The Chemical Fluctuation Theorem governing gene expression

Gene expression is a complex stochastic process composed of numerous enzymatic reactions with rates coupled to hidden cell-state variables. Despite advances in single-cell technologies, the lack of a theory accurately describing the gene expression process has restricted a robust, quantitative understanding of gene expression variability among cells. Here we present the Chemical Fluctuation Theorem (CFT), providing an accurate relationship between the environment-coupled chemical dynamics of gene expression and gene expression variability. Combined with a general, accurate model of environment-coupled transcription processes, the CFT provides a unified explanation of mRNA variability for various experimental systems. From this analysis, we construct a quantitative model of transcription dynamics enabling analytic predictions for the dependence of mRNA noise on the mRNA lifetime distribution, confirmed against stochastic simulation. This work suggests promising new directions for quantitative investigation into cellular control over biological functions by making complex dynamics of intracellular reactions accessible to rigorous mathematical deductions.

producing the product molecules can be arbitrary stochastic processes, which may be coupled to complex and dynamic cell states. The topology and regulatory mechanism of the reaction network producing the product molecules may be arbitrary as well. The product annihilation process can be a non-Poisson, renewal process, but the product creation process does not have to be a renewal process.  In terms of the mean product creation rate, ( ) R t 〈 〉, the mean product number, (0) ( ) n t 〈 〉, is given by

t t t t t t t n t t t t t t t t t
Equations (M-6) and (M- 19) hold even when the product creation process is a non-stationary stochastic process.
When the product creation process is a stationary process, we have As shown in the derivation, the chemical fluctuation theorem (CFT) is a general result that can be derived without making any assumption about the property of the product creation process. This means that it holds exactly for any intracellular regulatory network in which the product creation rate is modulated by product number or any other environmental variables.
The only assumption involved in our derivation of CFT is that the lifetimes of product molecules are identically distributed, independent random variables, so that the product lifetime distribution does not change over time. It is possible to think of a more general product decay process, but, in such a case, the CFT would lose its concise form and would become far more complicated. We believe that the current form of CFT is already general enough to

The Chemical Fluctuation Theorem from transient Little's law and the law of total variance
In this section, inspired by an anonymous reviewer's comment, we discuss the derivation of the CFT from transient Little's law and the law of total variance. In probability theory, the law of total variance states that, if X and Y are random variables on the same probability space and the variance of Y is finite, then where ( ) X f Y 〈 〉 denotes the average of ( ) f Y over the conditional probability ( | ) p Y X .
Given that the law of total variance could be extended to describe the variance in the product number for a non-stationary reaction system with rate being an arbitrary stochastic process, the variance in the product number is given by In addition, according to our CFT, the variance in the product number is equal to the mean for any particular realization of ( ) R t , i.e., On the other hand, if one were to mistakenly adopt equation (M-29b) for the definition of 0 0 ( ) ( ) R t t R t ′ + , one would obtain a different result, namely,

Derivation of equations (2) and (3)
In the steady-state where product creation is a stationary state, equation (1) reads as: for the transcription of a single gene copy. 1 ( ) R t and ( ) S t denote the rate of the transcription of the single gene copy and the time-dependent survival probability of mRNA, respectively.  ∫ . An alternative derivation of equation (2) can be found in ref.
2, but the derivation is only applicable when the product survival proability is an exponential function. Now let us move on to derivation of equation (3). For the transcription of g identical copies of the gene, equation (1) reads as where i R denotes the transcription rate of the i -th gene copy. The mean transcript number g n 〈 〉 created from g identical copies of the gene is related to the mean transcript number 1 n 〈 〉 created from the single gene copy by 1 1 because the mean transcription rate of one gene copy is the same as that of another gene copy, i.e., is the same as the mean lifetime of mRNA, i.e., In ( 1) From equations (M-38a) and (M-38b), we finally obtain equation (3): where 2 ,1 n η denotes the relative variance or noise in the copy number of mRNA produced by a single gene copy, i.e. 2 2 ,1 1 n n σ . n C denotes the mean-scaled correlation between mRNA levels produced from two gene copies, which is defined by In this note, we briefly mention the previous works that support our finding that the RNAP binding affinity of constitutive promoters fluatuates with a rate of about 100 Hz or greater. It was shown that the supercoiling of DNA affects the formation of the pre-initiation complex and the subsequent initiation process 3,4 . Such a tendency differs from gene to gene, depending on the promoter sequence 5,6 . The time scale associated to non-enzymatic supercoiling dynamics amounts to 10 ms or less, which is consistent with our estimation of the RNAP-

