Supporting Information: A quantitative comparison of srna-based and protein-based gene regulation

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1 Supporting Information: A quantitative comparison of srna-based and protein-based gene regulation Pankaj Mehta a,b, Sidhartha Goyal b, and Ned S. Wingreen a a Department of Molecular Biology, Princeton University, Princeton, NJ 8544, and b Department of Physics, Princeton University, Princeton, NJ 8544 I. MEAN STEADY-STATE BEHAVIOR In this section, we address the mean steady-state behavior of our kinetic models of gene regulation. We are concerned with two types of gene regulation, transcriptional regulation and post-transcriptional regulation by small non-coding RNAs (srnas). A. Transcriptional regulation First, we very briefly review the noise properties of transcriptional regulation. This topic has been extensively considered in the literature 5. The kinetics of ordinary transcription and translation in a cell are captured by the simple model 4 dm = α m τm m dp = α pm τp p (SI-1) with m the average number of mrna molecules, p the average number of proteins, α m the average rate of transcription, α p the average rate of translation, and τm and τp the first-order degradation rates of mrna molecules and proteins, respectively. Transcriptional regulators of genes such as activators (repressors) work by binding to the DNA in the vicinity of a gene and tuning the transcription rate α m by aiding (hindering) the recruitment of RNA polymerase. The steady-state protein number, p is obtained by setting the derivatives on the left hand side of (SI-1) equal to zero: p = α m τ p α p τ m. (SI-) Notice that the steady-state protein number depends linearly on the transcription rate α m. B. srnas as negative regulators srnas regulate genes by pairing with target mrnas via the RNA chaperone Hfq (see main text and Fig. ). Pairing results in rapid degradation of both the mrna and srna molecules. This stoichiometric regulatory process can be described quantitatively by the simple kinetic model 6,8 ds dm dp = α s τs s µms = α m τ m m µms = α p m τ p p (SI-3) with s the average number of srna molecules, m the average number of target mrna molecules, and p the average number of proteins; α s (α m ) is the average rate of transcription of srnas (mrnas); α p is the average rate of translation of the target mrna; τs, τm, and τp are the average rates of first-order degradation (and dilution) of the RNAs and protein, and µ is rate at which srna-mrna pairs form and degrade stoichiometrically. For future reference, we define a renormalized lifetime for both the srnas and the mrnas via τ sr τ mr = τ s + µm = τ m + µs (SI-4)

2 Notice that unlike the bare RNA degradation rates, τ sr and τmr generally depend on concentrations of the various RNAs. In steady state, proteins serve as a read-out for the mrna number since the average steady-state protein number, p, is directly proportional to the average mrna number, m, through the relation p = α p τ p m. Thus,itissufficientto study steady-state mrna levels. We solve for the equilibrium concentrations of the two RNA species by setting the time derivatives in (SI-3) to zero and obtain 7,8 s = (α s α m λ)+ (α s α m λ) +4α s λ τs m = (α m α s λ)+ (α m α s λ) +4α m λ τm λ = τ m τs µ (SI-5) The expressions have three qualitatively distinct behavioral regimes: repressed, α s α m λ, expressingα m α s λ, and crossover α m α s λ (see Fig. of the main text). Since srnas and mrnas bind stoichiometrically and degrade, the steady-state protein concentration depends only on the difference between the RNA transcription rates α m α s. Increasing µ increases the sharpness of the crossover between the repressed and expressing regimes. C. srnas as positive regulators srnas can positively regulate the expression of a protein by promoting ribosome binding to the mrna 1. For example, the small non-coding RNA DsrA prevents the formation of an inhibitory mrna secondary structure that occludes the ribosome binding site of the regulated mrna. We model positive regulation by an srna by assuming that mrnas can only be translated when they bind an srna and form an srna-mrna complex. This is described by the simple kinetic equations ds dm dc dp = α s τs s µms = α m τ m m µms = µms τc c = α p c τ p p, (SI-6) with s the average number of srna molecules, m the average number of target mrna molecules, c the average number of srna-mrna complexes, p the average number of proteins; α s (α m ) is the average rate of transcription of srnas (mrnas); α p is the average rate of translation of the target mrna; τs, τm, τc, and τp are the average rates of first-order degradation of the RNAs, complexes, and proteins, and µ is the binding rate between srnas and mrnas. We solve for the equilibrium concentration of protein by setting the time-derivatives in (SI-6) equal to zero. Since the the first two equations in (SI-3) and (SI-6) are identical, the steady-state number of srnas, s, and mrnas, m, are again given by (SI-5). However, the average steady-state protein number, p, is now related to s and m by the expression p = α p τ p τ c µ m s. (SI-7) II. NOISE: LINEAR-NOISE APPROXIMATION In this section, we derive analytical formulas for the steady-state noise in gene expression within the linear-noise approximation 9. The linear-noise approximation assumes that the noise in each physical process described in the differential equations (SI-3) is Poissonian, including: srna, mrna, and protein birth and death noise and srnamrna binding noise. The linear-noise approximation is exact for systems described by linear equations and is a good approximation for nonlinear systems with small fluctuations. Due to the threshold-linear behavior of srna regulation, one expects that the linear-noise approximation is well-suited to evaluate the noise in gene regulation via

