Modulation of first-passage time for bursty gene expression via random signals

Qiuying Li; Lifang Huang; Jianshe Yu; Qiuying Li; Lifang Huang; Jianshe Yu

doi:10.3934/mbe.2017065

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1261-1277. doi: 10.3934/mbe.2017065

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Modulation of first-passage time for bursty gene expression via random signals

. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received: 04 November 2016 Revised: 20 January 2017 Published: 01 October 2017
MSC : Primary: 34A30, 60G10, 60J20; Secondary: 60K05, 65D30

The stochastic nature of cell-specific signal molecules (such as transcription factor, ribosome, etc.) and the intrinsic stochastic nature of gene expression process result in cell-to-cell variations at protein levels. Increasing experimental evidences suggest that cell phenotypic variations often depend on the accumulation of some special proteins. Hence, a natural and fundamental question is: How does input signal affect the timing of protein count up to a given threshold? To this end, we study effects of input signal on the first-passage time (FPT), the time at which the number of proteins crosses a given threshold. Input signal is distinguished into two types: constant input signal and random input signal, regulating only burst frequency (or burst size) of gene expression. Firstly, we derive analytical formulae for FPT moments in each case of constant signal regulation and random signal regulation. Then, we find that random input signal tends to increases the mean and noise of FPT compared with constant input signal. Finally, we observe that different regulation ways of random signal have different effects on FPT, that is, burst size modulation tends to decrease the mean of FPT and increase the noise of FPT compared with burst frequency modulation. Our findings imply a fundamental mechanism that random fluctuating environment may prolong FPT. This can provide theoretical guidance for studies of some cellular key events such as latency of HIV and lysis time of bacteriophage $λ.$ In conclusion, our results reveal impacts of external signal on FPT and aid understanding the regulation mechanism of gene expression.
- First-passage time,
- gene expression,
- random input signal,
- burst frequency,
- burst size,
- regulation
Citation: Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1261-1277. doi: 10.3934/mbe.2017065

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Abstract

The stochastic nature of cell-specific signal molecules (such as transcription factor, ribosome, etc.) and the intrinsic stochastic nature of gene expression process result in cell-to-cell variations at protein levels. Increasing experimental evidences suggest that cell phenotypic variations often depend on the accumulation of some special proteins. Hence, a natural and fundamental question is: How does input signal affect the timing of protein count up to a given threshold? To this end, we study effects of input signal on the first-passage time (FPT), the time at which the number of proteins crosses a given threshold. Input signal is distinguished into two types: constant input signal and random input signal, regulating only burst frequency (or burst size) of gene expression. Firstly, we derive analytical formulae for FPT moments in each case of constant signal regulation and random signal regulation. Then, we find that random input signal tends to increases the mean and noise of FPT compared with constant input signal. Finally, we observe that different regulation ways of random signal have different effects on FPT, that is, burst size modulation tends to decrease the mean of FPT and increase the noise of FPT compared with burst frequency modulation. Our findings imply a fundamental mechanism that random fluctuating environment may prolong FPT. This can provide theoretical guidance for studies of some cellular key events such as latency of HIV and lysis time of bacteriophage $λ.$ In conclusion, our results reveal impacts of external signal on FPT and aid understanding the regulation mechanism of gene expression.

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