A matter of time: Using dynamics and theory to uncover mechanisms of transcriptional bursting
Nicholas C. Lammers, Yang Joon Kim, Jiaxi Zhao, Hernan G. Garcia
AA matter of time: Using dynamics and theory to uncovermechanisms of transcriptional bursting
Nicholas C Lammers , Yang Joon Kim ∗ , Jiaxi Zhao ∗ , Hernan G Garcia ∗∗ Abstract
Eukaryotic transcription generally occurs in bursts of activity lasting minutes tohours; however, state-of-the-art measurements have revealed that many of the molec-ular processes that underlie bursting, such as transcription factor binding to DNA,unfold on timescales of seconds. This temporal disconnect lies at the heart of abroader challenge in physical biology of predicting transcriptional outcomes and cel-lular decision-making from the dynamics of underlying molecular processes. Here,we review how new dynamical information about the processes underlying transcrip-tional control can be combined with theoretical models that predict not only averagedtranscriptional dynamics, but also their variability, to formulate testable hypothesesabout the molecular mechanisms underlying transcriptional bursting and control.
Keywords:
Live imaging, Transcriptional bursting, Gene regulation,Transcriptional dynamics, Theoretical models of transcription, Non-equilibriummodels of transcription, Waiting time distributions
1. A disconnect between transcriptional bursting and its underlying molec-ular processes
Over the past two decades, new technologies have revealed that transcription is afundamentally discontinuous process characterized by transient bursts of transcrip- ∗ These authors contributed equally to this work. ∗∗ For correspondence: [email protected] (HGG) Biophysics Graduate Group, University of California at Berkeley, Berkeley, California Department of Physics, University of California at Berkeley, Berkeley, California Department of Molecular and Cell Biology, University of California at Berkeley, Berkeley,California Institute for Quantitative Biosciences-QB3, University of California at Berkeley, Berkeley,California a r X i v : . [ q - b i o . S C ] A ug ional activity interspersed with periods of quiescence. Although electron microscopyprovided early hints of bursty transcription [1], the advent of single-molecule fluo-rescence in situ hybridization (smFISH) [2, 3], was key to establishing its centralrole in transcription. The single-cell distributions of nascent RNA and cytoplasmicmRNA molecules obtained using this technology provided compelling, if indirect,evidence for the existence and ubiquity of gene expression bursts, and indicated thattheir dynamics were subject to regulation by transcription factors [4, 5]. These fixed-tissue inferences have been confirmed with new in vivo RNA fluorescence labelingtechnologies such as the MS2/MCP [6] and PP7/PCP systems [7], which directly re-veal stochastic bursts of transcriptional activity in living cells in culture and withinanimals (Figure 1A-C) [8–11].What is the role of transcriptional bursting in cellular decision-making? One pos-sibility is that bursty gene expression is intrinsically beneficial, helping (for instance)to coordinate gene expression or to facilitate cell-fate decision-making [12]. Alterna-tively, bursting may not itself be functional, but might instead be a consequence ofkey underlying transcriptional processes, such as proofreading transcription factoridentity [13, 14].Bursting and its regulation are intimately tied to the molecular mechanisms thatunderlie transcriptional regulation as a whole. In this Review we argue that, tomake progress toward predicting transcriptional outcomes from underlying molecu-lar processes, we can start with the narrower question of how the burst dynamicsemerge from the kinetics of molecular transactions at the gene locus. To illustratethe importance and challenge of taking kinetics into account, we highlight two inter-related molecular puzzles that arise from new measurements of the dynamics of keytranscriptional processes in vivo .First, as illustrated in Figure 1D and reviewed in detail in Appendix Table A.1,despite qualitatively similar bursty traces from different organisms, bursts unfoldacross markedly distinct timescales ranging from several minutes [15, 16], to tens ofminutes [17, 18], all the way to multiple hours [19]. Is this wide range of burstingtimescales across organisms reflective of distinct molecular mechanisms or is it theresult of a common set of highly malleable molecular processes?Second, recent live imaging experiments have revealed a significant temporal dis-connect between transcription factor binding events, which generally last for seconds,and the transcriptional bursts that these events control, which may last from a fewminutes to multiple hours. The majority of the molecular processes underlying tran-scriptional control are highly transient (Figure 1E), with timescales ranging frommilliseconds to seconds (see Appendix Table A.2 for a detailed tabulation and dis-cussion of these findings). 2n this Review, we seek to address this second puzzle by surveying key theoreticaland experimental advances that, together, should shed light on the molecular originsof transcriptional bursting and transcriptional regulation. We leverage this frame-work to examine two kinds of molecular-level models that explain how slow burstdynamics could arise from fast molecular processes. Finally, we present concreteexperimental strategies based on measuring variability in the timing of bursts thatcan be used to discern between molecular models of transcriptional bursting.Overall, we seek to illustrate how iterative discourse between theory and ex-periment sharpens our molecular understanding of transcriptional bursting by re-formulating cartoon models as concrete mathematical statements. Throughout thisReview, we focus on illustrative recent experimental and theoretical efforts; we there-fore do not attempt to provide a comprehensive review of the current literature (see[20–25] for excellent reviews).
