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

This work was supported by the National Natural Science Foundation of China (11631005,11626246,11526203,11471085), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16).
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  • 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.

    Mathematics Subject Classification: Primary: 34A30, 60G10, 60J20; Secondary: 60K05, 65D30.


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  • Figure 1.  Schematic diagram for a gene model with burst manner. A: Constant input signal regulates gene expression. Transcription rate is denoted by $\lambda$. Protein count from a single mRNA and mRNA count from a transcription event are in the form of geometric burst and their means are denoted as $b_{2}=\frac{k_{p}}{\delta_{m}}, b_{1}$ respectively. B: Random input signal regulates gene expression. Here, signal regulation is distinguished into two cases. ① represents that input signal $z(t)$ regulates burst rate $\lambda z(t)$. ② represents input signal $z(t)$ regulates the mean $b_{1}z(t)$ of transcriptional burst

    Figure 2.  Effects of input signals on $FPT$ moments, where the blue solid lines represent the regulation on burst frequency by constant input signal; the black solid lines represent the regulation on burst size by constant input signal; the yellow dashed lines represent the regulation on burst frequency by random input signal; the red dashed lines represent the regulation on burst size by random input signal. A, B: The dependence of the mean and noise of FPT on the mean of input signal in the case of burst frequency modulation. C, D: The dependence of the mean and noise of FPT on the mean of input signal in the case of burst size modulation. It confirms that noisy signal regulations tend to increase the mean and noise of FPT compared with noiseless signal regulations. Here, the parameters value $\lambda=0.2, b_{1}=50, b_{2}=0.4, \beta=0.5, m=500.$

    Figure 3.  Effects of random input signal under different regulating ways, where the muddy circle dotted lines represent burst frequency regulation by random signal; the red square dotted lines represent the burst size regulation by random signal. A: A comparison between the effect of burst frequency regulation and the effect of burst size regulation on the mean of FPT. B: A comparison between the effects of burst frequency regulation and burst size regulation on the noise of FPT. The parameter values are the same as those used in Figure 2.

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