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

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.