Buffering effects of nonspecifically DNA-bound RNA polymerases in bacteria

RNA polymerase (RNAP) is the workhorse of bacterial gene expression, transcribing rRNA and mRNA. Experiments found that a significant fraction of RNAPs in bacteria are nonspecifically bound to DNA, which is puzzling as these idle RNAPs could have produced more RNAs. Whether nonspecifically DNA-bound RNAPs have any function or are merely a consequence of passive interaction between RNAP and DNA is unclear. In this work, we propose that nonspecifically DNA-bound RNAPs buffer the free RNAP concentration and mitigate the crosstalk between rRNA and mRNA transcription. We verify our theory using mean-field models and an agent-based model of transcription, showing that the buffering effects are robust against the interaction between RNAPs and sigma factors and the spatial fluctuation and temporal noise of RNAP concentration. We analyze the relevant parameters of Escherichia coli and find that the buffering effects are significant across different growth rates at a low cost, suggesting that nonspecifically DNA-bound RNAPs are evolutionarily advantageous.


I. INTRODUCTION
RNA polymerases (RNAPs) and ribosomes are finite resources that limit the overall rate of gene expression [1][2][3][4][5][6][7][8][9][10][11].As a result, genes compete for these resources, leading to inevitable crosstalk of gene expression [12][13][14].This competition is particularly significant in bacteria as all RNAs are transcribed using the same type of RNAP.Thus, increasing the transcription of rRNA will inevitably reduce the availability of RNAPs for transcribing mRNA.However, despite this competition, a significant proportion of RNAPs in bacteria (30% to 50%) are nonspecifically bound to DNA and do not participate in transcription [15][16][17][18][19].This phenomenon is puzzling as these idle RNAPs could have contributed to transcription, producing more mRNAs.Furthermore, producing these non-transcribing RNAPs may be a significant investment, and bacterial cells could have avoided producing these idle RNAPs and allocated the finite resources to other essential proteins to enhance their fitness.Indeed, eukaryotes have been found to suppress the nonspecific binding of RNAPs to DNA actively [20].Therefore, a puzzle emerges: why do bacteria keep so many RNAPs nonspecifically bound to DNA?
In this study, we propose that nonspecifically DNAbound RNAPs (which we refer to as nonspecific RNAPs in the following) in bacteria are crucial in regulating gene expression.These RNAPs mitigate the unwanted crosstalk between the expressions of different genes due to the limited availability of RNAPs.For example, when a subset of genes, such as the genes of rRNAs, are upregulated, more RNAPs participate in their transcription, reducing the concentration of free RNAPs.This reduction in free RNAPs decreases the initiation rates of other genes since the probability of an RNAP binding to a promoter depends on the free RNAP concentration [15,16,18].Nevertheless, nonspecific RNAPs help buffer the free RNAP concentration reduction and min-54 imize the unwanted impact of resource competition (see To strengthen our hypothesis, we also develop an agent-based model that explicitly incorporates spatial information, including the diffusion of RNA polymerases and their interactions with promoters and nonspecific binding sites.Importantly, the agent-based model naturally generates both spatial and temporal noise in the free RNAP concentration, from which we find that nonspecific RNAPs indeed suppress the correlation between the production rates of mRNA genes and rRNA genes, thereby providing further evidence supporting our predictions.
On the application side, our findings suggest a strategy for reducing the interference between the expressions of exogenous and endogenous genes in synthetic biology [22,23] by adding non-coding DNA sequences to the bacterial chromosome to increase the number of nonspecific binding sites.

A. Mean-field model
In the mean-field model, the probability of a promoter bound by an RNAP is a Hill function of the free RNAP concentration [15,16,18], where [n] f is the concentration of free RNAPs, and K i is the dissociation constant of the promoter i (Methods A).In the mean-field model, we model gene expression regulation by changing the dissociation constant K i .The smaller K i is, the stronger the promoter's binding affinity to RNAP.Once an RNAP binds to a promoter, it transitions to the elongating state with an initiation rate k ini i , moves along the operon, and eventually detaches from DNA after it finishes transcribing.The number of elongating RNAPs (n el,i ) on the operon following a promoter is proportional to the probability of the promoter bound by an RNAP: where Λ i ≡ k ini i L i (1+η i )/c i , the maximum possible number of elongating RNAPs on the operon, which we name as capacity in the following.Here, L i is the length of the operon following the promoter i, and c i is the elongation speed of RNAP.When the number of RNAPs on the operon reaches the steady state, the number of initiations per unit time must be equal to the number of RNAPs that detach from the end of the operon, that is, Here, we also include the pausing effect, that is, elongating RNAPs may pause for a finite duration, effectively increasing the capacity [18,24].η i is the ratio between the number of pausing RNAPs and actively elongating 137 RNAPs (Table S1).

