: see text] is the cutoff for the probability of the jth species of mRNA; and (b) the ME must be solved up to the cutoffs [Formula: see text], which must be selected a priori. In our approach, the equation for the probability to observe n mRNAs of any species depends only on the the probability of observing [Formula: see text] mRNAs of that species, thus yielding a correct probability distribution up to an arbitrary n. To demonstrate the validity of our derivations, we compare our results with Gillespie simulations for ten randomly selected system parameters.