Modeling stochastic gene expression: From Markov to non-Markov models

Zhenquan Zhang; Junhao Liang; Zihao Wang; Jiajun Zhang; Tianshou Zhou; Zhenquan Zhang; Junhao Liang; Zihao Wang; Jiajun Zhang; Tianshou Zhou

doi:10.3934/mbe.2020287

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5304-5325. doi: 10.3934/mbe.2020287

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Research article Special Issues

Modeling stochastic gene expression: From Markov to non-Markov models

Key Laboratory of Computational Mathematics, Guangdong Province, and School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received: 24 May 2020 Accepted: 12 July 2020 Published: 12 August 2020

Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.
- gene model,
- noise,
- Markov process,
- non-Markov process
Citation: Zhenquan Zhang, Junhao Liang, Zihao Wang, Jiajun Zhang, Tianshou Zhou. Modeling stochastic gene expression: From Markov to non-Markov models[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5304-5325. doi: 10.3934/mbe.2020287

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Abstract

Gene expression is an inherently noisy process due to low copy numbers of mRNA or protein. The involved reaction events may happen in a Markov fashion but also in a non-Markov manner, depending on waiting-time distributions for the occurrence of reaction events. In recent years, many mechanistic models of stochastic gene expression have been developed to forecast fluctuations in mRNA or protein levels. Here we discus commonalities between these models as well as their extensions from Markov to non-Markov models, focusing on the contributions of noisy sources to the protein level. We derive a useful formula for the protein noise quantified by the ratio of the variance over the squared mean. This formula, expressed in terms of the frequencies of the probabilistic events, can be used in the fast evaluation of fluctuations in the protein abundance. Although the detail of the formula may vary from gene to gene, it highlights sources of the protein noise, which can be decomposed into two parts: spontaneous fluctuations resulting from the birth and death of the protein and forced fluctuations originated from switching between the promoter states.

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