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Chapter 5

Eukaryotic Transcription: A Molecu-lar Ratchet Prone to Bursting

Eukaryotic transcription regulation depends on a multitude of protein complexesthat assemble on chromatin. Their activities are modulated by transcriptionfactors and the local chromatin state, which is dynamic during the course oftranscription initiation. We present a stochastic molecular-ratchet model of eu-karyotic transcription initiation based on the current knowledge. The ratchet iscomposed out of a sequence of transitions, each composed out of reversible pro-tein complex formation followed by irreversible histone modifications sensitiz-ing chromatin for the formation of the next protein complex. A coarse-graineddescription of this model describes a gene as switching between an on and offstate, both having non-exponential durations. Transcription activity proves ro-bust to the precise design of protein complex formation and histone modification.Transcription stochasticity (bursts and mRNA noise) is expressed in terms of thedesign and kinetics of transcription initiation. We discuss which experimentalapproaches can give most information about the transcription mechanism.

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5.1 Introduction

Eukaryotic cells, as well as prokaryotes, have shown a remarkable variability intheir transcription activities across isogenic populations. This was shown firstin single-cell studies of protein levels. The high cell-to-cell heterogeneity wasexplained in terms of a fluctuating transcriptional activity (Swain et al., 2002;Elowitz et al., 2002; Yu et al., 2006; Cai et al., 2006). Single mRNA count-ing confirmed this hypothesis, both for prokaryotes (Golding et al., 2005) andeukaryotes (Zenklusen et al., 2008; Raj et al., 2006). Since then, many studieshave considered the potential benefits and hazards of transcription stochasticityin environmental adaptation (Acar et al., 2008; Raser and O’Shea, 2005; Lehner,2008; van Hoek and Hogeweg, 2007; Fraser and Kaern, 2009) and decision-making (Tkacik et al., 2008; Maheshri and O’Shea, 2007).

In parallel, theory and models to describe the noise in gene expression were de-veloped, largely based on the design of prokaryotic transcription (Kepler andElston, 2001; Paulsson, 2004; Ko, 1991), for gradual (Thattai and van Oude-naarden, 2001) and bursty transcription activity (Raj et al., 2006; Ko, 1991).This has culminated in a near complete mechanistic understanding of transcrip-tion stochasticity and bursting of E. coli’s lac operon under repressed conditions(Golding et al., 2005; Choi et al., 2008; Yu et al., 2006; Elf et al., 2007). Themechanism for fluctuations in mRNA in this system turns out to be the switchingof the promoter between an on and off state due to diffusion-limited promotorbinding and stochastic dissociation of the repressor causing occasional bursts inmRNA production. The life times of the on and off states are adequately de-scribed in terms of exponentially distributed waiting times corresponding to acoarse-grained description of a single DNA-binding and dissociation event (Yuet al., 2006). At present such a level of understanding is lacking for eukaryotictranscription.

Many studies of transcription stochasticity are based on coarse-grained transcrip-tion models inspired by the design of prokaryotic transcription. Even thoughsuch models have been used for the analysis of eukaryotic transcription (Zen-klusen et al., 2008; Raj et al., 2006; Chubb et al., 2006), it remains doubt-ful whether a prokaryotic model of transcription can satisfactorily capture thestochastic properties of eukaryotic transcription. While prokaryotic transcrip-tion initiation typically relies on a single transcription factor (TF), sigma factorand RNA polymerase (Minchin and Busby, 2009; Reznikoff et al., 1985), its eu-

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karyotic counterpart has a much more intricate design. Eukaryotic transcriptionproperties are largely determined by the chromatin environment in which it takesplace. Transcription depends not only on the specific and general TFs but also ona multitude of co-regulators, covalent-histone modifying enzymes and chromatinre-modeling complexes (Hager et al., 2006; Metivier et al., 2003; Dundr et al.,2002; Darzacq et al., 2007; Gorski et al., 2008). At least dozens of proteins areimplicated in eukaryotic transcription.

Chromatin is a highly dynamic structure (Li et al., 2007). Its properties and struc-tural dynamics are determined by alterations in the position and the epigeneticstate of nucleosomes (Dinant et al., 2009; Hager et al., 2006) under the influenceof various protein complexes. The protein affinity of chromatin is influencedby its state and hereby affects gene activity (Lam et al., 2008). In particular,covalent-histone modifications play an important role in gene activity regulation(Barski et al., 2007; Valls et al., 2005; Bernstein et al., 2002; Roh et al., 2006);both in slow processes like cell differentiation, via cell-type specific gene silenc-ing and activation (Weishaupt et al., 2010; Aranda et al., 2009), and on short timescales during the progression of transcription initiation, elongation and mRNAprocessing (Reid et al., 2009). Recent studies of the basal design of eukaryotictranscription initiation suggest the existence of transcription cycles with periodsof about 30 to 60 minutes (Metivier et al., 2003), during which short-lived proteincomplexes (with life times of tens of seconds to minutes) are formed on chro-matin in a fairly strict sequential fashion (Reid et al., 2009; Dinant et al., 2009).Transcription initiation progresses through the sequential formation of proteincomplexes on chromatin and histone-modification-induced alterations in chro-matin structure (Reid et al., 2009). Chromatin modulation is believed to monitorthis progression and to sensitize the affinity of chromatin for the next batch ofproteins that need to perform their task (Reid et al., 2009; Dinant et al., 2009).Upon the formation of one or multiple elongating RNA polymerases, the genereturns to its initial state of transcription inactivity. This suggests a mechanismfor the occurrence of bursts in mRNA production in agreement with experimen-tal data on single-molecule mRNA counting, using either FISH (Raj et al., 2006;Zenklusen et al., 2008; Pare et al., 2009) or MS2-hairpin constructs (Chubb et al.,2006).

In this work, we propose a molecular-ratchet model of eukaryotic transcriptionas a coarse-grained model for the single-cell analysis of eukaryotic transcription.In this model the cyclic progression of transcription initiation is driven by irre-

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versible transitions between chromatin states that each involve reversible proteincomplex assembly followed by covalent histone modification. Since only a frac-tion of the states is transcriptionally permissive, this model naturally gives riseto bursts. The complexity of eukaryotic transcription allows for a richer reper-toire of stochastic behaviors and mechanisms for its control than for prokaryotes.The model we propose is highly flexible. It allows for extensions that incorpo-rate different transcription initiation, gene silencing and activation mechanisms.It should facilitate the analysis of experimental data on eukaryotic transcriptionstochasticity, bursts and mRNA distributions across cell populations.

5.2 Results

5.2.1 A molecular-ratchet model of eukaryotic transcription

Transcription is composed out of three processes that occur in sequence: initia-tion, elongation, and termination. The resultant transcript is then used for pro-tein translation or is degraded. In this section, we describe a model for eukary-otic transcription initiation up to promotor escape. We here interpret the experi-mental evidence of cyclic phenomena around eukaryotic transcription (Metivieret al., 2003; Saramaki et al., 2009) further and suggest that the eukaryotic tran-scription initiation mechanism resembles a molecular ratchet. The model wepropose is depicted in Figure 5.1. It is relatively easy for the ratchet wheel toundergo the stochastic reversible association and dissociation of a protein, butevery time it turns past the ratchet of irreversible covalent modification, it is pre-vented from turning back (counter clockwise). In the following we identify themolecular basis of the essential elements of the ratchet mechanism in what isnow known about eukaryrotic transcription initiation: eukaryotic transcriptioninitiation relies on a cyclic sequence of protein complex formations involved inpromoter (de-)activation and mRNA polymerase initiation complex formation.The first events involve the assembly of protein complexes at the TF bindingsite(s) (response element(s); RE(s)) and transcription start site(s) (TSS; onlyone such site is shown in Figure 1) induced by an active transcription factor.This is followed by covalent modification and remodeling of nucleosomes thatsensitize chromatin for the assembly of the next complex (involving differentproteins) in the sequence. The process of reversible protein complex formationand subsequent irreversible modification is repeated multiple times leading to aprogression through a number of distinct chromatin states in transcription initi-

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Figure 5.1: The eukaryotic transcription initiation mechanism resembles a molec-ular ratchet. Left: A gene switches between a transcriptionally permissive on stateand a non-permissive off state. Transitions between the states occur through a sequenceof protein complex formations on chromatin, intermitted by covalent-histone modifica-tions marking progress and sensitizing chromatin for the next protein complex assembly.Sequential protein complex formations occur on the response element and the transcrip-tion start site. This mechanism is in accordance with experimental data (see main text).Right-top: Ratchet modules are composed of reversible protein complex formations onchromatin and an irreversible histone modification step. A preferential random mecha-nism is depicted as an example for complex formation. Right-middle: The waiting timedistributions for individual ratchet modules (left) can be convoluted to give the net lifetime distributions for on and off states (right, dark green and black, respectively). Theconvoluted life time distributions can well be approximated with Gamma-distributionsthat have the same mean and variance (right, light green and gray). Right-bottom: Thelife time distribution is a measure for gene activity. Depending on the life time distri-butions of the mRNA, on- and off states, the molecular ratchet can display transcriptionbursts that result in various different shapes for the mRNA copy-number distribution.

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ation. We propose that a succession of Non of these states are transcriptionallypermissive (make up the on phase) and allow the multi-step formation of thepre-initiation complex and binding of elongation-competent polymerase. Theremaining number of (No f f ) states correspond to an off phase during which thesystem either restarts or gets stuck in the absence of active transcription factor.A single on and off state make up a transcription cycle. Transcription bursts(Raj et al., 2006) suggest that during the on phase of a single transcription cy-cle, multiple RNA polymerases can be assembled and prepared for promotorclearance; this has been suggested for yeast (Zenklusen et al., 2008). (For somegenes, promotor clearance has been suggested to be rate-limiting, i.e. the slow-est step (Kugel, 2000).) Another example of a molecular ratchet on chromatinappears to be the DNA repair system that essentially follows a similar mecha-nism of sequential protein complex formation, irreversible transitions, and activ-ity (Mone et al., 2004). In Figure 5.2, we show a literature-based reconstructionof a core set of nucleosome modification events that drive the transcription cycleas found in yeast. There appears to be evidence for the existence of a definedsequence of chromatin state transitions that sensitize protein binding and com-plex assembly. For instance, it has been shown that the histone modificationmark, H4R3me, created by the methyltransferase PRMT1, which acts as a co-factor to many specific TFs, increases the affinity of local chromatin for acetyl-transferase p300 (Li et al., 2010). This enzyme deposits lysine acetylations(H3K14Ac and H3K18Ac) which in turn attract a different methyltransferase(CARM1) that catalyzes H3R17 formation (Daujat et al., 2010). It has beenwell established that a number of acetylation modifications, including H3K14Acand H3K16Ac, increase the affinity of chromatin for the nucleosome remodel-ing complexes SWI/SNF and ISWI, which are each involved in gene activation(Corona et al., 2002; DiRenzo et al., 2000). The sequence of events that emergesis part of the creation of a permissive promoter state. This is further supported bythe experimental evidence that the general transcription factor, TFIID, has highaffinity for acetylated histones (Agalioti et al., 2002; Vermeulen et al., 2007;Sawa et al., 2004) and that the removal of nucleosomes from the TATA-box isassociated with transcription initiation (Metivier et al., 2003; Moreira and Holm-berg, 1998). In addition, the elongation-competent PolII in the vicinity of thepromoter is known to associate with the chromatin-modifying complex COM-PASS responsible for H3K4 methylation (Gerber and Shilatifard, 2003). In yeast,this mark has been shown to attract complexes containing histone-deacetylatingenzymes (Kim and Buratowski, 2009) capable of removing promoter-activating

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Figure 5.2: Core events of the transcription initiation cycle on a regulated yeastpromoter. The mechanism is a based on available experimental data for transcriptionfactor and covalent histone modification mediated progression of transcription initiationin yeast. More information on experimental data supporting the model can be found inthe Appendix.

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acetylation marks (Wang et al., 2002). The H3K4 tri-methylation also causes therecruitment of repressive nucleosome remodeling complexes (Wysocka et al.,2006; Santos-Rosa et al., 2003) to facilitate promoter deactivation. Taken to-gether, these covalent-modification mediated protein complex formations leadto a sequential and cyclic transcription initiation mechanism for yeast shown inFigure 5.2. This reconstruction is still very crude and will certainly lack com-ponents but it already indicates the basal cyclic design of transcription and itsprogression marking through covalent histone modifications. In the next sec-tion, we will describe and analyze single transitions within the molecular ratchetmechanism in more detail.

5.2.2 Waiting-time distributions for single transitions in the ratchet mech-anism

As outlined in the previous sections, transitions in the molecular ratchet each in-volve reversible events of protein complex formation followed by an irreversibleevent of histone modification marking progress and adaptation of the affinity oflocal chromatin for the assembly of the next protein complex (scheme in Figure5.1, Right-top). Since ratchet transitions depend on protein binding events atsingle chromatin sites (at REs and TSSs) they operate in a low molecule numberregime and may be highly stochastic. The duration of a single ratchet transi-tion then becomes a random variable following a (first passage time) distributionthat is specified by the kinetic mechanism of the transition (Figure 5.1, Right-middle). A first-passage time (or waiting time or duration) is a random variablethat specifies the time it takes for a single molecule to pass through some kineticmechanism from start to end. For a reversible assembly mechanism for a com-plex, composed out of two proteins (p1 and p2), on a single chromatin site s,i.e. p1 + s p1s and p1s + s2 p1 p2s, the first-passage time would capturethe assembly time for the p1 and p2 complex on a single DNA site s. Being arandom variable, the first-passage time derives from a distribution. First-passagetime distributions can be calculated by solving the master equation describingthe Markov chain description of the molecular mechanism as function of timeusing either the classical methods to solve differential equations (Qian and Elson,2002), Laplace transforms (Dobrzynski and Bruggeman, 2009) or a phase-typedistributions description. Each of the three methods is illustrated in the appendixfor a simple example.

In the simplest case, a single ratchet transition can be represented by a single

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first-order reaction. The transition duration then follows an exponential waitingtime distribution, with a probability density of f (t) = ke−k·t (with k as a first orderrate constant; unit s−1;

∫ ∞0

f (t)dt = 1). A probability for a duration to fall withinthe interval τ + dt is given by p(τ) = f (τ)dt. The mean waiting time wouldequal 〈t〉 = 1/k and the noise 〈δ2t〉/〈t〉2, defined as variance divided by the meansquared, would equal 1.

When a single ratchet transition would involve a sequence of n first-order reac-tions, each with rate constant k, the waiting time distribution would become apeaked distribution, known as the Erlang distribution, with mean time 〈t〉 = n/kand a waiting time noise of 1/n. In the latter case, the ratchet transition displaysa much more reproducible duration, becoming more and more precise when thenumber of sequential reactions, n, increases. A realistic mechanism for such atransition in the molecular ratchet will involve protein complex formation thatwill obey typically a much more complicated waiting time distribution, which inprinciple depends on all the kinetic parameters and the structure of the mecha-nism. The nature of those distributions may have large influences on the overallstochastic properties of the molecular ratchet leading to consequences for burstsstatistics and cell-to-cell mRNA copy number heterogeneity.

There exists limited experimental information on the mechanisms of proteincomplex assembly. Random (Sprouse et al., 2008; Dinant et al., 2009) as wellas sequential (Puigserver et al., 1999) assembly mechanisms have been reported.Therefore, we proceeded by first investigating the effects of the assembly mech-anism on the form of the transition waiting time distribution to determine theinfluence of the type of assembly mechanism, reversibility of binding events,and kinetics. We considered the assembly-time statistics of complexes of fourproteins on chromatin; following either a completely sequential, a fully random,or one out of six different partially random mechanisms. The assembly schemesare shown in Figure 5.3A. We took kinetic parameters from the literature.

Protein complex assembly rates depend on the accessibility of DNA, protein-protein interactions and protein-DNA interactions (Dasgupta et al., 2005; Sprouseet al., 2008; Yakovchuk et al., 2010; Hieb et al., 2007; Dundr et al., 2002). Theupper limit for an association rate constant is set by the diffusion limit, whichleads to a second-order rate constant of ≈ α109M−1s−1 (Berg et al., 1981), whereα is a factor denoting the probability that molecules hit each other in the rightorientation and that there is nothing in the way of the encounter. Measured dis-sociation rate constants (10−1 − 10−2s−1) (Phair et al., 2004), and equilibrium

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Figure 5.3: Assembly-time distributions for different protein complex formationmechanisms can be approximated by a simple generic model. A: Different assemblymechanisms: sequential (Seq), six partially random mechanisms (PR), and fully random(Ran). B. Assembly time distributions for the eight assembly mechanisms (modeled ir-reversible with equal effective rate constants k = 0.3 min−1: product of diffusion-limitedassociation rate of 3.6nM−1 min−1 and 0.083 nM protein concentration corresponding to50 molecules in 1pL nuclear volume). C. Mean assembly times for the eight mechanismsD. Noise in assembly time for the eight assembly mechanisms.

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constants for binding (10−7 − 10−9M) (Nalefski et al., 2006; Lavery and McE-wan, 2008; Neely et al., 1997) of specific proteins to chromatin suggest that theactual association rates are about 10-100 times below the diffusion limit. Theamount of free protein in the nucleus may vary between tens and thousands ofmolecules for each species (Lee and Young1, 1998). The values for the catalyticrate constants (kcat) of chromatin-modifying enzymes have been found to be inthe range of 100 − 10−2s−1 (Thompson et al., 2001; Schultz et al., 2004). Toevaluate the influences of assembly mechanisms on the distribution of ratchettransition duration times, we assumed: i. irreversible binding, ii. all rate con-stants equal to an apparent first order rate constant (10 times below the diffusionlimit and corresponding to a free protein concentration of 50 molecules per nu-cleus). Figure 5.3B and D indicate that all the calculated duration distributionsare peaked. For the fully sequential mechanism, the distribution corresponds toan Erlang distribution (Figure 5.3B). These calculations show that the mean oftransition times lie in the 5 - 10 min range in accordance with the expected totalduration of the transcription cycle consisting of five to six of such transitions insequence (Karpova et al., 2008; Metivier et al., 2003). The mean transition timedecreases from sequential to more random mechanisms (Figure 5.3C) whereasnoise shows an opposite trend, albeit less pronounced (Figure 5.3D).