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We categorize RNAPs into three types: ( RNAPs on rRNA genes RNAPs on mRNA genes On the right side of the above equation, the last three

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One can think of them as effective volumes that store  S1).The Vns = 2V line corresponds to the parameters of E. coli.(c) The absolute slopes of [n] f over vr vs. Kns (unit: µm −3 ) and gns.The slopes are equal as long as gns/Kns is the same (indicated by the dashed line with slope 1 in the logarithmic coordinates).The red star marks the corresponding values of E. coli at the 30-min doubling time (Table S1).(d) The estimated fractions of RNAPs on rRNA genes (fr), RNAPs on rRNA genes if the nonspecific binding is absent (fr,0), and nonspecific RNAPs (fns) vs. growth rate for E. coli.(e) The sensitivity of free RNAP concentration to the changes in rRNA production rates as a function of the nonspecific RNAP fraction under different growth rates.Here, T is the doubling time, the inverse of the growth rate.The stars mark the fns values of E. coli and the corresponding y-values.
per unit time, where i represents mRNA or rRNA genes.We define the sensitivity of [n] f against the change in the rRNA production rate due to a small change in ∂vr .Surprisingly, we find that s [n] f ←vr only depends on the fractions of different types of RNAPs, Here, f free , f r , f m , and f ns represent the fractions of free RNAPs, RNAPs bound to rRNA genes, mRNA genes, and nonspecific binding sites (see their values for E. coli at the doubling time 30 min in Table S1).Here, f r,0 represents the fraction of RNAPs bound to rRNA genes if there is no nonspecific binding, which satisfies f r,0 /f r = 1/(1 − f ns ).We note that f ns and f r,0 are growth-rate dependent (Fig. (6) holds as well (Eq.( 15) in Methods B).

C. The cost and benefit of nonspecific RNAPs
Although nonspecific RNAPs can attenuate unwanted crosstalk between genes, they also require the cell to spend additional resources to establish the reservoir of nonspecific RNAPs.To demonstrate whether nonspecific RNAPs are evolutionarily advantageous, we compare the cost and benefit of nonspecific RNAPs.To quantify the cost, we use the fraction of nonspecific RNAPs (which are considered as excess resources) in the entire proteome, Φ ex = Φ n f ns where Φ n is the fraction of total RNAPs in the proteome.We compute its value for E. coli for different growth rates and find that they are typically small (Fig. 2(a), and see the details of calculation in Methods C).
To quantify the benefit, we introduce perturbations to the dissociation constants K m and K r and calculate the relative changes of mRNA and rRNA production rate through the following sensitivity matrix (Methods D): where s mm and s rr are the self-sensitivity factors, and  (Fig. S3).

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We also study the buffering effects against random where i(j) denotes gene i recognized by ) is the effective volume of gene i that store holoenzymes on promoter i and the subsequent elongating core RNAPs, that is, [Eσ j ] f V i(j) is the total number of RNAPs on gene i.We have used the approxi- where 362 Eq. ( 9) and (10)

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We remark that the exact values of parameters, e.g.,

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the copy numbers of sigma factors, are uncertain.For total sigma factors is more than the total RNAPs, while others showed the opposite [40-44].The parameters under different conditions and growth rates are also distinct.
Therefore, we randomly sample the parameters from an extensive range to ensure that our conclusions are independent of specific parameters' values (Table S2).k ini i (Fig. 4(b)), where i is either "r" or "m", depending 459 on whether the promoter is followed by rRNA or mRNA Comparing Eq. (11) to Eq. (1), we have For nonspecific binding sites, there is no elongation; Considering a small change in K r (notice that V ns and 603 Rearranging the above 605 equation and using the approximation Pb ns ≪ 1, Pb m ≪ 606 1, which is biologically reasonable (Eq.( 1) and Table S1), 607 we obtain as they lead to the same g ns /K ns .We further rewrite Eq.

