Attenuation of transcriptional bursting in mRNA transport
aa r X i v : . [ q - b i o . S C ] J u l Attenuation of transcriptional bursting in mRNAtransport
Li-ping Xiong , Yu-qiang Ma and Lei-han Tang National Laboratory of Solid State Microstructures, Nanjing University, Nanjing210093, China Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong KongE-mail: [email protected]
Abstract.
Due to the stochastic nature of biochemical processes, the copy number ofany given type of molecule inside a living cell often exhibits large temporal fluctuations.Here, we develop analytic methods to investigate how the noise arising from a burstinginput is reshaped by a transport reaction which is either linear or of the Michaelis-Menten type. A slow transport rate smoothes out fluctuations at the output end andminimizes the impact of bursting on the downstream cellular activities. In the contextof gene expression in eukaryotic cells, our results indicate that transcriptional burstingcan be substantially attenuated by the transport of mRNA from nucleus to cytoplasm.Saturation of the transport mediators or nuclear pores contributes further to the noisereduction. We suggest that the mRNA transport should be taken into account in theinterpretation of relevant experimental data on transcriptional bursting. ttenuation of transcriptional bursting in mRNA transport
1. Introduction
Molecular binding and chemical modifications underlying intracellular processes areintrinsically stochastic. They give rise to temporal fluctuations and cell-to-cell variationsin the number of molecules of any given type, mask genuine signals and responses,and generally contribute to the phenotypic diversity in a population of geneticallyidentical individuals[1, 2]. Various characteristics of such noise have been under intensequantitative study in the past few years [3, 4]. One of the focal points of the discussionis how the noise propagates along a biological pathway. It has been shown that incases where the dynamics of the upstream molecules is not affected by the downstreamprocesses (e.g., in gene transcription and translation), a “noise addition rule” generallyapplies, i.e., each process in the pathway contributes to the overall noise strengthin a statistically independent fashion[5, 6]. Modifications to this rule in molecularcircuits with feedbacks or “detection” capabilities as in signalling have been examinedby T˘anase-Nicola and his colleagues[7].In metabolic pathways and other transport processes, however, the upstreammolecule is passed on to the downstream pool in a modified form. The conservation ofmass and/or number of molecules in the reaction sets new rules on noise propagation.In a recent work, Levine and Hwa considered this problem in the steady state of ametabolic network[8]. Their results show that fluctuations in the number of intermediatemetabolites are generally uncorrelated to each other, upstream or downstream along apathway or across branches. This, of course, does not exclude dynamic correlationswhich can be quite nontrivial in driven processes[9]. The full dynamic descriptionof stochastic transport through a network is a very challenging task which has notbeen fully solved even for a linear pathway[10]. A case of interest in the presentcontext is busty input, where the molecules to be transported are produced in batchesseparated by long silent intervals. Well-known examples of such behavior includethe transcriptional and translational bursting[1, 11] and vesicular transport[12]. Ineukaryotic gene expression, a large number of mRNAs are produced over a short periodof time and followed by a long silent period as a result of the chromatin remodeling. Inprokaryotes, burst of protein copy number can occur as the result of short lifetime of anmRNA transcript. Nutrient uptake in endocytosis can also be viewed as a burst event:each endocytic vesicle transports and releases a large number of extracellular moleculesto the target site.In this paper, we examine the attenuation of bursting noise in a two-compartmentmodel with a stochastic transport channel. The model can be viewed as an improvementover the one-compartment model used previously to analyze the mRNA copy numberfluctuations in single-molecule experiments carried out by Raj et al. [13] on mammaliancell gene expression. The experiments show that the number of mRNA transcriptsproduced in a single burst event ranges from a few tens to hundreds. It was arguedthat such large bursts, if unattenuated, could harm the progression of normal cellularactivity[14] due to their large perturbations to the cytoplasmic mRNA and protein levels. ttenuation of transcriptional bursting in mRNA transport