The calculations in Figure 5.3 all consider irreversible binding reactions, while inreality these are reversible. The effect of reversibility depends on the dissociationconstant (KD) defined as the ratio between the dissociation rate constant k− andthe apparent association rate constant k+. This equilibrium constant determinesthe percentage of bound DNA sites given a protein concentration. Already fordissociation constants smaller than 0.1 the difference in the waiting time distri-butions of a reversible and irreversible assembly mechanism is negligible as esti-mated by the Kullback-Leibler divergence index (Fig. S1). The Kullback-Leiblerdivergence index gives a non-symmetric measure of the difference between twoprobability distributions P and Q; calculated as DKL(P‖Q) =

∫ ∞−∞

p(x)log p(x)q(x) dx.

The significant differences occur as the dissociation constant approaches 1; thedistributions become more heavily tailed and the mean transition time increases.At dissociation constant approaching 2 the noise in the distribution becomesclose to 1, i.e the noise of the exponential distribution (Figure 5.5A). There isalso a possibility of the different steps in the transition having widely differentkinetic. To take this into account we explored effects of a single reaction(in thiscase modification) being much faster or slower than the rest of the reactions. As

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Figure 5.4: Waiting time distribution of ratchet transition can be successfully ap-proximated by Erlang distribution. The scheme of the protein assembly (PR3) ap-pended by an irreversible modification reaction is shown in the inset. Waiting time dis-tribution of the mechanism with indicated rate constants (blue). An Erlang distributionwith same mean and noise as the detailed mechanism is shown (red). The Kullback-Leibler divergence index is 0.042.

expected, a 10 times faster reaction rate constant does not significantly affect thenoise of the waiting time distribution. On the other hand the reaction rate con-stant 10 times slower results in an about 2-fold increase in noise and only causesthe noise to approach the levels of an exponentially distribution when the it isover 100-fold slower (Figure 5.5B).

Given that little is known about the exact assembly mechanisms and the kineticconstants of protein complex formation and covalent histone modification, weconsidered the possibility of coarse graining the ratchet transition models to amore simple and experimentally-testable model. As shown in Figure 5.1, a real-istic model for waiting time distributions for switching between active and inac-tive gene states, i.e. for the entire transcription cycle, comprises a sequence oftransitions. We can obtain the distribution of the duration of a single cycle of theratchet (or of its on and off period) by taking the convolution of the first-passagetime distributions of each of its composite transitions; this corresponds to deter-mining the probability of a sum of random variables (transition durations) havinga certain value.

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We next performed calculations of the overall waiting time distributions for se-quences of ratchet transitions (1, 3 and 5 ratchet modules) using any of the pos-sible combinations of four transition mechanisms (sequential, partially random:PR3, PR5, PR7) with uniformly random kinetic parameters within the ranges inaccordance with experimental measurements. We found that the overall wait-ing time distribution of these transitions could be satisfactorily approximated bya Gamma distribution, parameterized with the same mean and noise as the ex-plicit transition model as indicated by Kullback-Leibler divergence indices (Fig-ure 5.S.2). An example of distribution from these simulations is shown in Figure5.4E.

Gamma and Erlang distributions (used above) are very related. They each de-pend on a scale and shape parameter. The only difference is that the shape pa-rameter is an integer for the Erlang distribution and in case of the Gamma dis-tribution it is real-valued. As a consequence, the Erlang distribution models thedistributions of durations for a chain of identical first-order reactions, the shapeparameter then gives the number of reactions and the scale parameter the meanduration per reaction.

Collectively, the results of this section indicate that the distribution for a mul-tiple transitions in the ratchet, each of them composed out of protein complexevent followed by a histone-modification reaction, can be satisfactorily coarse-grained to a Gamma distribution with the same mean and noise as the morerealistic mechanism. In other words, the actual molecular events taking place inthe transitions of the molecular ratchet model for transcription initiation are notdetermining the statistics of a sequence of transitions under these conditions: 1)there is no single step that is orders of magnitude slower than the rest, 2) mostof the reactions are not highly reversible (Kd > 2). Both of these conditionsare expected to be fulfilled for actively transcribed genes. This suggests thatthe stochasticity of all the individual protein-chromatin binding events, protein-protein interactions and histone-modifications can average out by virtue of thoseevents occurring so often and fairly independently (in a sequence). Under con-ditions mentioned above the large number of stochastic phenomena of a similartype makes the ratchet display a random duration with a small dispersion. Thisalso means that fluctuations in protein levels, which affect binding rates, are notreadily causing changes in molecular ratchet statistics and hereby the mRNAstatistics should be fairly insensitive to fluctuations in transcription regulators atthe level of transcription initiation as long as they do not influence a great num-

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iseFigure 5.5: The noise of the waiting time distribution for one ratchet module tendsto 1 if either the reaction is highly reversible or if the modification reaction israte limiting, but can be considerably lower than 1 otherwise. A. noise of waitingtime distribution for one ratchet module with mechanisms "random"(blue), "preferen-tial random 4"(green), and "sequential"(red) as a function of the ratio of modificationrate constant and apparent association rate constant. Reactions are assumed to be irre-versible with equal apparent association rate constants k+. The inset shows the waitingtime distributions for the PR4 mechanisms at the kmod/k+ ratios marked with asterisks:kmod/k+ = 0.2 dashed, kmod/k+ = 1 solid, kmod/k+ = 5 dotted. B. noise of waiting timedistributions for a ratchet module with the PR4 mechanism with reversible reactions asa function of the ratio of apparent dissociation and association rates for three ratios ofkmod/k+: kmod/k+=0.2 dashed, kmod/k+=1 solid, kmod/k+=5 dotted. The apparent asso-ciation rate constants for all four proteins are taken as equal, as well as the dissociationrate constants. Inset: logarithmic plot

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ber of reactions in the ratchet. Below we will argue that the ratchet may evenoperate close to a deterministic regime.

In the next section, we will determine how the ratchet gives rise to bursts and howthe burst size statistics depend on the duration distributions of the on phase andthe net duration distribution for RNA polymerase initiation complex formationand promotor clearance.

5.2.3 Burst size distributions for genes differing in their coarse-grainedtranscription behavior

Bursts in transcription activity have been shown for bacteria (Golding et al.,2005; Choi et al., 2008; Yu et al., 2006; Elf et al., 2007; Cai et al., 2006), yeast(Zenklusen et al., 2008), and higher eukaryotes (Raj et al., 2006; Chubb et al.,2006). The prevailing mechanism for bursts is that multiple RNA polymerasesare prepared for elongation during the on state of the gene, while this does notoccur during the off phase. This was shown experimentally for E. coli’s lacoperon (Choi et al., 2008; Elf et al., 2007; Golding et al., 2005). An alternativemechanism would be the occurrence of pause-induced collisions of RNA poly-merase or the ribosome (Klumpp and Hwa, 2008; Dobrzynski and Bruggeman,2009; Mitarai et al., 2008). We show in this section that the burst size distribu-tion generated by the molecular ratchet models for transcription initiation can beobtained analytically for a number of relevant transcription models (for bacte-ria as well as eukaryotes). Those distributions can be measured experimentally(Chubb et al., 2006; Choi et al., 2008) and compared to theory.

We define the burst size distribution as the distribution (probability mass func-tion) of the number of transcription initiation events per single on-phase. Thisdistribution derives from the life time statistics of the on-phase and the kineticmechanism for RNA polymerase pre-initiation complex formation and promotorescape. In Figure 5.1 (Right-bottom), this corresponds to the statistics of thenumber of RNAs formed during the on intervals indicated in green. The ap-proach outlined in this section works for general life time distributions of the onstate and RNA polymerase assembly and promotor escape times. Assuming thatelongating RNA polymerases do not collide and hereby destroy bursts, generatedby transcription initiation and promotor clearance, we do not have to consider theelongation process in the determination of burst statistics.

We denote the first-passage time distribution of the molecular mechanism for

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transcription initiation and promotor clearance by fini(t). The probability that atleast b mRNA’s are formed during the life time of the on state, ton, is given by,

pb(B ≥ b|Ton = ton) =

∫ ton

0fini(t)(b)dt, (5.1)

where the probability density function fini(t)(b) equals the b-th convolution offini(t), given byL−1(L( f (t))b) andL(·) denotes the Laplace transform. The prob-ability that exactly b mRNA’s are formed in a time span ton is given by the prob-ability mass function,

pb(B = b|Ton = ton) = pb(B ≥ b|Ton = ton) − pb(B ≥ b + 1|Ton = ton) (5.2)

The expectation value and variance of this distribution as a function of ton areknown in queuing theory as the renewal function and the variance function, re-spectively (e.g. (Cinlar, 1975)). For general distributions of fini(t), their Laplacetransform can be expressed as (Parzen, 1962; Heyman and Sobel, 1982):

〈b〉(t) = L−1(L( fini(t))

1 − L( fini(t))

)(5.3)

and

〈δb2〉(t) = L−1(L( fini(t))(1 +L( fini(t)))

(1 − L( fini(t)))2

). (5.4)

The burst size distribution can be obtained after integrating over all Ton times toobtain the following probability mass function

pb(B = b) =

∫ ∞

0pb(B = b|Ton = ton) fon(ton)dton. (5.5)

Here fon(ton) is the probability density function for the duration or life time ofthe on-state, which was is analogous to the first passage distribution of the tran-scription activation times in the previous section.

Eukaryotic transcription initiation involves the assembly of the preinitiation com-plex composed out of the RNA polymerase, several general transcription factorsand additional proteins at the promotor. In addition, it requires phosphorylationof the resultant complex followed by promotor clearance. The first-passage time

242

distribution for this multi-step assembly process, fini(t), will generally be non-exponential and resemble one of the distributions we considered in the previoussection. To explore the effects of non-exponential first passage time distributionsfor fini and fon on the burst size distribution, we compare the four possible mod-els where these two distributions are described either as exponential or as Erlang(table 5.1, appendix sec. 5.S.3). In the most simple model, both distributionsare modeled as exponential (corresponding to mechanisms where a single step israte limiting). This results in a geometric burst size distribution characterized bya relatively high dispersion of 〈δb2〉/〈b〉2 = 1 + 1/〈b〉 that approaches 1 for highburst sizes. The mean 〈b〉 = 〈ton〉/〈ttr〉 corresponds to intuition: it is given by themean duration of the on phase, 〈ton〉 = k−1

o f f , divided by the mean waiting timefor the completion of RNA polymerase initiation, 〈ttr〉 = k−1

tr . Changing the firstpassage time distribution for transcription initiation and promoter clearance, fini,to an Erlang distribution does not change the shape of the distribution (geomet-ric), nor its dispersion, but note that the average number of transcripts producedduring an on phase is now lower than 〈ton〉/〈ttr〉 (table 5.1, appendix sec. 5.S.3).

For prokaryotic gene expression, the geometric burst-size distribution (Eq. 5.S.9)has been found experimentally in two separate studies on E. coli’s lac operon(Cai et al., 2006; Golding et al., 2005). These findings suggest that prokaryotictranscription bursts can be modeled by a gene that follows an exponential distri-bution for its on life time. This life time distribution corresponds to the searchtime for the lac repressor to find the promotor (Elf et al., 2007). This is indeedlikely an exponentially distributed waiting time (memoryless distribution) as thesearch trajectory is so long relative to the dimension of the cell that the initialposition of the lac repressor does not influence the search time.

As discussed in the previous section, in Eukaryotes the duration of the on stateis more likely to be described by a non-exponential waiting time distributionas it corresponds to traversing the on segment of the transcription cycle. As-suming an Erlang distribution for fon and modeling transcription initiation andpromoter clearance with an exponential waiting time leads to a negative bino-mial distribution for the burst size (Table 5.1). The average burst size for thismodel again corresponds to intuition (〈ton〉/〈ttr〉). Burst size noise then amountsto 〈δb2〉

〈b〉2 = 1Non

+ 1〈b〉 . It is lower than that of the geometric distribution and for

large burst sizes tends to 1/Non, where Non is the number of transitions in theratchet mechanism that together constitute the on state. Therefore such systemscan have a very reproducible and, hereby, precise burst size.

243

The most general ratchet model for burst size occurs when both the life time ofthe on state and the initiation times are non-exponentially distributed. The burstsize distribution of this model corresponds to a sum over elements of a negativebinomial distribution (table 5.1, appendix sec. 5.S.3). Again, non-exponentialityin the first passage time distribution for transcription initiation leads to an aver-age burst size that is lower than 〈ton〉/〈ttr〉. Though changing fini has no effecton the shape of the burst size distribution when the on state is described by anexponential distribution, it does change the shape of the burst size distributionfor systems with non-exponentially distributed life times of the on state.

Fig. 5.6 compares the effects of different numbers of steps for initiation andpromoter clearance (Ntr) and for the transition to the off state (Non). This com-parison shows that Non has a larger effect on the shape of the distribution as wellas its noise than Ntr. The differences in these burst characteristics for modelswith a multi-step mechanism for transcription initiation as compared to a singlestep reaction are largest for small average burst sizes and vanish in the limit ofinfinite burst sizes. As can be seen in Fig. 5.6 already for average burst sizesof 10 molecules, the effects of the multistep nature of the transcription initia-tion mechanism are almost negligible, while efficient noise reduction is possiblethrough a mechanism which leads to narrowly distributed life times of the onstate.

5.2.4 The effective burst size distribution: incorporation of mRNA degra-dation during the on phase

So far, we assumed that no degradation of mRNA occurred during the on phase.This would correspond to a situation when there is no degradation at all or itoccurs on a much slower timescale as compared to the duration of the on phase.The experimental data for prokaryotic genes (Golding et al., 2005; Yu et al.,2006; Cai et al., 2006) suggests that it is indeed the case for some genes forwhich the disappearance of mRNA occurs by way of dilution by cell growth. Ina general case we would expect mRNA degradation continuously, independentlyof whether the gene is in the on or off phase. In those situations, a better measurefor the burst size distribution is the distribution of the number of newly producedtranscripts at the end of a single on phase, taking into account mRNA synthe-sis and degradation during the on phase duration. This distribution we refer toas the effective burst size distribution. At any time t, the distribution of mRNAmolecules is described by a birth-death process. For exponentially distributed

244

waiting times for synthesis and degradation, this distribution has been solvedanalytically (Hemberg and Barahona, 2007) (Eq. 5.S.26). Substituting this re-lationship into equation 5.5 yields the effective burst-size distribution. With anErlang distribution for the life times of the on state, the mean and noise of theeffective burst size distribution are given by (where kdeg is degradation rate con-stant, ko f f is a single step promotor deactivation rate constant,Non - number ofsteps in promoter deactivation and ktr is the promoter deactivation constant):

〈be f f 〉 =ktr

kdeg

1 − (ko f f

kdeg + ko f f

)Non (5.6)

〈δb2e f f 〉

〈be f f 〉2 =

1〈be f f 〉

+

( ko f f

2kdeg+ko f f

)Non−

( ko f f

kdeg+ko f f

)2Non(( ko f f

kdeg+ko f f

)Non− 1

)2 (5.7)

At kdeg approaching zero the distribution in the effective burst size becomes1

〈be f f 〉ł + 1

Nonas shown above, i.e the noise without degradation during the burst.

Setting Non to 1 gives the following expression:

〈δb2e f f 〉

〈be f f 〉2 =

kdeg + ko f f

ktr+

ko f f

2kdeg + ko f f(5.8)

which as expected coincides with the noise in effective burst size when the onphase duration time is exponentially distributed, with kdeg+ko f f

ktr= 1〈be f f 〉

. The in-crease in degradation decreases noise for both Gamma and exponentially dis-tributed on times (Figure 5.S.3A). For the same mean on times as well as initia-tion and degradation rates noise in burst size is always lower in case of Gammadistributed on waiting time. However at high degradation rates, when a nearsteady state mRNA distribution is achieved during the on phase, this differencebecomes negligible (Figure 5.S.3B).

All burst size distributions described in this section could be used to analyzeexperimental data. Such data exists for prokaryotes (geometric burst size distri-bution), but is currently lacking for eukaryotes. Time resolved mRNA countingcould give rise to analyzable data as we will discuss in the last section.

5.2.5 Steady-state mRNA distributions: capturing cell-to-cell variabilityAt stationary state, the inevitably asynchronous transcription and mRNA degra-dation events cause fluctuations of the mRNA amount around its steady-state

245

A B

0 5 10 15 20 25 300.00

0.05

0.10

0.15

burstsize �b�

p b�B�b�

(a)

1 2 3 4 50.2

0.4

0.6

0.8

1.0

1.2

1.4

number of steps during on state �Non�

�∆b

2 ��b�2

(b)

Figure 5.6: The number of reactions in the transition from the on to the off state(Non) has a larger effect on the burst size distribution and its noise than the numberof steps for initiation (Ntr). All network parameterizations have the same mean burstsize (〈b〉 = 10 or 〈b〉 = 2). This can be achieved by adjusting either ko f f or ktr -the resulting distributions are the same. A. 〈b〉 = 10, Black: (overlaps with blue line)Non = 1, Ntr = 1 (geometric distribution), Red: Non = 5, Ntr = 1, Blue: Non = 1,Ntr = 5, and Green: Non = 5 and Ntr = 5. For Ntr = 1, Non = 5 effective burst sizedistributions are shown three different degradation rate constants; Dashed: tdeg = 2ton,Dotdashed: tdeg = ton, Dotted: tdeg = 0.5ton. B. Dashed: 〈b〉 = 2, Straight: 〈b〉 = 10,Black: Ntr=1,Green: Ntr=2, Orange: Ntr=5, Red: Ntr=10

246

OFF ON

mRNA

OFF ON

mRNA

OFF ON

mRNA

OFF ON

mRNA

fon(t) exponential exponential Erlang Erlang

fini(t) exponential Erlang exponential Erlang

〈b〉 ktrko f f

= τonτtr

(( ktr+ko f f

ktr

)Ntr− 1

)−1Non

ktrko f f

= τonτtr

NonktrNtrko f f

−Ntr−1

Ntr≤ 〈b〉 ≤ Nonktr

Ntrko f f

〈δb2〉

〈b〉2 1 + 1〈b〉 1 + 1

〈b〉1

Non+ 1〈b〉 ≥

ktr−ko f f

ktr

NtrNon

distribution geometric geometric negative binomial sum over elements

of a neg. binomial

Table 5.1: Comparison of burst characteristics for models with different distribu-tions for switching from the on state to the off state ( fon), and for the initiation oftranscription ( fini). Dashed lines indicate multistep mechanisms (modeled with an Er-lang waiting time distribution), solid lines indicate single step reactions (described byexponential waiting time distributions). The dashed arrow for the off -to-on transition isshown in gray as it does not influence the burst size statistics.