3 srnas. This is indeed the case as we have confirmed using the exact Gillespie algorithm to simulate the full stochastic equations. We start by introducing the linear-noise approximation and use it to derive the noise characteristics of gene regulation by a repressor. Next, we use the linear-noise approximation to derive an analytic expression for the noise in srna-mediated regulation. Finally, we analyze our expressions in the three relevant regimes: repressed, expressing, and crossover. 3 A. Transcriptional regulation (repressor) In this section we consider transcriptional regulation by a repressor. This topic has been studied extensively 4,5. We review it here to establish the notation and to demonstrate the linear-noise approximation on a well-studied system. Three noise sources are considered: binomial noise of repressor binding-unbinding, ˆη g, Poisson noise in mrna synthesis and degradation, ˆη m, and Poisson noise in protein synthesis and degradation ˆη p. As discussed in the main text, to model transcriptional bursting we assume the gene is in a transcriptionally active on state with probability g on. The kinetics of this model are described by the linear equations dg on = k (1 g on ) k + g on +ˆη g dm = α m τm m +ˆη m dp = α p m τp p +ˆη p. (SI-8) Here m and p are the number of mrna and protein molecules respectively, k + is the (repressor-concentration dependent) rate at which a repressor binds to the gene, k is the rate at which the repressor unbinds from the gene, and α m = αm on g on is the average transcription rate. In the absence of noise, the steady-state solution 13 of the system of Eqs. (SI-8) is k ḡ on = k + k + m = αm on ḡ on τ m p = α p τ p m. Since by assumption the mrna and protein synthesis have Poissonian noise, one has for j = m, p ˆη j (t)ˆη j (t ) =(α j + τ j j)δ(t t )=τ jδ(t t ) The noise due to repressor binding-unbinding is calculated using the fluctuation-dissipation theorem 1 ˆη g (t)ˆη g (t ) =ḡ on (1 ḡ on )(k + + k )δ(t t )=k + g on δ(t t ) j (SI-9) (SI-1) (SI-11) To calculate the noise around the mean steady state, we linearize the system around the mean and include the Langevin terms. Now, since the system is linear, it is helpful to solve the equations in Fourier space. The Fourier transform of the deviation δx(t) =x(t) x from the mean x is given by δ x(ω) = x(t)e iωt. Theresultis δ p(ω) = ˆη p iω + τ p α p +ˆη m (iω + τp )(iω + τm ) +ˆη α p αm g (iω + τp )(iω + τm )(iω + τg ) (SI-1) where τ g =(k + + k ) is the correlation time. (Note that we use the symbol ˆη j for noise both in the time domain and in the frequency domain, but it is always clear from the variables of the equations which domain pertains.) Since all the noise terms are of the form ˆη a (ω)ˆη a (ω ) = σaδ(ω ω ), we can readily solve for the variance in the protein number around the steady-state value, (δp) = p + m(α p τ p ) τ m τ + k (1 ḡ on )(α p α τ p + τ m) g τmτ p (τ g τ m + τ m τ p + τ p τ g ). (SI-13) m (τ g + τ m )(τ m + τ p )(τ p + τ g ) Using p max = α m τ m α p τ p and ḡ on = p/p max we can further simplify the above expression to (δp) τ m τ = p + p(α p τ p ) + p max k (p max g (τ g τ m + τ m τ p + τ p τ g ) p). (SI-14) τ p + τ m (τ g + τ m )(τ m + τ p )(τ p + τ g )

