Dimensionality reduction via path integration for computing mRNA distributions

TL;DR

This paper introduces a fast, efficient method for computing mRNA distribution profiles in cells by integrating over promoter states, outperforming traditional master equation approaches in speed and accuracy.

Contribution

The authors develop a novel linear ODE-based method for calculating mRNA distributions that avoids ad hoc cutoffs and reduces computational complexity compared to the master equation approach.

Findings

Method accurately matches Gillespie simulations.

Computational efficiency surpasses traditional master equation solutions.

Applicable to multiple mRNA species with different processing states.

Abstract

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.