247

average. The copy number of mRNA at stationary state follows a time-invariantprobability distribution that depends on the kinetics and mechanisms of tran-scription and degradation. An example of such a distribution can be found inFigure 5.1 (Right-bottom). Such distributions have been derived in closed formfor a number of gene systems, mostly dealing with the prokaryotic model withexponentially distributed waiting times for on and off state durations as wellas for transcription and degradation (Raj et al., 2006; Iyer-Biswas et al., 2009).Here we will extend those results to non-exponential waiting times in the switchto capture the molecular ratchet mechanism. We will compare two models, onewhere the switching times are deterministic and another where those times de-rive from a non-exponential distribution. The deterministic model is a limit ofthe non-exponential model with insignificant dispersion in its distributions forthe on and off times. We have derived an analytical equation for the mRNA dis-tribution for the deterministic model, which can be found in the appendix, andis discussed in more detail below. It is compared to an mRNA copy numberdistribution with exponential life times of the on and off states, which is alreadyknown (Raj et al., 2006; Iyer-Biswas et al., 2009).

The models we consider describe mRNA production and degradation as a zerothand first-order process, respectively; that is, each with exponentially distributedwaiting times. The switching between on and off states is described by gen-eral waiting time distributions, we consider Gamma distributed times, or assumethem deterministic. The timescales of mRNA synthesis and degradation (k−1

tr andk−1

deg) versus the timescales of the gene switch (for gamma distributed life times:No f f /kon and Non/ko f f ) determine into which class of shape the mRNA distri-bution can be categorized (Fig 5.7A). This has been analyzed for a model withexponential waiting times for switching (Iyer-Biswas et al., 2009): if switch-ing is fast in comparison to mRNA degradation, the mRNA distribution can bedescribed by a broadened “Poisson" distribution (e.g. a Gaussian distribution)(Figure5.S.6A), while a slow switch with comparable time scales for on and offstate leads to a bimodal distribution (Figure 5.S.6D). In the latter case, mRNAcan reach a quasi-steady state level during the duration of a single on and a dif-ferent quasi-steady state at a subsequent off state. If the time spent in the onstate is much longer than the duration of the off state, the distribution can beapproximated by a Poisson distribution (Figure 5.S.6B), while the opposite case,i.e. long times in the off state and short on periods, leads to spikes of mRNAsynthesis and at steady state has a power law distribution (Figure 5.S.6C). Next

248

we will explore the consequences of extreme non-exponential waiting times forthe on and off states, i.e. deterministic times.

A B

0 2 4 6 8 100

2

4

6

8

10

�ton��tdeg�

�t o

ff��t deg

�

incre

asing

burst

size

"spike regime" quasi steady-state reached during the off-state

"Poisson regime"quasi steady-state

reached during the on-state

"bimodal regime"quasi steady-statereached duringon- and off-states

increasing noise

0 2 4 6 8 100

2

4

6

8

10

�ton��tdeg��t o

ff��t deg

�

0.35

0.25

0.15

0.1

0.05

0.02

0.02

0.0750.2

0.3

Figure 5.7: Timescale separation in the model with genetic switch, transcriptionand degradation largely determines the shape of the steady-state mRNA distri-bution. A. Changes in qualitative behaviors at different parameter regimes. B. TheKullback-Leibler divergence index (in bits) between the steady state mRNA distribu-tions of the deterministic and the exponential switch shows that the effect of the preci-sion of the duration of the states has the largest effect on the mRNA distribution in anintermediate regime of 〈ton〉/〈tdeg〉 vs. 〈to f f 〉/〈tdeg〉. In the exponential switching model:kon = t−1

o f f and ko f f = t−1on . Darker colors indicate lower DKL. Contours are shown for

0.01, 0.02, 0.05, 0.075, 0.1, 0.15, 0.2, 0.25, 0.3 and 0.35. The mean mRNA level, 〈n〉was kept at 10 by adjusting the rate for transcription initiation, ktr. The rates for statetransitions in the exponential switching model were set to kon = 〈t〉−1

o f f and ko f f = 〈t〉−1on .

The degradation rate constant only changes the scaling of the plot and was therefore setto 1. Usage of other indices than the Kullback-Leibler index gave qualitatively similarresults.

The effects of deterministic life times of the on and off state is analyzed in Figure5.7B. This figure shows a contour plot of the Kullback-Leibler divergence index,DKL, between the steady-state mRNA distributions of a switch with deterministicand a corresponding switch with exponentially distributed life times for on- andoff -state over a range of mean switching time scales relative to the life time of

249

the mRNA. The Kullback-Leible divergence index measures the difference be-tween two distributions in terms of the entropy of the first distribution relative tothe second distribution. It equals zero for identical distributions. To allow for afair comparison of the distributions, the mean mRNA level was kept the same.As can be seen from figure 5.7B, the differences between the two mRNA dis-tributions are smallest if either ton is much larger than to f f (the system is almostcontinuously in the on state) or if ton and to f f are of comparable size and bothlarger than the average time for degradation. The former case can be understoodintuitively, this parameter combination leads to a distribution that is very similarto a Poisson distribution; the infrequent and short excursions to the off state haveonly a small effect on the distribution and the effect of whether these switches toand from the off state occur in more or less regular time intervals is also rathersmall.

An increase in the life time of the off state leads to a bimodal distribution. Forthe switch with exponential waiting times, one of the peaks of the bimodal dis-tribution always lies at zero mRNA molecules. This is not the case for the de-terministic switch in the region of large ton and intermediate to f f . Accordingly,the Kullback-Leibler divergence index is relatively high in this region. A furtherincrease of the life times of off state also causes the deterministic switch to haveits first peak at zero mRNA molecules, which is why in this regime the distribu-tions for deterministic and exponential switch become more similar again (Fig5.7C). The regularity of the timing of switching between the states has a minoreffect on the mRNA distribution in this regime. If the average time of the onstate is smaller than that for degradation, the distributions of the exponential andthe deterministic switch are dissimilar for a large range of values of 〈to f f 〉. For〈to f f 〉 ≤ 1/kdeg the mRNA distribution is a single-peaked distribution. In thisregime, the noise in the waiting time distributions for the on and off state havea large effect on the steady state mRNA distribution; the distribution for the ex-ponential switch is much broader than that for the deterministic switch (Figure5.7B; see Fig. 5.S.6 for example distributions). Greater durations of the off statewhile keeping 〈ton〉 smaller than 1/kdeg makes the two distributions even moredissimilar. While the distribution for the exponential switch is now in a regimethat can be described with a power law (Iyer-Biswas et al., 2009), the distribu-tion of the deterministic switch still has its maximum at n > 0. Increasing to f f

even more, i.e., increasing the burst size, leads to a regime where also the deter-ministic switch has a distribution with its maximum at zero mRNA molecules.

250

Nevertheless, the two distributions remain distinguishable over a wide range ofvalues for 〈to f f 〉.

The noise in mRNA in the deterministic switch model is always lower than forthe model with exponentially distributed life times of the on and off state. Thissuggests that eukaryotic cells can profit from the multiple step nature of theirtranscription initiation cycle in terms of reduced noise.

Elongation was not taken into account in either model. If either the first pas-sage time distributions for all processes from promoter clearance to the finishedtranscript (elongation, termination, poly-adenylation, etc.) have very low noiseor if the mean first passage time is short in comparison to the timescales of theswitch, these processes are expected to have only minor effects on steady statemRNA levels. We tested the effect of elongation on noise for a variety of pa-rameter combinations (including those where neither of these two assumptionsholds). Models that do take elongation into account explicitly (in a non-collisionregime) will generally lead to a steady-state mRNA distribution with lower noiseas compared to the models without elonation, especially for exponentially dis-tributed elongation durations (Figure 5.S.7).

5.2.6 mRNA noise for the ratchet model

We developed a method for calculating the moments of the steady-state mRNAdistribution for systems that switch between on- and off -states following anylife-time distribution (appendix sec. 5.S.4). Transcription initiation and mRNAdegradation are described always as single step reactions (exponential waitingtime distributions with rate constants ktr and kdeg). The model is considered inits stochastic hybrid system limit (Singh and Hespanha, 2009) where the dura-tions of the on state are infinitesimally short. Within such an on state, bursts areproduced that are solely characterized by a mean and noise level. For a gammadistributed life time of the off state, the noise in the steady-state mRNA distribu-tion, 〈δn2〉/〈n〉2, can be expressed as (see Appendix):

〈δ2n〉〈n〉2

=1

2〈n〉

〈b〉

1 +(

konkon+kdeg

)No f f

1 −(

konkon+kdeg

)No f f−

2kon

No f f kdeg

︸︷︷︸non-exponential effect

+〈δb2〉

〈b〉+ 1

(5.9)

251

This equation indicates the following about the noise in steady-state mRNA lev-els: 1) it increases with the mean burst size ( as can be seen by taking the limit ofburst size approaching infinity) 2) the dispersion in the burst size increases it aswell (can be shown similarly) and 3)noise is affected by the number of steps No f f ,in the transition from off to on as indicated by non-exponetial term. For exponen-tially distributed life times of the off state, the non-exponential term simplifiesto 1. Substitution of the geometric burst size statistic 〈δ2b〉/〈b〉2 = 1 + 1/〈b〉 (cf.equation 5.S.11 and 5.S.16) then gives for the noise in mRNA, 〈δ

2n〉〈n〉2 = 1

〈n〉 + 〈b〉〈n〉 .

This is consistent with earlier findings (Paulsson, 2004). In the limit of inifnitelylarge number of steps in the off transition the non-exponential term approaches afixed number the exact value of which depends on the relative timescale of the offphase and degradation. It is largest when degradation is slow (kdeg << kon). Inthe limit of a zero degradation constant, the non-exponential term becomes N−1

o f f ,which is equal to the noise of Gamma distributed life times for the off state,〈δ2to f f 〉/〈to f f 〉

2. In this limit, the equation reduces to the approximation given inPedraza & Paulsson (Pedraza and Paulsson, 2008). In contrast, in regimes whenthe half life of mRNA and the off state are comparable, the deviation of the non-exponential term from 1 is less pronounced. Overall, this suggests that eukary-otes can profit from a multi-step molecular ratchet mechanism for transcriptioninitiation in terms of decreasing mRNA noise levels. This design already provedbeneficial for the reduction of burst-size noise though only due to an increase inthe number of reactions for the on to off transition (Eq. 5.S.13).

The application of equation 5.9 is limited to systems with very short lived onstates, e.g. to genes which are supposed to be off but occasionally give rise toexpression due to leaky repression. This model can be extended to incorporatea more general switching description where the on state is characterized by aspecific life time distribution. This is explained in the appendix (Sec. 5.S.5).This allows for a comparison between the simpler instantaneous burst model (Eq.5.9) and the biologically more realistic full switch model. Figure 5.S.4 shows therelative deviation of the mRNA noise calculated according to the instantaneousburst model as compared to the switch model with gamma distributed life timesof on- and off -state as functions of the average life times of the on and off state.There are two causes for the differences between the two models, one being thedegradation of mRNA during the on state (both of transcripts that were presentbefore the onset of the on state and of transcripts that were produced during thispermissive period) and the other - the contribution of the mRNA distribution

252

produced during the on state to the total mRNA distribution. Consideration ofthe effective burst (Eq. 5.S.26) size in all cases improves the approximation.

We also derived the full steady-state mRNA distribution of a ’deterministic switch’,a model where the life times of both states are deterministic. For this model,probability generating functions can be used to determine the complete distribu-tions at the times of switching from on to off and vice versa, which are both Pois-son distributions (sec.5.S.6). While the analyses so far focused on the mRNAnoise, this result permits a comparison between full distributions which can beexpected to contain much more information than the first few moments alone.Figure 5.7B shows the Kullback-Leibler divergence index between the exponen-tial and deterministic switch models. Since the deterministic switch model canbe understood as a limit of the gamma switch with infinitely many steps for bothtransitions, the mRNA distribution of a gamma switch can be expected to bean intermediate between the distributions of the exponential and deterministicswitch models with the same average life times of both states. A stochastic sim-ulation algorithm (Gibson and Bruck, 2000) was used to obtain the full mRNAdistributions for different gamma switches in different parameter regimes. Acomparison of these distributions with the distributions of the correspondent ex-ponential and deterministic switch models using the Kullback-Leibler divergenceindex (appendix sec. 5.S.7) suggests that the steady-state mRNA distribution ofa gamma switch is similar to the deterministic model whenever the number ofsteps for both transitions exceeds four. This is presumably in the regime wherethe eukaryotic transcription mechanism operates, suggesting that the determin-istic switch model offers a promising coarse-grained model for eukaryotic tran-scription.

5.2.7 Time-resolved single-molecule mRNA counting allows for model dis-crimination

Mammalian genes can differ markedly in the number and design of their responseelements and transcription start sites (Davidson, 2006). Presumably, this allowsfor differences in the tuning of transcription activity to the levels of transcriptionfactors and epigenetic status. A study of the consequences of the design of thetranscription initiation mechanism for transcription activity should perhaps relatethose initiation mechanisms to the stochastic measures of transcription activity,such as mRNA burst size and stationary copy number statistics. Perturbationsat the input of a gene, i.e. at the response element or transcription start site,

253

need then to be read out in terms of those measures of transcription activity. Thequestion is, which experiments are most informative to do so?

A number of studies have used mRNA-FISH labeling (Raj et al., 2006; Zen-klusen et al., 2008; Femino et al., 1998; Pare et al., 2009) or RT-QPCR on indi-vidual cells (Bengtsson et al., 2005, 2008; Wagatsuma et al., 2005; Warren et al.,2006) to determine the copy number distributions of the transcripts of a certaingene. In some cases, the genes were found to display transcriptional bursts andgood quantitative fits to the experimental data could be obtained with a sim-ple ’exponential switch’ model. However, such fits may be misleading, as wasshown by Pedraza & Paulsson (Pedraza and Paulsson, 2008). They indicated thatdifferent models of gene expression and transcription can lead to distributionswith the same mean and noise. Even though the complete mRNA distributioncontains much more information that its first two moments alone, we show here,using the experimental data of Zenklusen et al. (Zenklusen et al., 2008), that thislack of discrimination between models also holds true for the complete mRNAdistribution.

We first show that discrimination between different models is in principle pos-sible when mRNA production events are followed in time. Figure 5.1 (Right-bottom) shows an example of such a time trace. Single-cell levels of mRNA canbe monitored using for instance mRNAs equipped with MS2-hairpin sequencesthat can specifically bind a phage-derived protein, MS2, fused to a fluorescentprotein. Individual mRNA molecules can then be observed as single fluorescentspots (Shav-Tal, 2004; Darzacq et al., 2007).

We fitted the mRNA distributions for the five genes measured by Zenklusen etal. (Zenklusen et al., 2008) to models with different numbers of steps in thetransitions from on and off states (No f f and Non ranged between 1 to 10). Therate constants for transcription initiation, ktr, and gene state switching, kon andko f f , were tuned such that the first three moments of the steady state distributionof the model were equal to the first three moments of the experimentally observeddata. Since we are dealing with time-independent stationary data, we choose adimensionless time by scaling it with respect to the degradation constant. ThemRNA distributions of the models were obtained through simulations with theGillespie algorithm. For all genes, the quality of the fit, as measured by the χ2

value, did not differ significantly between models with different values for thenumber of sequential steps in their on to off transitions (Non and No f f ; see thetable in the Appendix). In other words, the stationary distributions of mRNA do

254

not retain any information about the basal design of the transcription initiationmechanism responsible for them.

The failure to discriminate between alternative transcription mechanisms on thebasis of the stationary mRNA distribution was perhaps to be expected for thethree house-keeping genes (MDN1, DOA1, KAP104) used in the study by Zen-klussen et al (Zenklusen et al., 2008). Zenklussen et al already showed that themRNA distribution could be fitted by a Poisson distribution, suggesting that amodel with a short life time of the off state or a gene that is continuously in theon state applies. In this regime, the shapes of the distributions of the life timesof both states do not significantly affect the steady-state mRNA distribution asis indicated by Figure 5.7B. More surprisingly, we could not use the mRNA dis-tribution for the two bursting genes (PDR5, POL1) in the data set to distinguishdifferent transcription initiation models either (see Figure 5.8).

Then we examined which experiment would give discriminating data. It turnedout that time traces of mRNA copy numbers should allow for a discriminationof the different transcription mechanisms. The different ’gamma switch models’used in the fit vary vastly in the average fractions of time that the gene is in theoff state, in their total cycle duration relative to the average time for degradation,1/kdeg, and in their burst size distributions (Figure 5.8, Table 5.S.2, analogousfigures for the other four genes in appendix sec. 5.S.8). Even though theseparameters do also vary between the different models fitted to the three house-keeping genes, a closer look at mRNA time traces reveals that those differencesare less informative, as the on- and off -phases are not distinguishable from thetime traces of the mRNA production events alone. This was further confirmed bycalculations of the burst significance (Dobrzynski and Bruggeman, 2009). Forall models fitted to mRNA distributions of the three house keeping genes, theburst significance was below 0.1 (even though the fraction of time the gene is offaccording to the model can be significant, e.g. for DOA1, Table 5.S.3), while themodels for the two bursting genes showed high burst significances (>0.85).