4 Using τ g = k (p max / p) we can also write the equation as, (δp) τ m = p + p(α p τ p ) +(p max p) p τ p + τ m If we additionally assume τ m τ p, we obtain where b = α p τ m is the effective burst size. p(1 + τ m /τ p )+k τ m p max (1 + τ m /τ p )( p + k τ m p max )( p + k τ p p max ) (δp) = p(1 + b)+ ( pmax p) p p + k τ p p max 4. (SI-15) (SI-16) B. Post-transcriptional regulation 1. Without transcriptional bursting In this section, we consider post-transcriptional regulation by srnas in the absence of transcriptional bursting. We discuss transcriptional bursting in the next section. The steady-state noise is calculated using the linear-noise approximation. Protein noise is related to mrna noise in a simple manner as discussed below 11. For this reason, it is useful to separate the calculation into two parts. First, we calculate the noise in mrna levels and then we calculate the protein noise. We start by linearizing the steady-state equations (SI-3) for the RNAs around their steady-state values (Eqs. SI-5). The effects of noise are modeled by adding Langevin noise terms for each of the physical processes involved (see main text): ˆη s models srna birth and death noise, ˆη m models mrna birth and death noise, and ˆη µ models srnamrna mutual-degradation noise. Linearizing the equations (SI-3) around the average number of molecules in the steady-state, s and m yields, in the presence of Langevin noise, dδs dδm = τ sr µ s µ m τ mr δs δm ˆηs +ˆη + µ, ˆη m +ˆη µ with τ sr and τmr given in (SI-4) and δx = x(t) x denoting the deviation from the mean x for x = s, m, p. In our notation, the total number of molecules of species x = s, m, p is just x = x + δx Fourier transforming and solving for the RNA fluctuations yields δ s(ω) 1 iω + τ = δ m(ω) (iω + τ+ )(iω + τ mr µ s ˆηs +ˆη µ ) µ m iω + τ ˆη sr m +ˆη µ with τ ± =(τ sr + τ mr ) (τ sr τ mr ) +4µ m s Solving for the frequency-dependent fluctuations in the mrna number yields δ m(ω) = (iω + τ sr ) (iω + τ+ )(iω + τ ) (ˆη m +ˆη µ ) (SI-17) µ m (iω + τ + )(iω + τ ) (ˆη s +ˆη µ ) (SI-18). The noise in protein number due to protein birth and death is modeled by adding a Langevin term ˆη p to the third equation in (SI-3) (see main text). Fourier transforming this equation relates fluctuations in protein number to fluctuations in mrna number δ p(ω) = δ m(ω) (iω + τ p ) + ˆη p (iω + τ p ). (SI-19) Namely, the proteins act as a low-pass filter for the mrna noise with cut-off frequency τp, and the proteins add extra noise due to ˆη p. This yields for the frequency-dependent protein fluctuations δ p(ω) = (iω + τ p α p (iω + τ sr ) )(iω + τ+ )(iω + τ α p µ m ) (ˆη m +ˆη µ ) (iω + τp )(iω + τ+ )(iω + τ ) (ˆη s +ˆη µ )+ ˆη p (iω + τ p ). (SI-)

5 5 We calculate the noise in output-protein number using the expression (δp) = dω dp π δ p(ω), (SI-1) with the additional input that the noise terms are of the form ˆη a (ω)ˆη b (ω ) =πδ ab σaδ(ω ω )withσs = α s + τs s, σm = α m + τm m, and σµ = µ m s. The explicit expression for δp is quite cumbersome and not very illuminating so we do not give its explicit form here.. With transcriptional bursting We now consider the effect of transcriptional bursting. As discussed in the main text, to model transcriptional bursting we assume the average transcription rate of RNA species α j depends linearly on the probability, g on of the respective gene being on α j = αj on gj on (SI-) with the kinetics dg on j = k (1 gj on ) k + g on j +ˆη gj = k + τ g g on j +ˆη gj (SI-3) where τ g is the mean lifetime for a gene in the on state, τg = k + k +. We model the noise by adding a Langevin term ˆη g to (SI-3) with correlation given by (SI-11). We start by linearizing (SI-3) about the steady state and Fourier transforming to obtain For, convenience we define new renormalized Langevin noise terms δ g j on ˆη gj (ω) = iω + τg. (SI-4) j ˆη j R (ω) =ˆη j (ω)+αj on δĝj on (ω) (SI-5) with j = s, m. Rewritten with these new terms, which include transcriptional bursting, the linearized Fourier transformed equations for the srna and mrna numbers become δ s(ω) 1 iω + τ = δ m(ω) (iω + τ+ )(iω + τ mr µ s ˆη R s +ˆη µ ) µ m iω + τ sr ˆη m R. +ˆη µ Notice that (SI-6) is identical to (SI-17) except that ˆη j is replaced by ˆη j R. Thus, we can immediately conclude that the noise in protein number including transcriptional bursting is given by (SI-) and (SI-1) with the substitution ˆη j ˆη j R. Notice, that the noise due to trasncriptional bursting is amplified by srna-regulation through the second term in (SI-). This additional term leads to a strong dependence on transcriptional bursting of srna-based regulation. C. Asymptotic expressions In this section we compute asymptotic expressions for the noise in the repressed and expressing regimes using Eqs. (SI-) and (SI-1) in the case without transcriptional bursting. 1. Repressed regime In the repressed regime α s α m λ, there are many more srnas transcribed than mrnas, leading to a low average steady-state mrna number. The approximate numbers of RNAs are given by s (α s α m )τ s m α m τ m (SI-6)