613
(13) as which is Eq. ( 6).Here, f free,0 , f m,0 and f r,0 represent the 615 fractions of free RNAPs and RNAPs bound to mRNA 616 genes and rRNA genes if there is no nonspecific binding, respectively.The relationship f free,0 + f m,0 = 1 − f r,0 is 619 used to get the last line.Similarly, we can get 620 which is shown in Fig. S1(c).Here, Pb r is not neglected 621 as its value is comparable to other terms (Fig. S2).

649
The ratio between pausing RNAPs and actively elongat- , where i represents mRNA or rRNA genes.

672
The maximum transcription initiation rate k ini i and gene Combing Eq. ( 16) and Eq. ( 17) and using the approxi-686 mation Pb m ≪ 1 (Fig. S2(b)), we have Similarly, the concentrations of core RNAPs and holoen-715 zymes on nonspecific binding sites are: 716 The following equation describes the equilibrium con-717 dition of sigma-core binding and holoenzyme dissocia-718 tion: k ini i(j) [Eσ j ] i(j) .(24) Here, [σ j ] f is the concentration of free sigma factors un-concentrations in growing cells, Nature Communications

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terms represent the number of RNAPs binding to rRNA 163 genes, mRNA genes, and nonspecific binding sites, re-164 spectively.The 1 and Λ r,m in the 1 + Λ r,m term 165 correspond to promoter-bound RNAPs and elongating 166 RNAPs.g ns refers to the number of nonspecific bind-167 ing sites on DNA, and K ns represents the dissociation 168 constant of nonspecific binding.RNAPs nonspecifically 169 bound to DNA do not elongate along DNA; therefore, 170 there is no extra capacity in the nonspecific RNAPs term 171 of Eq. (3).Eq. (3) allows us to determine the free 172 RNAPs concentration self-consistently.

FIG. 1 .
FIG. 1. Nonspecific RNAPs buffer the free RNAP concentration upon gene expression changes.(a) The schematic diagram of the mean-field model.The nonspecific binding sites effectively enlarge the nucleoid volume, building a reservoir that exchanges RNAPs with the free RNAP pool.(b) The free RNAP concentration (unit: µm −3 ) vs. the rRNA production rates under different effective volumes of nonspecific binding.The free RNAP concentration before perturbation under different Vns is fixed at [n] f = 600 µm −3 (TableS1).The Vns = 2V line corresponds to the parameters of E. coli.(c) The absolute slopes of [n] f over vr vs. Kns (unit: µm −3 ) and gns.The slopes are equal as long as gns/Kns is the same (indicated by the dashed line with slope 1 in the logarithmic coordinates).The red star marks the corresponding values of E. coli at the 30-min doubling time (TableS1).(d) The estimated fractions of RNAPs on rRNA genes (fr), RNAPs on rRNA genes if the nonspecific binding is absent (fr,0), and nonspecific RNAPs (fns) vs. growth rate for E. coli.(e) The sensitivity of free RNAP concentration to the changes in rRNA production rates as a function of the nonspecific RNAP fraction under different growth rates.Here, T is the doubling time, the inverse of the growth rate.The stars mark the fns values of E. coli and the corresponding y-values.
1(d), see Methods C for the details of calculations.)Given a doubling time with its estimated f r,0 , we tune the fraction of nonspecific RNAPs.Interestingly, for E. coli, the sensitivity of free RNAP concentration to the change in rRNA gene expression is about half compared to the case where nonspecific RNAPs are absent (Fig 1(e)).This result suggests that the endogenous numbers of nonspecific binding sites and their dissociation constant significantly buffer the change of the free RNAP concentration across different growth rates.Similarly, nonspecific RNAPs significantly reduce the sensitivity of free RNAP concentration to the changes in mRNA gene expression (Fig. S1(c)) with an equation similar to Eq.
mr and s rm are the mutual-sensitivity factors.We quan-266 tify the benefit as the crosstalk factors: θ m = |s mr /s mm | 267 and θ r = |s rm /s rr |, which quantify the degree of gene 268 expression crosstalk (Methods D).

