arXiv:0711.1141v1 [q-bio.QM] 7 Nov 2007 l ftelte spoie ncssweeagraded of a behavior all-or-nothing where the cases hides in response provided average is exam- latter egregious the most of The ple in hidden studies. is population which cellular cell variability of a aspect stimuli, major to a num- response as protein recognized and now mRNA is in bers variability cell to Cell Introduction tion tcatcgn expression gene stochastic mRNA the from distribution. significantly the deviates which distribution for those protein and distribution mRNA distribu the protein mimics the tion which delineate for constants We rate of regimes distributions. the protein distributions corresponding mRNA the different the on w of proteins, effect by the determined investigate essentially Becaus is distributions. behavior the cellular of present evolution we time the experimentally dis- for reached results state be steady bi- not Since over may varied tributions scales. are time constants rate significant in the ologically occur as state distributions steady law) bi- the (power others, among long-tailed that, and show we modal results: of We plethora a distributions. find mRNA of for model equation pulsing Master transcriptional the a the investigating of by solution time-dependent questions exact these address transcription We to pulsing. attributed be experimentall can extent distributions what observed to reciprocally, dis- protein and tran- and tributions is mRNA of cellular episodes issue impact with bursting The associated scriptional constants cell numbers. time to mRNA the cell how in and role protein crucial of a variability plays and eukaryotes prokary- and both in otes observed been has pulsing Transcriptional 10029 NY York, New Place, Levy ∗ rvdaIyer-Biswas Srividya expression gene in stochasticity consequent and pulsing Transcriptional hoSaeUiest eateto hsc,WorffAe C Ave, Woodruff Physics, of Department University State Ohio | rti distribution protein | | ∗† rncitoa pulsing transcriptional ioa distribution bimodal .Hayot F. , ‡ n .Jayaprakash C. and | RAdistribu- mRNA lmu,O 31,and 43210, OH olumbus, al e e y - enosre nbt rkroe n eukaryotes. and prokaryotes both has in observed periods, been quiescent In with of alternate bursts process. transcription where Poisson pulsing, simple transcriptional a particular, by described be can what beyond noise shown transcriptional have for evidence distributions strong exper- mRNA Recent of studies [7]. imental and processes transcription Poisson as both translation consider to number protein is protein and distributions of mRNA model simplest cell The single experiments. in which measured of effects be the can both, or translation or scription 11]. 10, 9, 8, 7, 6, 5, fluctua- [4, extrinsic and tions intrinsic disentangle under- to been taken have theo- eukaryotes and to bacteria experimental from both retical studies, Many vol- nuclear umes. and cytoplasmic in fluctuations compo- or cascade nents, signaling or number molecules, the regulatory in cell of to cell from variations ori- as multiple such have gins, can fluctuations environ- Extrinsic fixed ments. identical, for in variability cells to identical lead genetically can that reactions oc- random of the currence is fluctuations intrinsic of fluctuations[4]. source or classi- The noise generally intrinsic and are extrinsic as which response fied origins, cellular many of Variability have can 3]. 2, [1, cells single † paper. the wrote and data the analyzed CJ and FH SIB, h uhr elr htte aen oflcso interests. of conflicts no have they that declare authors The research. the performed CJ and SIB ∗ owo orsodnesol eadesd E-mail:srivid addressed. be should correspondence whom To nrni utain rs rmete os tran- noisy either from arise fluctuations Intrinsic ‡ on ia colo eiieDprmn fNeurology, of Department Medicine of School Sinai Mount [email protected] 1 – 10 Raser and O’Shea [5], who studied intrinsic and ex- pects of it relating to bursts explored and discussed trinsic noise in Saccharomyces cerevisiae, showed that in reference [12]. Raj et al. [10] provided a steady the noise associated with a particular promoter could state solution to the Master Equation of the tran- be explained in a transcriptional pulsing model and scriptional model considered here, and analyzed it for confirmed it by mutational analysis. Transcriptional some ranges of the rate constants. Given the range of bursts were recorded in E. coli [12] by following time scales that can occur in transcriptional processes mRNA production in time, and their statistics com- in different organisms, it is imperative to highlight puted. Evidence for a pulsing model of transcription, the most significant behaviors that can arise in this obtained from fluorescent microscopy, has also been model and investigate how these depend on the many presented for the expression of the discoidin Ia gene time scales. In this paper, we provide a comprehen- of Dictyostelium [13]. Transcriptional bursts have sive analysis of a transcriptional pulsing model with as well been detected in Chinese hamster ovary cells an exact solution to the time-dependent Master Equa- [10]. In these experiments the production of mRNA tion for mRNA production. The advantage of this occurs in a sequence of bursts of transcriptional activ- model is that it is amenable to such an analytic de- ity separated by quiescent periods. Transcriptional termination of the probability distribution of mRNA bursting, an intrinsically random phenomenon, thus