6 6 where = λ (α s α m) 1 is a small parameter. In this limit the two time-constants in (SI-17) are given by, τ+ = τs Use of the approximate relations in Eqs. (SI-6) yields τ = µ(α s α m ) τ s m s α m µ p = α p τ p m = p max. (SI-7) (SI-8) (SI-9) Use of these expressions in Eq. (SI-1) yields for the protein noise (δp) p 1 p(1 + b p max )+ p3 + O( 4 ), p max λτ p where p max = α p τ p α m τ m. Practically, this asymptotic expression is valid for p.1p max. (SI-3). Expressing Regime In the expressing regime α m α s λ, there are many more mrnas transcribed than srnas, leading to a high average steady-state mrna level. By symmetry between the RNAs, the approximate numbers of the RNAs are s = α s τ s m = (α m α s )τ m (SI-31) where = λ (α m α s) 1 is a small parameter. In this limit the two time-constants in Eq. (SI-17) are: τ+ = τm Use of the approximate relations in Eqs. (SI-31) yields τ = µ(α m α s ) τ m m s α s µ Once again use of these expressions in Eq. (SI-1) yields p = α p τ p m = α pτ p µτ s 1. (SI-3) (SI-33) (SI-34) (δp) p(1 + b)+ 1 p p max α 3 pτ p µ + O( 3 ) (SI-35) III. SMALL-SIGNAL RESPONSE The response to a small time-varying input signal is evaluated by analyzing the linearized kinetic equations about some mean steady state or operating point. In the case of post-transcriptional regulation by srnas (Eqs. (SI- 3)), the mean steady-state level of output protein exhibits a threshold-linear behavior as a function of the mrna synthesis rate α m with the threshold at α s (see Fig. in main text). Due to the almost linear response in the repressed and expressing regimes, the linearized equations provide a good description for small input signals in these regimes. (Stronger nonlinearity near the threshold, i.e. in the crossover regime, limits the validity of the linearized equations to very small input signals, and in the µ case there is no valid linear description at the α s = α m threshold point.) Linear time invariant (LTI) system theory investigates the response of an LTI system to an arbitrary input signal. An LTI system is completely characterized by a function called the impulse response. Equivalently, for sinusoidal inputs the system is characterized by its frequency-dependent gain, which is the Fourier transform of the impulse response.