Supplementary Note 2 | Equation for the protein noise derived from CFT.
In this note, we apply equatin (1) to the simple vibrant translation process, the rate of which is given by ( ) The mRNA noise, 2 n η , appearing in equation (N2-2) is given by 2 1 n n n Q n η − =〈 〉 + 〈 〉 with n Q being given by equation (2) in the main text.
The noise susceptibility given in equation (N2-3) can be rewritten as

Supplementary Note 3 | Relationship between equation (1) and previously reported results.
In this note, we briefly discuss how equation (1) is related to the previously reported results for the product copy number variability.

Relation to results in refs. 2, 9, 10, 11 and 12
When the survival probability is a simple exponential function, i.e., when where the diagonal terms with i j = are separated from the off-diagonal ones with i j ≠ on the R.H.S. of the second equality. Let us consider ( ) ( ) n t R t′ in the limit where t t ε ′ → + with ε being positive and infinitestimal. Then each of the diagonal terms vanishes because By taking the average of equation (N3-2) over { , } in the steady-state or in long time limit. Equation (N3-5) is exact regardless of the time-profile of the survival probability and the stochastic properties of the steady-state transcription process, which may depend on the product number, in the presence of feedback regulation, as well as other cell-state variables. Equation (N3-5) is a new result, which has not been previsouly reported. Its application to biological systems is to be published separately.
In the special case where the product decay process is a simple Poisson process for which the survival probability of a product molecule is given by ( ) where nR χ is the same as that appearing in equation (N3-1). With equation (N3-6) at hand, equation (N3-1) can be rearranged to Performing the average of equations (N4-1) and (N4-2) over the distribution, ( ) p Γ , of cell state, we can easily obtain ( ) n t and 2 ( ) n t , respectively. By subtracting 2 ( ) n t from 2 ( ) n t , we obtain 2 1 2 1 2 2 1 0 0 where ( ) S t and and l m τ 〈 〉 is given by In Fig. 5f, we consider the model where the mRNA lifetime is statically heterogeneous dichotomous random variable among the cells. In this case, we have For this model, the R.H.S. of equation (N4-6) is given by and l m τ 〈 〉 is given by 5f, the non-Poisson mRNA noise for this model is greater than the non-Poisson mRNA noise of the transcription model with the mRNA survival probability that is not heterogeneous but the same across the cells and given by