copy number as a function of time. We find that the becomes an important element to consider when eval- system exhibits a surprising variety of distributions uating cell to cell variability. One can predict that in of mRNA number: this includes a bimodal distri- many cases it will be a significant part of overall noise, bution with power-law behavior between the peaks and most certainly of intrinsic noise. It has been spec- that evolves into a scale-invariant power-law distri- ulated that pulsatile mRNA production might permit bution as we vary the rates of activation and inacti- “greater flexibility in transcriptional decisions” [13]. vation. In some systems the mRNA distribution may Cook et al. [14] have argued that different aspects not reach steady state, and it is therefore necessary of haploinsufficiency can be connected to time scales to determine the time evolution of the distributions associated with transcriptional bursting. and characterize the time scales over which steady Our study focusses on the consequences of tran- state is attained. Our time-dependent analytic solu- scriptional bursting in a simplified model of transcrip- tion allows us to address these issues in detail, re- tion that has been the subject of many studies and vealing in particular that the mRNA lifetime plays a is believed to encapsulate the key features of burst- key role in shaping the mRNA distribution. Cellular ing [2, 5, 10, 12, 15, 16]. The complex phenomena behavior is however determined by proteins and not that can occur in transcription (chromatin remodel- the corresponding mRNA. Therefore, an important ing, enhanceosome formation, preinitiation assembly, question is to what extent the protein distributions etc.) are modeled through positing two states of gene follow the mRNA distributions obtained as a result activity: an inactive state where no transcription oc- of transcriptional pulsing. To answer this question, curs, and an active one, in which transcription occurs we have performed numerical simulations of a model according to a Poisson process. The production of using the Gillespie algorithm [17] in which proteins mRNA is thus pulsatile: temporally there are peri- are produced in a birth-death process from mRNA. ods of inactivity interspersed with periods or bursts When the protein decay rates are much larger than of transcriptional activity. Qualitative features of the mRNA decay rate the protein distributions re- this model were presented in reference [2], and as- flect the mRNA distributions; when the protein de-
2 cays more slowly the protein distribution can be very ing to three independent, dimensionless ratios. The different from that of the parent mRNA, even in the equation obeyed by the probability for the DNA steady state. We delve deeply into the structure of to be in the excited state, denoted by Q1(t) can the transcriptional bursting model, highlighting how be obtained directly from Equation (1): dQ1/dt = the shapes of mRNA distributions depend on the ra- cf − (cf + cb) Q1 . This shows that the effective DNA tios of time scales, determining over which time scales relaxation rate to the steady state is governed by distributions evolve to steady state, and stressing the cf + cb. The mean mRNA number obeys the equa- importance of considering the full distribution rather tion dhm(t)i/dt = −kd hmi + kb Q1(t). It is thus than characterizing it solely by its average and vari- clear that the temporal behavior of the mean mRNA ance. We thus provide an overview of possible be- number is determined by the rates kd and cf + cb, haviors which yield a framework for interpreting ex- the latter entering since it determines the dynam- perimental results on transcriptional bursting across ics of the transcriptionally active state. As long as prokaryotes and eukaryotes. kd < cf + cb, the mRNA decay rate sets the time scale over which relaxation to the steady state oc- Results curs. We find that the results of our exact solution We study a model of transcriptional pulsing described can be interpreted in the most natural and trans- ∗ −1 by the following reactions where D and D denote the parent way when we measure time in units of kd , gene in the inactive and active states respectively: i.e. in terms of the mRNA lifetime. Thus we will
cf use the three dimensionless ratios kb/kd, cf /kd, and D ⇋ D∗ [1] cb cb/kd to organize our results. The mean number of k D∗ −→b D∗ + M [2] mRNA in the steady state is given by the product of k M −→d ∅ . [3] cf /(cf +cb), the fraction of the time the gene is in the activated state and kb/kd, the mean value of mRNA The first equation describes the switching “on” and if the gene is always ‘on’. The ratio kb/kd clearly sets “off” of the gene at the rates c and c respectively. f b the scale for the number of mRNA and increasing The second and third equations describe transcrip- it extends the range over which P (m) is appreciable tion of the mRNA at a constant rate k in the active b without a significant change of shape. The remain- state and the subsequent degradation of the mRNA ing ratios cf /kd and cb/kd determine the shape of the at a rate k . We present results for P (m,t), the prob- d distribution. ability for the cell to contain m mRNA molecules at a time t that describes cell-to-cell variability of mRNA Superposition of Poisson distributions. We begin by copy number as a function of time. providing an intuitively appealing way to view our Time scales. The steady state and temporal behav- exact