Transient and delay chemical master equations

Abstract

The serial nature of reactions involved in the RNA life-cycle motivates the incorporation of delays in models of transcriptional dynamics. The models couple a bursty or switching promoter to a fairly general set of Markovian or deterministically delayed monomolecular RNA interconversion reactions with no feedback. We provide numerical solutions for the RNA copy number distributions the models induce, and solve several systems with splicing and degradation. An analysis of single-cell and single-nucleus RNA sequencing data using these models reveals that the kinetics of nuclear export do not appear to require invocation of a non-Markovian waiting time.

Accurate quantification of single-cell and single-nucleus RNA-seq transcripts using distinguishing flanking k-mers

“…The other one represents a single-nucleus RNA-seq experiment consisting of 4,000,000 nascent RNA reads and 1,000,000 mature RNA reads. The respective ratios of mature to nascent reads was estimated based on (Gorin, Yoshida, and Pachter 2022). The mature transcripts were obtained from version 104 of the Ensembl of GRCh38, and the nascent transcripts were taken to be the entire sequence from the start of the first exon through the end of the last exon.…”

“…The utility of single-cell RNA-seq measurements for defining cell types has represented a marked improvement over bulk RNA-seq, and is driving rapid adoption of single-cell RNA-seq assays (Zeng 2022). Another application of single-cell RNA-seq that is not possible with bulk RNA-seq is the study of cell transitions and transcription dynamics, even via snapshot single-cell RNA-seq experiments (Gorin et al 2022). This novel application of single-cell RNA-seq is based on the quantification of both nascent and mature mRNAs (Figure 1a), lending import to the computational problem of accurately quantifying these two modalities.…”

Biophysical modeling with variational autoencoders for bimodal, single-cell RNA sequencing data

“…However, in spite of its limitations and assumptions, the simple Markovian two-stage model has been successful in the past [61, 62]. Mechanistic evidence does not suggest that, e.g., deterministic delayed elongation is necessary to represent “unspliced” distributions [63]: under this model, we appear to be able to treat splicing and degradation as Markovian, without representing the elongation process at all, and obtain reasonable fits to the data. Somewhat surprisingly [64], the geometric-Poisson distribution, which describes bursty transcription coupled to deterministic elongation [65], is a particularly poor fit for unspliced RNA counts [63].…”

“…Furthermore, comparisons of paired single-cell and single-nucleus datasets are hampered by the limited characterization of the noise sources in the latter technology. Yet again, we have generally found that omitting this effect in single-cell data produces acceptable fits [61– 63].…”

The telegraph process is not a subordinator

“…To generate synthetic data for Figure 1, we simulated a system with 𝑘 𝑜𝑛 = 0.15, 𝑘 𝑜 𝑓 𝑓 = 0.1, 𝑘 𝑡 𝑥 = 20, and 𝛾 = 3.14 using Gillespie's stochastic simulation algorithm (17), as previously implemented for (18). We performed 1,000 simulations, run until 𝑡 = 5, with the system state stored at 200 uniformly spaced time points (Δ𝑡 = 0.025).…”