3A suggestion was made by Raj et al. that the latter could be avoided if the overallprotein copy number is kept high by a low protein degradation rate. Our analysis herepoints to a second possibility: a slow nuclear mRNA processing and export processcan also attenuate mRNA bursting and minimize its impact on the downstream proteinpopulation.Supporting this view, an earlier kinetic study of mammalian cell mRNA splicing andnuclear transport has shown that the nuclear dwelling (or retention) time of an mRNAmolecule can be comparable to its lifetime in the cytoplasm [15]. This is evidenced inthe time required to reach the respective steady state levels for the mRNAs residing inthe nucleus and in the cytoplasm, which were measured separately in the experiment.Consistent with this observation, both studies also concluded that, on average, about 10-40% of mRNA are retained in the nucleus. In this respect, the one-compartment modelof Raj et al. , which considers only the total mRNA copy number in a cell, overestimatesfluctuations in the actual number of mRNAs in the cytoplasm that participate in thetranslation process.To establish a reasonable model, let us first examine the typical fate of a singlemRNA: the mRNA is synthesized in bursts at the transcription site and almostsimultaneously processed into an mRNA-protein (mRNP) complex[16, 17]; the mRNPcomplex diffuses inside the nucleus[18, 19, 20], eventually reaches one of the nuclearpores and exits with the help of export mediators[21, 22]; the mRNA degrades in thecytoplasm. Therefore, our model includes three processes: the transcriptional burstingin the nucleus, the mRNA transport, and the mRNA decay in the cytoplasm. Weshall assume that the mRNA transport to the cytoplasm is much slower than thediffusion process, so that the spatial inhomogeneity of the mRNA molecules in thetwo compartments can be ignored[23].Both linear and nonlinear transport are considered. In linear transport, the exportmediators and nuclear pores are abundant as compared to the transported mRNA.Consequently, the export events are independent of each other. Nonlinear transportcorresponds to a situation where queuing of nuclear mRNA takes place due to a limitednumber of transport mediators or nuclear pores. Its mathematical description is identicalto that of the Michaelis-Menten (MM) reaction in enzyme kinetics.The stochastic bursting and transport processes can be described by the chemicalmaster equation governing the time-dependent distribution of mRNA copy numbers inthe two compartments. In the case of linear transport, exact expressions for the copynumber fluctuations in a steady-state situation are obtained. The MM transport inthe weak noise limit can be treated using the linear noise approximation (LNA)[24]. Anovel independent burst approximation (IBA) is introduced to treat the MM transportin the strong fluctuations regime. We also perform stochastic simulations to check theaccuracy of the analytic expressions.The main results of our calculation are summarized in Sec. 3. The noise strengthof cytoplasmic mRNA is expressed in terms of the average burst size and the ratioof the mean nuclear and cytoplasmic mRNA copy numbers, which are measurable in ttenuation of transcriptional bursting in mRNA transport et al. for the bursting and decay dynamics of mRNA in their experiments.
2. Methods
The transport event involves two compartments, usually with different volumes whichresult in different concentrations even when the number of molecules is the same. Tostep aside this problem and to be more clear, we measure the amount of mRNA in copynumber rather than concentration, and define all the reaction rates mesoscopically(byscaling the macroscopic counterparts with volumes). Such a treatment (we will notexplicitly refer to specific units for variables and parameters in the following calculations)also facilitates the application of chemical master equations. Then we can start safelyby introducing the methods for the simplest case: linear transport.
