Control of noise in gene expression by transcriptional reinitiation

TL;DR

This paper provides an analytical study of how transcriptional reinitiation influences noise in gene expression, showing it can reduce variability and serve as a fine-tuned regulatory mechanism.

Contribution

It offers exact analytical expressions for mRNA and protein noise, demonstrating reinitiation's role in noise reduction and regulation in gene expression.

Findings

Fano factor in proteins matches experimental data

Reinitiation can produce sub-Poissonian mRNA levels

Reinitiation allows increasing mRNA without raising noise

Abstract

Gene expression is a random or noisy process. The process consists of several random events among which the reinitiation of transcription by RNAP is an important one. The RNAP molecules can bind the gene only after the promoter gets activated by transcription factors. Several transcription factors bind the promoter to put the gene in the active state. The gene turns into inactive state as the bound transcription factors leave the promoter. During the active period of the gene, many RNAP molecules transcribe the gene to synthesize the mRNAs. The binding event of RNAP to the active state of the gene is a probabilistic process and therefore, introduces noise or fluctuations in the mRNA and protein levels. In this paper, we analytically calculate the Fano factor in mRNA and protein levels and also the probability distribution of mRNA numbers exactly with the binding event of RNAPs in gene transcription process. The analytically calculated expression of Fano factor of proteins shows excellent agreement with an experimental result. Then we show that the Fano factor in mRNA levels can be sub-Poissonian due to the reinitiation of transcription by RNAP and the mean mRNA level can be increased without increasing the Fano factor. Our study show that the Fano factor can also be reduced keeping mRNA levels fixed. We find that the reinitiation of transcription can behave as a fine-tuned control process to regulate the mRNA/protein level in the cell.

Equations

$$\begin{array}[]{ccc}\frac{\partial p(l_{2},l_{3},m,n,t)}{\partial t}&=&k_{1}[(l-l_{2}-l_{3}+1)p(l_{2}-1,l_{3},m,n,t)-(l-l_{2}-l_{3})p(l_{2},l_{3},m,n,t)]\\ &&+k_{2}[(l_{2}+1)p(l_{2}+1,l_{3},m,n,t)-l_{2}p(l_{2},l_{3},m,n,t)]\\ &&+k_{3}[(l_{2}+1)p(l_{2}+1,l_{3}-1,m,n,t)-l_{2}p(l_{2},l_{3},m,n,t)]\\ &&+k_{4}[(l_{3}+1)p(l_{2}-1,l_{3}+1,m,n,t)-l_{3}p(l_{2},l_{3},m,n,t)]\\ &&+J_{m}[(l_{3}+1)p(l_{2}-1,l_{3}+1,m-1,n,t)-l_{3}p(l_{2},l_{3},m,n,t)]\\ &&+k_{m}[(m+1)p(l_{2},l_{3},m+1,n,t)-mp(l_{2},l_{3},m,n,t)]\\ &&+J_{p}[mp(l_{2},l_{3},m,n-1,t)-mp(l_{2},l_{3},m,n,t)]\\ &&+k_{p}[(n+1)p(l_{2},l_{3},m,n+1,t)-np(l_{2},l_{3},m,n,t)]\end{array}$$

$$<m>=\frac{k_1{k}_3}{a_2}\frac{J_m}{k_m},F{F}_m=1+\frac{J_m{k}_3(a_2-k_1{a}_1)}{a_2(a_1{k}_m+a_2)}$$

$$<p>=\frac{<m>J_p}{k_p},F{F}_p=1+\frac{J_p}{k_p+k_m}+\frac{J_p{J}_m{a}_8(k_m+a_4)}{k_m(k_p+k_m)a_4}+\frac{J_p{J}_m{k}_3(a_6-k_1{k}_p{a}_8)}{k_p(k_p+k_m)a_5}-\frac{J_p{J}_m{k}_1{k}_3}{k_m{k}_p{a}_2}$$

$$a_2<k_1{a}_{1}or(J_m+k_4)<\frac{k_1}{k_2}(k_1+k_2+k_m)$$

$$J_m^c=\frac{k_1}{k_2}(k_1+k_2+1)-k_4$$

$$\frac{\partial p_1(m,t)}{\partial t}=k_2{p}_2(m,t)-k_1{p}_1(m,t)+k_m[(m+1)p_1(m+1,t)-m{p}_1(m,t)]$$

