Dimensionality reduction via path integration for computing mRNA distributions

• Main Author: Albert, J. (Author)
• Format: Article
• Language: English
• Published: Springer Science and Business Media Deutschland GmbH 2021
• Subjects: Gene regulatory networks; genetics; Master equation; messenger RNA; Path integral; probability; Probability; Probability distributions; Promoter; RNA, Messenger; Single-cell RNA data
• Online Access: View Fulltext in Publisher

 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× nj, where M is the number of available promoter states and nj 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× Λ 1× Λ 2× ⋯ × Λ L, where Λ j is the cutoff for the probability of the jth species of mRNA; and (b) the ME must be solved up to the cutoffs Λ j, 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 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. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.