The burst of mRNA arises from the transitions between active gene A and inactivegene I , while each active gene produces an amount of nuclear mRNA M n . To faithfullydescribe the bursting, we write the relevant reactions, as often done in many previousworks [1, 13], as: I λ −→ A, A γ −→ I, A µ −→ A + M n . (1)Here λ and γ are the rates of gene activation and inactivation respectively, and µ is thetranscription rate. Using M c to denote the cytoplasmic mRNA, we define the lineartransport and cytoplasmic decay of mRNA as: M n k −→ M c , M c δ −→ Ø . (2)Here k is the transport rate, and δ is the degradation rate of cytoplasmic mRNA. Thedecay of mRNA in nucleus is ignored [15].At a given time t , the gene can be in either active or inactive state, with theprobabilities given by P A ( m n , m c , t ) and P I ( m n , m c , t ), respectively. The probabilitiesdepend on the copy number of mRNA in nucleus m n and in cytoplasm m c . The timeevolution of the two probabilities are governed by the chemical master equations: dP A ( m n , m c , t ) dt = λP I ( m n , m c , t ) − γP A ( m n , m c , t )+ µ ( ε − − P A ( m n , m c , t )+ k ( ε n ε − − m n P A ( m n , m c , t )+ δ ( ε c − m c P A ( m n , m c , t ) , (3) ttenuation of transcriptional bursting in mRNA transport dP I ( m n , m c , t ) dt = γP A ( m n , m c , t ) − λP I ( m n , m c , t )+ k ( ε n ε − − m n P I ( m n , m c , t )+ δ ( ε c − m c P I ( m n , m c , t ) , (4)where ε is the step operator defined by its effect on arbitrary functions of m n and m c : ε ± f ( m n , m c , t ) = f ( m n ± , m c , t ) and ε ± f ( m n , m c , t ) = f ( m n , m c ± , t ).The first moments or mean values of m n and m c can be obtained by multiplyingequations (3) and (4) by m n and m c in turn, and summing over all m n , m c and genestates: d h m n i dt = µp A − k h m n i , (5) d h m c i dt = k h m n i − δ h m c i . (6)Here h·i denotes average over the distribution, and p A ( t ) ≡ P m n ,m c P A ( m n , m c , t ) is theprobability that the gene is in the active state. The following equation is easily seenfrom the gene activation dynamics (1): dp A /dt = λ (1 − p A ) − γp A . (7)Equations (5)-(7) are equivalent to the macroscopic rate equations given by the mass-action law due to the linearity of the process. Setting the right-hand-side of theseequations to zero, we obtain the steady-state relations for the average mRNA flux: J = µp ∗ A = µλ/ ( λ + γ ) = k h m n i = δ h m c i . In the following discussion we will focus on the burst limit where µ and γ aresignificantly larger than all other reaction rates. The steady-state probability p ∗ A goes to zero but the mRNA synthesis rate J = µp ∗ A ≃ λ ( µ/γ ) remains finite. Geneactivation in this case follows a Poisson process at a rate λ , but the number of mRNAcopies b produced in each burst event is a random variable that satisfies the geometricdistribution: G ( b ) = ( µ/γ ) b (1+ µ/γ ) − b − [25]. In terms of the mean mRNA copy numberproduced in each burst, h b i = µ/γ , we have J = λ h b i = k h m n i = δ h m c i . (8)A similar procedure as above yields the second moments of mRNA copy numbers in thesteady state: h m i = h m n i + ( h b i + 1) h m n i , (9) h m n m c i = h m n ih m c i + h b i h m n ih m c ih m n i + h m c i , (10) h m i = h m c i + ( h b i + 1) h m c i − h b i h m n ih m c ih m n i + h m c i . (11)It is customary to measure temporal variations of population size in a stationaryprocess using the noise strength (also known as the Fano factor), defined as the varianceover average [5, 11]: σ m n h m n i = h b i + 1 , (12) ttenuation of transcriptional bursting in mRNA transport σ m c h m c i = h b i + 1 − h b i h m n ih m n i + h m c i . (13)For the Poissonian fluctuation arising from the simplest case where molecules areproduced one by one with a constant probability and degraded linearly, the noisestrength is unity[26, 27]. When the synthesis is burst-like and the degradation is stilllinear, the noise strength becomes h b i + 1, which is much larger than the Poissonianfluctuation [28]. This is the case for the nuclear mRNA population [equation (12)],and for the mRNA without transport, as in the prokaryotic cells. Equation (13), on theother hand, shows that although the transport event follows a random process, the noisestrength of the cytoplasmic mRNA that propagates directly to protein noise, is actuallyreduced. The amount of reduction is controlled by the ratio h m n i / ( h m n i + h m c i ), whichincreases with decreasing transport rate k . Accumulation of the mRNAs in the nucleus may lead to a saturation