FIG. 2 .
FIG. 2. The cost and benefit of nonspecific RNAPs.(a) The cost of nonspecific RNAPs under different growth rates, quantified by the proteome fraction of nonspecific RNAPs Φex.(b-c) The crosstalk factors for mRNA genes (b) and rRNA genes (c) as a function of Φex under six different growth rates.Here, T is the doubling time, the inverse of the growth rate.The stars mark the Φex values of E. coli and the corresponding y-values.(d) The correlation coefficient of mRNA and rRNA production rates as a function of the proteome fraction of nonspecific RNAPs.The stars mark the values of E. coli.The detailed calculations of Φex, θm, θr and ρmr are shown in Methods C-E.Interestingly, we find that for E. coli, the crosstalk fac-270

282 gene production rates to mimic a bacterial cell exposed to 283 a
fluctuating environment.We assume that the dissoci-284 ation constants fluctuate with the following noise levels: 285 D m = ⟨( ∆Km Km ) 2 ⟩, D r = ⟨( ∆Kr Kr ) 2 ⟩.Using Eq. (7), the 286 Pearson correlation coefficient of the mRNA production 287 rate and rRNA production rate is analytically calculated 288 [Eq.(21)].Notably, the values of Φ ex of E. coli across different growth rates are again within the range where the buffering against temporal noise is significant with a small cost, showing the robustness of our results (Fig. 2(f)).Here, we use D m = D r = 0.2, and our conclusions remain the same under different values of D m and D r (Fig. S4).D. Effects of sigma factors 1.The RNAP partitioning rules considering sigma factors In bacteria, sigma factors bind RNAPs and are required for RNAPs to recognize promoters.RNAPs without sigma factor binding are called core RNAPs, and core RNAPs become holoenzymes only if they are bound by a sigma factor [29-31].While both core RNAPs and holoenzymes nonspecifically bind to DNA, only holoenzymes specifically bind to promoters and initiate transcription.Meanwhile, anti-sigma factors bind to sigma factors and prevent their binding with RNAPs, inhibiting transcription [31-34].It is unknown whether the buffering effects of nonspecific RNAPs are still valid in the presence of multiple types of sigma factors, and the existence of anti-sigma factors makes the conclusions even more unclear.To test the robustness of our results, we extend our model to include sigma and anti-sigma factors (Methods F).We use σ j to represent the type j sigma factor with its corresponding anti-sigma factors Anti j , E to represent the core RNAP, and Eσ j to represent the corresponding holoenzyme.The total RNAP number as the sum of core RNAPs and all types of holoenzymes is still represented by n t .The total free RNAP concentration as the sum of free core RNAP concentration ([E] f ) and all types of free holoenzyme concentration ([Eσ j ] f ) is still represented by [n] f (Eq.(29)).In this extended model with sigma factors, we assume that the transcription of each gene is initiated by a particular type of holoenzyme, which becomes a core RNAP during elongation because the sigma factor quickly dissociates from the holoenzyme early after transcription initiation [35-38].Therefore, the RNA production rate of gene i is determined by its corresponding free holoenzyme concentration [Eσ j ] f :

347
chemical reactions and conservation equations (Fig.3(a) 348 and Methods F) we derive two partition equations.The 349 first one tells us the fractions of different types of RNAPs 350 in the total pool of RNAPs (Fig. 3(b), left): 1+ j βj are the numbers 352 of free RNAPs, nonspecific RNAPs, and RNAPs bound 353 to genes recognized by σ j , respectively.Here, V j = 354 i(j) V i(j) is sum of effective volumes over all genes rec-355 ognized by σ j ; β j is the partition factor (see its detailed 356 expression in Methods F), which is positively related to 357 the total sigma factor number and negatively related to 358 the total anti-sigma factors and holoenzyme dissociation 359 constants.The second one tells us how the free RNAPs 360 are partitioned into different types of free holoenzymes 361 and free core RNAPs (Fig. 3(b), right):