2. The two-state model: a simple quantitative framework for burstingdynamics
To elucidate the disconnect between molecular timescales and transcriptionalbursting, we invoke a simple and widely used model of bursting dynamics: the two-state model of promoter switching. While the molecular reality of bursting is likelymore complex than the two-state model suggests [26–28], there is value in examiningwhere this simple model breaks down. This model posits that the promoter canexist in two states: a transcriptionally active ON state and a quiescent OFF state(Figure 2A). The promoter stochastically switches between these states with rates k on and k off , and loads new RNA polymerase II (RNAP) molecules at a rate r whenin the ON state [22, 29–31]. Figure 2B shows a hypothetical activity trace for a geneundergoing bursty expression, where a burst corresponds to a period of time duringwhich the promoter is in the ON state. The average burst duration, amplitude andseparation are given by 1 /k off , r and 1 /k on , respectively.Because the instantaneous transcription initiation rate during a burst is r andzero otherwise, the average initiation rate is equal to r times the fraction of time thepromoter spends in this ON state p on , (cid:68) initiation rate (cid:69) = r p on , (1)where brackets indicate time-averaging. As shown in Appendix B, in steady state, p on can be expressed as a function of the transition rates k on and k off : p on = k on k on + k off . (2)3 A) OFFON OFFON OFF ONONfly embryo (B)10 20 30 40time (min) M S - M C P s p o t fl u o r e s c e n c e ( A U ) M S - M C P s p o t fl u o r e s c e n c e ( A U ) M S - M C P s p o t fl u o r e s c e n c e ( A U )
10 811794 651 2 312
RNAP clusterdynamics10 5 s – 13 s mediator clusterdynamics11 ~10 s transcriptionfactor binding7 0.5 – 15 senhancer–promoter interaction8tens of seconds – minutes transcriptioninitiation9few secondsnucleosomal DNAwrapping/unwrapping4 10 ms – 5 s histonemodifications6~mins – daysnucleosome turnover5 ~mins – hoursfly embryo, yeast, bacteria 1 few minutes nematode2 tens of minutes human, mouse cells3tens of mins – hourspromoter–proximalpausing12 40 s – 20 min(D) B U R S T I N G T I M E S C A L E S M O L E C U L A R P R O C E SS E S T I M E S S C A L E S Figure 1:
Separation of timesscales between transcriptional bursting and its underlyingmolecular processes.
See caption in next page. igure 1: Separation of timesscales between transcriptional bursting and its underly-ing molecular processes. (A,B,C)
Transcriptional bursting in (A) an embryo of the fruit fly
Drosophila melanogaster , (B) the nematode
Caenorhabditis elegans , and (C) human cells. (D)
Inthese and other organisms, bursting dynamics (average period of ON and OFF) span a wide rangeof timescales from a few minutes to tens of hours. (E)
Timescales of the molecular processes behindtranscription range from fast seconds-long transcription factor binding to slower histone modifica-tions, which may unfold across multiple hours or days. A detailed summary of measurements leadingto these numbers, including references, is provided in Appendix Table A.1 and Appendix Table A.2.(A, adapted from [16]; B, adapted from [17]; C, adapted from [18]).
RNApolymerasepromoter k on k off r nascentRNA(A)(B) (C)(D)OFF ONOFFON timeseparation 1k on amplitude ≈ 1k off ≈ duration ≈ r t r a n s c r i p t i o n i n i t i a t i o n r a t e activator concentration a v e r a g e t r a n s c r i p t i o n i n i t i a t i o n r a t e activator concentration a m p li t u d e s e p a r a t i o n d u r a t i o n Figure 2:
The two-state model of transcriptional bursting . (A) A two-state model oftranscriptional bursting by a promoter switching between ON and OFF states. (B)
Mapping thebursting parameters k on , k off , and r to burst duration, separation, and amplitude, respectively. (C) The action of an activator results in an increase in the average rate of transcription initiation. (D)
In the two-state model, this upregulation can be realized by decreasing burst separation, increasingburst duration, increasing burst amplitude, or any combination thereof.