effect that changesthe transport dynamics when the number of export mediators or nuclear pores becomeslimiting. This prompts us to study a more general mechanism of transport that takesthe transport capacity into account. The resulting process can be cast in the form ofthe well-known Michaelis-Menten model.To simplify the discussion, we consider here only one source of constraint, say thelimited number of one kind of export mediator denoted by E , and treat the rest of theexport mediators (including the nuclear pores) in the process to be non-rate-limiting.Thus, the transport process of mRNA can be described as: M n + E k ⇋ k EM n k −→ E + M c , (14)where k and k are the binding and unbinding rates respectively, and k is the exportrate. This is similar to the MM model for enzymatic reaction.The analysis presented below is based on the “fast equilibration” approximation,in which EM n is treated as a transition state rather than an accumulation point inthe mRNA export. For this to be true, the lifetime of the complex EM n should besignificantly shorter than the total nuclear dwelling time, i.e., either the Michaelisconstant K m = ( k + k ) /k is much greater than one, or k is much smaller than the othertwo rates. Under this assumption, the EM n population remains in quasiequilibrium withthe nuclear population m n which varies on a much slower time scale as compared to thedecomposition time of the complex EM n .Due to the small copy number of the nuclear mRNA and the transporter E , wedistinguish the free M n from the bound ones, and define the total number of nuclearmRNA as m n = m nf + c , where m nf is the number of free M n and c is the number of thecomplexes. The total number of E , including both free and bound ones, is denoted by e t . The usual rate equation approach for the complex yields k m nf ( e t − c ) − ( k + k ) c = 0 ttenuation of transcriptional bursting in mRNA transport c = e t m nf K m + m nf . (15)On the other hand, through an exact analysis, Levine and Hwa[8] obtained a modifiedexpression in the limit k → c = e t m n K + m n , (16)where K = K m + e t . The two expressions converge to the exact result in both the linearregime m n ≪ K and the saturated regime m n ≫ K . They differ only in the crossoverregime m n ≃ K , where no exact result is available in the general case, though either ofthe two can be used as approximate expressions.Using equation (16), we write the transport flux as, v ( m n ) = k c = v max m n K + m n , (17)where v max = k e t . Following the fast equilibration assumption, we may now describethe mRNA export under the MM kinetics using the reduced description (2) with aneffective transport coefficient k = v max / ( K + m n ) that decreases with increasing m n .Again, taking the burst limit for the mRNA production, we arrive at the followingmaster equation for the joint distribution of m n and m c : dP ( m n , m c , t ) dt = m n X b =0 λG ( b ) P ( m n − b, m c , t ) − ∞ X b =0 λG ( b ) P ( m n , m c , t )+ ( ε n ε − − v ( m n ) P ( m n , m c , t )+ δ ( ε c − m c P ( m n , m c , t ) . (18)Since the nonlinear function v ( m n ) in the MM transport does not allow for the closureof equations for the second moments of the distribution, approximate treatment of themaster equation is necessary.The two limiting situations h m n i ≫ h b i and h m n i ≪ h b i call for differentconsiderations. In the former case, the contribution of each burst event on the totalnuclear mRNA population is small, so that perturbative treatment around the average h m n i is appropriate. The latter case, however, corresponds to the situation where thenuclear mRNA from each burst event is essentially cleared before the next one arrives.The two cases are treated separately below. We first consider the weak fluctuation case h m n i ≫ h b i .A general scheme to perform the noise calculations is the van Kampen’s Ω-expansionwhose lowest order terms reproduce the macroscopic rate equations and the next orderterms yield a linear Fokker-Planck equation (FPE), which is often called the linear noiseapproximation[24]. In Appendix A we derive the noise strengths under the LNA. Here,we outline a more direct yet equivalent way to obtain the results. ttenuation of transcriptional bursting in mRNA transport m n = h m n i : v ≃ k eff ( m n + m ) . (19)Here k eff = v max K/ ( K + h m n i ) and m = h m n i /K . Substituting (19) into (18), wemay compute moments of the