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 $M\times n_j$, where $M$ is the number of available promoter states and $n_j$ 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 $M\times\Lambda_1\times\Lambda_2\times ...\times\Lambda_L$, where $\Lambda_j$ is the cutoff for the probability of the $j^{\text{th}}$ species of mRNA; and b) 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 the probability to observe $n$ mRNAs of any species depends only on the the probability of observing $n-1$ 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.