$$\frac{\partial p_2(m,t)}{\partial t}=k_1{p}_1(m,t)+k_4{p}_3(m,t)-(k_2+k_3)p_2(m,t)+J_m{p}_3(m-1,t)+k_m[(m+1)p_2(m+1,t)-m{p}_2(m,t)]$$

$$\frac{\partial p_3(m,t)}{\partial t}=k_3{p}_2(m,t)-k_4{p}_3(m,t)-J_m{p}_3(m,t)+k_m[(m+1)p_3(m+1,t)-m{p}_3(m,t)]$$

$$p(m)=\frac{(-h_4{)}^m\Gamma (h_1+m)\Gamma (h_2-h_3)\Gamma (h_2+h_3)}{\Gamma (m+1)\Gamma (h_1)\Gamma (h_2-h_3+m)\Gamma (h_2+h_3+m)}{}_p{F}_q[\{h_1+m\},\{h_2-h_3+m,h_2+h_3+m\},h_4]$$

Full text

Key words: Stochastic gene expression, Reinitiation of transcription, Fano factor

Ramakrishna Mission Vidyamandira, Belur Math, Belur, Howrah, West Bengal, India

1 Introduction

Gene expression and its regulation is an inherently stochastic or noisy process and this stochasticity produces cell-to-cell variation in mRNA and protein levels. The variation or fluctuations in mRNA and protein levels is qualitatively known as noise. The genetically identical individuals can be very different even if they grow in an identical environmental condition because of noisy gene expression. There are large number of theoretical and experimental studies at the single cell level which establish the stochastic nature of gene expression [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The effect of noise or fluctuations in gene expression can be beneficial or detrimental to the progression of cellular activities [5, 9, 17, 18, 19, 20]. Variability in mRNA and protein levels can be beneficial to adapt to sudden changes in environmental conditions [17]. The noise in gene expression can also generate phenotypic diversity in genetically identical cells and may play an important role in the survival of cell colonies [5]. Fluctuations in mRNA and protein levels can also have detrimental role in functioning the fine-tuned cellular processes [19]. Different studies show that the cellular system has its own mechanism to reduce noise or fluctuations in gene expression. There is a study showing that noise levels in proteins from essential genes are lower than that from nonessential genes [21]. It has also been shown that organisms could evolve to minimize the variability in gene expression by adopting special technique e.g., high transcription rate and low translation rate. The noise strength does not remain fixed throughout the life of a cell rather it decreases during the normal aging of a cell [22]. Many other processes exist by which the noise in gene expression can be reduced [23, 24].

The gene expression and regulation consists of several important biochemical steps e.g., transcription, translation, degradation etc. The regulation can be achieved at the transcription or translation or degradation level and the control of transcription is the dominant form of regulation of gene expression. The transcriptional regulation is mostly done by DNA binding proteins known as transcription factors. A group of transcription factors (TFIIA, TBP and associated factors) bind the promoter sequences on the DNA and turn the gene into the active state or intermediate promoter complex [25, 26]. The transcription factors which modulate the transitions to the active (inactive) state are known as activators (repressors). Gene can either be in the active or inactive states in the cell and random transitions take place between the two states [6, 7, 8, 9, 10, 11]. In the active state, the RNA polymerase (RNAP) proteins bind the gene and form a preinitiation complex (including RNAP). The bound RNAP then initiate the transcription of the protein-coding gene to synthesize mRNAs [4, 5]. The RNAP molecules transcribe the gene several times and synthesize a burst of mRNA. That is known as reinitiation of transcription by RNAP. The initiation of transcription by RNAP molecules is an important part of the transcriptional regulation [27]. The control of transcription at the initiation stage is a key mechanism for the regulation of gene expression.