distribution P ( m n , m c , t ) in the same way as in Sec.2.1. In fact, equations (8)-(11) remain valid if we make the substitution k → k eff and m n → m n + m . After rearranging the terms, we obtain: σ m n h m n i = ( h m n i K + 1)( h b i + 1) , (20) σ m c h m c i = h b i + 1 − h b i h m n i K h m c i K + h m n i + h m n i . (21) In the case h m n i ≪ h b i , the mRNA producedin a given burst has sufficient time to exit the nucleus before the next burst arrives. Itis then appropriate to consider the independent burst approximation, where individualburst events contribute additively to m n ( t ) and m c ( t ): m n ( t ) = X t i 0) = b and y ( b, 0) = 0.As shown in Appendix B, the sum and integrals in equations (24) and (25) can beworked out exactly to give: h m n i = λv max h b i (cid:16) h b i + K + 12 (cid:17) , (31) σ m n ≡ h m i − h m n i = λv max h b i h h b i + ( K + 2) h b i + K i . (32)Hence, σ m n h m n i = h b i + 12 + h b i + h b i + K + + h b i . (33)The mean value of m c can be obtained from flux balance, i.e., λ h b i = δ h m c i or h m c i = ( λ/δ ) h b i . The calculation of h m i is a bit more involved which we relegate toAppendix B. Assuming δ ≪ v max , i.e., the decay time of an mRNA molecule is muchlonger than the fastest release time of one mRNA to the cytoplasm, we may write theresult in the form: σ m c h m c i = (cid:16) h b i + 12 (cid:17) Ψ( u, w ) , (34)where u = h b i δ/v max and w = Kδ/v max . The function Ψ is given byΨ( u, w ) = 2 Z dx Z x dx e w ln( x/x ) [1 + u ( x − x )] . (35)Since the integrand is less than 1, we have Ψ( u, w ) ≤ u, w ), but the integralcan be worked out in the two limiting cases: i) u = 0 , Ψ(0 , w ) = 1 / (1 + w ); and ii) w = 0 , Ψ( u, 0) = 1 / (1 + u ). These two expressions also set upper bounds for Ψ( u, w )in general. An approximate expression that is consistent with the two limits and alsoverified by numerical integration of (35) is given by:Ψ( u, w ) ≃ 11 + u + w . (36) ttenuation of transcriptional bursting in mRNA transport h m n i and h m c i and with the help of (31) and the flux balance condition,equation (34) can be rewritten as: σ m c h m c i ≃ h b i + h m n ih m c i K + h b i K + h b i + . (37) 3. Results and discussion Let us first summarize the analytical results derived in Section 2 when the average burstsize h b i ≫ 1. In general, noise strength of mRNA copy number in the two compartmentscan be expressed in the form, σ m n h m n i = α h b i + 1 , (38) σ m c h m c i = h b i β h m n i / h m c i + 1 . (39)For the linear model, we have α = β = 1. In the MM case where the transporter maybecome the bottleneck in the process, α = β = 1 + h m n i /K if the slow transport leads tothe nuclear accumulation of mRNA, i.e., h m n i ≫ h b i . In the opposite limit h m n i ≪ h b i ,where there is nuclear clearance between successive burst events, α ≃ h b i / ( K + h b i )and β ≃ h m n i to h m c i , further reduction of the noise is possible in the nonlinearMM transport when h m n i is greater than both the dissociation constant K and burstsize h b i . On the other hand, the independent burst approximation yields a β valueclose to one, extending the validity of the linear model when the noise strength of m c isconsidered as a function of the ratio of mean copy numbers h m n i / h m c i .Fluctuations in the nuclear mRNA level, on the other hand, exhibits a somewhatdifferent behavior. The parameter α that characterizes the noise strength of m n reachesthe minimum value 1 in the linear model, but increases when the saturation effect inthe MM transport kicks in. Thus queuing results in an enhanced fluctuation upstreamof the transport channel.To check the accuracy of the analytic results under parameter values that broadlycorrespond to the mammalian cell gene expression experiments mentioned above, wehave carried out simulations of the stochastic MM transport defined by (14), followingthe Gillespie’s exact algorithm[29]. The unit of time is chosen such that the mRNAdecay rate δ = 1. The number of transport channels is set to e t = 10. We fix theaverage mRNA production rate λ h b i relative to the mRNA decay rate δ such that themean cytoplasmic mRNA copy number h m c i = λ h b i /δ = 40. For easy comparisonwith the experimental measurements and analytic results, we also fix the mean nuclearmRNA copy number h m n i = 20. With these parameter values, the noise strengths in ttenuation of transcriptional bursting