Figure 5.9 demonstrates, that for genes that display significant bursting, the wait-ing time distributions of the underlying mechanism could in principle be deter-mined from data on the times of synthesis events (as would be observed duringa mRNA-counting experiment). The simulated synthesis times from Figure 5.8c(Non = 1, No f f = 10) were used to determine the average burst size (〈b〉) and du-ration (τb) from the sequence size function (Dobrzynski and Bruggeman, 2009).The sequence size function can be used to determine the mean burst size from

255

A B

0 10 20 30 400.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

mRNAs per cell

frequ

ency

PDR5

# mRNAs/cell

freq

uenc

y

0 10 20 30 400.00

0.05

0.10

0.15

burst size

frequ

ency

burst size [#transcripts/on-phase]

freq

uenc

y

C

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>0 2 4 6 8 100

10

20

30

402 3 1 7 1 62 8 8 9 0 2026 0 3 2 1114 1 4 11 21 07 8 4 0 3 3 3110 4 7 11 3 0 0 3 8 0 3 15 0 9

0 2 4 6 8 10010203040 6 17 0 19 9 41 0 20 14 0 1 1 1 15 13 13 16 10 24 1 13

0 2 4 6 8 100102030405060

9 19 16 9 15 17 13 28 7 7 19 11 13 25 28

0 2 4 6 8 10010203040506012 1315 13 14 4 7 125 9 11 10 12 7 5 9 3 9 4 11 10 9 7 8 11

0 2 4 6 8 100102030405018 29 12 19 14 15 16 19 28 21 11 21 28

Figure 5.8: Stationary single-cell mRNA data does not give information about tran-scription initiation mechanism. A) Experimentally determined mRNA distributions donot distinguish qualitatively-different transcription initiation mechanisms. Experimentaldata for PDR5 from (Zenklusen et al., 2008) (grey bars) and the mRNA distributions ofthe fitted models: i. red: Non = 1,No f f = 1, ii. blue: Non = 1,No f f = 10, iii. green:Non = 5,No f f = 5, magenta: Non = 10,No f f = 1, orange: Non = 10,No f f = 10. B)Burst size distributions vary greatly between the fitted models. Same color coding of themodels as in A. C) Time-traces for the models described in A. The colored lines show onand off states, the black lines are time-traces of simulations with the Gillespie algorithm,and the numbers on top indicate burst sizes.

256

a time series. All synthesis events separated by a period longer than the meanburst duration, τb, were scored as different on-phases. As shown in Figure 5.9a,this classification of on and off states based on the burst duration is correct morethan 95% of the time. Estimates of the distribution functions fon and fo f f (de-scribing the durations of on and off states, respectively) are very similar to theactual distributions used to simulate the time-traces (shown in Figure 5.9c and d).The advantage of using the sequence size function (Dobrzynski and Bruggeman,2009) to identify the on phases from the time-trace data is that no assumptionsare made about the shape of fon, fo f f and fini. Once the on and off phases areidentified, estimates for these distributions can then directly be obtained fromthe time-trace data. Since on phases which do not lead to initiation events cannotbe detected and since the beginning and end of an on phase are assigned to thetime of the first and the last initiation event of the phase, this methods tends toa slight underestimation of the duration of on phases and the mean burst sizeand an overestimate of the duration of the off phase. However, these effects arenegligible if on phases lead to many initiation events.

5.3 Discussion

Eukaryotic transcription regulation depends on dozens of proteins that assembleinto complexes and perform their activities on chromatin. During this processchromatin is highly dynamic, adapts and partially controls transcription initia-tion progression. On the basis of experimental evidence we here propose thattranscription initiation in eukaryotes resembles a molecular ratchet. The ratchetis composed out of a sequence of reversible protein complex formation eventsfollowed by irreversible histone modifications that sensitize chromatin for theformation of the next protein complex in line. This mechanism is constrainedby the chromatin environment and intrinsic limitations of single-molecule pro-tein complex formation. The molecular ratchet design facilitates transcriptionalregulation with high flexibility.

The default state of a regulated eukaryotic promoter is a repressed state (Schoneset al., 2008). Upon activation, several tasks need to be accomplished before tran-scription can commence, such as histone tail modification and remodeling of thenucleosomes around the promoter. This requires the involvement of multipleproteins with various activities. One possible mechanism would be to rely on asingle protein complex, as has been proposed in the transcription factory concept

257

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Figure 5.9: The life-time statistics of the switch can be obtained from time-tracedata of mRNA. A) The sequence size function of the stochastic time trace (blue) andthe fraction of time that is correctly classified as belonging to an on or off phase wherethe classification is based on the time scale (υ; dimensionless time as unit) (green). B-D:Distributions obtained from a classification into on and off states based on τb (blue) andthe exact distributions (red). B) burst size distribution. C) First-passage-time pdf of theon state duration, fon. D) First-passage-time pdf of the off state duration, fo f f .

258

(Jackson et al., 1993). The feasibility of this mechanism is questionable giventhe thermodynamic and kinetic constraints on higher order protein complex for-mation. For instance, Figure 5.3 indicates that the formation of a four proteincomplex already can take several minutes; a complex of tens of proteins wouldthen take easily more than 30 minutes to form. These long times derive fromthe fact that a single copy of the complex is being made. We cannot rule outthat chromatin folding could in some way stabilize protein complex formationbut we have found no evidence for that. The alternative explanation is that anumber of smaller complexes perform specific tasks. This mechanism has manyadvantages, some of them we will discuss in more detail. One requirement of thesequential design is that it introduces the necessity to have progression marks toguide the formation of the proper complexes in sequence. Those marks shouldbe long-lived, at least longer than the protein complexes. We proposed a ratchetdesign with histone tail modification for progression marking (cf. Figure 5.2 andmain text, see also (Reid et al., 2009)).

The cyclic nature of transcription initiation may have to do with updating thetranscription activity to changes in the transcription factor concentration. Al-though a single and prolonged event of chromatin opening upon transcriptionfactor activation could be beneficial and lead to a high transcription rate, it wouldalso decouple transcription rate from the actual concentration of these transcrip-tion factor. Thus, temporary deactivation of the promoter and a (partial) tran-scription reinitiation appears necessary to update the system to changes in tran-scription factor levels. This makes the transcription initiation mechanism cyclic.Such cyclic designs are prone to display bursty activity. This sets the eukary-otic transcription mechanism apart from the canonical prokaryotic design wherethe observed bursts typically arise under conditions of transcriptional repressionrather than activation (Yu et al., 2006; Cai et al., 2006). In agreement with this,it has been shown experimentally that active eukaryotic genes tend to be tran-scribed in large and infrequent bursts (Raj et al., 2006; Chubb et al., 2006). Infact, it has been already been suggested that the noise and bursting in eukary-otic transcription stem from the fluctuations in the chromatin state (Raj and vanOudenaarden, 2008), however, these changes have often been viewed as exter-nally imposed (Karpova et al., 2008; Raser and O’Shea, 2004) rather than anintegral part of transcription initiation itself.

A consequence of the sequential nature of the ratchet design is that it is highlymodularized, which allows for process decoupling and the integration of gene

259

specific and general transcription activities, e.g. RNA polymerase assembly.This modular design may also facilitate the evolution of the eukaryotic transcrip-tion regulation as single processes within the entire transcription mechanism canbe fine-tuned without affecting other functions.

A stochastic molecular ratchet has not been analyzed before in theoretical stud-ies of transcription as far as we know. It has however been explored in relationwith biological cellular motor protein complexes such as ATPase and actomyosincomplex (Takano et al., 2010; Jr. et al., 1998). The stochastic molecular ratchetin transcription initiation has different design properties compared to those sys-tems. The main consequence is that at a coarse-grained level of description theratchet model has state transitions with non-exponential waiting times. We foundthat the assembly time statistics for protein complex formation can often be ap-proximated by a Gamma distribution and parameterized by the mean time andnoise of the exact mechanism. Since, few of those mechanisms have been studiedexperimentally this approximation is very useful. In addition, it illustrates thatthe precise molecular mechanisms of complex formation are not so importantfor the basal assembly statistics. We did find that random assembly mechanismsare significantly faster than sequential mechanisms with only a relatively smallincrease in waiting time noise.

Regulation of eukaryotic transcription occurs on single sites on the chromosome,i.e. on response elements and transcription start sites. It relies on the activity ofsingle protein complexes on chromatin.

Most of the studies on transcription so far have considered gene switching withexponentially distributed waiting times between on and off states, such as thosethat describe prokaryotic transcription. Theory that deals with switches withnon-exponential waiting times and the consequences for noise in mRNA levelshas been lacking. Introductions of the non-exponential waiting time distributionsof on and off transitions in the mRNA noise description (Pedraza and Pauls-son, 2008; Singh and Hespanha, 2009) have been limited to certain parameterregimes. In this work, we extend those studies and provide a method to de-scribe the moments of steady-state mRNA distribution for arbitrary distributedon and off waiting times. We found that Gamma-distributed switching timesoffer a good approximation to the waiting time distributions for most of the rele-vant assembly mechanisms. We expect that many eukaryotic genes may functionin a near-deterministic regime for which we derived the stationary mRNA-copynumber distribution. This is a low mRNA-noise limit of the molecular ratchet

260

model. We also investigated the interplay between mRNA noise, burst statistics,and molecular ratchet design. More steps in the ratchet mechanism lead to areduction in mRNA and burst size noise. Noise in burst size increases mRNAnoise. These effects are most pronounced for parameter regimes expected tobe most relevant for active eukaryotic genes; at moderate off to on life timeratios. The methods we propose are very general and can be extended to spe-cific non-exponential waiting time distributions in cases where there is sufficientknowledge of the transcription initiation mechanism.

Despite the amount of indirect evidence for the ratchet-like mechanism of eu-karyotic transcription and resulting non-exponential waiting time distributions ofgene switching, measurements of the consequences of this design for transcrip-tion activity are still challenging. In addition, the stochastic nature of transcrip-tion, in some cases involving bursts, cannot be deduced from population levelexperiments, which further complicates experimental studies. At present, single-cell analysis of transcription is fairly limited. FRAP can be used to monitor pro-tein life times of chromatin and to capture qualitative aspects of protein complexformation (Phair et al., 2004). RNA FISH and single-cell qPCR allow for thedetermination of the distribution of mRNA across a cell population (Raj et al.,2006; Zenklusen et al., 2008; Femino et al., 1998; Pare et al., 2009; Bengtssonet al., 2005, 2008; Wagatsuma et al., 2005; Warren et al., 2006). Although, copynumbers of mRNA can be measured over time, using the MS2-hairpin RNA-labeling method, such studies are still limited to only a few systems and are notas straightforward as for instance RNA FISH, which does not require genetic en-gineering. The advantage of RNA FISH (Raj et al., 2006) is that it can be readilyapplied (Zenklusen et al., 2008). The question is what can be learned from RNAFISH data about transcription initiation and its regulation? Our analysis showedthat the experimental steady state mRNA distributions can be successfully fittedwith both exponential and non-exponential switches, indicating that RNA FISHcannot distinguish between alternate transcription initiation mechanisms. Ouranalysis suggests that real-time monitoring of mRNA copy numbers, using forinstance MS2-hairpin labeled RNAs, can provide information about switchingwaiting time and burst size distributions. To obtain robust statistics these studiesshould be continued for several hours. This could complicate the interpretationof the experimental data as it is doubtful whether the activity of the gene willremain the same over this long period because of system-level feedbacks on thegene via its cis-regulatory regions.

261

The proposed molecular ratchet model of transcription initiation has a numberof implications for regulatory mechanisms of gene expression. FRAP data sug-gests that transcription factors will stay on chromatin for several minutes at most(Phair et al., 2004). ChIP data indicates that transcription factors are not presenton chromatin throughout the entire transcription cycle (Metivier et al., 2003). Itappears that for transcription factors to have an influence on transcription activitythey should regulate rate-limiting steps in the off phase of the transcription cycle.In this manner, the (average) off duration would decrease causing an increase inthe average transcription rate; defined as 〈b〉/(τon + τo f f ). At low levels of thetranscription factor, the off duration will be long and limited by transcriptionfactor regulated ratchet transition(-s). If only one reaction is regulated by thetranscription factor, this reaction would become rate limiting under these con-ditions, resulting in a life time distribution of the off state that approximates anexponential waiting time distribution. This would lead to more noisy (leaky)gene expression. Under saturating transcription factor levels, when the rate ofthe regulated reaction becomes comparable to other transitions in the ratchet, thewaiting time distribution would become more peaked (and with decreased av-erage waiting time), and reduced mRNA noise would result. Variable kineticsof protein complex formation or protein levels could in principle influence thosetrends.

Another mode of transcription regulation is that transcription factors do not influ-ence one step in the ratchet but many steps. The data for this is scarce; it is knownthough that some TFs can bind multiple histone modifying and remodeling co-factors (Lee et al., 2002; Xing et al., 1995; Lee et al., 2009; Kwok et al., 1994;Feige et al., 2006). This would allow for frequent updating of the transcriptionactivity to actual changes in transcription factor levels and make the system morerobust to fluctuations in the concentrations of a single TF. Moreover, this designwould naturally allow the integration of regulation through multiple factors.Apart from regulation through modulation of the off duration, transcription ratecould in principle also be regulated through either the on duration or the rate oftranscription initiation during the on state. Regulation of the on duration wouldimply increased burst sizes caused by prolonged on durations for activated genes.Regulation of transcription rate through the rate of transcription initiation wouldlead to changes in burst size while total cycle duration would remain fixed. Thissuggests that silent genes can display transcription cycles with no or few mRNAproductions per cycle. We conclude that transcription factors have to at least ac-

262

tivate some step in the off phase of the cycle, and in addition could regulate burstsize and on duration.The molecular ratchet model is a step towards quantitatively modeling eukary-otic gene regulation. Many phenomena are not considered in it yet. For instance,we do not consider multi-step degradation of the mRNA which was shown insome cases for mRNAs with polyA tails (Tourriere et al., 2002), nor any of theeffects that stochasticity in the elongation process may introduce (Dobrzynskiand Bruggeman, 2009), i.e. pausing of RNA polymerases. TF binding sites areoften situated far from the TSS making transcription reliant on chromatin loop-ing (Sipos and Gyurkovics, 2005). The statistics of such looping events and theirtiming has not been considered but assumed comparable to average ratchet tran-sitions. Often multiple response elements, transcription start sites and transcrip-tion factors binding site are involved in the regulation of a single gene (Hallikaset al., 2006). This would lead to the interplay of multiple transcription cyclesthat compete for access to transcription start sites and hinder each other throughchromatin folding and looping. All of these phenomena will have their effectson transcription acitivity regulation and transcription stochasticity. The modularnature of the ratchet model should allow for extension in those directions by in-corporating ratchet transitions with mechanisms or coarse-grained waiting timedistributions corresponding to for instance chromatin looping and folding. In ad-dition, the refinement and application of existing real-time single-cell techniqueswill give more insight into the basal design of eukaryotic transcription, its regu-lation and consequences for cell function.

263

Appendix 5

5.S.1 Molecular ratchet: reconstruction from the literature

Here we describe in more detail the reconstructed cyclic sequence of nucleo-some modifications in the transcription cycle found in yeast. It is well estab-lished that many yeast transcription factors such as Gcn4p and Gal4 interact withcomplexes SAGA and NuA4 that contain histone acetylases Gcn5 and Esa1, re-spectively (Drysdale et al., 1998; Larschan and Winston, 2005; Herbig et al.,2010). This leads to the acetylation of several sites in the H3 histone tail, namelyH3K14, H3K9 and H3K18 by Gcn5, as well as acetylation of K5, K8 and K12of the H4 tail by Esa1 (Mitarai et al., 2008). Acetylation of H3K14 has beenshown to enhance binding and nucleosome-displacing activity of the remodel-ing complex SWI/SNF (Chandy et al., 2006). Both the acetylation of histonetails and the removal of nucleosomes from the TATA-box of the promoter con-tribute to establishing the initiation-permissive state. It was demonstrated thatthe basal transcription factor Bdf1, which promotes binding of TFIID, has highaffinity for acetylated H4 (Durant and Pugh, 2006), and that histone displace-ment accompanies transcription activation (Moreira and Holmberg, 1998). Oncethe promoter becomes accessible, the initiation complex containing polymeraseis assembled; the elongation competent PolII is known to associate with thechromatin-modifying complex COMPASS responsible for the H3K4 mono-, di-and tri-methylation (Gerber and Shilatifard, 2003). These histone marks pro-mote the deactivation of the promoter in several ways. H3K4 di-methylation hasbeen shown to attract the Set3 complex (Kim and Buratowski, 2009) containingHOS2 and Stp1 enzymes capable of deacetylating the H3 and H4 lysine residues(Wang et al., 2002). Both di- and tri-methylated H3K4 facilitates the binding ofthe nucleosome-remodeling complex Isw1 (Santos-Rosa et al., 2003), which hasbeen implicated in limiting gene activation - one might speculate through revers-ing the remodeling that caused exposing the TATA-box. In order to complete thetranscriptional cycle and return the promoter to the original state demethylationof H3K4 is required. An enzyme KDM1 has been demonstrated to have rele-vant activity (Ingvarsdottir et al., 2007), however, its specificity towards histonemodifications has not been described so far.

264

5.S.2 Waiting Time Distributions

For the example of the partially random mechanism 2 (Fig 5.3A) we demonstratehow the first passage time distribution for complex formation can be calculatedin different ways. Assuming that all reactions are irreversible, it is possible toderive this PDF as a convolution of PDFs for each binding step. The respectiveprobabilitie densities for each step are:

pstep1(t) = (k1a + k1b)e−(k1a+k1b)t

pstep2(t) =k1a

k1a + k1b× k2ae−k2at +

k1b

k1a + k1b× k2be−k2bt

pstep3(t) = (k3a + k3b)e−(k3a+k3b)t

pstep4(t) =k3a

k3a + k3b× k4ae−k4at +

k3b

k3a + k3b× k4be−k4bt (5.S.1)

The convolution of these PDFs equals the first passage time distribution of com-plex formation and can be expressed in form of its laplace transform:

L(p(t)) = f (s) = L(p(step1)) × L(p(step2)) × L(p(step3)) × L(p(step4))(5.S.2)

The moments of the first passage time distribution are then given by

〈ta〉 = (−1)a da f (s)dsa |s=0. (5.S.3)

If all the rate constants are equal the full PDF can be determined:

p(t) = 4 exp−2kt k × (2 + kt + expkt ×(kt − 2)) (5.S.4)

Alternativly the first passage time distribution can be described as a phase typedistribution. The continuos phase-type distribution for partially random mecha-

265

nism 2 can be written as

p(t) = αeS tS 0 (5.S.5)

S =

−k1a − k1b k1a k1b 0 0 00 −k2a 0 k2a 0 00 0 −k2b k2b 0 00 0 0 −k3a − k3b k3a k3b0 0 0 0 −k4a 00 0 0 0 0 −k4b

α = (1, 0, 0, 0, 0, 0, 0, 0)T

S 0 = S (1, 1, 1, 1, 1, 1, 1, 1)T ,

with the first two moments given by:

〈t〉 = −αS −11 (5.S.6)〈t2〉 = 2αS −21 (5.S.7)

The elements S i j of the matrix S equal the rate constants for the reaction leadingfrom state i to state j. This naturally allows including reversible reactions. Asshown in figures 5.S.1 and 5.5 reversibility of reactions has a significant effecton the shape of the first passage time distribution if the apparent association ratesare less than an order of magnitude larger than the dissociation rates.

