Effect of transcription factor resource sharing on gene expression noise

Gene expression is intrinsically a stochastic (noisy) process with important implications for cellular functions. Deciphering the underlying mechanisms of gene expression noise remains one of the key challenges of regulatory biology. Theoretical models of transcription often incorporate the kinetics of how transcription factors (TFs) interact with a single promoter to impact gene expression noise. However, inside single cells multiple identical gene copies as well as additional binding sites can compete for a limiting pool of TFs. Here we develop a simple kinetic model of transcription, which explicitly incorporates this interplay between TF copy number and its binding sites. We show that TF sharing enhances noise in mRNA distribution across an isogenic population of cells. Moreover, when a single gene copy shares it’s TFs with multiple competitor sites, the mRNA variance as a function of the mean remains unaltered by their presence. Hence, all the data for variance as a function of mean expression collapse onto a single master curve independent of the strength and number of competitor sites. However, this result does not hold true when the competition stems from multiple copies of the same gene. Therefore, although previous studies showed that the mean expression follows a universal master curve, our findings suggest that different scenarios of competition bear distinct signatures at the level of variance. Intriguingly, the introduction of competitor sites can transform a unimodal mRNA distribution into a multimodal distribution. These results demonstrate the impact of limited availability of TF resource on the regulation of noise in gene expression.

Models of transcription quintessentially hinge on the details of the promoter architecture such as the number and affinity of TF binding sites and relative binding positions on the gene [39,[49][50][51][52][53][54]. Most of these theoretical models implicitly assume that the number of TFs is in excess with respect to the number of its binding sites in the cell. However, inside the cell this assumption often breaks down. For example, TFs get shared by multiple gene copies, in highly replicated viral DNA genes [55], genes expressed on plasmids [56], and in multiple identical copies on the chromosome [57][58][59][60][61]. Furthermore, the majority of TFs are entrusted with the regulation of multiple genes; for instance, cAMP receptor protein (CRP) is reported to have nearly 400 binding sites per E. coli genome copy [62,63]. It is therefore crucial to predict gene expression noise due to limited availability of TF resource, in order to dissect how the sharing of TFs among multiple genes contributes to noise.
Some studies have explored the interplay of TF sharing between genes and other competing binding sites in different scenarios [44,[64][65][66][67][68][69][70][71][72]. For instance, in a recent study, Rydenfelt et al. [73] have theoretically explored the effect of TF sharing on the mean level of gene expression, which has since been verified experimentally [65,67]. However, few studies have systematically explored how competition, for TF binding influences gene expression noise [68][69][70][71]. In one such study, it was found that in the context of a single auto-regulated gene copy, the addition of decoy binding sites can lead to bimodal protein distributions in an isogenic population. Although these studies provide useful insights, nevertheless a general theoretical framework of understanding how TF resource sharing affects noise in gene expression remains in its infancy. The goal of this study is to develop such a framework.
Here we examine a simple model of transcription, where a number of TFs are shared between a comparable number of gene copies and other competitor sites. We explicitly envisage two different scenarios of competition, demonstrated schematically in Fig 1: (i) multiple identical promoters sharing their TFs (Fig 1B), and (ii) promoters sharing their TFs with other competitor sites ( Fig 1C). These competitor sites can represent 'decoy sites' or promoters driving expression of other gene species. We find that competition enhances noise across an isogenic population, i.e. the Fano factor (defined as the variance divided by the mean) of the mRNA distribution shows a peak when the TF number becomes comparable to the promoter number. Interestingly when a single promoter copy shares TFs with multiple competitor sites in a cell, we find that there exists a master curve for the mRNA variance as a function of its mean that is independent of the binding, unbinding rates and the number of competitor sites. However, this result does not hold when a number of promoter copies compete for a pool of TFs. In other words, these different scenarios of TF competition bear distinct signatures at the variance level. This finding stands in sharp contrast to the behavior at the mean level [67], which follows a universal master curve irrespective of the nature of competition. We further find that a unimodal mRNA distribution without any competitor sites can be transformed into a multimodal distribution when competitor sites are introduced. For such a multimodal distribution, different modes arise and diminish as the number of competitor sites is systematically tuned. While our model predictions are consistent with recent experimental [67] and theoretical [73] studies at the mean level of expression, we also find that TF competition can have a complex impact on the noise in expression Competitor sites can correspond to 'decoy sites' or promoters driving expression of other gene species. Therefore, although the competitor sites bind to the TFs, they do not regulate the production of mRNA molecules from the target promoters. https://doi.org/10.1371/journal.pcbi.1005491.g001 Effect of transcription factor resource sharing on gene expression noise which results in multimodality and a scaling with mean expression level that depends on the source of the competition.

