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Transcriptional Pulsing and Consequent Stochasticity in Gene Expression

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Transcriptional Pulsing and Consequent Stochasticity in Gene Expression Transcriptional pulsing and consequent stochasticity in gene expression Srividya Iyer-Biswas ∗†, F. Hayot ‡ and C. Jayaprakash ∗ ∗Ohio State University Department of Physics, Woodruff Ave, Columbus, OH 43210, and ‡Mount Sinai School of Medicine Department of Neurology, Levy Place, New York, NY 10029 Transcriptional pulsing has been observed in both prokary- single cells [1, 2, 3]. Variability of cellular response otes and eukaryotes and plays a crucial role in cell to cell can have many origins, which are generally classi- variability of protein and mRNA numbers. The issue is fied as extrinsic and intrinsic noise or fluctuations[4]. how the time constants associated with episodes of tran- The source of intrinsic fluctuations is the random oc- scriptional bursting impact cellular mRNA and protein dis- tributions and reciprocally, to what extent experimentally currence of reactions that can lead to variability for observed distributions can be attributed to transcriptional genetically identical cells in identical, fixed environ- pulsing. We address these questions by investigating the ments. Extrinsic fluctuations can have multiple ori- exact time-dependent solution of the Master equation for gins, such as variations from cell to cell in the number a transcriptional pulsing model of mRNA distributions. We find a plethora of results: we show that, among others, bi- of regulatory molecules, or signaling cascade compo- modal and long-tailed (power law) distributions occur in nents, or fluctuations in cytoplasmic and nuclear vol- the steady state as the rate constants are varied over bi- umes. Many studies, both experimental and theo- ologically significant time scales. Since steady state dis- retical from bacteria to eukaryotes have been under- tributions may not be reached experimentally we present taken to disentangle intrinsic and extrinsic fluctua- results for the time evolution of the distributions. Because cellular behavior is essentially determined by proteins, we tions [4, 5, 6, 7, 8, 9, 10, 11]. investigate the effect of the different mRNA distributions Intrinsic fluctuations arise from either noisy tran- on the corresponding protein distributions. We delineate scription or translation or both, the effects of which the regimes of rate constants for which the protein distribu- can be measured in single cell mRNA and protein tion mimics the mRNA distribution and those for which the protein distribution deviates significantly from the mRNA experiments. The simplest model of protein number distribution. distributions is to consider both transcription and translation as Poisson processes [7]. Recent exper- stochastic gene expression | transcriptional pulsing | mRNA distribu- imental studies of mRNA distributions have shown tion | protein distribution | bimodal distribution strong evidence for transcriptional noise beyond what can be described by a simple Poisson process. In arXiv:0711.1141v1 [q-bio.QM] 7 Nov 2007 Introduction particular, transcriptional pulsing, where bursts of Cell to cell variability in mRNA and protein num- transcription alternate with quiescent periods, has bers is now recognized as a major aspect of cellular been observed in both prokaryotes and eukaryotes. response to stimuli, a variability which is hidden in cell population studies. The most egregious exam- SIB and CJ performed the research. ple of the latter is provided in cases where a graded SIB, FH and CJ analyzed the data and wrote the paper. average response hides the all-or-nothing behavior of The authors declare that they have no conflicts of interests. †To whom correspondence should be addressed. E-mail:[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.
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