Supplementary Note 5 | Brief review on several approaches dealing with reaction networks in dynamic environments.
Here, we present a brief review on several approaches dealing with reaction networks in dynamic environments, which are mentioned in the main text. Combining equations (2) and (3), we obtain the following expression of the non-Poisson for Model III. The non-Poisson noise, To provide a quantitative explanation of the experimental data shown in Fig. 2 That is to say, in terms of x , we have 2 (1 ) / x x ξ η = − , which vanishes in the limit where the gene is always in the unrepressed state, i.e., in the 1 x → or on off k k → ∞ limit. Therefore, the first two terms on the right-hand side of equation (N6-1) vanish in the On the right-hand side of this equation, 2 nκ κ χ η and g F originate from the fluctuation in transcription rate κ of the gene in the unrepressed state and the variation in the gene copy number, but they are independent of the gene-state switching dynamics or on k , off k , and x .
In addition, for Model II and III, n C is independent of on k , off k , and x , as we will show in Supplementary Note 10. Since is not unique; instead, it is sensitive to the transcriptional regulation mechanism. When the value of off k is modulated while the value on k is held constant through the transcriptional regulation, we have on off with α being on k γ . On the other hand, when the value of on k is modulated while the value with β being off k γ . As mentioned before, we have in any case. Therefore, the dependence of the non-Poisson mRNA noise in equation (N6-4) on the maximum-scaled mean mRNA level, x , is given by Note that the non-Poisson noise is divergent in the small x limit for the on k modulation mechanism, whereas it approaches a finite value for the off k modulation mechanism. The experimental data shown in Supplementary  The dependence of ∆ on κ φ for Model III can be obtained from equation (N6-1) as where The explicit expression of ( ) x ∆ can be obtained by substituting equation (N6-5) into equation (N6-9): . The corresponding inverse Laplace transform, for the case where the mean mRNA level is close enough to the maximum value, and for the case where the mean mRNA is small enough. As shown in Supplementary Figure  . To do this, we first make use of the substitution, 1 / x s α , in equation (N6-10a) and then make a simple rearrangement of the resulting equation to obtain Since the inverse Laplace transform, Substituting equation (N6-13) into equation (N6-12), we obtain where the value of ( ) Here, 0 J denotes the zeroth order Bessel function of the first kind 24 , which is one of the wellknown oscillatory functions frequently encountered in physics and chemistry. As shown in Fig.   2b, the TCF given in equation (N6-15) exhibits an oscillatory behavior.
According to an anonymous reviewer's suggestion, we repeated our analysis using a nonparametric interpolation of the raw data version of ( ) x ∆ to confirm the oscillatory feature in the resulting TCF of the transcription rate. We interpolate the raw data points for ( ) x ∆ , or We have also used the Stehfest method for the numerical inverse Laplace transform to extract the TCF of the transcription rate from the non-parametric interpolation of the raw data version of ( ) x ∆ . In contrast with the TCF obtained from the Durbin-Crump method, the TCF extracted from the Stehfest method has a noisy shape, and the details of the shape depend on which option was chosen for the numerical inverse Laplace transform routine in use. However, we find that the noisy TCF extracted from the Stehfest method also shows an oscillatory feature in qualitative agreement with the TCF extracted with use of the Durbin-Crump method, or the result of our analysis that relies on equation (N6-13), the smooth function version of ( ) x ∆ , which are presented in Supplementary Figure 5 and Fig. 2, respectively. However, we do not present the unnaturally irregular TCFs extracted using the Stehfest method. Given that the variance in the copy number of mRNA is a slowly varying function of the mean mRNA number, and hence the TCF of the transcription rate should be smooth functions.
We find that the oscillatory time dependence of ( ) t  Figure 6). For the model, ( ) T t ψ is given by the convolution, which is in excellent agreement with the experimental data for the dependence of the non- as the initial binding of RNAP to promoter DNA becomes sluggish, and it gets smaller as the initial binding of RNAP to promoter DNA becomes faster, making the subsequent successful initiation step the rate determining step. This is because the randomness of the initial binding of RNAP to promoter is far greater than that of the successful initiation step composed of a number of consecutive reaction processes, which is also expected to be the case in living cells.
The cells with a smaller number of RNAP and sigma factors would have a slower rate for the initial binding of RNAP to promoter DNA compared to cells with a greater number of the proteins. In comparison, the rate of the successful initiation step is expected to be far less In the present work, instead of separating product noise into intrinsic and extrinsic noise, we have factored the product creation rate into two factors: the control variable dependent factor and the environmental variable dependent factor. The former takes into account the rate of the chemical process that is coupled to the experimentally controlled variable as well as to the environmental variables. On the other hand, the latter takes into account the rate of the remaining chemical processes in the network, which are coupled to the environmental variables, but not to the control variable. Using the factorized form of the product creation rate in CFT, we can obtain the relationship between the product noise and the noise in both rate factors, as demonstrated in equations (2) and (3) for both the single gene transcription and multi gene transcription versions of Model III, respectively. As shown in equations (2) and (3) ( 1)  n 〈 〉 has a significant contribution from , which makes it difficult to extract information about the transcription dynamics from the dependence of Fanofactor on the mean mRNA level, either.
By dividing equation (N8-1) by ( ) 1 n g n 〈 〉 =〈 〉〈 〉 , one can easily obtain the analytic result for the mRNA noise or the relative variance in the mRNA number, given by 2 , ( 1)