Model
In order to investigate how TF sharing affects cell-to-cell variability at the mRNA level across a population of genetically identical cells, we examine a simple model of transcription, where a number of TFs are shared between a comparable number of identical promoter copies and competitor sites (see Fig 1B and 1C). Moreover, as case studies, we consider two regulatory motifs of transcription in E. coli [74]: 1) a promoter containing a single binding site where a repressor molecule can bind and hinder transcription, and 2) a promoter consisting of a single binding site where an activator molecule can bind and increase the rate of transcription. These regulatory architectures have been extensively explored in different studies, using the implicit assumption that TFs are in abundance [38,64,[75][76][77][78][79].
We construct a general model of transcription, that explicitly incorporates the interplay between TF copy number and its binding sites. To elucidate the model, we first consider a number of identical promoters sharing a pool of TFs, as shown in Fig 2A in the absence of any competitor sites. We can easily incorporate the presence of competitor sites in our model, as shown in the later sections. For the sake of generality we discuss the process of TF binding and unbinding generically without regard to their regulatory functions; later we will outline the influence of activators and repressors separately. Let us consider N TF number of TFs being shared by N P number of promoters, where each promoter contains a single TF binding site. A TF binds to a promoter at a rate k on , and it unbinds from it at a rate k off (see Fig 2A). For mathematical convenience we view the process of TF binding to a promoter as the formation of a 'TF-promoter' complex, and the unbinding of the TF as dissociation of the complex [80]. Consequently, complex formation rate per TF per promoter is k on , while dissociation rate per complex is k off (see Fig 2B).
The number of TF-promoter complexes determines the transcriptional output. In the case of activators, mRNA production takes place when the activator is bound to a promoter (see Fig 2C). Each of the activator-promoter complexes produces mRNA molecules at a rate r, which then subsequently degrades at a rate γ. Here we have assumed for simplicity that the basal transcription rate, when the promoter is not bound to an activator, is zero, as it is negligible for many activator regulated genes [62]. However, all our results still qualitatively hold even when we incorporate a non-zero basal rate of transcription (see Fig D in S1 Text). On the other hand, since repressor binding to a promoter hinders the attachment of RNA polymerases to the promoter, each promoter produces mRNA molecules at a rate r, only when it is not bound to a repressor, as shown in Fig 2E. In the limit of one promoter copy, our model reduces to the well known ON-OFF model [75,[81][82][83][84], where the promoter switches between an active and an inactive state, and transcription can ensue only in the active state.