result for P (m,t). The key conclusion is that iors of the mRNA distribution depend on the rate P (m,t) can be pictured as a superposition of Poisson constants that control the time scales of various pro- distributions with different mean values. If the gene cesses. Therefore, we begin by briefly summarizing is always “on” the mRNA distribution in steady state the different time scales that arise in the model. The is Poisson with the Poisson parameter, λ, given by the model has four rate constants, the forward and back- mean kb/kd, the ratio of transcription and degrada- ward rates for the gene to switch between the ac- tion rates. Since the gene flips between the “on” and tive and inactive states and the transcription and “off” states with the rates determined by cf and cb, degradation rates that govern mRNA numbers, lead- the mRNA distribution is determined by a stochas-
3 tic transcription rate kbζ(t) where ζ(t) is a dichoto- ral to classify the distributions by plotting cf /kd and mous noise that assumes values 0 or 1 corresponding cb/kd along the x- and y-axes for fixed kb/kd. The re- to the gene being in the inactive or active state re- sults for fixed value kb/kd = 100 are displayed in Fig- spectively. The dynamics of the random variable ζ ure 1 and provide a bird’s eye view of the strikingly is determined by the stochastic chemical reaction de- different mRNA distributions that arise in different scribed by Equations (1). Thus the distribution of regions of parameter space. We recall that the exper- the mRNA number is described by a Poisson process iments are performed on a variety of organisms both in which the parameter λ itself is stochastic, a process prokaryotic and eukaryotic. While rate constants are called a doubly stochastic Poisson process [18]. not known, given the different time scales involved Consider observing a particular cell at a time in the experiments, we have chosen to investigate a
T . The number of mRNA at time T is distributed range of values of cf /kd and cb/kd that encompass according to a Poisson distribution with parameter different biologically significant cases: for example, λ(T ) that depends on the time history of ζ(t) from 0 our choices include the vastly different rate constant to T describing the sequence of flips between the on values in the experiments of Raser and O’Shea [5] and and off states in that cell. This time history corre- Raj et al. [10]. sponds to a series of pulses of unit height with both We start with the interesting case displayed in the the widths of the pulses and the intervals between bottom left figure in Figure 1, when the mRNA half- pulses independently and exponentially distributed life is shorter than the the time scales over which the with parameters cf and cb respectively. Different cel- gene turns on or off, i.e., cf , cb < kd. In the steady lular behaviors correspond to different realizations of state at any given time, the gene is off in some cells. the random sequence of pulses. Thus cell to cell vari- Since the mean duration of the pulse 1/cb > 1/kd ability in mRNA copy number is given by a superpo- the transcripts produced in the previous occurrence sition of Poisson distributions with parameter λ that of the on state would probably have decayed and so is itself a random variable: the number of transcripts will usually be small in m λ λ these cells. This causes a peak in the mRNA distri- P (m,t) = dλρ(λ,t) e− [4] Z m! bution near m = 0. In those cells in which the gene where ρ(λ,t) is the probability density of the random is on at the time of observation the number of tran- variable λ. The fraction of cells with m copies of scripts can display a broad range of values depend- mRNA at time t is determined by ρ(λ,t). Such su- ing on how long the gene was active as compared to perpositions have been considered in the context of the mRNA lifetime. Thus, we expect to observe a bi- stochastic processes, for example, in [19]. This rep- modal distribution, as was qualitatively argued in [2]. resentation provides an attractive conceptual frame- The result is shown in the lower left quadrant of Fig- work for understanding the transcriptional pulsing ure 1. One finds a peak at m = 0 (with a weight that problem. We shall elaborate elsewhere on how the can be computed analytically) and another peak at different forms of ρ(λ,t) in the different regions of pa- large (∼ kb/kd) m values. If the values of cf and cb rameter space allow us to interpret the corresponding are such that the peaks are well-separated, much of behaviors of P (m,t). the intermediate region displays a power-law behav- Steady state distributions. We now describe the vari- ior. This reflects the broad range of times for which ety of steady state distributions that occur in differ- the gene has been active in different cells at the time ent regions of parameter space. In view of the discus- of observation. It is useful to remark that bimodal sion of time scales in the previous section, it is natu-
4 distributions have been obtained in models with feed- tial state with no mRNA and the gene in its inactive back [20]. In contrast, in the transcriptional pulsing state. Using the time-dependent result for the distri- model bimodality is obtained without the presence of bution (see Material and Methods, equation (9)), we a feedback loop. evaluate the evolution using Mathematica and plot