FIG. 3 .
FIG. 3. The buffering effects of nonspecific RNAPs in the model with sigma factors.(a) The schematic of the mean-field model with sigma and anti-sigma factors included.Both core RNAPs and holoenzymes nonspecifically bind to DNA with the same dissociation constant Kns.(b) The schematic of the partitioning of total RNAPs and free RNAPs based on Eq. (9) and Eq.(10).(c-f) Nonspecific RNAPs can buffer the free RNAP concentration regardless of the types of regulation.The y-axis shows the sensitivity of free RNAP concentration to the changes in the RNA production rate of the regulated gene, which can be gene B or C, depending on the regulation type.The values under different sets of parameters are scaled to −1 at Vns = 0.The curves with random parameters are shown in the background, and the black lines are their means.The error bars are the standard deviation of the background curves.The dashed line marks the Vns/V value of E. coli, which is approximately constant across growth rates (Methods C). (g-j) The same as (c-f), but the y-axis shows the sensitivity of the RNA production rate of genes A to the changes in the RNA production rate of genes B or C, depending on the regulation type.Nonspecific RNAPs attenuate gene expression crosstalk significantly in type I (dissociation-constant regulation), but non-significantly in type II (sigma competition).

FIG. 4 .
FIG. 4. Simulations of the agent-based model.(a) A schematic of the agent-based model of bacterial transcription.We model the three-dimensional nucleoid with multiple binding sites corresponding to promoters and nonspecific binding sites.RNAPs diffuse until they hit any free binding site.(b) If an RNAP hits a promoter, it will start transcription with a rate k ini i or hop off the promoter with a rate k off i .Here, i can be "r" or "m", depending on whether rRNA or mRNA genes follow the promoter.If an RNAP hits a nonspecific binding site, it will only hop off with a rate k off ns .(c) We relocate an RNAP that just left a binding site inside a shell with inner diameter a and outer diameter R = γ × a.(d) The scatter plot of mRNA and rRNA production rates (unit: 1/s) under different Vns.The negative correlation between vm and vr across time is strong under weak nonspecific binding (small Vns) while weak under strong nonspecific binding (large Vns) (Methods H).Here, each point represents one time point in each repeat, and the straight line is the linear fit of all points of the corresponding Vns.(e) The absolute value of the negative correlation coefficient of mRNA and rRNA production rates decreases with the effective volume of nonspecific binding.The dashed line marks the Vns/V value of E. coli, which is approximately constant across growth rates (Methods C).

475
and fixed positions of binding sites, and the binding of 476 RNAPs to promoters and nonspecific binding sites are 477 diffusion-limited.More details of the agent-based model 478 are included in Methods G-H.Intriguingly, the simulated free RNAP concentration 480 (averaged over binding sites) vs. the distance to the 481 binding site center can be well fitted by our analytical 482 predictions (Fig. S5(a-b)), and the agent-based simula-483 tions agree well with the mean-field predictions regarding 484 the sensitivity of average free RNAP concentration to the 485 changes in gene expression (Fig. S5(c)) despite the spa-486 tial heterogeneity in the free RNAP concentration.We 487 find a strong negative correlation between the mRNA 488 and rRNA production rates given weak nonspecific bind-489 ing (small V ns ) and an almost zero correlation between 490 them given strong nonspecific binding (large V ns ) (Fig. 491 4(e-f)).Therefore, we conclude that the buffering effects 492 of nonspecific RNAPs on gene expression crosstalk are 493 robust against the spatial heterogeneity of free RNAP 494 concentration and noises in random processes.495 III.DISCUSSION 496 Previous studies have shown that nonspecific tran-497 scription factor-DNA binding can buffer against gene ex-498 pression noise in eukaryotes [46-48].Similarly, it has 499 been suggested that bacteria produce extra DnaA (mas-500 ter regulator of replication initiation) to suppress noise in 501 initiation [49].Recent studies highlight the importance 502 of a trade-off between protein overabundance and cell 503 fitness, especially for transcription machinery [50].Nev-504 ertheless, the benefit compared to the cost of nonspecific 505 binding has yet to be estimated as far as we realize, nei-506 ther for transcription factors in eukaryotes nor RNAPs 507 in bacteria.One should note that nonspecific binding of 508 RNAPs appears absent in eukaryotes [20], which means 509 that our conclusions mostly apply to bacteria.510 In this work, we demonstrate that the nonspecifically 511 DNA-bound RNAPs are natural buffers to mitigate the 512 unwanted crosstalk between the transcription of rRNA 513 and mRNA, and the buffering effects are significant in E.514 coli.Importantly, we mathematically prove that our con-515 clusions are parameter-insensitive across different growth 516 rates.We find that the buffering effects of nonspecific 517 RNAPs are more significant against rRNA regulation in 518 rapid-growth conditions (Fig. 1(e)).In contrast, the 519 buffering effects are more significant against mRNA reg-520 ulation in slow-growth conditions (Fig. S1(c)).We argue 521 that this growth rate dependence is due to the growth 522 rate dependence of the fractions of different types of 523 RNAPs (Eqs.(14, 15), Fig. 1(d) and Fig. S2).We 524 note that an additional benefit of nonspecific binding at 525 slow growth can be providing a reservoir for transitions 526 to fast growth with high transcription rates of rRNA.527 Interestingly, the number of nonspecific binding sites is 528 much more than that of promoters, and the nonspecific 529 binding affinity is much weaker than those of promoters, 530 similar to pH buffers made of weak acid-base pairs ( 596 therefore, Pb ns = [n] f [n] f +Kns with K ns = sensitivity of [n] f to the changes in rRNA 598 (mRNA) production rates 599 We replace the number of RNAPs on rRNA genes term 600 ([n] f V r ) in Eq. (4) by the rRNA production rate v r such 601 that 602