7 7 A. Frequency-dependent gain To calculate the frequency-dependent gain, we start by linearizing and Fourier transforming the kinetic equations (SI-3) around an operating point. The gain in general depends on the operating point and expresses the frequencydependent change in output protein δ p(ω) resulting from a small change in the input TF concentration δ c(ω) g(ω) = δ p(ω) δ c(ω) = δ gon (ω) δ c(ω) I δ m(ω) δ g on (ω) II δ p(ω) δ m(ω) III. (SI-36) The frequency-dependent gain is a combination of gains from different intermediate steps (see main text): I from gene on-off dynamics (SI-3) yields δ g on (ω) δ c(ω) = ḡ on dk + iω + τg dc (SI-37) where τ g =(k + + k ) is the correlation time, II from transcription depends on the mode of regulation (see below), and III from translation yields δ p(ω) δ m(ω) = α p iω + τ p Gain from transcription depends on the mode of regulation. TF-based regulation yields δ m(ω) δ g on (ω) = αon m iω + τ m. (SI-38). (SI-39) For srna-based regulation, the input protein regulator can change the synthesis rates of either RNA. The linear equations for the case in which the protein regulator controls srna synthesis are iω + τ sr µ s δ s(ω) α on µ m iω + τ = s δ g on. (SI-4) δ m(ω) mr These can be readily solved to yield The case of the protein regulator controlling mrna synthesis yields δ m(ω) δ g on (ω) = µ mα on s (iω + τ+ )(iω + τ ). (SI-41) δ m(ω) δ g on (ω) = (iω + τ sr )αon m (iω + τ + )(iω + τ ). (SI-4) B. Minimum input signals and logarithmic gains This section briefly reviews the relationship between gain, logarithmic gain, and the minimum signal defined above. Consider a signaling system that takes as its input some concentration c and outputs a protein level p = f(c). Then, as discussed above, we can define the linear gain at low frequencies, g = g(ω ), as g = dp dc. Alternatively, one can define the low-frequency logarithmic gain by the relation g log = d log p d log c. The two gains are related by the relation g log = c p g. (SI-43) and, in general, the magnitudes of these gains will differ. In contrast, the minimum signal is invariant under changes of convention. The relationship defining the minimum signal (see above) for the linear and logarithmic conventions of the signal are given by (respectively) σ (δlog c ) min log p = g log σp = δc min = g (SI-44)

8 where we have defined σ log p as the output noise in log p. Since σ log p is related to σ p, the output noise in p, bythe relation 8 σ log p = 1 p σ p. (SI-45) we conclude that the minimum signal, δc, does not depend on how the input and output signals are defined. IV. LARGE SIGNAL RESPONSE: COMPARISON WITH EARLIER WORK Large-signal response via srnas has also been modeled recently by (Shimoni et al, 7). They found that regulation by srnas is faster than regulation by proteins in all cases, which differs from our results. The reason for this discrepancy is a difference in how the comparisons are made. In their model of protein-based gene regulation, input signals couple to the system by regulating the transcription of the protein regulator (repressor). The protein regulator needs to be transcribed/translated to turn genes on and needs to be degraded to turn genes off. On the other hand, in their model of srna-based regulation, the external signal directly modulates the transcription of srnas, i.e. their model does not include the TF necessary for coupling the input signal to the srnas. The model of Shimoni et al effectively introduces an additional layer of transcriptional regulation to the genetic network in Fig. 3. Consequently, they find srnas to be faster than protein regulators. In contrast, in our model we compare proteinbased and srna-based regulation on the same footing, with signal coupling to a TF that regulates transcription of either an mrna or an srna. V. DEPENDENCE ON CHOICES OF PARAMETERS As described above, for principled comparison between the two regulatory schemes the upstream components are assumed to have identical properties. In particular, the components that control the level of the upstream protein regulator and the kinetics of this regulator are assumed to be identical. Within this assumption of identical upstream components, the three main conclusions made in the paper regarding comparison between srnas and TFs do not depend on the choice of parameters. Protein expression can be kept off more reliably with srnas. This conclusion is a result of both the low noise for srna regulation at low output-protein levels (repressed regime) and the large range of inputs (α s α m λ) for which the level of output protein is low. The reason for low noise is the reduced lifetime of mrnas when regulated by srnas. The range of output-protein levels for which this is true could change when RNA bursting is included (Fig. 4 of main text), but the repressed regime would still exist for large range of inputs (α s ). Thus, this conclusion is robust with respect to parameters. srnas can more quickly turn off negatively regulated genes and more quickly turn on positively regulated genes. The rate of switching from a high level to a low levels of mrna is limited by the mrna degradation time. For direct TF-based regulation the output protein is switched off by stopping further synthesis of mrna. Consequently, the large pool of mrna degrades at a fixed rate τm. However, for srna-based regulation the output protein is switched off by increasing the srna rate of synthesis. These additional srnas increase the rate of degradation of mrnas resulting in a faster decrease in the mrna pool in comparison to direct TF regulation. Thus, while the switching rate is parameter dependent, the switching rate for srna-based regulation can be tuned to be faster than for TF-based regulation. Even for moderate levels of transcriptional bursting, srna-based regulatory circuits are worse at transducing small input signals. The minimum detectable signal for a signal-processing system depends both on the noise and the gain in the system (see main text). For srna-based regulation, due to near-critical fluctuations, the intrinsic noise overwhelms the gain even for moderate levels of transcriptional busting (see Fig. 4 in main text). This claim is robust with respect to parameters as it only relies on the presence of a near-critical regime. VI. STOCHASTIC SIMULATIONS Stochastic simulations were implemented using a Gillespie algorithm 1 for the coupled srna-mrna system with protein level determined by the mrna level (Eq. 1 of main text). We did not include RNA bursting. The results from stochastic simulations agree well with the linear-noise-approximation (LNA) results, however, the peak in the stochastic simulation is a shifted to lower values of α m (Fig. 3(a)). The shift in the stochastic simulations, however, depends on the system size (volume). At small volumes, the behavior of the mean is significantly larger than the