Now imagine that we keep cf fixed and vary cb the complete probability distribution as a function of so that it is larger than the decay rate. This leads to time. Consider the case cf , cb < kd, (bottom left in a scale-invariant, power-law behavior over a signifi- Figure 1), where the mRNA distribution displays bi- cant range of mRNA values. This simple power-law modality. Here the mRNA decay rate sets the scale decay obtained in the case cf < kd and cb > kd is for approach to the steady state. Figure 2(a) shows illustrated in the lower right quadrant of Figure 1. the evolution of the bimodal distribution as a func- This case has been treated analytically in [15, 10]. tion of time. For the given initial condition the sec- The mRNA distribution can be fitted by a Gamma ond peak away from zero develops after a period of distribution, which for appropriate values of the rate roughly twice the mRNA half-life. Steady state be- −1 constants, shows power-law behavior over a substan- havior sets in at about 4 to 5 times kd . It is clearly tial range. possible that, depending on the relative values of the When both the activation and inactivation rates cell cycle time and the mRNA half-life, steady state are rapid, i.e., cf , cb ≫ kd, eliminating the fast re- and therefore, full bimodality may not be observable. actions naively yields a simple birth-death process Consider now the time evolution of the distribu- for the mRNA with an effective transcription rate tion that evolves into the“pure” power law behavior kb×cf /(cf +cb). This would lead one to expect a Pois- featured in the bottom right of Figure 1. In Fig- son distribution for the mRNA number. However, in ure 2(b) we plot P (m,t) vs m on a double logarithmic this ‘quadrant’, i.e. for cf , cb > kd, the observed plot. We have chosen cf = 0.25kd and cb = 2.5kd to distribution has a broad single-humped shape as dis- illustrate this case. Larger values of the transcription played in the upper right quadrant of Figure 1, much rate will lead to a larger range over which the power broader than a Poisson distribution. This broaden- law behavior obtains. It is clear that the exponent of ing occurs because the parameter λ itself is stochas- the power law increases in magnitude with time and tic. When cf > kd > cb the gene is on most of the saturates at the steady state value for t greater than −1 time. Not surprisingly, the distribution is Poisson about 4kd . Thus the shape the distribution depends to a very good approximation as seen in the upper crucially on the time (measured in units of the decay left quadrant of Figure 1. In the intermediate re- time) when experimental measurements are made. gion when cf , cb ∼ kd the distribution interpolates Mean and variance versus full distribution. From the between these different possibilities and is rectangu- examples given in Figure 1 it is clear that the com- lar when they are equal (see Figure 1, center). plete probability distribution of mRNA number is re- quired to characterize the behavior of the transcrip- Time evolution of probability distributions. Because tional pulsing model. Nevertheless, for completeness, of the range of possible time scales, it can happen we make some remarks concerning attempts to repre- that that the time when measurements are made, sent a mRNA distribution by its mean and variance the biological system has not attained steady state. only. For this reason, we now present results for how the We recall the expressions for the mean and vari- distributions evolve to a steady state from an ini- ance in mRNA number in the transcriptional puls-
5 ing model reported in [5]. The mean is given by this region being straight lines with slope 1. For the
µ = kbcf /(kd(cf + cb)) and the variance by region with cf , cb > kd, there is rapid switching be-
k c k c tween on and off states and the Fano factor depends σ2 = b f 1+ b b [5] kd cf + cb „ (cf + cb)(kd + cf + cb) « weakly on the rates cf and cb. There is danger in characterizing distributions The first term in (5) is equal to the mean while the solely by their mean and variance. The variety of second term arises from the stochasticity in the puls- possible mRNA distributions across cells shown in ing process. It can be shown [18] that Figure 1, demonstrates this. One of the interest- 2 2 σ = hλi + σλ [6] ing results in Reference [5] showed the decrease in the noise strength (Fano Factor) with increase in the Thus, in a doubly stochastic birth-death process the mean for genes with different activation rates. Here variance in mRNA number has an additional contri- we show that a wide variety of distributions under- bution due to the stochasticity of gene activation and lies this correlation between the noise strength and inactivation. the mean. The increase in the mean can be ob- There are two popular measures of noise in terms tained in the model through an increase in the ac- of the first two moments of a probability distribution: tivating rate, namely the rate cf , and experimentally the coefficient of noise, ξ, defined as the ratio of the through mutations of an appropriate promoter [5]. standard deviation σ to the mean µ, and the noise Even though a smooth curve is obtained for the de- strength or Fano factor, η, defined as the ratio of the crease of noise strength with the mean, we illustrate variance σ2 to the mean . The latter has the value of how the full mRNA distribution can differ for differ- unity for a Poisson distribution and is therefore con- ent points along the curve. For specificity, we choose venient for describing deviations from Poisson behav- parameter values kb = 200kd