650ing
RNAPs on mRNA and rRNA genes (η m and η r ) are 651 negatively related to growth rates due to ppGpp regu-652 lation[24, 60].For simplicity, we assume that η m and 653 η r are the same, and both linearly decrease with growth 654 rates and approach 0 when the doubling time T = 12 655 min.This leads to η r = η m = 1.5 − 0.5(µ − 2), and the 656 detailed choices of η r and η m do not affect our main con-657 clusions (Fig.S7).We then obtainf m = f a m (1 + η m ) 658 and f r = f a r (1 + η r ).The ratio of f ns to f free is 659 V ns /V = (g ns /V )/K ns since K ns ≫ [n] f , which is approx-660imately constant across growth rates given a constant 661 density of DNA and a constant K ns .Thus, we set this 662 constant ratio as 0.3 0.15 = 2 using the data of the 30-min 663 doubling time and we obtainf free = 1 3 (1 − f m − f r ) and 664 f ns = 2 3 (1 − f m − f r ).f free,0 , f m,0 and f r,0 are computed 665 from f free /f free,0 = f m /f m,0 = f r /f r,0 = 1 − f ns .f r , f r,0 666 and f ns are shown in Fig. 1(d), and the other fractions 667 are shown in Fig. S2(a).668Weestimate the probabilities of the promoter bound 669 by an RNAP for mRNA genes and rRNA genes from the 670 fractions of actively elongating RNAPs: Pb i =