However, even though reversibility can significantly change the first passage timedistributions for individual ratchet transitions, the convolution of several suchdistributions (describing the distribution of the life time of an on or off state)can for biologically realistic parameters be approximated reasonably well by agamma distribution that has the same mean and variance (fig 5.S.2).

5.S.3 Burst-Size Distributions

The simplest model for a switching gene is one that has an exponential distri-bution of the duration of its on-state as well as for its time to prepare a RNApolymerase for promotor escape. If the latter event is modeled with a first-orderreaction, with a rate constant ktr, the probability that b mRNAs are producedduring a time span ton is given by a Poisson distribution (from equation 5.2),

pb(B = b|Ton = ton) =(ton · ktr)be−ktrton

b!(5.S.8)

266

In this model, the distribution of life times for the on-state is also modeled withan exponential distribution, fon(t) = ko f f e−ko f f t. The resulting burst-size distribu-tion becomes a geometric distribution (using equation 5.5),

pb(B = b) = q(1 − q)b, (5.S.9)

with q = ko f f /(ko f f + ktr) as the probability of an on to off transition.. The meanburst size and its noise (squared coefficient of variation) are given by:

〈b〉 =ktr

ko f f=τon

τtr(5.S.10)

〈δb2〉

〈b〉2= 1 +

1〈b〉

(5.S.11)

As discussed in section 5.2.2, a more realistic description of the distribution oftimes spent in the on state by an eukaryotic gene takes into account the multi-step nature of the segment of the molecular ratchet model that corresponds to thetransitions from the onset of its transcription-permissive states (green region infigure 5.1) to its first transcription-silent state (off state). Readily interpretableanalytical results can be obtained if the transition from on to off is modeled asa sequence of Non irreversible first order reactions with equal rate parameters.Such a chain of reactions has an Erlang distributed waiting time. This model isthe simplest non-exponential waiting time distribution and so should highlightthe effects of non-exponential waiting times most clearly. On top of that, in sec-tion 5.2.2, we concluded that a gamma distribution, closely related to the Erlangdistribution, approximates the overall waiting time distribution of a sequence ofrealistic ratchet transitions. The main consequence of those distributions is thatthey can have a much smaller dispersion than exponential distributions.

Assuming an Erlang distributed life time for the on state, fon(t), and substitutingequation 5.S.8 into equation 5.5, yields the following burst size distribution:

pb(B = b) =

(b + Non − 1

b

)qNon (1 − q)b (5.S.12)

which is a negative binomial distribution. This distribution is qualitatively dif-ferent from equation 5.S.9. It is a peaked distribution. Non-exponential waitingtimes for the lifetime of the on state introduce completely new burst behavior. Its

267

mean burst-size and noise are given by

〈b〉 = Nonktr

ko f f

〈δb2〉

〈b〉2=

1Non

+1〈b〉

. (5.S.13)

The mean burst size can again be understood in terms of the mean life time of theon state (Non/ko f f ) and the mean initiation time (k−1

tr ). The noise in the burst sizedecreases with the number of steps in the Erlang distribution. This indicates thatthe cell-to-cell heterogeneity of burst size across an isogenic population of cellsis small for genes that display little dispersion of the duration of the on state.If a single reaction is rate limiting in the overall transition mechanism from theon-to-off state this model reduces to the previous model.

To highlight the main difference of incorporating a non-exponential initiationtime distribution we again take the simplest non-exponential distribution, the Er-lang distribution. Instead of calculating the probability of the number of mRNAproduction events that occur before the gene switches to the off state, the proba-bility of a certain number of synthesis steps is calculated (using equation 5.S.8).The probability for b mRNA production events can then be calculated from,

pb(B = b) =

(b+1)Ntr−1∑i=bNtr

(∫ ∞

ton=0

(ton · ktr)ie−ktrton

i!f (ton)dton

), (5.S.14)

where Ntr is the number of steps in the transcription initiation mechanism. Foran exponential waiting time distribution for the on-state life time, the burst sizedistribution equals

pb(B = b) = (1 − q)bNtr (1 − (1 − q)Ntr ), (5.S.15)

where q is again defined as ko f f /(ktr + ko f f ). This is again a geometric dis-tribution: changing the number of steps of mRNA synthesis alone does notchange the shape of the burst size distribution in comparison to a model withonly exponential waiting times (equation 5.S.9). The probability that the nextevent is transcription initiation (and not switching to the off state) changes to

268

(1− q)Ntr = (ktr/(ktr + ko f f ))Ntr . The mean and noise of the burst size are given by

〈b〉 =1( ktr+ko f f

ktr

)Ntr− 1

〈δb2〉

〈b〉2=

(ktr + ko f f

ktr

)Ntr

= 1 +1〈b〉

. (5.S.16)

The burst noise for this model is the same as the fully exponential model. It onlydepends on the mean burst-size and does not vary with the number of steps inthe first-passage time distribution for the transcription initiation up to promoterclearance, Ntr. Eukaryotes do not gain much more control over their burst sizestatistics using this molecular ratchet design. Multiple steps in the on to offtransition are a much more potential mechanism for burst size control.

The most general ratchet model for burst size is when both the life time of the onstate and the initiation times are non-exponentially distributed. For this case, theburst size distribution obeys

pb(B = b) =

(b+1)Ntr−1∑i=bNtr

(i + Non − 1

i

) (ktr

ktr + ko f f

)i (1 −

(ktr

ktr + ko f f

))Non

(5.S.17)

with Non sequential steps for the transition from on to off and Ntr steps for ini-tiation of mRNA synthesis. Note that even though changing only the waitingtime distribution for initiation from exponential to Erlang does not change theshape of the distribution in comparison to both distributions being exponential,the combination of both Erlang distributions results in a different shape than ei-ther one alone. The effect of Ntr is largest for small average burst sizes. Thiscan be understood from Equation 5.S.17: the number of elementary synthesissteps follows a negative binomial distribution, the probability distribution for thenumber of mRNAs is derived from the negative binomial distribution by sum-ming the probabilities for the Ntr possible number of initation steps that result inthe same number of mRNAs, i.e the number of steps i, for which the floor of i

Ntr

equals b. If the average number of initiation steps is large, the floor function canbe approximated by i

Ntrand then the burst size distribution is well described by

a negative binomial distribution with 〈b〉 = NonktrNtrko f f

. The noise in burst size thenequals 1/〈Non〉 + 1/〈b〉. In general, the noise for this model is always larger thanthe (not so stringent) lower bound, ktr−ko f f

ktr

NtrNon

.

269

5.S.4 Moments of steady-state mRNA distributions - instan-taneous burst model

First we consider a simplified model for a system with very pronounced burststhat can be approximated as producing a number of B transcripts instantaneously,where B is drawn from a general burst size distribution. The time intervals be-tween bursts are described by the distribution of the life times of the off state andassumed to be independent of the burst size (in queuing theory this descriptioncorresponds to a G/M/∞ queue with batch arrival). For exponentially distributedlife times, the noise in mRNA levels for such a burst model can be describedby the use of a stochastic hybrid system and Dynkin’s formula (Singh and Hes-panha, 2009) (which allows solving for steady state moments as well as describ-ing the time evolution of moments) or through the use of the chemical masterequation in combination with moment equations (Pedraza and Paulsson, 2008)(exact solution for the steady-state mRNA noise and an approximation for sys-tems with non-exponentially distributed life times of the off state).

The method presented here consists of two steps: first, solving a linear system ofk equations to determine the first k raw moments of the mRNA distribution at thetime of a burst and second deriving the moments of the full steady-state mRNAdistribution from this. The evolution of the moments of the mRNA distributionduring the off state can be described relatively easily, because degradation is theonly process that can occur. Since each mRNA molecule has the same probabil-ity to be degraded within a certain time t, the number of mRNAs, Nt, after timet given that no new burst has occured yet, is distributed according to a binomialdistribution, where the initial number of molecules N0 that are present at the be-ginning of the off state and the duration t of the off state are random variablesthemselves, described by pini(N0 = n0) and fo f f (t) respectively:

pdeg(Nt = nt|N0 = n0) =

(n0

nt

)e−ntkdegt(1 − e−kdegt)n0−nt (5.S.18)

and therefore the average distribution at the end of an off state is given by


∫ ∞

t=0pdeg(Nt = nt|N0 = n0) fo f f (t)dt. (5.S.19)

270

At steady state the convolution of this distribution with the burst-size distribu-tion must equal pini(N0 = n0). However, for most combinations of burst-sizedistributions and distributions for the life time of the off state pini(N0 = n0) canonly be approximated numerically from this equation, whereas a linear systemof k equations can be obtained for the first k moments of pini. Using Burgess’variance theorem (and its extension to higher moments), the first k moments ofpdeg can be expressed as a function of the first k moments of pini:

〈nαt 〉(t) =

∞∑n0=0

n0∑nt=0

nαt pini(N0 = n0)pdeg(Nt = nt|N0 = n0) (5.S.20)

〈nt〉(t) = e−kdegt〈n0〉

〈n2t 〉(t) = e−2kdegt〈n2

0〉 + e−kdegt(1 − e−kdegt)〈n0〉

Assuming a gamma distribution with shape parameter No f f and rate parameterkon for the distribution of the life time of the off state, fo f f (t), the time averagedmoments at the end of an off state equal:

〈nt〉 =

(kon

kon + kdeg

)No f f

〈n0〉

〈n2t 〉 =

(kon

kon + kdeg

)No f f

〈n0〉 +

(kon

kon + 2kdeg

)No f f (〈n2

0〉 − 〈n0〉). (5.S.21)

Moments of convolutions of distributions can be calculated by first convertingthe moments of the individual distribution to cumulants, adding the cumulants toobtain the cumulant of the convolution and then converting the cumulant back tothe moment (Mueller, 1982). Denoting the first and second moment of the burstsize distribution with 〈b〉 and 〈b2〉, the first two moments of the convolution ofthe mRNA distribution at the end of an off state with the burst size distribution(which at steady state have to equal 〈n0〉 and 〈n2

0〉) are given by:

〈n0〉 = 〈nt〉 + 〈b〉〈n2

0〉 = 〈n2t 〉 + 〈b

2〉 + 2〈b〉〈nt〉 (5.S.22)

271

Substituting equations 5.S.21 into eq. 5.S.22 allows solving for the first twomoments of the mRNA distribution at the time of a burst, pini:

〈n0〉 =〈b〉

1 −(

konkon+kdeg

)No f f

〈n20〉 =

〈b2〉 + kNo f f((1 + 2〈b〉)(kon + kdeg)−No f f − (kon + 2kdeg)−No f f

)〈n0〉

1 −(

konkon+2kdeg

)No f f

(5.S.23)

Equations 5.S.21, 5.S.21 describe how these moments change over time. Themoments of the full mRNA distribution can therefore be obtained by averagingover the moments at times t weighted according to the probability that the nextburst has not yet occured at t, i.e. the survival probability ps(t) = 1−Fo f f (t) (withFo f f (t) as the cumulative distribution function of the distribution of off -state lifetimes):

〈n〉 =1〈to f f 〉

∫ ∞

t=0〈nt〉(t)(1 − Fo f f (t))dt

〈n2〉 =1〈to f f 〉

∫ ∞

t=0〈n2

t 〉(t)(1 − Fo f f (t))dt (5.S.24)

5.S.5 Moments of steady-state mRNA distributions - "Gammaswitch”

The simplified instantaneous burst model can be extended to one that explicitelytakes into account the duration(distribution) of the on state as well: transcriptioninitiation and mRNA degradation are described as first-order processes with rateconstants ktr and kdeg, and the distributions for the life times of on and off statesare described by general waiting time distributions. The reason for modelingtranscription initiation with an exponential distribution is that as was shown insection 5.2.3 the exact distribution for transcription initiation has a minor effecton the burst-size distribution and is therefore also expected to have little effect onthe steady state mRNA distribution. Depending on the time scales, the waitingtime distribution for degradation of mRNA can have large effects (simulations,results not shown), that can be difficult to predict intuitively. However, many

272

measurements of mRNA turnover in mammalian cells can well be fit with anexponential decay model (Lam et al., 2001).

Due to the switching between on and off states, the mRNA numbers fluctuateperiodically even when the system has reached steady state. By definition ofthe steady state the first and second moment of the mRNA distribution is thesame at all points where the system switches from on to off. Consider how thesemoments evolve between two consecutive points, i.e. within one cycle - theresulting moments have to equal the initial moments because both initial and finaltime points are at the time when the system switches from on to off (Fig S5(a)).The evolution of the moments of the mRNA distribution during the off statecan be described relatively easily, because degradation is the only process thatcan occur. Since each mRNA molecule has the same probability to be degradedwithin a certain time t, the number of mRNAs, Nt, after time t during an off stateis distributed according to a binomial distribution, where the initial number ofmolecules N0 is itself a random variable described by pini(N0 = n0). In FigureS5(b) the initial distribution is shown in red, the magenta distribution is obtainedfrom this through degradation during the off state.


(n0

nt

)e−ntkdegt(1 − e−kdegt)n0−nt (5.S.25)

During an on period degradation of the mRNA molecules that have been presentat the beginning of the burst continues and can simply be described by Equation5.S.25 (Fig. S5(b) green curve), but in addition mRNAs are also created anddegraded according to a birth-death-process with exponential waiting times (Fig.S5(b) blue curve). This can be described as an effective burst size, Be f f , since thisonly takes into account those mRNA molecules synthesized that survive at leastuntil the end of the burst. The transient distribution for this birth-death processas described in (Hemberg and Barahona, 2007) is given by:

pbirth/death(Be f f = be f f , t) =1

be f f !

(ktr

kdeg

(1 − e−kdegt

))be f f (e

ktrkdeg

(e(−kdegt)

−1))

(5.S.26)

In principle, it is possible to obtain an equation that defines the distribution ofN0: based on an initial distribution pini(N0 = n0) the distribution at the end of one

273

complete cycle can be computed as the convolution of Eq. 5.S.25 and 5.S.26 tak-ing into account the probabilities for the duration of each state. According to thethe definition of steady state, this convolution has to equal pini(N0 = n0). How-ever, in general it is quite difficult to obtain an analytical solution for pini(N0 =

n0) from this equation (for the special case of deterministic switching times, seeEqs. 5.S.42 to 5.S.49)). Therefore, instead of trying to calculate the completedistribution pini(N0 = n0), we derive here equations for the first two moments ofit, which can be solved analytically (the approach is also applicable to higher mo-ments in an analogous way). The moments of the distribution generated throughdegradation alone after one complete cycle are given by Burgess’ Variance theo-rem and equal

〈nαt 〉(ton, to f f ) =

∞∑n0=0

n0∑nt=0

nαt p(N0 = n0)pdeg(Nt = nt|N0 = n0, t = ton + to f f )

〈nt〉(ton, to f f ) = e−kdeg(to f f +ton)〈n0〉 (5.S.27)

〈n2t 〉(ton, to f f ) = e−2kdeg(to f f +ton)〈n2

0〉 + e−kdeg(to f f +ton)(1 − e−kdeg(to f f +ton))〈n0〉 (5.S.28)

These moments are a function of to f f and ton, however at this point only thedependence on to f f can be removed by multiplication with the probability densityfunction f (to f f ) of the waiting time distribution of the off -state and integration:

〈nt〉(ton) =

∫ ∞

0f (to f f )〈nt〉(ton, to f f )dto f f

〈n2t 〉(ton) =

∫ ∞

0f (to f f )〈n2

t 〉(ton, to f f )dto f f (5.S.29)

Moments for the birth-death-process do not depend on the distribution of N0 andare given by

274

〈be f f 〉(ton) =

∞∑be f f =0

be f f p(Be f f = be f f , ton) =ktr

kdeg(1 − e−kdegton) (5.S.30)

〈b2e f f 〉(ton) =

∞∑be f f =0

b2e f f p(Be f f = be f f , ton)

= −ktr

k2deg

(e−2kdegton

(ekdegton − 1

) (ktr − ekdegton(kdeg + ktr)

))(5.S.31)

The total distribution of mRNAs at the time of switching to the off state is givenby the convolution of the distributions that are generated by these two processes(degradation and the effective burst). The moments of the convoluted distributioncan be calculated from the moments of the individual distributions (Mueller,1982) and have to be equal to the moments of pini:

〈n0〉(ton) = 〈be f f 〉(ton) + 〈nt〉(ton)〈n2

0〉(ton) = 〈b2e f f 〉(ton) + 〈n2

t 〉(ton) + 2〈be f f 〉(ton)〈nt〉(ton) (5.S.32)

Substituting Equations 5.S.29 to 5.S.31 into Eq. 5.S.32 and integrating out thedependence on ton allows solving for 〈n0〉 and 〈n2

0〉. For models with with Gammadistributions for both transitions these equations can be solved analytically, how-ever the approach is not limited to any particular type of waiting time distribu-tion. With gamma distributions for both fo f f and fon one obtains:

275

〈n0〉g =

ktr

(1 −

( ko f f

kdeg+ko f f

)Non)

kdeg

(1 −

( ko f f

kdeg+ko f f

)Non ( konkdeg+kon

)No f f) (5.S.33)

〈n20〉g =

( ko f f

kdeg+ko f f

)Non(1 −

( ko f f

kdeg+ko f f

)Non) (

konkdeg+kon

)No f f(−1 +

( kdeg+ko f f

2kdeg+ko f f

)Non ( kdeg+kon

2kdeg+kon

)No f f)

ktr

kdeg

(1 −

( ko f f

kdeg+ko f f

)Non ( konkdeg+kon

)No f f) (−1 +

( ko f f

2kdeg+ko f f

)Non ( kon2kdeg+kon

)No f f)