and cb = kd, and vary ior. In Figure 3 we display constant η contours as a the forward rate cf for gene activation which changes function of c /k and c /k for a fixed value of k /k f d b d b d the mean value. The result is shown in Figure 4(a) on a logarithmic scale to encompass a broad range of and is similar to that obtained experimentally. Now parameter variation. When c 6 non entropy for different values of the rate constants. modal for this case. Similarly, in Figure 5(b), a long While the flat distribution clearly corresponds to high tailed mRNA distribution with a power law region entropy, the power-law distribution also yields large gives rise to a similar protein distribution when the values of the entropy indicating that greater informa- other rate constants are appropriately chosen. tion content than the other distributions. The results The other important case is when the proteins are presented in Supplementary Section B. are relatively stable compared to the mRNA. This Protein distributions.Given the variety of mRNA case is more complex; however, there are ranges of distributions that can result from genes undergoing rate constants for which the protein and mRNA dis- transcriptional pulsing, it is important to understand tributions are vastly different. We illustrate this with how this affects the probability distributions for the two examples. If the rate constants are chosen such corresponding protein. While a careful answer to this that the mRNA distribution is bimodal (cf. Figure question would require detailed modeling of mRNA 1), then P (p) is qualitatively different from P (m) translocation and translation, we address this issue in if pd < cf + cb. The protein distributions for this a model in which translation is treated as a Poisson case may be either monotonically decreasing or bell- process [7]. Thus, we use shaped, depending on whether pd is less than one or both of cf , cb. This contrast between the protein and pb M −→ M + P [7] the mRNA distributions is illustrated in Figure 6(a). p P −→d ∅ . [8] A second case is displayed in Figure 6(b) for values of the rate constants that lead to a power-law distribu- The effective protein degradation rate would include tion for the mRNA. Here a bell-shaped distribution is contributions from cell division, dimerization and obtained for P (p) when pd << cf < kd < cb. These other gene specific processes involving the loss of pro- examples demonstrate that one has to be careful in teins. From our results we identify two regimes, one inferring the shape of one of the mRNA or protein in which the protein distribution is similar to the distributions from the other. mRNA distribution and another in which they are It is worth noting that even if mRNA numbers different. are small, even as low as O(10), the above conclusions The protein distribution qualitatively mirrors the continue to hold. Thus, a wide variety of protein dis- mRNA distribution if the protein dynamics is faster tributions that may or may not reflect the underlying than the mRNA dynamics: translation and protein mRNA distribution could be realized in real biologi- degradation occur at a rate higher than the mRNA cal systems when the gene undergoes transcriptional degradation rate k (with c + c of the order of k ). d f b d pulsing. In this case, bimodals give rise to bimodals, power- It has been argued recently [22] that if the ef- laws give rise to power-laws and so on. The results fective protein degradation is very slow compared to are displayed in Figure 5, where the protein distribu- that of the mRNA, the protein distribution can be ap- tions for two of the cases illustrated in Figure 1 are proximated by a gamma distribution. While gamma shown along with the corresponding mRNA distribu- distributions do provide a good fit for some regions of tions. We show results for the protein degradation parameter space, none of the distributions obtained rate p set to twice k , the mRNA degradation rate. d d in the bimodal quadrant can be reasonably approxi- Figure 5(a) has rate constants c , c such that the f b mated by a gamma distribution. mRNA distribution exhibits bimodality, as seen in the inset. The protein distribution is also clearly bi- 7 Discussion Materials and Methods In this paper, we have presented results for the time- The results presented and discussed here for mRNA dependent and steady-state probability distributions are based on the exact form of the distribution func- for mRNA in a transcriptional pulsing model. A vari- tion P (m,t) of the mRNA number, m at time t. We ety of mRNA distributions occur in different regimes have solved the Master Equation [19] for reactions of rate constants. Our aim is to provide a guide for (1)-(3) that describes the time evolution of the dis- the interpretation of data on cell-to-cell variability tribution to obtain these results. We found it con- that could arise from transcriptional pulsing. Tran- venient to work with the generating function defined ∞ m scriptional pulsing, entailed by the dynamics of chro- by G(z,t) = m=0 z P (m,t). If we can evalu- matin remodeling, reinitiation and similar processes, ate G(z,t) exactly,P then the probability of having appears as a straightforward mechanism leading to m mRNA transcripts at time t can be obtained by bimodality and also to mRNA distributions with long extracting the coefficient of the zm term. We have tails. Long-tailed distributions of mRNA have been computed the generating function exactly (See Sup- seen in a