∆v r v r = −( 1 −+( 1 − 1 + 714 [
18)where we introduce the excess factor ε = fns 1−fns .Simi-688 larly, for the rRNA production rate we have689 Pb r )[1 − f r,0 (1 − Pb r ) 1 + ε − f r,0 Pb r ] ∆K r K r Pb r ) f m,0 1 + ε − f r,0 Pb r ∆K m K m .(19)Writing the above two equations in matrix form, we fi-690 nally get Eq.(7), and the sensitivity matrix is:s mm s mr s rm s rr = −Here, w 1 = 1 + ε − f r,0 Pb r , and w 2 = 1 − Pb r .Notice 692 that w 1 increases with ε linearly, so with the increase of 693 nonspecific binding, the self-sensitivity factors (s mm and 694 s rr ) approach their maximum absolute values, while the 695 mutual sensitivity factors (s mr and s rm ) decrease to 0. 696 We define the crosstalk factors for mRNA and rRNA 697 genes as θ m = |s mr /s mm | and θ r = |s rm /s rr |, and com-698 pute their values using the fractions of different types of 699 RNAPs and probabilities of promoter binding estimated 700 in section C of Methods.701 E. The correlation coefficient between the mRNA 702 and rRNA production rates 703 We compute the covariance of v m and v r as 704 Cov(v m , v r ) = (s mm s rm D m + s rr s mr D r )v m v r , and 705 the product of the variances of v m and v r as 706Var(v m )Var(v r ) = (s 2 mm D m + s 2 mr D r )(s 2 rr D r + 707 s 2 rm D m )v 2 m v 2 r .Here, Eq. (7) and ⟨∆K r ∆K m ⟩ = 0 708 are used.Finally, we get the correlation coefficient 709 between mRNA and rRNA production as710 ρ(v m , v r ) = Cov(v m , v r ) Var(v m )Var(v r ) = s mm s rm D m + s rr s mr D r (s 2 mm D m + s 2 mr D r )(s 2 rr D r + s 2 rm D m ) .(21)F.Mean-field model including sigma factors711The concentration of holoenzymes on promoter i is re-712 lated to the corresponding free holoenzyme concentration 713 by: Eσ j ] i(j) = V i(j) (1 + Λ i(j) )V [Eσ j ] f .
Eσ j ] f V ns and [E] f V ns are the numbers of holoen- )), which is biologically reasonable since the probability of a promoter bound by a holoenzyme is typically low (Fig.S2(b)).Moreover, 337 this approximation significantly simplifies the analytical 338 derivations and numerical simulations (Methods F).Sim-339 ilarly, [340 zymes and core RNAPs nonspecifically bound to DNA, 341 respectively.Here, we approximate V ns = gns Kns because 342 K ns ≫ [Eσ j ] f , and K ns ≫ [E] f in typical biological sce-343 narios (Table S1), where we assume core RNAPs and 344 holoenzymes have the same K ns because their nonspe-345 cific binding affinities with DNA are comparable [39].346 Intriguingly, based on several equilibrium equations of

Table 531
S1).The buffering effects of nonspecific RNAPs to main- 544 ity through evolution.We cannot exclude the possibility 545 that the origin of nonspecific binding may result from 546 passive interaction between RNAP and DNA.But still, 547 whether it is passive or selected actively, one cannot ig-548 nore the function of nonspecifically DNA-bound RNA 549 polymerases as a buffer of resource competition.550 In growing cells, the number of σ 70 is significantly 551 larger than other alternative types of sigma factors in 552 E. coli [40, 41, 52], and most genes are controlled by 553 σ 70 .However, in the presence of multiple types of sigma 554 factors [41, 53-55], different types of sigma factors can 555 compete for the pool of core RNAP such that the up-556 regulation of one group of genes can result in reduced

)
Eq. (13) expresses s[n] f ←vr as a function of V ns .Because 609 in typical biological scenarios, [n] f ≪ K ns (TableS1), we 610 have V ns = g ns /K ns , which means that changing K ns or The growth-rate-dependent parameters for E. coli RNAPs at the 30-min doubling time.f free , f ns , and 625 (f m + f r ) at the 30-min doubling time are measured 626 by single-molecule tracking due to the differences in 627 the diffusion coefficients among free RNAPs, nonspecific 628 RNAPs, and specifically-bound RNAPs [17].For sim-629 plicity, we neglect a small fraction of RNAPs (about 630 0.06) unable to bind to DNA, which is likely immature 631 RNAPs [16, 18].To ensure that f free + f r + f m + f ns = 1, 632 the fractions we estimate are slightly larger than the re-633 ported mean values but still within the ranges of exper-634 imental errors [17], which are f free = 0.15, f ns = 0.3, 635 f m + f r = 0.55.The ratio of actively elongating RNAPs 636 transcribing rRNA to those transcribing mRNA is about 637 2.2 [59]; therefore, we obtain f r = 0.38 and f m = 0.17.638Theratio between pausing RNAPs and actively elongat-641 S1(9)).For simplicity, we assume the ratios for rRNA 642 and mRNA genes are the same and equal to 1.5.643Next,we determine the RNAP fractions across differ-644 ent growth rates.We calculate the fractions of actively 645 elongating RNAPs on mRNA and rRNA genes across dif-