Supplementary Note 9 | Comparison between the static and dynamic models of replication.
In this note, the estimations of the first two moments of gene copy number for the static and dynamic models of replication are compared. The non-Poisson mRNA noise, equation (3), accounting for the effect of gene copy number variation is obtained by combining equations (M-38a) and (M-38b). For convenience, both equations are reproduced below: These equations are valid irrespective of the explicit time dependence of the slow gene copy number variation. When the gene copy number, g, is either 1 or 2, Jones et al. obtained the following equations for g 〈 〉 and 2 g 〈 〉 15 : where In this note, we show that the mean scaled correlation n C between copy numbers of mRNAs produced from different gene copies is independent of the mean mRNA level for Model III, as stated before. For Model III, the mean-scaled mRNA correlation, n C , is independent of the mean mRNA level, which can be shown as follows. According to ref. 2, n C can be decomposed into the three terms: ( , ) where q C denotes the mean-scaled correlation between fluctuations of q for genes A and B, explicitly, The analytic expression of susceptibility C nq χ is given by where γ is defined as denoting the decay rate of mRNA produced from the gene X.
To show that n C is constant in on k , off k , and is the four-dimensional column vector defined as Substituting the result into equation (N4-1), one can obtain ν ∈ ), respectively. Here, ν is equivalent to ξ . Therefore, like 0 C ξ = , we also have 2) The number Rp N of RNAP is a static random variable, i.e. 3) The transcription rates of different gene copies are uncorrelated, i.e., 0 n C = .
When a single gene transcription rate is given by where the RNAP-promoter binding affinity, ( ) K p , is dependent on the protein copy number, p, given by Here, 0 K , p K , and h denote the RNAP-promoter binding affinity in the small p limit, the binding affinity of protein to the operator site, and the Hill exponent, respectively. 0 K and p K are not just constants but stochastic variables that are coupling to the cellular environment.
A positive h would then indicate negative feedback, and a negative h, positive feedback.
In the actual application of the CFT to the quantitative analysis of the chemical fluctuation resulting from a regulatory network, it is necessary to calculate the TCF of the product creation rate, which depends on the product number.
A simple but general method is to use a perturbative expansion of the transcription rate in terms of the protein number around its mean value, which was proposed by Tattai respectively. Substituting equations (N13-3) and (N13-4) into equation (N13-1), we obtain to the leading-order approximation neglecting the higher-order relative fluctuation terms with Similarly, we consider C θ and ( ) XY t θ φ simultaneously as one single factor in equation Applying equation (N7-2) separately to ( ) 1 (1 ) Substituting equations (N13-7) and (N13-8) into equation (N13-6) gives to the leading-order approximation neglecting the higher-order relative fluctuation terms with We assume that identical copies of the target gene have the same binding affinity to RNAP, i.e., , when equation (N13-9) assumes a simpler form: We note here that ( ) XY t ζ φ is not the same as

Estimation of the environment-induced correlation between transcription levels of different gene copies
The mean scaled correlation, ( ) n i j i j C n n n n δ δ = 〈 〉 〈 〉〈 〉 , is a measure of the environment-induced correlation between the transcription levels of different gene copies. In the present work, we assume that n C is the same for any pair of gene copies. As detailed below, the maximum value of n C is estimated to be about 0.2 and n C is proportional to the square of the probability that the promoter is not occupied by RNAP so that it has a smaller value for strong promoters. (1 ) where the meanings of the symbols are the same as those in Supplementary Note 15. Equation  (1 ) According to our analysis of the experimental data shown in Fig. 4