variety of systems: the experiments of Raser plementary Section A for the details) for the initial and O’Shea [5] show evidence for long tails which condition with zero mRNA, i.e., P (m, 0) = δm,0, and they attribute to transcriptional pulsing. In experi- find ments on the gene ActB in cells from mouse pancre- G(z,t)= Fs(t)Φ(cf , cf + cb; −kb(1 − z)) atic islets the distribution of mRNA number, m, was + Fns(t)(1 − z)Φ(1 − cb, 2 − cf − cb; −kb(1 − z)) found to be consistent with a log-normal distribution [9] that would correspond to P (m) ∼ m−1 over some where Φ is the (Kummer) confluent hypergeometric range of m [23]. More recently, a similar distribution function [24, 25] and the coefficients Fs(t) and Fns(t) of the Interferon-β gene transcripts has been found can be calculated explicitly in terms of confluent hy- in human dendritic cells [11]. For the latter two ex- pergeometric functions. The results are displayed in periments our results should help clarify whether the Supplementary Section A. At large times Fs = 1and origin of the mRNA behavior lies in transcriptional Fns = 0. Thus in the steady state the generating pulsing. However, cell behavior is controlled not by function is given by( see also [10]) mRNA but by the proteins they encode. Therefore, it is crucial to determine whether the protein distri- Gs(z) = Φ(cf , cf + cb; −kb(1 − z)) , [10] bution follows the corresponding mRNA distribution. We can use the exact solution in Equation (9) We have determined when the two distributions are to extract the time-dependent behavior of P (m,t) or similar but also identified situations when the pro- Equation (10) for the steady state for different ranges tein distributions are strikingly different from those of values of the rate constants. The results presented of the mRNA. In general, and for these situations in were obtained by extracting the coefficients of zm in particular, our results on the range of cell-to-cell vari- the expansion of the generating function using Math- ability of mRNA and protein responses due to tran- ematica[26]. While standard numerical simulations scriptional pulsing should provide significant help in based on the Gillespie algorithm [17] can be employed interpreting experiments. to study both the steady state and the time evolution, the exact solution allows us to extract the results much more efficiently and explore the behavior of the system systematically in the space of rate constants 8 without statistical errors, especially, when there are of Allergy and Infectious Diseases through contract long tails present. The results for the protein dis- HHSN266200500021C. We thank Stuart Sealfon, James Wetmur, and Jianzhong Hu for discussions that stimu- tributions were obtained from numerical simulations lated this work and Stuart Sealfon for a careful reading of using the Gillespie algorithm. the manuscript and comments. S. I. B. thanks Rudro R. Biswas for productive discussions. This work was supported by the National Institute 1. Fiering S, Northrop JP, Nolan GP, Mattila PS, Crabtree GR, 14. Cook DL, Gerber AN, Tapscott SJ (1998) Modeling stochastic Herzenberg LA (1990) Single cell assay of a transcription fac- gene expression: Implications for haploinsufficiency. Proc Natl tor reveals a threshold in transcription activated by signals Acad Sci USA 95:15641-15646 emanating from the T-cell antigen receptor. Genes & Devel- 15. Karmakar R, Bose I (2006) Graded and binary responses in opment 4:1823-1834. stochastic gene expression. Physical Biology 1:197-204 2. Hume DA (2000) Probability in transcriptional regulation and its implications for leukocyte differentiation and inducible 16. Kepler TB, Elston TC (2001) Stochasticity in transcriptional gene expression. Blood 96:2323-2328. regulation: origins, consequences, and mathematical represen- tations. Biophys J 81:3116-3136 3. Ko MS (1992) Induction mechanism of a single gene molecule: stochastic or deterministic? Bioessays 14:341-346. 17. Gillespie DT (2001) Exact stochastic simulation of coupled 4. Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochas- chemical reactions. J Chem Phys 115:1716–1733. tic gene expression in a single cell. Science 297:1183-1186; 18. Cox DR, Isham V (1980) in Point Processes (Chapman and Swain PS, Elowitz MB, Siggia ED (2002) Intrinsic and ex- Hall, London) trinsic contributions to stochasticity in gene expression. Proc Natl Acad Sci USA 99:12795-12800. 19. Gardiner CW (1997) Handbook of Stochastic Methods 5. Raser JM, O’Shea EK (2004) Control of stochasticity in eu- (Springer Berlin) karyotic gene expression. Science 304:1811-1814. 20. Ferrell JE (2002) Self-perpetuating states in signal transduc- 6. Blake WJ, Kaern M, Cantor CR, Collins JJ (2003) Noise in tion: positive feedback, double-negative feedback and bista- eukaryotic gene expression. Nature 422:633-637. bility. Curr. Opin. Cell Biol. 14:140-148. 7. Thattai M, van Oudenaarden A (2001) Intrinsic noise in gene 21. Bar-Even A, Paulsson J, Maheshri N, Carmi M, O’Shea E, regulatory networks. Proc Natl Acad Sci USA 98:8614-8619. Pilpel Y, and Barkai N (2006) Noise in protein expression 8. Ozbudak EM, Thattai M, Kurtser I, Grossman AD, van Oude- scales with natural protein abundance. Nat Genet 38:636-643. naarden A (2002) Regulation of noise in the expression of a single gene. Nat Genet 31: 69-73. 22. Friedman N, Cai L Xie SX (2006) Linking stochastic dynam- 9. Paulsson J (2004) Summing up the noise in gene networks. ics to population distribution: an analytical framework of gene Nature 427:415-418 expression. Phys Rev Lett 97:168302. 