in mRNA transport 〈 b 〉 =4 N o i s e s t r eng t h o f m n K m N o i s e s t r eng t h o f m c 〈 b 〉 =20 K m 〈 b 〉 =80 K m (f)(a) (b) (c)(e)(d) Figure 1. Noise strengths (normalized by their corresponding values in the linearmodel) of the nuclear (upper panel) and the cytoplasmic (lower panel) mRNA copynumbers against the Michaelis constant K m which controls saturation of transportchannels. Here e t = 10, h m n i = 20 and h m c i = 40. Results are shown for stochasticsimulation (open circles), LNA (solid line) and IBA (dashed line). The overall noisestrength is set by the mean burst size h b i which is chosen to be (in units of mRNAcopy number): 4 [(a) and (d)]; 20 [(b) and (e)]; 80 [(c) and (f)]. See text for the choiceof other model parameters. the linear model are given by σ m n / h m n i = h b i + 1 and σ m c / h m c i = h b i + 1, respectively.Simulations are then performed to examine the effect of channel saturation on the noisestrengths by varying the MM parameters k and k in such a way that the mean nuclearmRNA copy number stays at the value set above. The unbinding rate of the mRNA-transporter complex EM n is fixed at a low value k = 0 . K m = ( k + k ) /k . Fluctuations in m n grow as the saturation effect becomes moreprominent on the low K m side. An opposite trend is seen in the fluctuations of thecytoplasmic mRNA copy number m c shown in plots on the lower panel. As expected,the LNA results (solid line) agree well with the simulation data (circles) in the weakburst regime [(a)and (d)], in which case the overall noise strength (as compared to themean mRNA copy number) is weak. On the other hand, the IBA results (dashed line)represent a better approximation in the strong burst regime [(c)and (f)][30]. Thereforeeach of the two approximations perform reasonably well in their respective regimes ofvalidity, and are complementary to each other.Figure 2 shows the actual distribution [(a) and (c)] and sample time course [(b) and(d)] of the nuclear and cytoplasmic mRNA copy number, respectively, generated from ttenuation of transcriptional bursting in mRNA transport m n F r equen cy m c F r equen cy Time m n Time m c (d)(b)(c)(a) Figure 2. Distributions of mRNA copy numbers in the nucleus (a) and in thecytoplasm (c), obtained from 500 000 repeated runs of the Gillespie exact simulation.Sample time courses of m n and m c in a single simulation are shown in (b) and (d).Parameter values are: λ = 2, h b i = 20, k = 1 . k = 0 . k = 10, e t = 10 and δ = 1. simulations at K m = 6 . 5. Other model parameters are the same as that of figure 1 (b)and (e), which represents a borderline case for the two approximate treatments. It isseen from figure 2 (b) that m n falls below K = 16 . m n = h m n i is too crude. On the other hand,there are occasional overlaps of the mRNA produced in successive bursts. This has theeffect of slowing down the mRNA transport than what is assumed in the IBA, leadingto a somewhat lower m c noise as seen in the left part of figure 1 (e).We have also examined the validity of equation (17) for the mean transport flux. Ata given number c of complexes EM n , the expected mRNA export flux is k c . Therefore v ( m n ) can be obtained in the simulations from the conditional average of c at a given m n . As shown in figure 3, equation (17) (dashed line) fits the simulation data (dots)very well when K m ≥ h m n i [(c)], but quite poorly in the regime K m ≪ h m n i [(a)].Surprisingly, the classic MM equation (15), suitably modified to be considered as afunction of m n , agrees with the numerical result extremely well for both large andsmall values of K m . Note that equation (17) was derived under the quasi-equilibriumapproximation k ≪ k , k which does not hold in the present case. This inaccuracymay also contribute to the discrepancy between our analytic results and simulation dataat small K m in figure 1. ttenuation of transcriptional bursting in mRNA transport m n T r an s po r t f l u x v (a) (b) m n (c) m n Figure 3. Transport flux as calculated from the stochastic simulations (dots), equation(15) (solid line) and equation (17) (dashed line). The transport parameters are: (a) k = 8 . K m = 0 . 1; (b) k = 14, K m = 24 . 5; (c) k = 30, K m = 102 . 5. Otherparameter values are given by λ = 2, h b i = 20, k = 0 . e t = 10 and δ = 1. Note that h m n i = 20 and h m c i = 40 are the same as before. 