9 mean-field calculation (see Fig. 3(c)) in the crossover regime. This early increase in mean is due to the long tails in the distributions (see Fig. SI-) and also leads to near-critical fluctuations. As shown in Fig. 3(c), the transition from repressed to expressing regime starts early at α m.5, where the peak in noise is also seen in the stochastic simulations. As expected, the peak in noise from stochastic simulations moves closer to the peak from LNA as we increase the volume of the system 3(b). We also plot the output protein distributions for typical points in the three regimes (shown in Fig. SI-). For the repressed and crossover regimes, as expected due to the low means, we obtain skewed distributions that are not well described correctly by Gaussians. In the crossover regime, the standard deviation (relative to mean) is large due to the long tail in the distribution of output protein. The presence of long tail also shifts the mean, given by p/ p = 1, significantly to the right of the peak of the distribution. 9 1 Storz, G., Opdyke, J., and Zhang. A. (4) Curr. Opinon Microbiology Elowitz M. B., Levine A. J., Siggia E. D. and Swain P.S. () Science Thattai M. and van Oudenaarden A. (1)Proc Natl Acad Sci U S A Paulsson, J. (4) Nature Paulsson J. (5) Phys. Life Rev Lenz D. H, Mok K.C., Lilley B. N., Kulkarni R. V., Wingreen N. S., and Bassler B. L. (4) Cell Elf J., Paulsson J., Berg O.G. and Ehrenberg M. (3)Biophys. J Levine E., Kuhlman T., Zhang Z., and Hwa T. et al. Preprint. 9 van Kampen, N.G. (1) Stochastic Processes in Physics and Chemistry (North Holland) 1 Bialek, W. and Setayeshgar, S. (5) Proc Natl Acad Sci U S A Detwiler PB, Ramanathan S, Sengupta A and Shraiman BI. () Biophys J Gillespie, D. T. (1977) J. Phys. Chem. 81, The steady-state solutions may not be the true means if the equations are non-linear and there is noise.

10 μ=. μ=. μ=.5 σ p p α s α m FIG. SI-1: Protein noise depends on srna-mrna interaction strength µ. Noise in protein expression σp/ p (variance divided by mean squared) as a function of the ratio of the srna and mrna transcription rates, α s/α m, for different values of the srna-mrna binding interaction µ, in the limit where transcriptional bursting is negligible. The noise is peaked in the crossover regime, α s α m. A larger µ results in stronger binding between mrnas and srnas. Parameters as in Fig. 3 of the main text: (in min ): α m =3,αm on =1,αs on =3,τ m =1,τ s =3,µ =., α p =4,τ p =3.

11 αm = 1.8 σp p =.4 p = αm =.5 σp p =.61 p = 8.9 P(p).5.4 P(p) p p (a)repressed regime (b)crossover regime αm = 5. σp p =.3 p = 47 P(p) p (c)expressing regime FIG. SI-: Results of stochastic simulations. Probability distributions of output protein as functions of protein number relative to its mean p/ p, for the three different regimes. Note that the all three plots are for the same range of relative protein level. The relative standard deviation σ p/ p of the distribution in the crossover regime is the largest due to its long tail. Parameters as in Fig. SI-1 and Fig. 3 of the main text.

12 σ p p 1.5 σ p p α m α m (a)v =1 (b)v = p α m (c)v =1 FIG. SI-3: Comparison of the linear-noise approximation with exact stochastic simulations. Noise in protein expression σ p/ p as a function of the ratio of the mrna transcription rates, α m, for different system sizes (volumes). The peak in the stochastic simulations is shifted to lower values of α m. (a) Parameters as Fig. SI-1 and Fig. 3 of the main text with µ =. (b) The volume is 1 times that in (a). (c) Mean as function of α m for parameters as (a).