10. Raj A, Peskin CS, Tranchina D, Vargas DY, Tyagi S (2006) 23. Bengtsson M, St¨ahlberg A, Rorsman P, Kubista M (2005) Stochastic mRNA synthesis in mammalian cells. PLoS Biology Gene expression profiling in single cells from the pancreatic 4:1707-1719 islets of Langerhans reveals lognormal distribution of mRNA 11. Hu J, Sealfon SC, Hayot F, Jayaprakash C, Kumar M, Pendle- levels. Genome Res 15: 1388-1392. ton AC, Ganee A, Fernandez-Sesma A, Moran TM, Wetmur 24. Abramowitz M, Stegun IA (1964) Handbook of Mathematical JG (2007) Nucleic Acids Res doi:10.1093/nar/gkm557 Functions (US Govt. Printing, Wahington DC) 12. Golding I, Paulsson J, Zawilski SM, Cox EC (2005) Real-time kinetics of gene activity in individual bacteria. Cell 123:1025- 25. Slater LJ (1960) in Confluent Hypergeometric Functions 1036. (Cambridge, England) 13. Chubb J, Trcek T, Shenoy S, Singer R (2006) Transcriptional 26. Wolfram Research, Inc.,(2005) Mathematica, Version 5.2, pulsing of a developmental gene. Current Biology 16:1018- Champaign, IL 1025. 9 (1.5, 0.5) (1.2, 1.5) 0.024 0.016 0.012 0.008 0 0 0 50 100 150 (1, 1) 0 50 100 150 0.012 0.006 0 (0.5, 0.5) 0 50 100 150 (0.5, 2) 0.02 0.032 0.01 0.016 0 0 0 50 100 150 0 50 100 150 Fig. 1. Steady state mRNA distributions P (m) vs. m, labeled by (cf ,cb) in units of kd. The mRNA transcription rate kb is 100kd for all the distributions. The figure shows prototype distributions for the five major regimes cf ,cb ≶ kd in parameter space. The distribution for cf ,cb = kd is flat. Bimodals are obtained in the lower left panel for cf ,cb (a) 0.015 0.01 0.005 0 0 50 100 150 (b) 0.1 0.01 0.001 0.0001 1 10 100 Fig. 2. Time evolution of P (m, t) towards steady state as a function of m at different time points (a) in the bimodal regime for cf = 0.75kd, ∞ −1 −1 cb = 0.5kd and kb = 100kd at times t = 0.5, 1, 2, 3, 4 and in units of kd . The steady state with bimodality is reached around 4kd . ∞ −1 (b) Log-Log plot of P (m, t) for cf = 0.25kd, cb = 2.5kd and kb = 100kd at times t = 1, 2, 3, 4 and in units of kd . The figure shows the time evolution of the power law region of P (m, t). For the different curves time increases from bottom to top with the slope increasing to its steady-state value. 10 Log10 cb 1 0 Log10 cf -1 -2 -3 -3 -2 -1 0 1 Fig. 3. Contour plot of the noise strength(Fano factor) η as cb and cf (in units of kd) are varied, for kb = 1000kd, in the steady state; cb and cf are varied on a log10 scale over 5 decades. 9 contours for different values of η are placed at intervals of 100, from 1 to 1001 with η increasing from light to dark values. (b) 0.024 (a) 0.012 150 0 0 50 100 150 200 250 0.006 100 0.003 P(m) 50 Noise Strength 0 0 50 100 150 200 250 0.01 0 0 50 100 150 200 Mean 0.005 0 0 50 100 150 200 250 m Fig. 4. Smoothly varying noise strengths as activation rate is varied can correspond to different probability distributions. (a) Variation of noise strength (Fano factor) with activation rate for kb = 200kd and cb = kd (b) The steady state distributions P (m) corresponding to the (diamond- shaped) points marked in (a). The points in (a) going from right to left and the corresponding figures from top to bottom in (b) are for cf /kd = 10, 1, and 0.1 respectively. Figure (b) illustrates how different points on the same curve (a) can be associated with dramatically different mRNA distributions. 11 (a) 0.002 0.02 0.01 0 0.001 0 50 100 150 0 0 500 1000 1500 (b) 0.005 0.05 0.025 0 0.0025 0 50 100 0 0 500 1000 p P(p) Fig. 5. Examples of steady state distributions of proteins reflecting those of mRNA. Protein distributions (a) for cf = 0.5kd, cb = 0.5kd and kb = 100kd, pb = 20kd and pd = 2kd (b) for cf = 0.5kd, cb = 2kd and kb = 100kd, pb = 20kd and pd = 2kd. The insets show the corresponding mRNA distributions. When the protein degrades faster than the mRNA, its distribution qualitatively mirrors the corresponding mRNA distribution. (a) 0.002 0.02 0.01 0.001 0 0 50 100 0 0 500 1000 1500 2000 (b) 0.004 0.06 0.03 0.002 0 0 50 100 0 0 500 1000 1500 p P(p) Fig. 6. Examples of steady state distribution of proteins differing from those of mRNA. Protein distributions for (a) for cf = 0.6kd, cb = 0.2kd and kb = 100kd, pb = 1kd and pd = 0.05kd (b) for cf = 0.5kd, cb = 2kd and kb = 100kd, pb = 1kd. The protein lifetime is 20 times longer than the mRNA lifetime. The insets show the corresponding mRNA distributions. When the protein degrades slowly compared to the mRNA, its distribution can be qualitatively different from the corresponding mRNA distribution. 12 arXiv:0711.1141v1 [q-bio.QM] 7 Nov 2007 P nteiatv n ciesae epciey ti straig probabilities: is two It the respectively. for states Equation active and inactive the in ytesum the by for ucin rmteMse qain(ihtm esae by re-scaled time Equations(with Master the from functions l h aecntnsaemaue nuisof units in measured are constants rate the All hneo variables of change aibe,w have we variables, oethat Note nit h qain for equations the entiate of digtetoeutosw aeteueu relation useful the have we equations two the Adding 1 ( w iedpnetslto oteMse qainfrtasrpinlbursts transcriptional for Equation Master the to solution Time-dependent α ,t m, o h e fratosdsrbdb qain - ntetex the in 1-3 Equations by described reactions of set the For edfietegnrtn functions generating the define We esmlf h qain sn naao fteGlla tra Galilean the of analog an using equations the simplify We ti ovnett eieascn-re ieeta equati differential second-order a derive to convenient is It = dP dP n .TemN itiuin(needn ftesaeo t of state the of (independent distribution mRNA The 1. and 0 = ∂ ∂ ob h rbblt hta time at that probability the be to ) k t t 1 0 G G b dt dt ( ( (1 ,t m, t m, 0 1 G ( ( 0 ,t z, ,t z, − G ( ,t z, ) ) z = ) = ) ) ≡ e and ) = = − t G v∂ v h eedneon dependence the ; 0 c − + ≡ v 2 − c f G + c k G f P c f b G k 1 1 0 f P G [ G ( v∂ b ( G 0 ,t z, P (1 0 1+ (1 + ,t m, ( 1 0 ( 0 ,t z, 1 v ti ayt eueteeutosoee ytegenerating the by obeyed equations the deduce to easy is It . 