Stationary moments, distribution conjugation and phenotypic regions in stochastic gene transcription

Jiajun Zhang; Tianshou Zhou; Jiajun Zhang; Tianshou Zhou

doi:10.3934/mbe.2019307

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 6134-6166. doi: 10.3934/mbe.2019307

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Stationary moments, distribution conjugation and phenotypic regions in stochastic gene transcription

Jiajun Zhang ^1,2,
Tianshou Zhou ^{1,2
,
,}

1.
Guangdong Province Key Laboratory of Computational Science, Guangzhou 510275, P.R. China
2.
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R. China

Received: 10 January 2019 Accepted: 02 April 2019 Published: 03 July 2019

Transcription is a pivotal step in gene expression yet a complex biochemical process. It occurs often in a bursty manner, leading to cell-to-cell variability important for the cells surviving in complex environments. Quantitative experiments of such stochastic transcription call for model analysis in a systematic rather than case-by-case fashion. Here we analyze two general yet biologically-reasonable classes of stochastic transcription models: the first class called promoter models considers that promoter structure is general, and the second class called queuing models considers that waiting-time distributions are general. For the former, we show that there are conjugate relationships between models with different transcription exits. This property well reveals the mechanic principle of stochastic transcription in complex cases of transcription factor regulation. For the latter, we establish an integral equation for the mRNA moment-generating function, through which stationary mRNA moments of any orders can be analytically derived. Finally, we analyze parametric regions for robust unimodality and bimodality. The overall analysis not only lays a foundation for quantitative analysis of stochastic transcription but also reveals how the upstream promoter kinetics impact the downstream expression dynamics.
- stochastic transcription,
- chemical master equation,
- unimodal distribution,
- bimodal distribution,
- cellular phenotype
Citation: Jiajun Zhang, Tianshou Zhou. Stationary moments, distribution conjugation and phenotypic regions in stochastic gene transcription[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 6134-6166. doi: 10.3934/mbe.2019307

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Abstract

Transcription is a pivotal step in gene expression yet a complex biochemical process. It occurs often in a bursty manner, leading to cell-to-cell variability important for the cells surviving in complex environments. Quantitative experiments of such stochastic transcription call for model analysis in a systematic rather than case-by-case fashion. Here we analyze two general yet biologically-reasonable classes of stochastic transcription models: the first class called promoter models considers that promoter structure is general, and the second class called queuing models considers that waiting-time distributions are general. For the former, we show that there are conjugate relationships between models with different transcription exits. This property well reveals the mechanic principle of stochastic transcription in complex cases of transcription factor regulation. For the latter, we establish an integral equation for the mRNA moment-generating function, through which stationary mRNA moments of any orders can be analytically derived. Finally, we analyze parametric regions for robust unimodality and bimodality. The overall analysis not only lays a foundation for quantitative analysis of stochastic transcription but also reveals how the upstream promoter kinetics impact the downstream expression dynamics.

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