There are some gene regulatory systems where activators and repressors are both involved in transcriptional regulation [4, 5, 12, 13]. It has been shown that the gene can be in more than two states under the regulation of activators and repressors [4, 5, 13, 29]. It has also been shown that the stochastic effects due to random transitions between active and inactive states of a gene are much stronger than the stochastic effects caused by random production and degradation of single mRNA and protein molecules [4, 5, 7, 9, 10, 29, 30]. In the course of time, different model systems have been developed to explain the experimentally observed variability in gene expression. The simplest model is the constitutive gene expression in which gene is always in the active state and mRNAs are synthesized at constant rate from that state [2, 11, 30, 31]. In such case, the fluctuations in mRNA level arise from the random birth and death of mRNAs. The study of cell-to-cell variation in mRNA and protein levels at the single cell level is mostly done by assuming two states of a gene under the regulation of activators [6, 7, 8, 9, 10, 11, 14, 15]. In the two-state model, the randomness in mRNA levels comes from the random transitions between the gene states and from the random birth and death of mRNAs from the active gene [15, 30, 31]. But, the stochasticity due to reinitiation of transcription by RNAP is ignored in both constitutive and two-state gene activation models. Blake et al. experimentally study the synthetic GAL1* yeast promoter and identify the regulatory mechanism using stochastic simulations that agrees well with their experimental data. They observed that the pulsatile mRNA production through reinitiation is crucial for the dependence of noise or fluctuations on transcriptional efficiency [4, 5]. It has been shown that the reinitiation also enhances the reliability of transcriptional response in eukaryotic systems [28]. Sanchez et al. studied the GAL1* yeast promoter also and analytically calculate the various moments neglecting the reinitiation step in gene transcription [29]. Though it has been identified that the reinitiation of transcription by RNAP plays crucial role in the phenotypic variability in cells but its functional role in noisy mRNA or protein levels is very poorly understood.

The stochasticity in gene expression is generally quantified by calculating the variance or coefficient of variation (CV) or Fano factor [2, 3, 4, 7, 15, 29, 30, 31]. The mRNA synthesis from a gene constitutively is a Poisson process with unity Fano factor ($FF$). The two-state model gives rise to an increased FF in mRNA level ($FF_{m}$) due to random transitions between gene states. The $FF_{m}$ now becomes super-Poissonian (greater than unity) [15, 30]. It is observed that only negative feedback can reduce the $FF_{m}$ and can move that into the sub-Poissonian (less than unity) regime [14]. With the two-state gene activation model, we cannot think of the sub-Poissonian Fano factor regime at the mRNA level without the negative feedback in gene expression. In this article, we show that the reinitiation of transcription by RNAP in two-state gene activation process can move the $FF_{m}$ into the sub-Poissonian regime. We observe that the mean mRNA level can be increased without increasing the $FF_{m}$ by controlling the reinitiation of transcription. We also observe that $FF_{m}$ can be decreased to sub-Poissonian regime keeping average mRNA level fixed. Again, the knowledge of FF may not always give the complete information of the proteins. The distribution of mRNA/protein level helps to identify the more important information about responses of gene expression. Obtaining an exact analytical expression for the distribution of mRNA/protein has been a challenging task and that always gives some extra insights. There are many works that have already been done on the exact solution of distribution of mRNA and protein levels with and without the feedback but excluding the reinitiation process [8, 11, 13, 14]. We also find out the exact steady state distribution of mRNA levels with reinitiation of transcription by RNAP.

2 The model and analysis

2.1 The model

We consider the gene regulation model where only activators regulate the gene transcription. In this model, the activators bind the promoter region of the gene and turn the gene into active state ($G_{2}$) from the inactive state ($G_{1}$). Once the gene turns into active state, the RNAP then binds the gene and form an initiation complex ($G_{3}$). Then, bound RNAP can follow two paths: either it unbinds and leaves the $G_{3}$ state or it starts transcription. As it starts transcription, it moves forward along the gene and the $G_{3}$ state turns into the $G_{2}$ again. Another RNAP can bind the gene again to form an initiation complex and then starts transcription. So, RNAP can bind multiple times in the active state of the gene and a burst of mRNAs are synthesized before the gene turns into the inactive state ($G_{1}$) from the active state ($G_{2}$). The biochemical reactions and the corresponding rate constants are shown in figure 1. The model considered here is a part of the network identified in synthetic GAL1* promoter by Blake et al. [4]. The GAL1* promoter is regulated by activator and repressor both and their concentrations are further controlled by external inducer GAL and ATC respectively. In experiment, the activation (repression) probability of the promoter increases (decreases) with the increase of GAL (ATC) concentrations [4]. The GAL1* promoter may behave as an activator-only system with the full induction of ATC ([ATC] = 500ng/ml). Thus, the promoter architecture proposed by them has the resemblance with our model given in figure 1 with full ATC induction.