Fluctuation in RNAP binding affinity is essential for quantitative explanation of experimentally measured mRNA noise for constitutive promotors as well.
Without taking into account the on-and-off fluctuation in RNAP binding affinity of the constitutive promoters, we cannot provide a quantitative explanation of the experimental data shown in Fig. 4  As shown in the figure, the prediction of (N14-5) is found to be qualitatively different from the experimental data. This observation shows that the fluctuation in the binding affinity is an important source of the non-Poisson mRNA noise for the constitutive promoters as well in E.
coli. 15 Here, we present the details of the quantitative analysis of lacZ gene mRNA copy number variation measured for various constitutive promoters in E. coli, shown in Fig. 4 15 . Here, we assume that the lacZ mRNA in this system also shows the same exponential decay as the lacZ mRNA in the system investigated in ref. 15. As shown in Supplementary Figure 12, these experimental data can be explained moderately well by Model III. This suggests that, under the constitutive promoters as well, the gene expression turns on and off in E. coli, which may be ascribed to the conformation dynamics of DNA and conformation dependent RNAP binding affinity of the promoter 30,31 . However, the quality of agreement between theory and experiment can be significantly improved by using a more accurate model for the experimental system considered in Fig. 4, which is described below.

Supplementary Note 15 |Analysis of the copy number variation of lacZ gene mRNA expressed through various constitutive promoters among a clonal population of E. coli
In the experiment shown in Fig. 4, the lacZ gene is expressed under various constitutive promoters in E. coli. Therefore, the control variable in the experiment can be identified as the RNAP binding affinity, K, of promoter. According to the present approach, we use an explicit model for the control variable dependent part of the transcription rate only. To explain the experimental data shown in Fig. 4, for example, we model the single-gene transcription rate as

Relationship between non-Poisson mRNA noise and the control variable, ζ
By applying equation (1) to the transcription model in Fig. 4, we obtain the expression for the mRNA noise produced by a single gene, which has exactly the same mathematical structure as equation (2) can be related to the control variable, ζ , as follows: ( ) ( 1) As we shall see shortly, for the model shown in Fig. 4b Using equation (N15-9a), those terms involving 2 ζ η in equation (N15-6) can be rewritten as where the trilinear susceptibility, where the trilinear susceptibility, (1 ) Substituting equations (N15-12) and (N15-14) into equation (N15-6), we obtain ( 1) ( )(1 ) For the model for RNAP-promoter binding affinity fluctuation described in the subsection 15.2, we have 0 K C = . In this case, equation (N15-15) reduces to equation (N15-7).

Model for RNAP-promoter binding affinity fluctuation
A change in the promoter architecture can give rise to changes not only in the mean binding affinity, K , but also in the magnitude and dynamics of the fluctuation in K. The former affects the mean transcription level, while the latter affects the transcription level variability.
The way in which the fluctuation in binding affinity K is dependent on the mean binding affinity determines the dependence of the mRNA level variability on the mean mRNA level, which depends on the mechanism and dynamics of the bacterial transcription.
Constitutive promoters also undergo an on-and-off state switching in the RNAP binding affinity, K, of promoter. If the fluctuation in the RNAP binding affinity is negligible for the constitutive genes, i.e., where ( ) θ ζ is approximated by   Table 3.

