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Inherent stochasticity in gene expression leads to distributions of mRNA copynumbers in a population of identical cells. These distributions are determinedprimarily by the multitude of states of a gene promoter, each drivingtranscription at a different rate. In an era where single-cell mRNA copy numberdata are more and more available, there is an increasing need for fastcomputations of mRNA distributions. In this paper, we present a method forcomputing 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 promoterstates, which we cast into a set of linear ordinary differential equations ofdimension $M\times n_j$, where $M$ is the number of available promoter statesand $n_j$ is the mRNA copy number of species $j$ up to which one wishes tocompute the probability distribution. This approach is superior to solving theMaster equation (ME) directly in two ways: a) the number of coupleddifferential equations in the ME approach is$M\times\Lambda_1\times\Lambda_2\times ...\times\Lambda_L$, where $\Lambda_j$is the cutoff for the probability of the $j^{\text{th}}$ species of mRNA; andb) the ME must be solved up to the cutoffs $\Lambda_j$, which are {\it ad hoc}and must be selected {\it a priori}. In our approach, the equation for theprobability to observe $n$ mRNAs of any species depends only on the theprobability of observing $n-1$ mRNAs of that species, thus yielding a correctprobability distribution up to an arbitrary $n$. To demonstrate the validity ofour derivations, we compare our results with Gillespie simulations for tenrandomly selected system parameters.

Dimensionality reduction via path integration for computing mRNA distributions