2.2 Analysis

Let us consider that there are $l$ copy number of a particular gene exist in the cell. Let $p(l_{2},l_{3},m,n,t)$ be the probability that at time $t$, there are $m$ number of mRNAs and $n$ number of proteins molecules with $l_{2}$ number of genes in the active state ($G_{2}$) and $l_{3}$ number of genes in the initiation complex ($G_{3}$). The number of genes in the inactive states are $l_{1}=(l-l_{2}-l_{3})$ as the gene copy number is conserved. The time evaluation of the probability is given by the Master equation [32]

[TABLE]

The standard generating function technique is now applied for the expression of means and the variances of mRNA and protein in the steady state [32].

The expressions for mean and FF of mRNAs ($<m>,\,FF_{m}$) and proteins ($<p>,\,FF_{p}$) in terms of the rate constants (figure. 1) at the steady state are given by (for $l=1)$

[TABLE]

[TABLE]

where $a_{1}=k_{m}+J_{m}+k_{1}+k_{2}+k_{3}+k_{4},\,a_{2}=k_{1}J_{m}+k_{2}J_{m}+k_{2}k_{4}+k_{1}k_{3}+k_{1}k_{4},\,a_{3}=k_{m}+J_{m}+k_{4}$, $a_{4}=k_{p}+J_{m}+k_{4}$, $a_{5}=a_{4}\{a_{2}+k_{p}(a_{1}+k_{p}-k_{m})\}$, $a_{6}=(k_{p}a_{7}+k_{1})a_{4}+k_{p}a_{8}(J_{m}+k_{4})$, $a_{7}=a_{3}(k_{m}+k_{1})/(a_{1}k_{m}+a_{2})$ and $a_{8}=k_{3}(k_{m}+k_{1})/(a_{1}k_{m}+a_{2})$.

Blake et al. measured the $FF_{p}$ as a function of transcriptional efficiency by varying the inducer [GAL] (0 to 2%) with full induction of [ATC] (500ng/ml) and also by varying [ATC] with full induction of [GAL] (2%) [4]. They found maximum $FF_{p}$ at the intermediate values of transcriptional efficiency. We compared our analytical result (eq.(3)) with the experimental observations of Blake et al. with full [ATC] induction. The plot of $FF_{p}$ with transcriptional efficiency along with the experimental data points is shown in figure 2(a). We also plot the mean protein $<p>$ level with GAL concentration (inset) with the same rate constants as given in Blake et al. [4]. We also plot the $FF_{p}$ with transcriptional efficiency and $<p>$ with GAL concentration (inset) without the reinitiation of transcription by RNAP (red dotted curves in figure 2(a)). We plot the percentage relative error between the experimental data points and analytically calculated values with transcriptional efficiency in figure 2(b). That shows the excellent agreement between the experimental data in Blake et al. [4] and our analytically calculated curve for intermediate and higher values of transcriptional efficiency and a little disagreement at the lower values of transcriptional efficiency. The initial disagreement may be due to the presence of repressors in the experimental system. We also observe the large deviation of $FF_{p}$ and $<p>$ from the experimental data points without the reinitiation of transcription by RNAP.

The transcriptional reinitiation by RNAP affects the FF at mRNA level first. Post transcription processes then further amplify that effect. So, we study the $FF_{m}$ to explore the role of reinitiation of transcription. In the expression of $FF_{m}$ in equation (2), the first term arises due to the random birth and death of mRNA molecules and the second term arises due to the random transitions between different gene states, $G_{1}$, $G_{2}$ and $G_{3}$ (figure 1). Eq. (2) shows that the reinitiation can drive the $FF_{m}$ to the sub-Poissonian level depending on the condition, given by

[TABLE]

And there is a critical value $J_{m}^{c}$ for the given value of $k_{1}$, $k_{2}$ and $k_{4}$ and that is given by