Supplementary Note 16 | Estimation of the Fano factor g F of gene copy number g.
Here, we present a detailed description of the method used to estimate the first two moments, Let us first examine the respective contribution of the first two terms on the right-hand side of equation (N6-1) to the total mRNA noise. The sum of the first two terms is designated by ∆, whose expression is given in equation (N6-8) for Model III. Its dependence on the mean mRNA level is given in equation (N6-10a), which is found to be in excellent agreement with experimental data (Fig. 2a & Supplementary Figure 2a). As shown in equation (N6-10a), the mean mRNA level dependence of the first term originating from the gene-state switching process is simply given by (1 ) ( ) In contrast, the mean mRNA level dependence of the second term originates both from the transcription rate fluctuation of the unrepressed gene and the gene-state switching process, which assumes a more complicated form, The second term or the bilinear term is sensitive to the transcription dynamics of unrepressed genes. Note that the second term extracted from the entire data, represented by the blue curve in Supplementary Figure 16b, diminishes the mRNA noise as long as the mean copy number 1 n 〈 〉 of mRNA generated from the single gene copy is greater than roughly 0.3. As shown in Fig. 3, this means that the transcription of the unrepressed gene is a highly sub-Poisson process, for which the TCF of the transcription rate κ has an oscillatory function of time. In contrast, the TCF of the transcription rate κ for the slowly growing cells with division time greater than 45 minutes is the simple exponential function, for which case the bilinear coupling term increases the mRNA noise at any value of the mean mRNA level. Our analysis indicates that the fluctuation in the transcription waiting time or the intermittent time between successive transcription events among the slowly growing cells is greater than the fluctuation in the transcription waiting time among the entire cells, even if they have the same mean mRNA level. Now let us now discuss the non-Poisson mRNA noise originating from sources other than the gene-state switching process. According to equation (N6-1), the non-Poisson mRNA noise is composed of the mRNA noise solely arising from the fluctuation in transcription rate κ of the unrepressed gene, the mRNA noise originating from gene-copy number variation, and the mRNA noise originating from the correlation between the number of mRNA produced from one gene copy and the number of mRNA produced from another gene copy. These sources correspond to the three terms in the bracket on the right-hand side of equation (N6-1). We can estimate the sum of the non-Poisson mRNA noise originating from the sources independent of the gene-state switching process by the high expression limiting value of the non-Poisson noise (see equation (N6-2)), which is found to be about 0.38, which is smaller than the magnitude of mRNA noise originating from the gene-state switching process when 1 n 〈 〉 is smaller than roughly 10 (see Supplementary Figure 16a). However, mRNA noise originating from sources other than the gene-state switching process makes the major contribution in the high expression limit, where the gene is almost always in the unrepressed state, so that the mRNA noise term caused by the gene-state switching process is negligible.

Supplementary Note 18 |Extraction method of the time correlation function of product creation rate from the time series of reaction events
We present here the method for extracting the TCF of the creation rate from the time series of reaction events, or the series of times at which reaction events occur. This method is used in In terms of the Dirac delta functions, the number ( ) n t of reaction events occurring in time interval (0, ) t can be written as To calculate the variance in the number of reaction events, we need the expression for the square 2 ( ) n t of the number of reaction events as well, which is given by By performing the average of equations (N18-2) and (N18-3) over a large number of reaction time sequences, we obtain 0 ( ) 〉 is the TCF of the reaction rate defined by Note here that j t is the time at which the j-th reaction event is completed, so j t increases with j. With this notation, one can obtain the following equation from equation (N18-5) by noting that Using the following property of the Dirac delta function, can rewrite equation (N18-6) as ( ) Let us confine ourselves into the case where our reaction process is a stationary process, for Noting that the correlation between 0 ( ) R t t + and 0 ( ) R t vanishes in the long time limit, i.e., From equation (N18-9), we obtain By dividing equation (N18-11) by R and taking the Laplace transform on both sides of the resulting equation, we get 1( When the reaction process is a renewal process 36 , the reaction time elapsed for a pair of successive reaction events is statistically independent of the reaction time elapsed for another pair of reaction events. For a renewal reaction process, ˆ( ) l s ψ can be replaced by 1 ( ) l s ψ , which is well-known, so that equation (N18-12) simplifies to This equation enables us to obtain the TCF R φ of the reaction rate fluctuation from the distribution, 1 ψ , of reaction waiting time or the intermittent times between consecutive reaction events, which is applicable to a renewal process.
When our reaction process is not a renewal process, equation (N18-13) is no longer exact.
However, for any stationary reaction process, one can obtain the TCF Let the asterisk signify the particular initial condition that any one of { } i t is set to time 0, after which the reaction event counting begins. With this notation, we can rewrite equation (N18-8) as Noting that equation (N18-2) can be used for any initial condition, i.e., In the long time limit, equation (N18-16) yields makes sense because the initial condition is irrelevant in the long time limit. By subtracting the latter equation from equation (N18-15), we obtain By dividing both sides of equation (N18-16) by R 〈 〉 , we get A similar result can be found in refs.22 and 37 but equation (N18-19) is more general in that it is applicable to the case that 1 ( ) t ψ is a sub-Poisson distribution.
Thanks to equation (N18-19), one can obtain the TCF of the product creation rate from the mean number