The above processes can be depicted as trajectories in the space of TF-promoter complex number and mRNA copy number, as shown in Fig 2B, 2D and 2F. By employing a stochastic framework, we monitor the time evolution of two random variables: the instantaneous number of TF-promoter complexes, n(t) and the instantaneous number of mRNA molecules, m(t). Evidently for activators, the instantaneous mRNA production rate is proportional to the number of complexes (n); while for repressors, the mRNA production rate is proportional to the number of unbound promoters (N P − n) (compare Fig 2D and 2F). Using the master equation approach, we write down the time-evolution equation for the joint probability distribution P(n, m, t) of having n complexes and m mRNA molecules at a time t. For instance, the master equation for activators is given by dPðn; m; tÞ dt ¼ k on ðN TF À n þ 1ÞðN P À n þ 1ÞPðn À 1; m; tÞ þ k off ðn þ 1ÞPðn þ 1; m; tÞ þrnPðn; m À 1; tÞ þ gðm þ 1ÞPðn; m þ 1; tÞ À ½k on ðN TF À nÞðN P À nÞ þ k off n þ rn þ gmPðn; m; tÞ: The rate of binding of a TF to a promoter is k on and the rate of unbinding of the TF from the promoter is k off . (B) List of possible stochastic transitions leading to either formation or dissociation of 'TF-promoter' complexes, and their respective statistical weights. The weights represent the probability that each transition will occur during a time interval, Δt. TF copy number, promoter copy number, and the instantaneous number of complexes are denoted by N TF , N P and n respectively. (C) While an activator is bound to a promoter, it promotes the binding of RNAP molecules to the promoter, which subsequently leads to the production of an mRNA molecule at a rate r. Each mRNA molecule then decays at a rate γ. Here, m denotes mRNA copy number. The basal transcription rate, when the activator is not bound to the promoter, is assumed to be zero (see S1 Text for a more general model for activators with a nonzero basal rate). (D) List of stochastic transitions leading to changes in mRNA copy number by the action of activators and their respective weights. (E) When a repressor is bound to a promoter, it hinders RNAP binding to the promoter, subsequently blocking mRNA production. However when a promoter is not bound to any repressor molecule, it produces mRNA molecules at a rate r, which again subsequently decay at a rate γ. (F) List of stochastic transitions leading to changes in mRNA copy number by the action of repressors, with respective weights.
The above equation is built by counting all possible steps that lead to either a change in complex number (n) or in mRNA copy number (m) by one (see trajectories in Fig 2B and 2D). Similarly, we could also write the master equation for repressors by combining the trajectories depicted in Fig 2B and 2F (see S1 Text for details). In principle, we can obtain the relevant moments of the mRNA distribution from these master equations. However, exact expressions of mean (hmi) and variance (var(m)) of the mRNA distribution can only be obtained for the simplest case of one promoter. For the general case of multiple promoters, deriving exact closed-from expressions for the moments is challenging, because the system of equations for the moments do not close. Nonetheless, we can obtain approximate analytical solutions (see S1 Text for details).
Since we do not have exact analytical solutions for the moments of the mRNA distribution, we simulate the kinetic processes defined above using the Gillespie algorithm [85]. This allows us to numerically generate multiple time-traces of the random numbers n and m, using realistic values of the kinetic rates suitable for E. coli promoters (see Table 1). From these different time traces, we calculate the values of the mean and variance of the steady state mRNA distribution. The approximate analytical solutions match reasonably well with our simulation data (see Fig A in S1 Text). While there is a systematic deviation which results from ignoring higher order terms (beyond second order), these closed form relations can be useful in understanding the qualitative behaviors of these curves, which the analytical solutions do capture (see S1 Text for details). Below we summarize our main results, obtained from the Gillespie simulations.

Multiple identical promoters competing for a pool of TFs
Armed with the framework established above, we first investigate the scenario when multiple identical promoter copies compete for a pool of TFs (as shown in Fig 1B). Our model predicts the effect of TF sharing on the mean (hmi) and variance (var(m)) of the steady-state mRNA distribution. We shall first discuss the case when the TFs act as activators, and then shift our focus to the case of repressors.