2 ,t m, ( − ( and G adhne hi u)aefntosof functions are sum) their hence, (and ) ,t z, v∂ v∂ m ) z ) 0 upeetr eto A Section Supplementary and ) v v G + ) − − + ) − 1+ (1 + G G G α c 1 c 0 1 f ∂ ( 1 c t , b ,t z, b v c + P n bin(7) obtian and c G ( ) b w b = = 1 G P G 1 c ) ( − ( 1 b 0 ,t m, c w ≡ 1 ,t z, ≡ ( f + + ( − c ,t m, P ,t z, + f sdtrie ytebudr conditions. boundary the by determined is t ve 1 c v (1 + ) m G + ) G X ( f h elhas cell the ∞ ) c ,t m, =0 − (1 + ) 1 1 + ) ∂ G 0 b = ) t v + 0 − k G z k = d ] ) + m d 1 v . k c − ) [( b d − k P 1+ (1 + ∂ c G m − b [( v α z b G (1 G 1 G ) m ( z 1 1) + ,t m, ∂ − − 1 0 ) towr owiedw h Master the down write to htforward m z . 1) + ∂ G z + vG z ) ) c RAmlclsadtegn is gene the and molecules mRNA P G 1 c f e ( 1 1 k f P − 0 ) ,t z, ( G . G d ( nfor on 0 t m ,t z, ): ntrso h transformed the of terms In . ( 1 0 ) m 1 + somto ymkn the making by nsformation − (3) ) 1 + (8) 0 = 0 = edefine we t t , k G v b egn)i determined is gene) he t , ) hrfr ediffer- we Therefore . (1 nyadindependent and only − ) . − − mP z mP ) 1 G ( 0 P ,t m, 1 ( ( 0 ,t m, ,t z, ( ,t m, ] ) ) (1) ] ) . and ) (2) (4) (5) (7) (6) (9) We add the two equations and use Equation (7) to obtain 2 v∂v G + (cf + cb + v)∂vG + cf G = 0 . The substitution G(v) = e−vF (v) shows that F (v) satisfies the confluent hypergeometric equation in the canonical form. The solution is given by 1−cf −cb F = A(w) Φ(cb, cf + cb; v) + B0(w) v Φ(1 − cf , 2 − cf − cb; v) . (10) Upon using the Kummer transformation, e−vΦ(α, γ; v) = Φ(γ − α, γ; −v), we obtain 1−cf −cb G = A(w)Φ(cf , cf + cb; −v) + B0(w)v Φ(1 − cb, 2 − cf − cb; −v) . (11) In order to obtain a well-defined power series in v = kb(1 − z) for the generating function we must impose cf +cb cf +cb −(cf +cb)t B0(w) = w B(w) = v e B(w) . This yields the form −(cf +cb)t G = A(w) Φ(cf , cf + cb; −v) + B(w) e vΦ(1 − cb, 2 − cf − cb; −v) . (12) We impose the boundary conditions at t = 0 which corresponds to w = v. The initial condition P (m,t =0) = δm,0 leads to G(w = v, v) = 1 . (13) We assume that the gene is initially in the inactive state and thus G1(z,t = 0) = 0. The additional condition that arises from Equation (7)implies ∂vG(w, v)|w=v = 0 . (14) Imposing these conditions we determine the unknown functions A and B0. This involves judicious use of the Wronskian identity α Φ(α − γ + 1, 1 − γ; z)Φ(α, γ; z) − zΦ(α − γ + 1, 1 − γ; z)Φ(α + 1, γ + 1, z) = ez (15) γ(1 − γ) that follows from from results in Ref. [1] and other identities to found there. The final result is G(z,t) = Fs(t) Φ(cf , cf +cb; −kb(1−z)) + Fns(t) (1−z) Φ(1−cb, 2−cf −cb; −kb(1−z)) . (16) where −t Fs(t) = Φ(−cf , 1 − cf − cb; kbe (1 − z)) and (17) cf kb(1 − z) −(cf +cb)t −t Fns(t) = − e Φ(cb, 1+ cf + cb; kbe (1 − z)) . (18) (cf + cb)(1 − cf − cb) 2 5 4 3 cb/kd 2 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 cf/kd Figure 1: Contour Plot of the Shannon entropy defined in Equation 19 of P (m) for kb = 100kd; separation between contours is 0.05 and the Shannon entropy decreases outward from the central contour. Supplementary Section B Characterization of noise in terms of Shannon Entropy In information theory the Shannon entropy serves as a measure of the average information content or the uncertainty of a random variable. We now present results for this measure of the noise in the mRNA distribution. Given the exact solution we can directly evaluate the Shannon entropy associated with the steady state distribution P (m) defined by S = − X P (m) log2 P (m) . (19) i In Figure 1 we display contours of constant entropy as a function of the forward and backward rates cf and cb. Not surprisingly, the values cf /kd, cb/kd ≃ 1 yield the largest en- tropy since this choice leads to a uniform distribution in P (m). The power law distribution also provides a range of values for the rates in which larger values of entropy can be obtained. Since the Shannon entropy is a measure of the amount of information required to describe the random variable on average it can be helpful in the interpretation of data. For example, con- sider dendritic cells involved in providing innate immunity to an organism against pathogens. Assume that the mRNA or the protein produced by the cell to overcome a viral antagonist has a broad distribution. It is plausible that the greater the amount of information required to describe the distribution the lesser the chances of the pathogen being able to overcome the 3 organism’s immune system by random mutations. This can confer greater immunity against mutations in the virus that could evade the defence mechanisms of the organism. It is known that the interferon-β mRNA distribution is broad in human dendritic cells [2]. References [1] Slater LJ (1960) in Confluent Hypergeometric Functions (Cambridge, England) [2] Hu J, Sealfon SC, Hayot F, Jayaprakash C, Kumar M, Pendleton AC, Ga- nee A, Fernandez-Sesma A, Moran TM, Wetmur JG (2007) Nucleic Acids Res doi:10.1093/nar/gkm557 4