−

(1 − 2

( ko f f

kdeg+ko f f

)Non+

( ko f f

2kdeg+ko f f

)Non)

k2tr(

−1 +( ko f f

2kdeg+ko f f

)Non ( kon2kdeg+kon

)No f f)

k2deg

−

ktr

(1 −

( ko f f

kdeg+ko f f

)Non+ 2

(( ko f f

kdeg+ko f f

)Non−

( ko f f

2kdeg+ko f f

)Non) (

konkdeg+kon

)No f f〈n0〉

)(−1 +

( ko f f

2kdeg+ko f f

)Non ( kon2kdeg+kon

)No f f)

kdeg

(5.S.34)

Having solved for the moments at the time of switching it is most convenient todetermine the moments of the steady state mRNA distribution during the on andoff state separately (Fig. 5(c)). To do so Eqs. 5.S.27 to 5.S.31 are used againto describe how the moments evolve during time. Each time point contributes tothe distribution according to the probability that it occurs before the transition tothe other state, therefore the time points are weighted according to their survivalprobability, ps(t) = (1 − F(t)) and the moments of the distribution of the numberof mRNA molecules during the off -state, No f f , are calculated as:

〈no f f 〉 =1〈to f f 〉

∫ ∞

0e−kdegto f f 〈n0〉ps(to f f )dto f f

〈n2o f f 〉 =

1〈to f f 〉

∫ ∞

0

(e−2kdegto f f 〈n2

0〉 + e−kdegto f f (1 − e−kdegto f f )〈n0〉)

× ps(to f f )dto f f (5.S.35)

276

With gamma distributed life times for both states this becomes:

〈no f f 〉g =kon

No f f

〈n0〉g

kdeg−

(kon

kdeg+kon

)No f f〈n0〉g

kdeg

(5.S.36)

〈n2o f f 〉g =

kon(〈n0〉g + 〈n20〉g)

2kdegNo f f

−

kNo f f +1on

(2(2kdeg + kon)No f f 〈n0〉g + (kdeg + kon)No f f (−〈n0〉g + 〈n2

0〉g))

2kdegNo f f (kdeg + kon)No f f (2kdeg + kon)No f f

(5.S.37)

The moments of the distribution of the number of mRNA molecules during theon-state, are given through integration of the convolution of the moments for thedegradation process and the birth-death process:

〈non〉 =1〈ton〉

∫ ∞

0(〈be f f (ton〉 + 〈nt(ton)〉) f (to f f )ps(ton)dton

〈n2on〉 =

1〈ton〉

∫ ∞

0(〈b2

e f f (ton)〉 + 〈n2t (ton)〉 + 2〈be f f (ton)〉〈nt(ton)〉)

× f (ton)ps(to f f )dton (5.S.38)

and with gamma distributed life times for on and off state:

277

〈non〉g =

−ko f f

(( ko f f

kdeg+ko f f

)Non− 1

) (kon

kdeg+kon

)No f f〈n0〉g

kdegNon

+ktr

(k1+Non

o f f + (kdeg + ko f f )Non(−ko f f + kdegNon))

k2degNon

(5.S.39)

〈n2on〉g =

ko f f k2tr

2k3degNon

−ko f f

(kon

kdeg+kon

)No f f〈n0〉g

2kdegNon+

ko f f kNo f fon (kdeg + kon)−No f f 〈n0〉g

2kdegNon

−ko f f

(kon

kdeg+kon

)No f fktr〈n0〉g

k2degNon

+

ko f f (kdeg + 2ktr)(−ktr + kdeg

(kon

kdeg+kon

)No f f〈n0〉g

)k3

degNon

− (ko f f kNo f fon (2kdeg + kon)−No f f (〈n0〉g − m2))/(2kdegNon)

+2kdegko f f (ko f f (2kdeg + ko f f ))Nonktr − ko f f (ko f f (kdeg + ko f f ))Nonk2

tr

2k3degNon((kdeg + ko f f )(2kdeg + ko f f ))Non

+4ko f f (ko f f (2kdeg + ko f f ))Nonk2

tr + 2kdeg((kdeg + ko f f )(2kdeg + ko f f ))Nonk2trNon


−2k2

degko f f (ko f f (2kdeg + ko f f ))Non(

konkdeg+kon

)No f f〈n0〉g


+k2

degko f f (ko f f (kdeg + ko f f ))Non(

kon2kdeg+kon

)No f f〈n0〉g


+2kdegko f f (ko f f (kdeg + ko f f ))N

on(kon/(kdeg + kon))No f f ktr〈n0〉g


−4kdegko f f (ko f f (2kdeg + ko f f ))Non

(kon

kdeg+kon

)No f fktr〈n0〉g


−k2

degko f f (ko f f (kdeg + ko f f ))Non(

kon2kdeg+kon

)No f f〈n2

0〉g


+2k2

deg((kdeg + ko f f )(2kdeg + ko f f ))NonktrNon


(5.S.40)

278

The mean and second moment of the complete steady state distribution can beobtained from this as

〈n〉 =〈to f f 〉〈no f f 〉 + 〈ton〉〈non〉

〈to f f 〉 + 〈ton〉

〈n2〉 =〈to f f 〉〈n2

o f f 〉 + 〈ton〉〈n2on〉

〈to f f 〉 + 〈ton〉(5.S.41)

5.S.6 Steady State mRNA Distribution of a “Deterministic Switch”

As a limit to very precise life-time distributions for both states, these durationscan be considered deterministic. The noise in the steady-state mRNA distributionthen derives from the birth-death process that occurs during the on-state and theongoing degradation during the off state. Modeling synthesis and degradation ofmRNA again with exponential waiting times, the steady-state mRNA distributionof this ’deterministic switch’ can be solved: the probability distribution of theeffective burst size is given by equation 5.S.26; together with equation 5.S.18one can calculate the probability distribution of the number of mRNAs, Nx, thatare produced during one burst and are still present after X complete cycles afterthe end of that burst:

px(Nx = nx|X = x) =e−e−kdeg(to f f +ton)x

ktr+e−kdeg(ton+(to f f +ton)x)

ktr−k2degn(to f f +ton)x

kdeg

n!(

ktr−e−kdegton ktrkdeg

)−n (5.S.42)

The steady state distribution at the time of switching to the off state can be cal-culated as a convolution of these probability distributions, px, for x from 0 toinfinity. To calculate this convolution it is helpful to calculate the probabilitygenerating function, G(n,x)(z) of px first and then take the product of the generat-ing functions to obtain the generating function of N0.

G(n,x)(z) = ee−kdeg(ton+(to f f +ton)x)(

−1+ekdegton

)ktr (−1+z)

kdeg (5.S.43)

Gn0(z) = Π∞x=0G(n,x)(z) = e

ekdegto f f

(−1+e

kdegton)ktr (−1+z)(

−1+ekdeg(to f f +ton))

kdeg (5.S.44)

279

This is the generating function of a Poisson distribution with rate

λ1 =ekdegto f f

(−1+ekdegton

)ktr(

−1+ekdeg(to f f +ton))kdeg. Similarly, the distribution at the time of switching form

the off to the on state is also a Poisson distribution with λ2 =

(−1+ekdegton

)ktr(

−1+ekdeg(to f f +ton))kdeg.To

obtain the steady state mRNA distributions during off and on state, again oneneeds to consider how these distributions evolve and average over all possibletimes during on- and off -state. During the off state the distribution evolves ac-cording to Eq. 5.S.25. With pini(N0 = n0) described by a Poisson distributionwith λ1 the distribution at time t is given by:

pdeg(N = n, t) =

∞∑n0=0

λn01

n0!e−λ1

(n0

nt

)e−ntkdegt(1 − e−kdegt)n0−nt

=λn

1

n!Exp

[−λ1ekdegt − nkdeg

](5.S.45)

The distribution during the on state is given as the convolution of the effectiveburst size distribution at time t and the distribution of mRNA that were presentat the beginning of the on state and are not yet degraded. The former distributionis described by Eq. 5.S.26, while the latter can be expressed as (analogously toEq. 5.S.45):

pdeg2(N = n, t) =λn

2

n!Exp

[−λ2ekdegt − nkdeg

](5.S.46)

Their convolution equals:

pconv(N = n, t) =e(−1+e−kdegt) ktr

kdeg−e−kdegt

λ2−kdegnt

n!(

ktrkdeg

(ekdegt − 1

)+ λ2

)−n (5.S.47)

Since the life times of both states are deterministic each time between 0 and to f f ,or between 0 and ton occurs once before the system switches. The distributionsduring on and off states are therefore described by:

280

po f f (N = n) =1

to f f

∫ to f f

t=0pdeg(n = n, t)dt

pon(N = n) =1ton

∫ ton

t=0pconv(N = n, t)dt (5.S.48)

The complete mRNA distribution is obtained as an average of the distributionsduring on- and off -states:

p(N = n) =po f f (n)to f f + pon(n)ton

to f f + ton(5.S.49)

5.S.7 Comparison of different switch models

Since the exponential and deterministic switch models are opposing limits tomodels with Erlang (or gamma) distributions for the durations of the on and offstates, the steady-state mRNA distribution of any Erlang switch model can beexpected to be an intermediate between the two distributions for the exponentialand deterministic switch models, that have the same average life time for eachstate as well as the same rate constants for transcription initiation and mRNAdegradation. For three different Erlang switch models, each with a total of Non +

No f f = 10 steps and eight different combinations of ton and to f f that exemplifythe different possible shapes of mRNA distributions (Fig. 5.7), the steady-statemRNA distributions were obtained from simulations with the Gillespie algorithmand compared to the distributions for the exponential and deterministic switch interms of the Kullback-Leibler divergence index:

With at least five steps for each transition the resulting mRNA distribution ismore similar to the distribution generated by a system with deterministic switch-ing times than that of a system with exponential switching times. If one of thetransitions occurs in only one step, the resulting distribution is much better de-scribed by an exponential rather than by a deterministic system.

5.S.8 Comparison to Experimental Data

The mRNA distributions measured by Zenklusen et al. (Zenklusen et al., 2008)were fit to different Erlang switch models (Non = 1,No f f = 1: red, Non =

281

1,No f f = 10: blue, Non = 5,No f f = 5: green, Non = 10,No f f = 1: magenta,Non = 10,No f f = 10: orange) by calculating setting kdeg = 1 and choosing ktr,kon, and ko f f to give rise to a steady-state mRNA distribution with the first threemoments equal to the experimentally observed distribution. The full mRNAdistributions at steady-state were then obtained for these models through simu-lations. The parameters used for these simulations can be retrieved from Tables5.S.2 and 5.S.3 (kon = No f f (1 + 〈ton〉)/〈to f f 〉(〈ton〉 + 〈to f f 〉), ko f f = Non〈ton〉, ktr =

〈b〉/〈ton〉). Figures 5.8 and S5.S.8 to 5.S.11 show these distributions in com-parison to the experimental data, the burst size distribution for each model andsimulated time traces of mRNA copy numbers. Even though the steady-statedistributions are remarkably similar for all models, the underlying mechanisms,burst-sizes and time traces differ significantly.

282

A B

0 20 40 60 80 1000.00

0.02

0.04

0.06

0.08

t,min

k =k+ -k =2k+ -k =10k+ -k =0-

p(t)

Figure 5.S.1: Effect of reversibility on the waiting time distribution. A. waiting timePDFs for preferentially random mechanism 2, effective k+ = 0.3, irreversible case (red)and k+/k− ratios 0.1 (orange), 0.5 (green) and 1 (blue). B. Kullback-Leibler divergenceindex for comparing irrversible case with three k+/k− ratios (red), 0.5 (orange) and 1(yellow) for three meachanisms Seq, PR2 and Ran.

A B

Seq

PR 2

PR 4PR 6

Seq

PR 2

PR 4PR 6

Figure 5.S.2: Waiting time distributions for multiple transition mechansims canbe successfully approximated by Erlang distribution with same mean and noiseproperties A. Schemes of four mechanism used for simulations; all protein reactionswere taken reversible, parameters generated by random sampling from uniform distri-bution with the following ranges: k+, k− and kmod - 10−4 − 100s−1, 10−2 − 10−1s−1

and 10−2 − 10−1s0 , respectevly. B. Mean and standard deviation of Kullback-Leiblerdivergence index distribution resulting from comparing the true distributions and approx-imations by Erland distribution. Simulation were performed for single transition and aconvolution of 3 and 5 transitions allowing random combination of four mechanisms in(A). For each case 1000 simulations were run.

283

A B

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

toff�tdeg

X∆b

2

eff

\�Xb

eff

\2

0 10 20 30 40 50 60

0.00

0.02

0.04

0.06

0.08

effective burst size HbeffL

PM

F

Figure 5.S.3: Effect of degradation on the noise in burst size. A. Dependence ofthe burst size noise on the degradation rate. The noise was calculated for the gamma-distributed on time (Non = 5) (red), to f f = 1 with kdeg increasing from 0 to 10, ktr

adjusted to keep the burst size at 25. The calculation was also done for the exponen-tially distributed on time (black) with the same parameters. B. Examples of burst sizedistributions from A - with gamma (red) or exponentially (black) distributed on timeand to f f /tdeg=1; with gamma (orange) or exponentially (grey) distributed on time andto f f /tdeg=10.

〈ton〉/〈to f f 〉 0.5/0.5 0.5/2.5 0.5/5.0 0.5/8.0 4.0/0.5 4.0/2.5 4.0/5.0 4.0/8.0DKL(Non = 1,No f f = 9, exp) 0.04 0.11 0.07 0.04 0.01 0.05 0.04 0.02DKL(Non = 1,No f f = 9, det) 0.20 0.67 0.56 0.49 0.01 0.04 0.03 0.01DKL(Non = 9,No f f = 1, exp) 0.03 0.06 0.04 0.04 0.00 0.01 0.01 0.01DKL(Non = 9,No f f = 1, det) 0.24 0.72 0.33 0.15 0.03 0.16 0.07 0.04DKL(Non = 5,No f f = 5, exp) 0.15 0.23 0.14 0.07 0.01 0.07 0.05 0.03DKL(Non = 5,No f f = 5, det) 0.04 0.19 0.11 0.07 0.00 0.02 0.01 0.00

Table 5.S.1: KL-divergence indices between the steady state mRNA distributions shownin figure 5.S.6 and either a system with deterministic switching times (det) or one withexponential switching times (exp). The total number of steps for both transitions equalsten for all the three systems.

284

A B

0 2 4 6 8 100

2

4

6

8

10

�ton��tdeg�

�t o

ff��t deg

�

10�25�50�100�

0 2 4 6 8 100

2

4

6

8

10

�ton��tdeg��t o

ff��t deg

�

Figure 5.S.4: For average life times of the on state that are considerably shorterthan the average degradation time, the instantaneous burst model is a good ap-proximation of the gene switch modelled with gamma distributed life times. Therate of transcription initiation ktr was adjusted to keep the average steady-state mRNAlevel constant at 〈n〉 = 10. Red lines indicate the use of the effective burst size in theinstantaneous burst model, whereas for the blue lines the usual burst size was used. A:both on and off state modeled with exponentially distributed life times. B: life times ofon and off state modeled with Erlang(5) distributions.

PDR5 MDN1Non No f f 〈b〉〈ton〉

〈to f f 〉

〈ton〉+〈to f f 〉

〈tdeg〉〈b〉〈ton〉

〈to f f 〉


〈tdeg〉

1 1 4.8 0.0215 0.36 1.6 2.1 0.281 10 12.4 0.2364 0.93 5.2 3.0 0.865 5 16.2 0.0007 1.20 6.4 2.2 1.05

10 1 8.6 0.0009 0.64 1.8 1.4 0.3210 10 21.4 0.0005 1.60 9.0 2.3 1.48

Table 5.S.2: Comparison of the models with different numbers of steps (Non, No f f ) forPDR5 and MDN1.

285

A

0 2 4 6 8 100

5

10

15

20

25

30

35

mR

NA

t/<tdeg>

ton ton tontoff toff

nd(toff + ton)

neffnd(toff)

no

B C

�

�

�

�

��

��

�

��

��

��

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�

�

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�

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0 10 20 30 40 50 600.00

0.02

0.04

0.06

0.08

0.10

mRNA

frequ

ency

0 10 20 30 40 50 600.00

0.01

0.02

0.03

0.04

0.05

0.06

mRNA

freq

uenc

y

Figure 5.S.5: Visualization of how to calculate the moments of the steady statemRNA distribution. A. Example time-trace from a simulation with the Gillespie algo-rithm. Effective burst sizes (be f f ), values at the times of switching (n0 and nt(to f f )) andvalues of the mRNA numbers after one cycle of degradation (nt(to f f + ton)) are indicated.B. Steady-state mRNA distributions i) at the time of switching to the off state (red), ii) atthe time of switching to the on state (magenta), iii) of remaining mRNAs after one cycleof degradation (green), and iv) of the effective burst size (blue). C. Steady-state mRNAdistribution (grey bars), distribution during the on state (green), and distribution duringthe off state (black). The mixture of the green and black curves yields the steady-statemRNA distribution given by the grey bars.

286

A

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

mRNA

p�mRNA�

0 5 10 15 200

1020

t��tdeg�

mRN

A

01020

mRN

A

010

22

mRN

A

B

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

mRNA

p�mRNA� 0

1020

mRN

A0 5 10 15 200

1020

t��tdeg�mRN

A

01020

mRN

A

C

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

mRNA

p�mRNA� 0

2550

mRN

A

0 5 10 15 2002550

t��tdeg�

mRN

A

02550

mRN

A

D

0 10 20 30 400.00

0.05

0.10

0.15

0.20

mRNA

p�mRNA�

0102030

mRN

A

0 5 10 15 200

102030

t��tdeg�

mRN

A

0102030

mRN

A

Figure 5.S.6: Steady state mRNA distribution for different parameter regimes anddifferent waiting time distributions for switching: BLUE: exponential waiting times,GREEN: deterministic waiting times, BLACK: Non : 5, No f f : 5. 〈n〉 = 10, kdeg = 1.The 〈ton〉/〈tdeg〉 and 〈to f f 〉/〈tdeg〉 are as follows: 0.5 and 0.5 (A), 4 and 0.5 (B), 0.5 and4 (C), 4 and 4 (D). The kon, ko f f and ktr were adjusted accordingly.

