Gene transcription is a stochastic process, as the mRNA copies of the same gene in a population of isogeneic cells are often distributed unevenly. The fluctuation has been attributed to the random transition of system states and random production or degradation of transcripts, as characterized by the prevailing two-state model. In addition, as cells live in heterogeneous environments, noisy signals provide a further source of randomness for transcription activation. In this paper, we study how the coupling of random environmental signals and the core transcription system coordinates transcriptional dynamics and noise by extending the two-state model. One of our major concerns is whether noisy signals activate noisier transcription. We find the exact forms for the steady-states of the mean mRNA level and its noise and clarify their dynamical behavior. Our numerical examples strongly suggest that the randomness of the signals inducing a positive or negative regulation does not make significant impact on transcription. Corresponding to each noisy signal, there is a deterministic signal such that the two signals generate nearly identical temporal profiles for the mean and the noise. When transcription is regulated by pulsatile signals, the mean and the noise exhibit damped but almost synchronized oscillations, indicating that noisy pulsatile signals may even reduce transcription noise at some time intervals. Our further analysis reveals that the transition rates in the core transcription system make more notable impacts on creating transcription noise than what the randomness in external signals may contribute.
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Figure 1. The two-state transcription model. The transcription system randomly switches between the inactive (gene OFF) state and the active (gene ON) state with rates $\lambda>0$ and $\gamma>0$. When the gene is active, mRNA molecules are produced with a rate $\upsilon>0$. Independent of system states, mRNA molecules are degraded with a rate $\delta>0$
Figure 2. The model of transcription in a random environment. The production and degradation of mRNAs follow the same mechanism as in the two-state model. The transcription is initiated by parallel pathways $O_1$ and $O_2$, along which the durations for initiation are independent and exponentially distributed with rates $\lambda_1>0$ and $\lambda_2>\lambda_1$, respectively. The pathway selection probabilities are determined by the random processes $Q_1(t)$ and $Q_2(t) = 1-Q_1(t)$
Figure 3. The temporal profiles of the expected pathway selection probability $q_2(t)$ in pulsatile environments. The probability $q_2(t)$ given in Example 1 exhibits distinct dynamics, with zero, one, two or three critical points as shown in the panels a, b, c and d where we set $a_- = 0$, $a^+ = 1$. The vector ($\kappa_1$, $\kappa_2$, $\kappa_3$, $\kappa^+$, $\kappa_-$) takes values a: (0.2, 0.3, 0.25, 0.3, 0.25), b: (5, 0.3, 0.25, 0.3, 0.25), c: (5, 0.8, 0.25, 0.3, 0.25), d: (10, 1, 0.45, 0.3, 0.25)
Figure 4. Minimal impact of noisy signals in positively regulated genes. In each panel, the solid curve represents the transcription activated by a stable signal with $Q_2(t)\equiv 0.6$, and the three dashed-curves represent the transcription activated by random signals with an increasing $Q_2(t)$ that approaches $0.6$ as $t\to \infty$. These curves are generated by applying the parameter values in (56), $\lambda_2 = 5\lambda_1$, and the data specified within each panel to Eqs. (28)-(35). In a-d, the temporal profiles of the mean mRNA level, the variance, the noise strength $\phi(t)$, and the noise $\eta^2(t)$ are shown, respectively. Panels e and f show the dependance of $\phi(t)$ and $\eta^2(t)$ on the mean level. As the dashed-curves do not deviate from the solid curve considerably, noisy signals exhibit only a minimal impact on the mean mRNA level and the transcription noise in positively regulated genes
Figure 5. Weak impact of random signals in negative regulations. In all panels, the solid curves represent the transcription profiles for the system activated by deterministic signals, and the dash-dot curves are for the system with random signals. The selection probability $Q_2(t)$ decreases with $a_i = 0.6-0.1(i-1)$ for $i = 1, 2, \cdots, 6$, and $a_i = 0.1$ for $i>6$. The waiting time $T_i-T_{i-1}$ follows an exponential distribution with a rate $\kappa_i>0$ for random signals, and equals $1/\kappa_i$ for deterministic signals. The activation strengths $\kappa_i = \kappa_2$ for $i>2$, and $(\kappa_1, \kappa_2) = (1, 2)$, $(0.5, 1)$, and $(0.25, 1)$, respectively. In a-c, the temporal profiles of the mean mRNA level, the noise and the noise strength are shown, respectively. Panel d shows the dependance of $\eta^2(t)$ on the mean level. The six curves in each panel are apparently clustered in three groups, indicating that noisy signals make only a weak impact on transcription