[TABLE]

The transcriptional reinitiation process in figure 1 is mainly controlled by the rate constants $k_{3}$ and $J_{m}.$ We observe the variation of $FF_{m}$ with $k_{3}$ for the rate constants $k_{1}=6.02$ and $k_{2}=3.012$ (i.e., high $GAL$ concentration ($30\%$)) with $J_{m}$ as parameter (figure 3(a)). For the rate constant considered in figure 3, the critical value of $J_{m}$ is $J_{m}^{c}=10.048$ for which the $FF_{m}$ is Poissonian (green, solid line in figure 3(a)). For $J_{m}$ greater than $J_{m}^{c}$, the $FF_{m}$ always lies in the super-Poissonian phase (blue dashed line for $J_{m}=13.048$ in figure 3(a)) whereas for $J_{m}$ less than $J_{m}^{c}$ the $FF_{m}$ always lies in the sub-Poissonian phase (red dotted line for $J_{m}=1.048$ in figure 3(a)). The green solid line ($FF_{m}$=1) in figure 3(a) clearly separates the sub-Poissonian and super-Poissonian phases. The degree of deviation of $FF_{m}$ towards sub-Poissonian phase increases with the decreases in $k_{2}$ and $k_{4}$. The variation of $FF_{p}$ is plotted against $k_{3}$ with above mentioned three different values of $J_{m}$. Translation in gene expression produces burst of proteins from each mRNAs. That bursting process adds some extra amount of fluctuations in protein levels (eq.(3)). The straight solid line (for $J_{m}=10.048$) and curved dot-dashed line (for $J_{m}=8.048$) in figure 3(a) becomes curved and straight respectively in figure 3(b) due to translation in protein synthesis. In the previous models and calculations, the $FF_{m}$ is always found to be greater than unity [9, 15, 29, 30]. But, with the inclusion of reinitiation processes in gene transcription, we observe that two distinct phases viz. super-Poissonian and sub-Poissonian Fano factors separated by Poissonian one are possible. Depending on the values of different rate constants, the $FF_{m}$ can be in any one of the phases.

The rate constants for gene expression for different gene are not unique. The different literature on gene expression pointed out different values of rate constants for different steps. Figures 4(a) and 4(b) show the more intense variation of $FF_{m}$ with $k_{3}$ for the rate constants given in Kaern et al. [9]. We also calculate the $FF_{m}$ and $FF_{p}$ from stochastic simulation using Gillespie algorithm with the rate constants corresponding to the curves in figures 4(a) and 4(b) [33]. Figures 4(c) and 4(d) show that our analytical results agree well with the simulation results.

To observe the more clear variation of $FF_{m}$ with other rate constants over wide range, we use surface plots with different rate constants (figures 5-7). Figures 5-7 show that higher values of $k_{1}$ and lower values of $k_{2}$, $k_{4}$ and $J_{m}$ favour sub-Poissonian $FF_{m}$. At a very high value of $k_{1}$, the $FF_{m}$ becomes independent of $k_{2}$ (figure 6(a)). We see from eq.(5) that the critical value of $J_{m}$ increases with the increase of rate constant $k_{1}$ and decrease with rate constants $k_{2}$ and $k_{4}$. That is reflected in figures 5(b) and 7(b). We observe from eq.(2) that $k_{1}$, $k_{3}$ and $J_{m}$ play crucial role for the determination of mean mRNA level. Figure 8 shows that the mean mRNA level does not increases much with the increase of $k_{1}$ and $k_{3}$ but increases more with $J_{m}$. Therefore, the variation of $k_{1}$ or $k_{3}$ is sufficient to change the mean mRNA level very little but the variation of $J_{m}$ is necessary for more change in mRNA level. This behaviour is similar like ’coarse’ and ’fine’ control knob of electronic devices. Now, if $k_{1}$ and $J_{m}$ are increased (keeping $J_{m}<J_{m}^{c}$) simultaneously to increase the mean mRNA number, the $FF_{m}$ moves to the sub-Poissonian regime (figures 5(b) and 8(b)).