4. Conclusions and outlook The main conclusion of our work is that the nuclear envelope, which sets a naturalbarrier for the exodus of mature mRNA to the cytoplasm in an eukaryotic cell, cansignificantly attenuate the effect of transcriptional bursting on the downstream proteinpopulation. The extent of the noise reduction on the cytoplasmic mRNA copy numberis controlled by the transport efficiency. A high transport rate has a weak effect in noisereduction and essentially brings one back to the one-compartment model consideredpreviously. On the other hand, a low transport rate turns the nucleus into a bufferfor the bursting noise, thereby reducing the temporal variation of the mRNA copynumber in the cytoplasm. This effect is more dramatic in the saturated regime under theMichaelis-Menten dynamics where, due to the limited availability of transport channels,the mRNA export becomes a Poisson process unaffected by the bursty input.Our results can be used to re-estimate the mean mRNA burst size in the experimentby Raj and his colleagues. Take the linear transport as an exmaple, the noise strengthof the total mRNA copy number in the cell can be obtained from equations (9), (10)and (11): σ m n + m c h m n + m c i = h b i + 1 + h b i h m n ih m c i ( h m n i + h m c i ) . (40)Thus the average burst size h b i can be obtained from the measured total mRNAfluctuations and the ratio h m n i / h m c i of nuclear to cytoplasmic mRNA. If 30% of themRNA accumulate in the nucleus, the burst size should be 83% of that estimated fromthe model without transport. With the help of the two-compartment models, it wouldbe interesting to revisit the single-molecule experiment data of Raj et al. to gain amore complete view of the role of transport on the characteristics of the mRNA noisegenerated by transcriptional bursting.Previous studies suggest that for eukaryotes, such as S. cerevisiae [1], Dictyostelium [31] and mammalian cells[13], transcriptional bursting is a dominant source ttenuation of transcriptional bursting in mRNA transport et al [26, 27]. SR is usually related to periodic signal detection, where the signal noise istypically external, while SF exploits signal noise to make a gradual response mechanismwork more like a threshold mechanism. Here, the interesting phenomenon we observe isdue to a quite different mechanism: the degradation of upstream species and productionof downstream species share a common reaction (here transport), whose effective orderat steady state can be tuned from zero to one by a combination of reaction parameters. Acknowledgments We thank HG Liu and XQ Shi for helpful discussions. LPX would like to thank thePhysics Department, Hong Kong Baptist University for hospitality where part of thework was carried out. This work was supported by the National Natural ScienceFoundation of China under grant 10629401, and by the Research Grants Council ofthe HKSAR under grant HKBU 2016/06P. Appendix A. The Ω -expansion Equations (20) and (21) can be equivalently derived using the more formal Ω-expansionwhich applies when fluctuations of m n and m c are weak. (Ω here stands for the systemvolume.) To set the notation straight, the rate equation for the copy number x i ofmolecule i (i.e., m n or m c in the present case) is given by, dx i /dt = X q S iq V q = ( S · V ) i , (A.1)where S iq is the stoichiometric coefficient of molecule i in reaction q , and V q (whichdepends on the copy numbers in general) is the propensity of reaction q . Stochasticity ttenuation of transcriptional bursting in mRNA transport V q with a strengthset at V / q due to the underlying Poisson process.In the large volume limit and for the steady state, the joint distribution of the x i ’s,which satisfies the Fokker-Planck equation, can be approximated by a gaussian centeredat the solution to the equation d x /dt = S · V = 0. The width of the distribution isparametrized by a covariance matrix C which satisfies the equation [33]: AC + CA T + B = . (A.2)Here A ij = ∂ ( S · V ) i ∂x j , (A.3) B ij = X q V q S iq S jq , (A.4)all evaluated at the steady state. Specializing on our two-component problem, thereare three reactions: the bursting reaction leading to the production of m n , the MMtransport reaction for nuclear export, and the mRNA decay in the cytoplasm. Simplecalculations yield: A = − v max K ( K + h m n i ) v max K ( K + h m n i ) − δ . (A.5)Application of (A.4) yields: B = v max h m n i K + h m n i + X b λG ( b ) b = v max h m n i