Figure 6. Pulsatile signals induce oscillatory dynamics and noises. The transcriptional dynamics of a constantly activated system (the solid curves) and a system activated by pulsatile signals (the dash-dot curves) are shown. In the pulsatile system, $Q_2(t)$ oscillates between $a_{2i-1} = 0$ and $a_{2i} = 1$ for $i\ge 1$. The inter-arrival time $T_i-T_{i-1}$ follows independent and exponential distribution with rate $\kappa_i>0$, $(\kappa_1, \cdots, \kappa_7) = (20, 1, 0.06, 0.09, 0.007, 0.015, 0.0011)$, $\kappa_{8} = \kappa_{10} = \cdots = \kappa^+ = 0.001$, and $\kappa_{9} = \kappa_{11} = \cdots = \kappa_- = 0.0003$. In the constant system, the selection probability for the signal pathway is the limit of the pulsatile system $\kappa_-/(\kappa_-+\kappa^+) = 3/13$. As shown in Fig. 6a, the mean transcription level oscillates around the mean level of the constant system. In Fig. 6b and c, the noise strength $\phi(t)$ and the noise $\eta^2(t)$ also oscillates with some reduced magnitudes. The two curves for the noise against the mean level in Fig. 6d are almost identical, indicating that the oscillations of the mean transcription level and the noise are almost synchronized
Figure 7. The dominant roles of the transition rates. The temporal profiles for the transcription activated by deterministic signals (solid curves) vs. that activated by noise signals (dash-dot curves) with varying activation and inactivation rates are shown in Panels a-d. We fix $\delta = 0.173$, $\upsilon = 118\delta$, and $\lambda_2 = 5\lambda_1$ as in Fig. 4-6. $\lambda_1/\gamma = 0.1$, and $\gamma = 0.94\delta$ or $1.2\delta$ in a and c, and $\lambda_1/\gamma = 2$, and $\gamma = 0.94\delta$ or $1.2\delta$ in b and d. The systems are regulated negatively with $Q_2$ and the inter-arrival time $T_i-T_{i-1}$ defined as in Fig. 5 with $\kappa_1 = 0.5$ and $\kappa_2 = \kappa_3 = \cdots = 1$. In a and b, the mean mRNA levels are clustered and increase rapidly in the first 20 hours to reach the unique peaks at about the same time, and then decay to the same limit. The maximum mRNA level increases about 7-fold from a to b, in response to the 20-fold increase in the activation strength. In Panels c and d, the temporal profiles for the noise strengthes are clearly clustered in two groups. As the inactivation rate $\gamma$ increases from $0.94\delta$ to $1.2\delta$, the noise strengthes shifted down by 6.5 in Panel c and by 1.7 in Panel d at each time $t$. Corresponding to the 20-fold increase in the activation strength from Panels c to d, the noise strength has about 5-fold reduction
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The two-state transcription model. The transcription system randomly switches between the inactive (gene OFF) state and the active (gene ON) state with rates
The model of transcription in a random environment. The production and degradation of mRNAs follow the same mechanism as in the two-state model. The transcription is initiated by parallel pathways
The temporal profiles of the expected pathway selection probability
Minimal impact of noisy signals in positively regulated genes. In each panel, the solid curve represents the transcription activated by a stable signal with
Weak impact of random signals in negative regulations. In all panels, the solid curves represent the transcription profiles for the system activated by deterministic signals, and the dash-dot curves are for the system with random signals. The selection probability
Pulsatile signals induce oscillatory dynamics and noises. The transcriptional dynamics of a constantly activated system (the solid curves) and a system activated by pulsatile signals (the dash-dot curves) are shown. In the pulsatile system,
The dominant roles of the transition rates. The temporal profiles for the transcription activated by deterministic signals (solid curves) vs. that activated by noise signals (dash-dot curves) with varying activation and inactivation rates are shown in Panels a-d. We fix