Supplementary Note 19 | Stochastic simulation methods used in Figs. 3 and 5
Stochastic simulation method used in Fig. 3 Here we provide a detailed description of the stochastic simulation method used in Fig. 3c.
The transcription model shown in Fig. 3a consists  Fig. 3b), whose Laplace transform is given by  Figure 18 for the dependence of 2 T η on 2 1 τ τ , which is the parameter determining which step between the first and second steps in Fig. 3a is the ratedetermining step.

Stochastic simulation method used in Fig. 5
Here we provide a detailed description of the stochastic simulation method used in Figs.    Fig. 5f.

Supplementary Note 20 | Robustness of the quantitative analysis given in Supplementary
Note 15 for the experimental data shown in Fig. 4.
In the transcription model shown in Fig. 4b, the RNAP binding affinity 0 K of the promoter in the active state has been regarded as a constant. However, even if we take into account the fluctuation in 0 K , the quality of the agreement between the theoretical model and experiment does not significantly improve.
To show this, let us consider the case where 0 K is a stochastic variable. For this case as well, the non-Poisson mRNA noise is given by equation (N15-6) with TCF given by equations (N15-9a) and (N15-9b). However, the first term on the R.H.S. of the latter equations has an additional contribution from the fluctuation in 0 K : On the right-hand side of equation (N20-1b), only the first term survives because 0 C ν = for our model, i.e., where we have three adjustable parameters, explicitly,  (1 ) 1 const 1 1 (1 ) 1 In equation ( From the quantitative analysis, we find that the global trend in the dependence of non-Poisson mRNA noise on the mean mRNA level is more consistent with the on k regulation mechanism, in which the gene-to-gene variation in the transcription dynamics is primarily achieved by changing, on k , the rate at which the gene state switches from the inactive state to the active state, rather than the off k regulation mechanism, as shown in Supplementary Figure   3a. This is also the case when we assume the TCF for the transcription rate is the oscillatory function whose Laplace transform is given in equation (N6-14).
for the on k modulation scheme and The fact that the range of off k value extracted using the exponential TCF is comparable to the reference value, while the range of off k value extract using the oscillatory TCF is not, is consistent with our analysis shown in Fig. 2, according to which the TCF of transcription rate κ was found to be an exponential rather than the oscillatory function, for the slowly cells The deviation of the experimental data from the global trend curve obtained assuming the on k modulation mechanism in Supplementary Figure 3 indicates that this mechanism is not the only transcriptional control mechanism in E. coli. There exist other transcriptional control mechanisms in E. coli 6,47 , and our analyses do not exclude them from the control mechanisms of E. coli's transcription. Nevertheless, our analysis clearly shows that the off k modulation mechanism is not the universal transcription-control mechanism of E. coli as suspected in refs.
The quantitative information extracted from the analysis of experimental data shown in Supplementary Figure 3 is presented in Supplementary Table 3.

SUPPLEMENTARY FIGURES
with 1 q = saturates to a plateau as x approaches unity, whereas for the on k modulation scheme, ( )(1 ) q x x − ∆ − with 2 q = saturates to a plateau in the high induction limit. As shown in (c), which suggests validity of the off k modulation scheme for the experimental system. (solid line) given by equation (N6-13).