In the experiment of Blake et al., the rate constants $k_{1}$ and $k_{2}$ are function of GAL concentrations. The rate constant $k_{1}$ increases monotonically with the increase of GAL concentration but the rate constant $k_{2}$ initially decreases (up to 3%) and then increases with the increase of GAL concentration (beyond 3%). Figure 9(a) shows that average mRNA level does not increase with the increase of GAL concentration beyond 2%. In figure 9(b), we see that the $FF_{m}$ can be sub-Poissonian with different values of GAL concentration with the given reinitiation rate constant $J_{m}$. Higher GAL concentration is required for sub-Poissonian $FF_{m}$ for higher $J_{m}$. That is, the $FF_{m}$ can be decreased without changing the mean mRNA level by varying GAL concentration only (keeping other rate constants fixed). Sanchez et al. observed that the FF can be decreased substantially with fixed mean by varying [GAL] only beyond 2% in GAL1* promoter [29].

3 Steady state probability density function for mRNA

For a single copy number of the gene ($l=1$), the equation (1) can be decomposed into three different equations corresponding to the three different gene states from which the

[TABLE]

[TABLE]

[TABLE]

where $p_{i}(m,t)$ ($i=1,2,3$) be the probability that at time $t,$ gene is in the $G_{i}$($i=1,2,3$) state and the number of $mRNA$ molecule is $m$.

Now using the standard generating function technique [13, 32], we have the total probability density function from the Master equations (6) and (7)

[TABLE]

where $h_{1}=\frac{s_{4}}{s_{3}}=k_{1}$ $h_{2}=-{\displaystyle\frac{1}{2}+{\displaystyle\frac{s_{1}}{2}}}$, $h_{3}={\displaystyle\frac{1}{2}}{\displaystyle\sqrt{1-2s_{1}+s_{1}^{2}-4s_{2}}}$, $h_{4}=s_{3}$ and $s_{1}=1+J_{m}+k_{1}+k_{2}+k_{3}+k_{4}$, $s_{2}=k_{1}J_{m}+k_{2}J_{m}+k_{2}k_{4}+k_{1}k_{3}+k_{1}k_{4}$, $s_{3}=-J_{m}k_{3}$, $s_{4}=-J_{m}k_{1}k_{3}$.

The plot of $p(m)$ versus $m$ is shown in figure 10 (a) (figure 10(b)) corresponding to the different $FF_{m}$ regimes shown in figure 3(a) (figure 4(a)). In figure 10(b), both the curves, the red-dotted ($J_{m}$= $1.0$) and the cyan dot-dashed ($J_{m}$= $10.0$), are in sub-Poissonian $FF_{m}$ but with different mean value. The dot-dashed curve for $J_{m}$= $10.0$ has higher mean but lower $FF_{m}$. The mean mRNA level, thus, can be increased and at the same time, the $FF_{m}$ can be decreased with reinitiation in gene transcription.

In figure 11(a), we draw two curves by changing the GAL concentrations only for same mean ($<m>=15.5$) but with different $FF_{m}$. The red-dotted curve is for 10% [GAL] with super-Poissonian $FF_{m}$ and the blue-dashed curve is for 60% [GAL] with sub-Poissonian $FF_{m}$. In figure 11(b), we see that mean mRNA level can be increased without increasing the $FF_{m}$. The red-dotted curve, blue-dashed curve and cyan-dot-dashed curves have the same $FF_{m}$ but they are with increasing mean respectively. The green-solid curve in figure 11(b) has higher mean but lower $FF_{m}$ compared to the red-dotted and blue-dashed curves. That kind of behaviour of mean and FF can also be observed with lower values of GAL concentration but in narrow range of $k_{3}$ and $J_{m}$. With higher GAL concentrations, the $J_{m}^{c}$ becomes high and the $FF_{m}$ become sub-Poissonian over a wide range of $k_{3}$ and $J_{m}$.

4 Conclusion

Noise or stochasticity in gene expression produces fluctuations in mRNA and protein levels. The fluctuations in protein levels can corrupt the quality of intracellular signals, thereby, affect the cellular functions negatively. Different studies show that the cellular system has different mechanisms to control the inherent stochasticity. Negative feedback is one such mechanism by which noise (FF) reduction takes place [14]. The network architecture, consisting of four regulatory genes, helps to reduce the FF and CV during aging [22]. A coherent feed forward network with three genes can also show least variance level in proteins [34]. In this study, we observed that the reinitiation of transcription by RNAP can also reduce the FF during gene transcription.