We explore the effect of activator sharing in the fold-change of mean expression, a quantity often measured in bulk assays (see Fig 3A) [67,73]. Here we define the fold-change as the ratio of the mean expression hmi to its maximum value N P (r/γ). The mean of the mRNA distribution, increases steadily with activator number, and asymptotically goes towards the value of N P (r/γ) (see Fig F in S1 Text). In the limit of activator copy number being much greater than the promoter copy number, all the promoters essentially remain bound to activators. Accordingly each of these activator-promoter complexes contributes, on average, r/γ mRNA molecules, resulting in the production of N P (r/γ) mRNA molecules on average, in the steady state. The qualitative behavior of fold-change, which first monotonically increases Table 1. Kinetic rates that are used in simulations.
In contrast to the fold-change, variance is a non-monotonic function of the number of activator molecules, which peaks when the number of activators equals the promoter copynumber (see Fig 3B and also see Fig F in S1 Text). The origin of the peak in variance lies in the activator-promoter complex number variability. In fact the variance in complex number also peaks when the promoter copy number equals activator copy number (see Fig B in S1 Text). In other words, when activator number is comparable to promoter copy-number, there is enhanced variability in complex number stemming from the stochastic complex formation and dissociation processes, which subsequently gets reflected at the level of mRNA distribution. On the contrary, if the numbe of activators is either much greater or much less than the promoter number (in the limits of N TF ) N P and N TF ⪡ N P ), the effect of competition disappears, as all the promoters remain either bound or unbound to showing peaks when the repressor number becomes comparable, but slightly higher than the promoter number. In all the plots, symbols represent data from numerical simulations, which are connected by smooth dashed lines to guide the eyes. The parameter values are as specified in Table 1. activators. If we consider the example of activator number being much higher than promoter number, all the identical gene copies driven by the promoters will be expressed all the time. This will lead to each gene copy producing mRNA molecules which is Poisson distributed across an isogenic population [54]. Hence the distribution of all the mRNAs produced from all the identical gene copies will be given by the sum of the numbers of mRNA produced from every gene, leading to a reduction in mRNA variability and a sharply peaked distribution. We further show in Fig 3C the effect of activator sharing on the Fano factor (var(m)/hmi). Just like the mRNA variance, the Fano factor also exhibits non-monotonicity as a function of activator number, and peaks when the activator and promoter numbers become equal.
Turning our attention to repressors, we find an analogous result where instead of gene expression being proportional to the number of activator-promoter complexes; it is now proportional to the number of promoters not bound to repressors. As such, we find that the fold-change, defined by hmi/(N P r/γ), monotonically decreases with increasing repressor number, and asymptotically approaches zero (see Fig 3D). Moreover, the variance is a nonmonotonic function of repressor number, and exhibits a sharp peak when the number of repressors equal promoter copy number, as was the case for activators (see Fig 3E). As before the appearance of these peaks can be understood from a corresponding enhancement in complex number variability (see Fig B in S1 Text). Finally, the non-monotonic behavior of Fano factor, found in the case of activators, also persists for repressors, when the Fano factor is plotted as a function of the repressor number (see Fig 3D). However, there is a distinction between the cases of activators and repressors at the level of Fano factor. Unlike the activators, the Fano factor for repressors do not peak exactly when the repressor number equals the promoter number; rather the peaks appear when the repressor number is slightly greater than (although comparable to) the promoter number (see Fig 3D). These behaviors are very robust with respect to changes in the different parameter values associated with the model (for details see the S1 Text, Figs G-I).
In summary, a key prediction of our model is that TF sharing among identical promoters leads to a substantial increase in gene expression noise when TF and promoter copy number become comparable to each other. Sharp peaks arise in mRNA variance and Fano factor as a function of TF copy number, which bears the signature of pronounced competition.

Promoters sharing TFs with other competitor sites
Inside single cells, TFs often promiscuously bind to different competitor sites [42,88,89] such as 'decoy sites' or promoters driving expression of other gene species, compelling a promoter of interest to compete with these sites for a limiting pool of TFs (as shown in Fig 1C). Recent bulk studies have theoretically [68,73] and experimentally [67,90] studied this scenario inside bacterial cells, by synthetically integrating multiple competitor sites at several chromosomal locations or on plasmids.