K + h m n i + λ h b i = 2 λ ( h b i + h b i ) . (A.6)In the last step, we have used the flux-balance condition and h b i = 2 h b i + h b i . Othermatrix elements of B can be obtained as: B = B = − λ h b i , B = 2 λ h b i . (A.7)Substituting the values of A and B into equation (A.2) gives: σ m n = C = h m n i ( h m n i + K ) K ( h b i + 1) , (A.8) h m n m c i − h m n ih m c i = C = C = h b i h m n ih m c i K h m c i K + h m n i + h m n i , (A.9) σ m c = C = h m c i ( h b i + 1) − h b i h m n ih m c i K h m c i K + h m n i + h m n i , (A.10)from which equations (20) and (21) follow. ttenuation of transcriptional bursting in mRNA transport Appendix B. Integrals in the independent burst approximation A convenient way to carry out the integrals in (24)-(27) is to convert them intointegration over x , which decreases monotonically from its initial value b to 0 in asingle burst event, with the help of (29) and (30). Following this procedure, we maywrite, Z ∞ dtx ( b, t ) = − Z b xdxdx/dt = Z b dx K + xv max = 1 v max ( Kb + 12 b ) . (B.1) Z ∞ dtx ( b, t ) = − Z b x dxdx/dt = Z b xdx K + xv max = 1 v max ( 12 Kb + 13 b ) . (B.2)To perform the averaging over b , we make use of the following results for the geometricdistribution, h b i = h b i (1 + 2 h b i ) , h b i = h b i (1 + 6 h b i + 6 h b i ) . With the help of these results, (31) are (32) are readily obtained.The dependence of y on x follows the equation, dydx = dy/dtdx/dt = − δv max (1 + Kx ) y, (B.3)which can be integrated to give, y ( x ) = Z bx dx e δ ( x − x ) /v max + w ln( x/x ) , (B.4)where w ≡ δK/v max .As a check, let us first consider Z ∞ dty = Z b dx K + xv max x y. (B.5)Using equation (B.3) and noting that y ( x = b ) = y ( x = 0) = 0, we obtain, Z b dx K + xv max x y = δ − Z b dx (1 + dydx ) = b/δ. (B.6)Hence, h m c i = λ h b i /δ, (B.7)which is nothing but the conservation law.We now consider Z ∞ dty = Z b yδ − ( dydx + 1) dx = δ − Z b ydx. (B.8)Using equation (B.4) and perform the substitution x → bx , we obtain, Z ∞ dty = b δ Z dx Z x dx e b ( δ/v max )( x − x )+ w ln( x/x ) . (B.9) ttenuation of transcriptional bursting in mRNA transport b can now be readily carried out. Using the result P ∞ b =0 b a b = a (1 + a ) / (1 − a ) , we obtain, X b G ( b ) Z ∞ dty ( b, t ) = δ − Z dx Z x dx e w ln( x/x ) h b i a (1 + a )(1 − a ) . (B.10)Here a = h b i e ( x − x ) δ/v max / (1 + h b i ).To avoid run-away accumulation of mRNAs in the nucleus, we require λ h b i < v max .Therefore h m c i < v max /δ . Note that h m c i > δ/v max to be asmall quantity. In this case, we can approximate a ≃ h b i [1 + ( x − x ) δ/v max ] / (1 + h b i ).Consequently, X b G ( b ) Z ∞ dty ( b, t ) = h b i ( h b i + ) δ Ψ( u, w ) , (B.11)where u = h b i δ/v max and Ψ( u, w ) is given by (35).Finally, the integral in equation (28) can be rewritten in the form, Z ∞ dtx ( b, t ) y ( b, t ) = Z b xδ − ( dydx + 1) dx = b δ − δ − Z b ydx. (B.12)Comparing with (B.8) and using (B.11), we obtain, X b G ( b ) Z ∞ dtx ( b, t ) y ( b, t ) = h b i ( h b i + ) δ [1 − Ψ( u, w )] . (B.13) References [1] Raser J M and O’Shea E K 2004 Control of stochasticity in eukaryotic gene expression Science Science Nature Proc. Natl.Acad. Sci. USA Nature Science Phys. Rev. Lett. Proc. Natl. Acad. Sci.USA Phys. Rev. Lett. Phase Transitions and Critical Phenomena Nat. Rev. Genet. ttenuation of transcriptional bursting in mRNA transport [12] Miaczynska M and Stenmark H 2008 Mechanisms and functions of endocytosis J Cell Biol. PLoS Biol. e309[14] Fraser H B, Hirsh A E, Giaever G, Kumm J and Eisen M B 2004 Noise minimization in eukaryoticgene expression PLoS Biol. e137[15] Audibert A, Weil D and Dautry F 2002 In vivo kinetics of mRNA splicing and transport inmammalian cells Mol. Cell Biol. Cell Nat. Rev. Mol. Cell Biol. Science Proc. Natl. Acad. Sci. USA Curr. Opin. Cell Biol. RNA Curr. Opin. Cell Biol. D = 0 . − . µ m /sec [19]. Given that the diameter of CHO nucleus is about 5 µ m, the timeneeded for the mRNP complexes to disperse throughout the nucleus is of the order of a fewminutes, much shorter than the nuclear dwelling time of several hours.[24] Van Kampen N G 1992 Stochatic processes in physics and chemistry (North-Holland-Elsevier)[25] S´anchez ´A and Kondev J 2008 Trancriptional control of noise in gene expression Proc. Natl. Acad.Sci. USA Proc. Natl. Acad. Sci. USA Phys. Rev. Lett. Nature Genet. J. Phys. Chem. Curr. Biol. Nat. Genet. Genome Res.13