To study the contribution of reinitiation in the variability of mRNA and protein levels, we considered a simple two-state model with reinitiation of RNAP (figure 1) and calculated the Fano factor in mRNA and protein levels using simple mathematics. Our model network is similar to the regulatory network of GAL1* yeast promoter with full ATC induction [4]. So, we compared our analytically calculated FF of protein levels with the experimental results of Blake et al. at full ATC induction and found excellent agreement. Then we analyzed the Fano factor in mRNA levels and observed three different phases viz., Poissonian, sub-Poissonian and super-Poissonian phases’. We found that there exists a critical value $J_{m}^{c}$ for which the $FF_{m}$ is always Poissonian. For $J_{m}<J_{m}^{c}$ ($J_{m}>J_{m}^{c}$) the $FF_{m}$ lies in the sub-Poissonian (super-Poissonian) phase. That result is distinct from the well-studied constitutive and two-state gene activation model. In those two models, the FF in mRNA levels is Poissonian and super-Poissonian respectively [15, 30]. The minimum value of $FF_{m}$ in two-state model can be very close to unity but never less than unity.

In our model network, we included the reinitiation step into the two-state gene activation model and found sub-Poissonian Fano factor regime in mRNA levels. The reinitiation process, therefore, not only removed the extra additive term in $FF_{m}$ due to random transitions between the gene states in two-state model completely rather decreased it further below unity. That shows that the reinitiation step in gene expression has the strong ability to control the fluctuations in mRNA and protein levels. In the constitutive and two-state model, the mRNA synthesis takes place with constant rate from the active state of the gene. But, the reinitiation process in gene transcription adds a refractory period after each mRNA synthesis and that reduces the effective rate of mRNA synthesis and variance though the mean mRNA level increases. This results in the** $FF_{m}$** going below unity in gene transcription with reinitiation.

In the two-state gene activation model, the activation ($k_{1}$) and deactivation ($k_{2}$) rate constants mostly regulate the FF in mRNA level. The fast (slow) transition between the active and inactive states decreases (increases) the FF in mRNA level [6]. Thus, sub-Poissonian $FF_{m}$ in our model network is favorable with higher values of $k_{1}$ and $k_{2}$ (figure 6(a)). It can be shown that the mean mRNA level decreases (increases) with the increase of $k_{2}$ ($k_{1}$). But with high $k_{1}$, the $k_{2}$ has the least effect on mean mRNA levels. In the experiment of Blake et al., $k_{1}$ and $k_{2}$ are both functions of GAL concentrations and can be made high. The high $k_{1}$ and $k_{2}$ leads to mean mRNA level going up to almost saturation but the $FF_{m}$ decreasing to the sub-Poissonian regime (figures 9(a) and 9(b)). We can visualize that result from the distributions of mRNA levels in figure 11(a). Therefore, one can adjust the $FF_{m}$ to sub-Poissonian regime without changing the mean mRNA by regulating the GAL concentrations in the system.

In our model network, the desired mean and $FF_{m}$ can also be obtained by choosing the rate constants $k_{3}$ and $J_{m}$ properly (figure 8(c), figure 5(a) and figure 11(b)). The $FF_{m}$ shows a dip when plotted with $k_{3}$ and $J_{m}$ whereas the mean increases with those rate constants (figure 5(a) and 8(c)). Therefore, mean can be increased keeping $FF_{m}$ fixed (the red-dotted line, the blue-dashed line and cyan-dot-dashed line) or mean can be increased while $FF_{m}$ decreases (red-dotted line, the green solid line) (figure 11(b)). Like $k_{1}$ and $k_{2}$ in the experiment of Blake et al., the rate constants $k_{3}$ and $J_{m}$ are not controlled externally. But cell can change their values to fulfill its requirements. There are several instances that the cellular system has the ability to adapt to the unpredictable environmental changes by regulating its internal states, mean protein levels, the fluctuations about the mean levels etc. [17, 18, 20, 21, 22, 35, 36]. This ability is crucial for survival and proper functioning of cells.