To explore how this interplay affects the cell-to-cell variability at the mRNA level, we incorporate TF binding to competitor sites, and extend our model as described before to consider N P number of promoters competing for N TF number of TFs in the presence of N C number of competitor sites (shown in Fig 4A). Consequently, every TF can bind either to a promoter or to a competitor site. The rates at which a TF binds to a competitor site and unbinds from it are given by k C on and k C off respectively. We first discuss how the competition of a single promoter copy (N P = 1) with multiple competitor sites affects the variability at the mRNA level.
A key prediction of this model is that for both activators and repressors, the variance as a function of the mean remains unaltered, regardless of the number and the strength of the competitor sites, as shown in Fig 4B and 4C (also see Fig E in S1 Text). In other words, when we tune the variance as a function of the mean by altering TF number N TF , all the data points for variance, generated from numerical simulations using different parameter values for the number of competitor sites collapse onto a single master curve. This result holds true even for different TF binding/unbinding rates to the competitor sites (please see Fig E in S1 Text). In order to develop an intuition for this result, we employ already existing analytical results involving TF regulating a single promoter copy [37]. In fact, when the number of competitor sites is zero, it is easy to obtain exact analytical expressions for the variance and Effect of transcription factor resource sharing on gene expression noise the mean from the master equation (Eq 1), which are given by, It is conceivable from the above analytical expressions, that both the variance and the mean can be expressed as simple functions of an 'effective' binding rate of the TFs to the promoter, k eff on ¼ k on N TF (also see Eq. 14-15 in S1 Text). The presence of the competitor sites essentially modifies the TF pool as seen by the promoter i.e. the effective binding rate of the TFs to the promoter without altering any of the other parameters such as, k off , r and γ. Hence we can express the variance as a function of the mean by eliminating the dependence of these functions on the effective binding rate, k eff on . For both the activators and repressors, we can thus express the variances as functions of the means given by, These are the functional forms of the master curves, which exactly match the collapsed data, as shown in Fig 4B and 4C. In other words, for a single promoter copy, noise at the mRNA level changes only as a function of the mean, irrespective of whether the promoter of interest is competing for a pool of TFs with competitor sites or not. Moreover, this result holds true even when there exists a basal rate of transcription for promoters not bound to activators (see Fig E in S1 Text). This highlights an intriguing prediction for transcriptional noise in the face of competition; the noise depends only on the mean, and it is determined solely by an effective binding rate of the TFs to that promoter, irrespective of the interaction of the TFs with the competitor sites. In fact this result also implies that variance as a function of the mean cannot be used as a signature to distinguish scenarios where a single promoter copy is in competition for a pool of TF with other competitor sites.
Next, we investigate the case of multiple promoter copies competing for a pool of TFs with a number of competitor sites. Our model predicts that the collapse, as observed in the variance as a function of the mean for a single promoter copy does not hold anymore. To explicate this further, we consider the hypothetical situation of two promoter copies, competing with other competitor sites. When we systematically tune the number of competitor sites and using gillespie simulations obtain the variance as a function of the mean, we find that the different curves, defining the variance per promoter as a function of the mean per promoter, no longer collapse onto a single master curve, as shown in Fig 4B and 4C (open symbols). We note that the TF copy number variation, affects mRNA production from both the promoter copies. Hence the numbers of mRNAs produced from the two copies become correlated [73]. Owing to this correlation, the mean and variance cannot be simply expressed in terms of some effective variable, such as k on N TF , as before (see Eq. 17 in S1 Text, where nonlinear terms, like k 2 on N 2 TF , appear in the expressions). However, in the limit of the TF copy number or the number of competitor sites becoming much greater than one (i.e. N TF ) 1 or N C ) 1), the correlation between the number of mRNAs produced from different promoter copies become negligible. In both these limits the data for variance per promoter versus mean per promoter again fall on the master curve obtained for a single promoter copy, signifying that the promoters behave independently (see Fig 4B and 4C).