287

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5 10 15

- 60

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- 20

0

tcyc telodeviation

%

0- 6 6- 12 12 - 18 18 - 24 24 - 300

10

20

30

40

% deviation

%ofsimulations

0- 15 15 - 30 30 - 45 45 - 60 60 - 750

10

20

30

40

% deviation

%ofsimulations

/ /

A B

C D

Figure 5.S.7: Comparison of simulated noise values with and without elongation.The simulations were made for different parameter sets produced by random samplingof the follwong parameter ranges: ton = 10 − 250min, to f f = 0.8 − 15min, ttr = 0.05 −10min, tdeg = 30 − 550min, Non = 1 − 5, No f f = 1 − 5. The resulting mRNA steadystate values lay between 0.3 to 50. Elongation was modeled as either a multi-step withN=50 (A,C) or a sigle step process with (B,D), complete elongation time (telo) 60min.TOP: percent difference between simulation with and without elongation plotted againstrelative timescale of the gene switching ton+to f f

telo. BOTTOM: Barplots of percentage of

simulations with percentage difference between 0 and 75 between noise in the systemwith/without elongation.

288

A B

0 5 10 150.00

0.05

0.10

0.15

0.20

0.25

0.30

mRNAs per cell

Freq

uenc

yPOL1

# mRNAs/cell

freq

uenc

y

0 10 20 30 400.000.020.040.060.080.100.120.14

burst size

frequ

ency


freq

uenc

y

C

0 2 4 6 8 10051015202530

7 13 10 6 4 4 6 9 7 7 7 2 7 2 8 14 6 4 5 9 7 3 11 8 4 12 6 3 6 7 5 6

0 2 4 6 8 10051015202530

6 12 10 10 14 9 10 12 21 24 11 12 6 9 3 15 22 10 19

mRNA

mRNA

mRNA

mRNA

mRNA

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

0 2 4 6 8 100510152025 26 6 4 36 3 2 1 3 29 71 11 6 16

0 2 4 6 8 100

10

20

30

4021 12 19 1 0 6 2 0 3 1 10 0 2 1 2 1 5 23 6 10 48 11 11 3 18 7

0 2 4 6 8 10051015202530

13 5 17 17 9 9 29 22 13 4 17 5 10 8 11 4 4 7 11

Figure 5.S.8: Fitting POL1 mRNA distribution with different transcription initia-tion mechanisms. A. Experimental data for POL1 from (Zenklusen et al., 2008) (greybars) and the mRNA distributions of the fitted models: i. red: Non = 1,No f f = 1, ii.blue: Non = 1,No f f = 10, iii. green: Non = 5,No f f = 5, magenta: Non = 10,No f f = 1,orange: Non = 10,No f f = 10. B. Burst size distributions vary greatly between the fittedmodels. Same color coding of the models as in A. C. Time-traces for the models de-scribed in A. The colored lines show on and off states, the black lines are time-traces ofsimulations with the Gillespie algorithm, and the numbers on top indicate burst sizes.

289

A B

0 2 4 6 8 10 12 140.00

0.05

0.10

0.15

mRNAs per cell

Freq

uenc

yMDN1

# mRNAs/cell

freq

uenc

y

0 5 10 15 20 250.0

0.1

0.2

0.3

burst size

frequ

ency


freq

uenc

y

C

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

0 2 4 6 8 1002468

1012 1 5 8 2 5 6 2 0 8 3 0 5 2 6 5 10 8 3 6 11 3 19 14 0 1 8 2 6 3 17 30 2 8 0 0 5 10 0 0 3 0 3 1 13 2 1 2 6

0 2 4 6 8 1005

1015 7 10 4 9 4 10 9 6 7 11 2 12 10 9 1 7 6 5 6 7 7 9 7 8 11 8 8 8 3 5 6 1 11 3 9 8 2 8

0 2 4 6 8 1002468

101200 0 0 1 0 1 2 4 2 31 0 2 3 2 1 10 0 1 2 0 2 1 3 2 1 3 0 2 1 0 5 1 2 3 3 2 42 1 5 1 2 5 2 3 2 4 4 4 41 1 0 1 3 5 3 1 1 0 4 1 3 0 3 3 1 1 3 0 1 6 0 1 0 0 0 3 1 3 1 1 0 1 1 2 40 2 0 1 0 0 3 1 0 0 4 2 3 3 4 4 0 1 1 3

0 2 4 6 8 1002468

101214

7 7 5 10 13 7 9 22 6 5 11 10 4 16 8 3 8 4 10 6 7 7 5 12 5 6 7 9 5 12 14 5

0 2 4 6 8 1002468

1012 20 0 0 1 0 2 0 0 2 9 1 51 2 10 30 0 4 3 00 13 10 0 0 50 0 31 0 000 0 5 1 0 1 10000 62 0 1 8 4 2 31 5 0 5 7 310001 500 70 0 1 3 20 1 4 6 1 2 70 30 4 2 5 200 1 2 4 0 10 1 8 1 700 0 21 1

Figure 5.S.9: Fitting MDN1 mRNA distribution with different transcription initia-tion mechanisms. A. Experimental data for MDN1 from (Zenklusen et al., 2008) (greybars) and the mRNA distributions of the fitted models: i. red: Non = 1,No f f = 1, ii.blue: Non = 1,No f f = 10, iii. green: Non = 5,No f f = 5, magenta: Non = 10,No f f = 1,orange: Non = 10,No f f = 10. B. Burst size distributions vary greatly between the fittedmodels. Same color coding of the models as in A. C. Time-traces for the models de-scribed in A. The colored lines show on and off states, the black lines are time-traces ofsimulations with the Gillespie algorithm, and the numbers on top indicate burst sizes.

290

A B

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

mRNAs per cell

Freq

uenc

yDOA1

# mRNAs/cell

freq

uenc

y

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

burst size

frequ

ency


freq

uenc

y

C

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

0.0 0.5 1.0 1.5 2.0 2.502468

100 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 2 0 1 0 1 0 0 1 0 0 1 0 1 1 0 2 2 1 2 0 1 1 0 0 0 2 1 0 2 0 0 1 0 1 0 2 0 3 0 1 0 0 2 1 1 0 0 3 0 1 0 1 2 0 0 2 2 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 2 0 1 1 2 0 1 0 0 0 0 0 0 0 0 2 0 1 1 0 2 1 2 0 0 0 0 1 2 0 1 1 0 0 2 0 0 1 2 1 0 1 1 1 1 0 0 0 0 0 2 0 1 0 3 0 1 0 0 1 1

0.0 0.5 1.0 1.5 2.0 2.502468 0 00 0 0 0 00 0 000 0 1 0 00 0 00 00 001 0 0 0 0 0 02 0 0 0 0 0 0 0 1000 0 00 0000 0 0 0 0 0 00 1 2 0 1 000 0 0 0 00 0 0 0 0 00 0 0 0 0 100 00 0 0 0 00 2 0001 00 0 0 0 0 0001 00 0 00 0 1 0 20 0 0 0 0 00 0 00 000 01 1 0 000 0 0 0 001 0 0 00 00 0 0 0 000 000 0 0 0 40 0 01 0 0 0 0000120 00 1 0 000 00 0 0 0 1 1 0 0 01 00 0 0 0 0 0 0 0 00 0 110 0000 0 00 1 0 0 0 0

0.0 0.5 1.0 1.5 2.0 2.502468

1012 2 2 0 1 1 1 0 3 0 1 0 1 0 0 1 0 3 1 0 1 0 1 1 1 4 3 2 2 0 3 2 0 0 2 2 1 0 2 3 0 2 3 2 3 2 1 1 1 1 2 0 2 1 0 0 0 0 0 4 0 0 1 2 0 0 3 1 1 1 2 1 0 3 0 2 0 1 1 0 1 0 1 2 0 3 0 0 3 0 0 0 0 0 2 1 1 1 0 1 0 1 1 0 2 3 1 1 3 1 0 0 0 1 1 0 1 1 1 1 1 2 0

0.0 0.5 1.0 1.5 2.0 2.50

5

10

150 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 2 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 3 0 2 1 0 0 2 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0.0 0.5 1.0 1.5 2.0 2.502468

1000 1 0 0 00 1 0 1 0 0 0 0 0 00 1 00 0 0 0 0 0 0 0 0 0 00 0 0 10 0 0 0 00 00 0 0 00 0 1 0 0 1 0 0 001 0 0 0 1 0 0 1 00 0 0 00 1 0 0 00 0 10 0 0 0 01 0 0 0 1 01 01 0 1 0 000 1 0 0 0 0 00 1 0 0 0 1 0 00 0 0 00 0 0 0 0 00 1 0 0 0 00 1 0 10 0 0 0 1 0 1 0 0 10 1 0 0 1 00 0 1 0 00 0 0001 0 0 00 0 0 0 0 0 0 00 0 0 1 00 0 0 2 11 0 0 0 0 0 1 0 001 0 0 0 0 0 0001 01 02 0 0 2 0 0 0 0

Figure 5.S.10: Fitting DOA1 mRNA distribution with different transcription initi-ation mechanisms. A. Experimental data for DOA1 from (Zenklusen et al., 2008) (greybars) and the mRNA distributions of the fitted models: i. red: Non = 1,No f f = 1, ii.blue: Non = 1,No f f = 10, iii. green: Non = 5,No f f = 5, magenta: Non = 10,No f f = 1,orange: Non = 10,No f f = 10. B. Burst size distributions vary greatly between the fittedmodels. Same color coding of the models as in A. C. Time-traces for the models de-scribed in A. The colored lines show on and off states, the black lines are time-traces ofsimulations with the Gillespie algorithm, and the numbers on top indicate burst sizes.

291

A B

0 2 4 6 8 10 120.00

0.05

0.10

0.15

0.20

0.25

mRNAs per cell

Freq

uenc

yKAP104

# mRNAs/cell

freq

uenc

y

0 20 40 60 800.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

burst size

frequ

ency


freq

uenc

y

C

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

# m

RN

As/

cell

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

t/<tdeg>

0 2 4 6 8 1002468101214

5 12 23 5 12 7 89 46 4 2 28

0 2 4 6 8 100

5

10

15 34 42 27 28 31 32 15

0 2 4 6 8 100

5

10

15

2012 13 9 15 15 13 9 11 7 10 20 7 6 16 4 16 6 13 10 18

0 2 4 6 8 100

5

10

15 29 27 38 26 48 44

0 2 4 6 8 100

5

10

15 3 2 7 6 22 18 9 0 27 1 2 16 6 27 23 6 15 8 13 9 2

Figure 5.S.11: Fitting KAP104 mRNA distribution with different transcription ini-tiation mechanisms. A. Experimental data for KAP104 from (Zenklusen et al., 2008)(grey bars) and the mRNA distributions of the fitted models: i. red: Non = 1,No f f = 1, ii.blue: Non = 1,No f f = 10, iii. green: Non = 5,No f f = 5, magenta: Non = 10,No f f = 1,orange: Non = 10,No f f = 10. B. Burst size distributions vary greatly between the fittedmodels. Same color coding of the models as in A. C. Time-traces for the models de-scribed in A. The colored lines show on and off states, the black lines are time-traces ofsimulations with the Gillespie algorithm, and the numbers on top indicate burst sizes.

292

POL1 DOA1 KAP104Non No f f 〈b〉〈ton〉

〈to f f 〉



〈to f f 〉



〈to f f 〉


〈tdeg〉

1 1 5.7 0.20 1.81 0.14 0.010 0.0620 13.6 6.7 2.841 10 10.1 0.29 3.23 0.24 0.0010 1.0 35.2 8.3 7.325 5 11.4 0.14 3.61 0.62 0.0004 2.6 34.1 7.5 7.0810 1 6.4 0.05 2.05 0.25 0.0012 1.1 13.4 5.6 2.7910 10 11.9 0.09 3.77 0.99 0.00002 4.1 40.3 7.9 8.38

Table 5.S.3: Comparison of the models with different numbers of steps (Non, No f f ) forPOL1, DOA1 and KAP104.

293

References

Acar, M., Mettetal, J. T. and van Oudenaarden, A. (2008). Stochastic switching as asurvival strategy in fluctuating environments. Nat. Genet. 40, 471–475.

Agalioti, T., Chen, G. and Thanos, D. (2002). Deciphering the Transcriptional HistoneAcetylation Code for a Human Gene. Cell 111, 381 – 392.

Aranda, P., Agirre, X., Ballestar, E., Andreu, E. J., Roman-Gomez, J., Prieto, I., Martin-Subero, J. I., Cigudosa, J. C., Siebert, R., Esteller, M. and Prosper, F. (2009). Epige-netic signatures associated with different levels of differentiation potential in humanstem cells. PLoS ONE 4, e7809.

Barski, A., Cuddapah, S., Cui, K., Roh, T. Y., Schones, D. E., Wang, Z., Wei, G., Chep-elev, I. and Zhao, K. (2007). High-resolution profiling of histone methylations in thehuman genome. Cell 129, 823–837.

Bengtsson, M., Hemberg, M., Rorsman, P. and Ståhlberg, A. (2008). Quantification ofmRNA in single cells and modelling of RT-qPCR induced noise. BMC Mol Biol 9,63.

Bengtsson, M., Ståhlberg, A., Rorsman, P. and Kubista, M. (2005). Gene expressionprofiling in single cells from the pancreatic islets of Langerhans reveals lognormaldistribution of mRNA levels. Genome Res 15, 1388–92.

Berg, O. G., Winter, R. B. and von Hippel, P. H. (1981). Diffusion-driven mechanismsof protein translocation on nucleic acids. 1. Models and theory. Biochemistry 20,6929–6948.

Bernstein, B. E., Humphrey, E. L., Erlich, R. L., Schneider, R., Bouman, P., Liu, J. S.,Kouzarides, T. and Schreiber, S. L. (2002). Methylation of histone H3 Lys 4 in codingregions of active genes. Proc. Natl. Acad. Sci. U.S.A. 99, 8695–8700.

Cai, L., Friedman, N. and Xie, X. S. (2006). Stochastic protein expression in individualcells at the single molecule level. Nature 440, 358–362.

Chandy, M., Gutierrez, J. L., Prochasson, P. and Workman, J. L. (2006). SWI/SNFdisplaces SAGA-acetylated nucleosomes. Eukaryotic Cell 5, 1738–1747.

Choi, P. J., Cai, L., Frieda, K. and Xie, X. S. (2008). A stochastic single-molecule eventtriggers phenotype switching of a bacterial cell. Science 322, 442–446.

Chubb, J. R., Trcek, T., Shenoy, S. M. and Singer, R. H. (2006). Transcriptional pulsingof a developmental gene. Curr. Biol. 16, 1018–1025.

Cinlar, E. (1975). Introduction to stochastic processes. Prentice-Hall, Englewood Cliffs,NJ.

Corona, D. F., Clapier, C. R., Becker, P. B. and Tamkuna, J. W. (2002). Modulation ofISWI function by site-specific histone acetylation. EMBO Rep. 3, 242–247.

Darzacq, X., Shav-Tal, Y., de Turris, V., Brody, Y., Shenoy, S. M., Phair, R. D. andSinger, R. H. (2007). In vivo dynamics of RNA polymerase II transcription. Nat.

294

Struct. Mol. Biol. 14, 796–806.Dasgupta, A., Juedes, S. A., Sprouse, R. O. and Auble, D. T. (2005). Mot1-mediated

control of transcription complex assembly and activity. EMBO J 24, 1717–29.Daujat, S., Bauer, U.-M., Shah, V., Turner, B., Berger, S. and Kouzarides, T. (2010).

Crosstalk between CARM1 Methylation and CBP Acetylation on Histone H3. Cell12, 2090 – 2097.

Davidson, E. H. (2006). The regulatory genome: Gene regulatory networks in Develop-ment and Evolution. Academic Press.

Dinant, C., Luijsterburg, M. S., Hofer, T., von Bornstaedt, G., Vermeulen, W.,Houtsmuller, A. B. and van Driel, R. (2009). Assembly of multiprotein complexesthat control genome function. J. Cell Biol. 185, 21–26.

DiRenzo, J., Shang, Y., Phelan, M., Sif, S., Myers, M., Kingston, R. and Brown, M.(2000). BRG-1 Is Recruited to Estrogen-Responsive Promoters and Cooperates withFactors Involved in Histone Acetylation. Mol. Cell. Biol. 20, 7541–7549.

Dobrzynski, M. and Bruggeman, F. J. (2009). Elongation dynamics shape bursty tran-scription and translation. Proc Natl Acad Sci USA 106, 2583–8.

Drysdale, C. M., Jackson, B. M., McVeigh, R., Klebanow, E. R., Bai, Y., Kokubo, T.,Swanson, M., Nakatani, Y., Weil, P. A. and Hinnebusch, A. G. (1998). The Gcn4pactivation domain interacts specifically in vitro with RNA polymerase II holoenzyme,TFIID, and the Adap-Gcn5p coactivator complex. Mol. Cell. Biol. 18, 1711–1724.

Dundr, M., Hoffmann-Rohrer, U., Hu, Q., Grummt, I., Rothblum, L. I., Phair, R. D. andMisteli, T. (2002). A kinetic framework for a mammalian RNA polymerase in vivo.Science 298, 1623–1626.

Durant, M. and Pugh, B. F. (2006). Genome-wide relationships between TAF1 andhistone acetyltransferases in Saccharomyces cerevisiae. Mol. Cell. Biol. 26, 2791–2802.

Elf, J., Li, G. W. and Xie, X. S. (2007). Probing transcription factor dynamics at thesingle-molecule level in a living cell. Science 316, 1191–1194.

Elowitz, M. B., Levine, A. J., Siggia, E. D. and Swain, P. S. (2002). Stochastic geneexpression in a single cell. Science 297, 1183–1186.

Feige, J. N., Gelman, L., Michalik, L., Desvergne, B. and Wahli, W. (2006). Frommolecular action to physiological outputs: Peroxisome proliferator-activated recep-tors are nuclear receptors at the crossroads of key cellular functions. Progress inLipid Research 45, 120 – 159.

Femino, A. M., Fay, F. S., Fogarty, K. and Singer, R. H. (1998). Visualization of singleRNA transcripts in situ. Science 280, 585–590.