Introduction of competitor sites leads to multimodal mRNA distribution
The observations in the last two sections suggest that the mRNA distribution passes through an interesting regime when the TF number is comparable to the number of binding sites (promoter copies and competitor sites). In order to examine the features of mRNA distribution, we consider activator sharing between multiple promoter copies driving the expression of the same target gene. We find that multimodal mRNA distributions arise as activator number is tuned for a given number of promoter copies (see Fig 5A), when the binding and unbinding rates of activators (k on and k off respectively) to the promoters are much slower compared to the mRNA production and degradation rates (r and γ respectively)(see Table 1). Multi-modal mRNA distribution implies that mRNA molecules are present in multiple distinct abundances across an isogenic cell population. The appearance of multimodality at the mRNA level can be intuited by examining the steady state activator-promoter complex number distribution, since mRNA production is directly proportional to the number of activator-promoter complexes. In order to elucidate this point further, we consider the scenario when there are three activator copies, shared by two promoters. As evident, we can have three possible values of the complex number: zero, one or two. Since the binding and unbinding of the activator molecule to the promoters are much slower than the mRNA production and degradation rates, the promoters remain in activator bound and unbound states for a long time, such that sampling of mRNA molecules across a population can capture the effect of distinct relative abundances. Consequently in a genetically identical cell population, we may find three sub-populations, with one population having one complex producing *r/γ amount of mRNA molecules on average, another population having two complexes producing *2r/γ amount of mRNA molecules on average; and the other population having zero mRNA molecules on average due to the absence of any activator-promoter complexes. However, for our particular choice of parameters, the relative probability of having zero complexes becomes much smaller than having one or two complexes. Consequently the peak at zero vanishes, and we obtain a bi-modal mRNA distribution with two peaks, approximately at 2r/γ and r/γ (see the blue curve in Fig 5A). Nevertheless, with suitable choices of parameters one could recover all the three modes (see Fig J in S1 Text). In the limit of activator copy number being much greater than the promoter copy number, all the promoters are essentially occupied and the mRNA distribution approaches a uni-modal distribution, sharply peaked around N P r/γ (N P being the number of promoter copies). For identical promoter copies competing for repressor molecules, we again find bi-modal mRNA distribution in certain parameter regime, as shown in Fig 5B. The appearance of bimodal distribution is known theoretically for a single 'two-state' promoter switching between 'on' and 'off' states, when the state switching rates are slower than mRNA production and degradation rates [37,91]. The fact that we consider more than one competing identical promoters leads to the possibility of obtaining more than two modes as we tune the TF copy number, since it allows for the formation of more than one TF-promoter complex. However, it is to be noted that multimodality can be observed only for a certain parameter regime, namely when the binding and unbinding rates of the TFs are much slower than mRNA production and degradation rates ({r, γ} > {k on , k off }). When this stringent condition is met, sharing of TFs among promoters alters the relative proportion of the different  Table 1. Insets: The mRNA distributions with a faster TF binding rate (k on = 0.027s -1 ) than the mRNA degradation rate (γ = 0.011s −1 ). Other parameters are as specified in Table 1. (C-D) The steady-state mRNA distributions for two identical promoters are shown in presence of competitor sites, when the TFs either act as activators (C) or repressors (D). For these plots, we choose a faster TF binding rate than mRNA degradation rate (as in the insets of A, B) such that the distributions are unimodal in absence of any competitor sites. The number of competitor sites is increased systematically keeping a fixed number of TFs (N TF = 3). For simplicity we assume that the TF binding and unbinding rates to the promoters are the same as to the competitor sites. https://doi.org/10.1371/journal.pcbi.1005491.g005 Effect of transcription factor resource sharing on gene expression noise modes, making them either more or less prominent. On the other hand, if the condition is not met, the distribution becomes unimodal. For example, see the inset of Fig 5A and 5B, where we choose the TF binding rate k on to be higher than the mRNA degradation rate γ.
Next, we investigate how the presence of competitor sites affects the distribution of mRNAs produced by multiple copies of a promoter of interest. In the case of activator sharing, one of the key findings is that the introduction of competitor sites can lead to multi-modality even when the binding rate of TFs to the promoter is faster than the degradation rate of the mRNA. In order to explain this further let us consider the example discussed above, where two competing promoters of interest produce a unimodal mRNA distribution in the absence of any competitor sites (Fig 5A, inset). By systematically increasing the number of competitor sites, we find that the unimodal mRNA distribution transforms into a multimodal distribution (see Fig 5C). This is because the activators can now also bind to the competitor sites, restricting the pool of freely available activators (ones that are not bound to any binding sites). Since the free activator number is restricted due to competition, the promoters can stay in unoccupied states for sufficiently long time, even when the binding rate of the activators are faster than the mRNA degradation rate. Hence the only requirement for observing multimodality is that the unbinding rate of activators to the promoter has to be slower than mRNA degradation rate, such that promoters can also stay occupied for a sufficiently long time, allowing the system to again sample the relative abundances of mRNA molecules. It should be noted that for quite a few known TFs, such as LacI, or lambda CI repressors in E. Coli, the unbinding rate from the promoter is slower or comparable to the mRNA degradation rate [42,92]. This is also true for many TFs in yeast [93][94][95]. Please see the Methods section for a comprehensive discussion of the rates we choose.
Similarly, for two promoter copies sharing repressors with a number of competitor sites, our model predicts emergence of multi-modal mRNA distribution, as shown in Fig 5D. Just like the case of activators, different modes of mRNA distribution arise and diminish as the number of competitor sites is tuned. In the results above, we assumed for simplicity that the TFs have the same binding and unbinding rate to both the promoters and competitor sites. Nevertheless, our conclusions remain intact even when the TFs bind or unbind to the competitor sites with a different rate than to the promoters (see Fig K in S1 Text).
To further validate our model prediction that multi-modal mRNA distribution stems from the promoters of interest competing with the competitor sites for a pool of TFs, we also look at the Pearson correlation coefficient between the numbers of mRNA molecules produced from two promoters of interest. If the number of competitor sites is much smaller than the promoter number, and the TF number is much higher than the promoter number, effectively each promoter is expected to express independently. Thus, a nonzero correlation coefficient can be regarded as a signature of competition. For our choice of parameters, we see that the correlation starts from zero, reaches a minimum and then asymptotically approaches zero as the number of competitor sites is increased (See Fig L in S1 Text). Hence the rise of multimodality at the mRNA level coincides with nonzero values of correlation coefficient, signifying that this effect stems from the competition between the promoters and competitor sites. One central finding of this section is that altering the number of competitor sites and changing the number of TFs in the absence of competitor sites yield qualitatively distinct responses at the level of the mRNA distribution and correlation (see Fig M in S1 Text).

Discussion
The impact of transcriptional dynamics on gene expression noise has been a topic of intense enquiry, in the field of quantitative regulatory biology [33][34][35][36][37]39]. Most of these studies have investigated models of transcription, hinging on the properties of the cis regulatory elements, such as the number of TF binding sites, their binding strength, etc [39,49,54]. The key assumption these studies make is that we can treat each gene in isolation while dissecting their transcriptional dynamics and its impact on noise in expression. In reality this assumption often breaks down when a TF for a gene of interest gets shared by other genes or competitor sites on the chromosomal DNA or plasmids [55][56][57][58][59][60][61]. Hence in order to understand how transcriptional dynamics influences noise in gene expression, theoretical studies need to include the global effect of TF sharing.