Fraser, D. and Kaern, M. (2009). A chance at survival: gene expression noise andphenotypic diversification strategies. Mol. Microbiol. 71, 1333–1340.

Gerber, M. and Shilatifard, A. (2003). Transcriptional Elongation by RNA Polymerase

295

II and Histone Methylation. Journal of Biological Chemistry 278, 26303–26306.Gibson, M. A. and Bruck, J. (2000). Efficient exact stochastic simulation of chemical

systems with many species and many channels. J. Phys. Chem. A 104, 1876–1889.Golding, I., Paulsson, J., Zawilski, S. M. and Cox, E. C. (2005). Real-time kinetics of

gene activity in individual bacteria. Cell 123, 1025–1036.Gorski, S. A., Snyder, S. K., John, S., Grummt, I. and Misteli, T. (2008). Modulation

of RNA polymerase assembly dynamics in transcriptional regulation. Mol. Cell 30,486–497.

Hager, G. L., Elbi, C., Johnson, T. A., Voss, T., Nagaich, A. K., Schiltz, R. L., Qiu, Y.and John, S. (2006). Chromatin dynamics and the evolution of alternate promoterstates. Chromosome Res. 14, 107–116.

Hallikas, O., Palin, K., Sinjushina, N., Rautiainen, R., Partanen, J., Ukkonen, E. andTaipale, J. (2006). Genome-wide Prediction of Mammalian Enhancers Based onAnalysis of Transcription-Factor Binding Affinity. Cell 124, 47 – 59.

Hemberg, M. and Barahona, M. (2007). Perfect sampling of the master equation forgene regulatory networks. Biophys J 93, 401–10.

Herbig, E., Warfield, L., Fish, L., Fishburn, J., Knutson, B. A., Moorefield, B., Pacheco,D. and Hahn, S. (2010). Mechanism of Mediator recruitment by tandem Gcn4 ac-tivation domains and three Gal11 activator-binding domains. Mol. Cell. Biol. 30,2376–2390.

Heyman, D. and Sobel, M. (1982). Stochastic Models in Operations Research. McGraw-Hill, New York.

Hieb, A. R., Halsey, W. A., Betterton, M. D., Perkins, T. T., Kugel, J. F. and Goodrich,J. A. (2007). TFIIA changes the conformation of the DNA in TBP/TATA complexesand increases their kinetic stability. J Mol Biol 372, 619–32.

Ingvarsdottir, K., Edwards, C., Lee, M. G., Lee, J. S., Schultz, D. C., Shilatifard, A.,Shiekhattar, R. and Berger, S. L. (2007). Histone H3 K4 Demethylation during Ac-tivation and Attenuation of GAL1 Transcription in Saccharomyces cerevisiae . Mol.Cell. Biol. 27, 7856–7864.

Iyer-Biswas, S., Hayot, F. and Jayaprakash, C. (2009). Stochasticity of gene productsfrom transcriptional pulsing. Phys. Rev. E 79, 9.

Jackson, D. A., Hassan, A. B., Errington, R. J. and Cook, P. R. (1993). Visualization offocal sites of transcription within human nuclei. EMBO J. 12, 1059–1065.

Jr., K. K., Yasuda, R., Noji, H., Ishiwata, S. and Yoshida, M. (1998). F1-ATPase: ARotary Motor Made of a Single Molecule. Cell 93, 21 – 24.

Karpova, T. S., Kim, M. J., Spriet, C., Nalley, K., Stasevich, T. J., Kherrouche, Z., Heliot,L. and McNally, J. G. (2008). Concurrent fast and slow cycling of a transcriptionalactivator at an endogenous promoter. Science 319, 466–469.

Kepler, T. B. and Elston, T. C. (2001). Stochasticity in transcriptional regulation: origins,

296

consequences, and mathematical representations. Biophys. J. 81, 3116–3136.Kim, T. and Buratowski, S. (2009). Dimethylation of H3K4 by Set1 Recruits the Set3

Histone Deacetylase Complex to 5’ Transcribed Regions. Cell 137, 259 – 272.Klumpp, S. and Hwa, T. (2008). Stochasticity and traffic jams in the transcription of

ribosomal RNA: Intriguing role of termination and antitermination. Proc. Natl. Acad.Sci. U.S.A. 105, 18159–18164.

Ko, M. S. (1991). A stochastic model for gene induction. J. Theor. Biol. 153, 181–194.Kugel, J. F. (2000). A Kinetic Model for the Early Steps of RNA Synthesis by Human

RNA Polymerase II. Journal of Biological Chemistry 275, 40483–40491.Kwok, R. P., Lundblad, J. R., Chrivia, J. C., Richards, J. P., Bachinger, H. P., Brennan,

R. G., Roberts, S. G., Green, M. R. and Goodman, R. H. (1994). Nuclear protein CBPis a coactivator for the transcription factor CREB. Nature 370, 223–226.

Lam, F. H., Steger, D. J. and O’Shea, E. K. (2008). Chromatin decouples promoterthreshold from dynamic range. Nature 453, 246–50.

Lam, L. T., Pickeral, O. K., Peng, A. C., Rosenwald, A., Hurt, E. M., Giltnane, J. M.,Averett, L. M., Zhao, H., Davis, R. E., Sathyamoorthy, M., Wahl, L. M., Harris, E. D.,Mikovits, J. A., Monks, A. P., Hollingshead, M. G., Sausville, E. A. and Staudt,L. M. (2001). Genomic-scale measurement of mRNA turnover and the mechanismsof action of the anti-cancer drug flavopiridol. Genome Biol 2, RESEARCH0041.

Larschan, E. and Winston, F. (2005). The Saccharomyces cerevisiae Srb8-Srb11 com-plex functions with the SAGA complex during Gal4-activated transcription. Mol.Cell. Biol. 25, 114–123.

Lavery, D. N. and McEwan, I. J. (2008). Functional Characterization of the NativeNH2-Terminal Transactivation Domain of the Human Androgen Receptor: BindingKinetics for Interactions with TFIIF and SRC-1a. Biochemistry 47, 3352–3359.

Lee, C. W., Arai, M., Martinez-Yamout, M. A., Dyson, H. J. and Wright, P. E. (2009).Mapping the interactions of the p53 transactivation domain with the KIX domain ofCBP. Biochemistry 48, 2115–2124.

Lee, D., Kim, J. W., Seo, T., Hwang, S. G., Choi, E. J. and Choe, J. (2002). SWI/SNFcomplex interacts with tumor suppressor p53 and is necessary for the activation ofp53-mediated transcription. J. Biol. Chem. 277, 22330–22337.

Lee, T. I. and Young1, R. A. (1998). Regulation of gene expression by TBP-associatedproteins. JGenes & Dev 12, 1398–1408.

Lehner, B. (2008). Selection to minimise noise in living systems and its implications forthe evolution of gene expression. Mol. Syst. Biol. 4, 170.

Li, B., Carey, M. and Workman, J. L. (2007). The Role of Chromatin during Transcrip-tion. Cell 128, 707 – 719.

Li, X., Hu, X., Patel, B., Zhou, Z., Liang, S., Ybarra, R., Qiu, Y., Felsenfeld, G., Bungert,J. and Huang, S. (2010). H4R3 methylation facilitates beta-globin transcription by

297

regulating histone acetyltransferase binding and H3 acetylation. Blood 115, 2028–2037.

Maheshri, N. and O’Shea, E. K. (2007). Living with noisy genes: how cells function re-liably with inherent variability in gene expression. Annu Rev Biophys Biomol Struct36, 413–434.

Metivier, R., Penot, G., Hubner, M. R., Reid, G., Brand, H., Kos, M. and Gannon, F.(2003). Estrogen receptor-alpha directs ordered, cyclical, and combinatorial recruit-ment of cofactors on a natural target promoter. Cell 115, 751–763.

Minchin, S. D. and Busby, S. J. (2009). Analysis of mechanisms of activation andrepression at bacterial promoters. Methods 47, 6 – 12.

Mitarai, N., Sneppen, K. and Pedersen, S. (2008). Ribosome collisions and translationefficiency: optimization by codon usage and mRNA destabilization. J. Mol. Biol.382, 236–245.

Mone, M. J., Bernas, T., Dinant, C., Goedvree, F. A., Manders, E. M., Volker, M.,Houtsmuller, A. B., Hoeijmakers, J. H., Vermeulen, W. and van Driel, R. (2004). Invivo dynamics of chromatin-associated complex formation in mammalian nucleotideexcision repair. Proc. Natl. Acad. Sci. U.S.A. 101, 15933–15937.

Moreira, J. M. and Holmberg, S. (1998). Nucleosome structure of the yeast CHA1promoter: analysis of activation-dependent chromatin remodeling of an RNA-polymerase-II-transcribed gene in TBP and RNA pol II mutants defective in vivo inresponse to acidic activators. The EMBO Journal 17, 6028 – 6038.

Mueller, J. (1982). The moments of self-convolutions. Rapport BIPM-82 15, 7.Nalefski, E. A., Nebelitsky, E., Lloyd, J. A. and Gullans, S. R. (2006). Single-Molecule

Detection of Transcription Factor Binding to DNA in Real Time: Specificity, Equilib-rium, and Kinetic Parameters. Biochemistry 45, 13794–13806.

Neely, L., Trauger, J. W., Baird, E. E., Dervan, P. B. and Gottesfeld, J. M. (1997).Importance of minor groove binding zinc fingers within the transcription factor IIIA-DNA complex. Journal of Molecular Biology 274, 439 – 445.

Pare, A., Lemons, D., Kosman, D., Beaver, W., Freund, Y. and McGinnis, W. (2009).Visualization of individual Scr mRNAs during Drosophila embryogenesis yields evi-dence for transcriptional bursting. Curr. Biol. 19, 2037–2042.

Parzen, E. (1962). Stochastic Processes. Holden-Day, San Francisco.Paulsson, J. (2004). Summing up the noise in gene networks. Nature 427, 415–418.Pedraza, J. M. and Paulsson, J. (2008). Effects of molecular memory and bursting on

fluctuations in gene expression. Science 319, 339–343.Phair, R. D., Scaffidi, P., Elbi, C., Vecerova, J., Dey, A., Ozato, K., Brown, D. T., Hager,

G., Bustin, M. and Misteli, T. (2004). Global Nature of Dynamic Protein-ChromatinInteractions In Vivo: Three-Dimensional Genome Scanning and Dynamic InteractionNetworks of Chromatin Proteins. Mol. Cell. Biol. 24, 6393–6402.

298

Puigserver, P., Adelmant, G., Wu, Z., Fan, M., Xu, J., O’Malley, B. and Spiegelman,B. (1999). Activation of PPAR coactivator-1 through transcription factor docking.Science 286, 1368–1371.

Qian, H. and Elson, E. L. (2002). Single-molecule enzymology: stochastic Michaelis-Menten kinetics. Biophys Chem 101-102, 565–76.

Raj, A., Peskin, C. S., Tranchina, D., Vargas, D. Y. and Tyagi, S. (2006). StochasticmRNA synthesis in mammalian cells. PLoS Biol. 4, e309.

Raj, A. and van Oudenaarden, A. (2008). Nature, nurture, or chance: stochastic geneexpression and its consequences. Cell 135, 216–226.

Raser, J. M. and O’Shea, E. K. (2004). Control of stochasticity in eukaryotic geneexpression. Science 304, 1811–1814.

Raser, J. M. and O’Shea, E. K. (2005). Noise in gene expression: origins, consequences,and control. Science 309, 2010–2013.

Reid, G., Gallais, R. and Metivier, R. (2009). Marking time: the dynamic role of chro-matin and covalent modification in transcription. Int. J. Biochem. Cell Biol. 41, 155–163.

Reznikoff, W. S., Siegele, D. A., Cowing, D. W. and Gross, C. A. (1985). The Regulationof Transcription Initiation in Bacteria. Annual Review of Genetics 19, 355–387.

Roh, T. Y., Cuddapah, S., Cui, K. and Zhao, K. (2006). The genomic landscape ofhistone modifications in human T cells. Proc. Natl. Acad. Sci. U.S.A. 103, 15782–15787.

Santos-Rosa, H., Schneider, R., Bernstein, B. E., Karabetsou, N., Morillon, A., Weise,C., Schreiber, S. L., Mellor, J. and Kouzarides, T. (2003). Methylation of Histone H3K4 Mediates Association of the Isw1p ATPase with Chromatin. Molecular Cell 12,1325 – 1332.

Saramaki, A., Diermeier, S., Kellner, R., Laitinen, H., Vaisanen, S. and Carlberg, C.(2009). Cyclical chromatin looping and transcription factor association on the regula-tory regions of the p21 (CDKN1A) gene in response to 1alpha,25-dihydroxyvitaminD3. J. Biol. Chem. 284, 8073–8082.

Sawa, C., Nedea, E., Krogan, N., Wada, T., Handa, H., Greenblatt, J. and Buratowski,S. (2004). Bromodomain Factor 1 (Bdf1) Is Phosphorylated by Protein Kinase CK2.Mol. Cell. Biol. 24, 4734–4742.

Schones, D. E., Cui, K., Cuddapah, S., Roh, T.-Y., Barski, A., Wang, Z., Wei, G. andZhao, K. (2008). Dynamic Regulation of Nucleosome Positioning in the HumanGenome. Cell 132, 887 – 898.

Schultz, B. E., Misialek, S., Wu, J., Tang, J., Conn, M. T., Tahilramani, R. and Wong, L.(2004). Kinetics and Comparative Reactivity of Human Class I and Class IIb HistoneDeacetylases. Biochemistry 43, 11083–11091.

Shav-Tal, Y. (2004). Dynamics of Single mRNPs in Nuclei of Living Cells. Science

299

304, 1797–1800.Singh, A. and Hespanha, J. P. (2009). Optimal feedback strength for noise suppression

in autoregulatory gene networks. Biophys. J. 96, 4013–4023.Sipos, L. and Gyurkovics, H. (2005). Long-distance interactions between enhancers and

promoters. FEBS Journa; 272, 3253–3259.Sprouse, R. O., Karpova, T. S., Mueller, F., Dasgupta, A., McNally, J. G. and Auble,

D. T. (2008). Regulation of TATA-binding protein dynamics in living yeast cells.Proc Natl Acad Sci U S A 105, 13304–8.

Swain, P. S., Elowitz, M. B. and Siggia, E. D. (2002). Intrinsic and extrinsic contribu-tions to stochasticity in gene expression. Proc. Natl. Acad. Sci. U.S.A. 99, 12795–12800.

Takano, M., Terada, T. P. and Sasai, M. (2010). Unidirectional Brownian motion ob-served in an in silico single molecule experiment of an actomyosin motor. Proceed-ings of the National Academy of Sciences 107, 7769–74.

Thattai, M. and van Oudenaarden, A. (2001). Intrinsic noise in gene regulatory networks.Proc. Natl. Acad. Sci. U.S.A. 98, 8614–8619.

Thompson, P. R., Kurooka, H., Nakatani, Y. and Cole, P. A. (2001). TranscriptionalCoactivator Protein p300. Journal of Biological Chemistry 276, 33721–33729.

Tkacik, G., Callan, C. G. and Bialek, W. (2008). Information flow and optimization intranscriptional regulation. Proc. Natl. Acad. Sci. U.S.A. 105, 12265–12270.

Tourriere, H., Chebli, K. and Tazi, J. (2002). mRNA degradation machines in eukaryoticcells. Biochimie 84, 821 – 837.

Valls, E., Sanchez-Molina, S. and Martinez-Balbas, M. A. (2005). Role of histone modi-fications in marking and activating genes through mitosis. J. Biol. Chem. 280, 42592–42600.

van Hoek, M. and Hogeweg, P. (2007). The effect of stochasticity on the lac operon: anevolutionary perspective. PLoS Comput. Biol. 3, e111.

Vermeulen, M., Mulder, K. W., Denissov, S., Pijnappel, W., van Schaik, F. M., Varier,R. A., Baltissen, M. P., Stunnenberg, H. G., Mann, M. and Timmers, H. (2007). Se-lective Anchoring of TFIID to Nucleosomes by Trimethylation of Histone H3 Lysine4. Cell 131, 58–69.

Wagatsuma, A., Sadamoto, H., Kitahashi, T., Lukowiak, K., Urano, A. and Ito, E.(2005). Determination of the exact copy numbers of particular mRNAs in a singlecell by quantitative real-time RT-PCR. J. Exp. Biol. 208, 2389–2398.

Wang, A., Kurdistani, S. K. and Grunstein, M. (2002). Requirement of Hos2 HistoneDeacetylase for Gene Activity in Yeast. Science 298, 1412–1414.

Warren, L., Bryder, D., Weissman, I. L. and Quake, S. R. (2006). Transcription fac-tor profiling in individual hematopoietic progenitors by digital RT-PCR. Proc. Natl.Acad. Sci. U.S.A. 103, 17807–17812.

300

Weishaupt, H., Sigvardsson, M. and Attema, J. L. (2010). Epigenetic chromatin statesuniquely define the developmental plasticity of murine hematopoietic stem cells.Blood 115, 247–256.

Wysocka, J., Swigut, T., Xiao, H., Milne, T. A., Kwon, S. Y., Landry, J., Kauer, M.,Tackett, A. J., Chait, B. T., Badenhorst, P., Wu, C. and Allis, C. D. (2006). A PHDfinger of NURF couples histone H3 lysine 4 trimethylation with chromatin remod-elling. Nature 442, 86–90.

Xing, L., Gopal, V. K. and Quinn, P. G. (1995). cAMP response element-binding protein(CREB) interacts with transcription factors IIB and IID. J. Biol. Chem. 270, 17488–17493.

Yakovchuk, P., Gilman, B., Goodrich, J. A. and Kugel, J. F. (2010). RNA polymerase IIand TAFs undergo a slow isomerization after the polymerase is recruited to promoter-bound TFIID. J Mol Biol 397, 57–68.

Yu, J., Xiao, J., Ren, X., Lao, K. and Xie, X. S. (2006). Probing gene expression in livecells, one protein molecule at a time. Science 311, 1600–1603.

Zenklusen, D., Larson, D. R. and Singer, R. H. (2008). Single-RNA counting revealsalternative modes of gene expression in yeast. Nat. Struct. Mol. Biol. 15, 1263–1271.

301

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