The model of transcription developed here offers a way to incorporate the interplay of TF copy number and its binding sites and make predictions for how this interplay impacts gene expression noise. Predictions of this model, for the mean mRNA expression as a function of TF copy number is in agreement with recent bulk studies [67] (see Fig C in S1 Text). Moreover, for identical promoters competing for a pool of TFs, we find that both the variance and Fano factor of the mRNA distribution are non-monotonic functions of the TF copy number and show a peak when the TF and promoter copy numbers are comparable.
The incorporation of competitor sites into this model gives rise to intriguing predictions at the noise level. As has been expounded in recent studies [65], at the mean level, gene expression profiles from a wide range of competition scenarios in the presence of competitor sites can be collapsed onto a single master curve by considering the natural variable of the problem. However, at the level of variance this elegant universality ceases to exist. Although there exists a master curve in the variance as a function of the mean for a single promoter copy competing with multiple competitor sites, for more than one promoter copy, this result no longer holds. In other words, for more than one promoter copy, the simple picture of each promoter seeing an effective pool of TFs fails. Although introduction of competitor sites would imply a change in free TF concentration, one key implication of our results is that altering the number of competitor sites has a qualitatively different impact than directly changing the number of TFs in the absence of competitor sites.
As is evident, one major outcome of this study is that different scenarios of TF sharing lead to qualitatively distinct noise characteristics at the mRNA level. In a recent set of experiments in E.coli, Jones et al. [44] systematically tuned the number of LacI repressors and counted mRNA molecules across an isogenic population in order to demonstrate the effect of promoter architecture on gene expression noise. Similar strategies could be adopted to systematically test the different scenarios of TF resource sharing by altering the number and strength of competing binding sites, as we have outlined above.
We also find that the presence of competitor sites can further lead to a multi-modal distribution, so long as unbinding rates of TFs to the promoters of interest are slower than the mRNA production and degradation rates. Multi-modal distributions in expression, across an isogenic population can provide a fitness advantage in a changing environment, by giving rise to phenotypic diversity [96]. Certain cis regulatory elements have been demonstrated to lead to bimodal expression patterns [71,91,97,98]. Here we demonstrate that a generic feature of promoters competing for a finite TF resource in the presence of competitor sites lead to a multi-modal distribution at the mRNA level. Whether such differential control is used by cells to control the shape and modality of the mRNA distributions is a fascinating question.
In general, the impact of global properties of transcription, such as the limited availability of TF [99] or sigma factor [100,101] resources on gene expression noise remain poorly understood. In this light our theoretical framework provides a way of deciphering the global effect of genetic resource sharing on expression noise.

Materials and methods Parameter selection
For E.Coli, there are several experimental studies that measure the degradation rates of the different mRNAs. In particular genome wide studies in E.Coli have shown that the average lifetime of mRNAs in E.coli is 2.5 minutes in the exponential phase and 4.5 minutes in the stationary phase respectively, as shown in Figure 3A of [95]. On the other hand numerous studies have measured the binding and unbinding rates of TFs to the promoters, as we have cited in the manuscript. For a lot of these TF such as the well-known Lac and Lambda cI repressors, the average residence times of minutes up to tens of minutes [42,92,102]. In other words the residence time is longer or comparable than the degradation rates of the mRNA molecules, justifying the assumption we make.
For eukaryotes, genome wide studies (see Figure 2A of [93]) in yeast have measured the lifetime of mRNA molecules from 4687 genes. These studies found that around 1700 genes have an average lifetime of less than 10 minutes. Although we do not know of genome wide studies that characterize the unbinding rates of all TFs to all binding sequences, we can find examples where the residence time of different TFs to the DNA were measured or inferred [94]. For TFs such as Rap I and Gcr I, the residence times are of the order or 10 minutes. As binding and unbinding rates of different TFs are discovered more of them might have a long residence time at the promoter.

Computational tools
In order to compute the mRNA distribution across an isogenic population, we performed Gillespie simulations [85] using codes written in C++.