Plugging this solution into Equation 1 results in the average rate of mRNA produc-tion as a function of the bursting parameters given by (cid:68) initiation rate (cid:69) = r (cid:124)(cid:123)(cid:122)(cid:125) transcription ratein ON state k on k on + k off (cid:124) (cid:123)(cid:122) (cid:125) probability ofON state . (3)Equation 3 shows that, within the two-state model, transcription factors caninfluence the mean transcription rate by modulating any one of the three burst5arameters (or a combination thereof). For example, consider an activator that canincrease the mean transcription rate (Figure 2C) by decreasing k off , increasing k on or r , or any combination thereof (Figure 2D). Both live-imaging measurements andsmFISH have revealed that the vast majority of transcription factors predominantlymodulate burst separation by tuning k on [5, 11, 15, 16, 32–35]. There are alsoexamples of the control of burst amplitude and duration [17, 33, 36].Yet although experiments have identified which bursting parameters are underregulatory control, the question of how this regulation is realized at the molecularlevel remains open (with one notable exception in bacteria [37]). This is becausethe two-state model is a phenomenological model: we can use it to quantify burstdynamics without making any statements about the molecular identity of the burstparameters. Nonetheless, by putting hard numbers to bursting and identifying whichparameter(s) are subject to regulation, this framework constitutes a useful quantita-tive tool to formulate and test hypotheses about the molecular mechanisms under-lying transcriptional control.For instance, consider the observation that many activators modulate burst sep-aration. This observation can be explained if transitions between the ON and OFFstates reflect the binding and unbinding of individual factors to regulatory DNA.Here, k off would be the activator DNA-unbinding rate and k on would be a functionof activator concentration [ A ], k on ([ A ]) = [ A ] k b , (4)where k b is the rate constant for activator binding.The two-state model highlights the absurdity of this proposition: if k off were anactivator unbinding rate, then it would be on the order of 1 s − (Figure 1D and E,box 7). However, measurements of burst duration reveal that k off must be ordersof magnitude smaller ( (cid:46) .
01 s − , Figure 1D). Thus, the two-state model lendsa quantitative edge to the disconnect in Figure 1, confirming that transcriptionalbursting cannot be solely determined by the binding kinetics of the transcriptionfactors that regulate it. We must therefore extend our simple two-state frameworkto incorporate molecular mechanisms that allow rapid transcription factor bindingand transcriptional bursts that are orders of magnitude slower.
3. Bridging the timescale gap: kinetic traps and rate-limiting steps
Recent works have considered kinetic models of transcription that describe tran-sition dynamics between distinct microscopic transcription factor binding configu-rations. These models make it possible to investigate how molecular interactions6acilitate important behaviors such as combinatorial regulatory logic, sensitivity tochanges in transcription factor concentrations, the specificity of interactions betweentranscription factors and their targets, and transcriptional noise reduction [13, 14,38–41].We illustrate how these kinetic models can shed light on the disconnect betweenthe timescales of transcription factor binding and bursting using the activation of the hunchback minimal enhancer by Bicoid in the early fruit fly embryo as a case study[34, 38, 41–44]. Recent in vivo single-molecule studies have revealed that Bicoidspecifically binds DNA in a highly transient fashion ( ∼ − hunchback tran-scriptional bursts, which happen over minutes [34]. We seek molecular models thatrecapitulate two key aspects of bursting: (1) the emergence of effective ON and OFFtranscriptional states, and (2) “slow” ( > hunchback minimalenhancer consists of six binding sites, we first use a simpler version with three bind-ing sites to introduce key features of our binding model before transitioning to themore realistic six binding sites version when discussing our results. We capture thedynamics of activator binding and unbinding at the enhancer by accounting for thetransitions between all possible binding configurations (Figure 3A). Our assumptionof identical activator binding sites leads to two simplifications: (1) the same rate, k i,j governs the switching from any configuration with i activators bound to any config-uration with j bound and, (2) all binding configurations with the same number n ofactivators bound have the same rate of transcription, r n = r n , which we posit tobe proportional to the number of bound activators. As a result we need not trackspecific binding configurations and may condense the full molecular representation inFigure 3A into a simpler four-state chain-like model with one state for each possiblevalue of n (Figure 3B).Transitions up and down the chain in Figure 3B are governed by the effectivebinding and unbinding rates k + ( n ) and k − ( n ). To calculate these rates from themicroscopic transition rates k i,j , consider, for example, that there are three possibleways of transitioning from the 0 state to the 1 state, each with rate k , . Thus, theeffective transition rate between states 0 and 1 is given by 3 k , . More generally, inthe effective model, activator binding rates are k + ( n ) = ( N − n ) k n,n +1 , (5)7 C) (D)(E) (F)(G) (H) i n d e p e n d e n t b i n d i n g c oo p e r a t i v e b i n d i n g t r a n s c r i p t i o n r a t e r a t e - li m i t i n g s t e p ( s ) t r a n s c r i p t i o n r a t e t r a n s c r i p t i o n r a t e k k − (1)k + (0) k − (2)(1) k − (3)(2)k k k k k k k k k k k k k k activatorbinding sites promoteractivator k + k + k k fraction oftime in state0 11 12 13 14 15 16 k + ( / s ) k − ( / s ) − fraction oftime in state0 11 12 13 14 15 16 k + ( / s ) k − ( / s ) − − fraction oftime in state0 11 12 13 14 15 160 11 12 13 14 15 16M off ratelimiting steps M on ratelimiting stepsk on1 k on2 k on3 k onM on k off1 k off2 k off3 k offM off ONOFF10 − number ofbound sites, n Figure 3:
Using theoretical models to understand the origin of ON/OFF bursting dy-namic . See caption in the next page. igure 3: Using theoretical models to understand the origin of ON/OFF bursting dy-namic . (A) Model with three activator binding sites. The transition rates between states with i and j activators are given by k i,j . (B) The model in (A) can be simplified to an effective four-state chain model in which each state corresponds to a certain number of bound molecules and thetranscription rate is proportional to the number of bound activators. (C)
Independent activatorbinding model with effective binding and unbinding rates plotted above and below, respectively.Shading indicates the fraction of time that the system spends in each state. (D)
Stochastic simula-tions indicate that rapid activator binding alone drives fast fluctuations about a single transcriptionrate. (E)
Cooperative binding model in which already-bound activators enhance the binding rateof further molecules. (F)
Simulation reveals that cooperativity can cause the system to exhibitbimodal rates of transcription and slow fluctuations between effective ON and OFF states. (G)
Rate-limiting step model in which several molecular steps can connect a regime where binding isfavored (ON) and a realization where binding is disfavored (OFF). (H)
Simulations demonstratethat rate-limiting steps can lead to bimodal transcriptional activity reminiscent of transcriptionalbursting. Simulation results were down-sampled to a resolution of 0.5 s to ensure plot clarity in D,F, and G. (Parameters: C, D, k b = k u = 0 . − ; E, F, k b = 0 .
004 s − , k u = 0 . − ; and ω = 6 . k uon = k uoff = 0 . − , k boff = 0 .
01 s − , k bon = 21 s − , M off = 1, M on = 2, k off = 0 . k on = k on = 0 . − .) where n indicates the current number of bound activators and N is the total numberof binding sites. Similarly, activator unbinding rates are given by k − ( n ) = nk n,n − . (6)These transition rates allow us to generalize to the more realistic enhancer with sixbinding sites.We first examine a system in which activator molecules bind and unbind inde-pendently from each other (Figure 3C). There are only two unique microscopic ratesin this system: activator molecules bind at a rate k i,i +1 = k b = k b [ A ], with [ A ] beingthe activator concentration and k b the binding rate constant, and unbind at a rate k i,i − = k u . We fix the unbinding rate k u = 0 . s − to ensure consistency withrecent experimental measurements of Bicoid in [45, 46]. For simplicity, we also set k b = 0 . − (see Appendix C.2.3 for details).To gain insight into the model’s transcriptional dynamics, we employ stochasticsimulations based on the Gillespie Algorithm [47]; however a variety of alternativeanalytic and numerical approaches exist [40, 41, 44]. Our simulations reveal thatindependent binding leads to a unimodal output behavior in which the transcriptionrate fluctuates rapidly about a single average (Figure 3D). This result is robust to ourchoices of k b or k u , as well as the number of binding sites in the enhancer (AppendixC.2.2). The observed lack of slow, bimodal fluctuations leads us to conclude that theindependent binding model fails to recapitulate transcriptional bursting.9 thought-provoking study recently suggested that protein-protein interactionsbetween transcription factors near gene loci could generate burst-like behavior [48].Inspired by this work, we extend the independent binding model to consider cooper-ative protein-protein interactions between activator molecules [49] that catalyze thebinding of additional activators. Here, the activator binding rate is increased by afactor ω for every activator already bound, leading to k i,i +1 = k b ω i . (7)Because we assume that activator unbinding still occurs independently, the effectiveunbinding rates remain unchanged (Equation 6).Stochastic simulations of the cooperative binding model in Figure 3F reveal thatthe output transcription rate now takes on an all-or-nothing character, fluctuatingbetween high and low values that act as effective ON and OFF states. Further,our simulation indicates that these emergent fluctuations are quite slow (0.13 tran-sitions/min for the system shown), despite fast activator binding kinetics. Both ofthese phenomena result from large imbalances between k + ( n ) and k − ( n ) that act as“kinetic traps”.Consider the case with five bound activators. If k + (5) (cid:29) k − (5), then the enhanceris much more likely to bind one more activator molecule and move to state six thanto lose an activator and drop to state four. For instance, if k + (5) /k − (5) = 23(Figure 3F), then the system will on average oscillate back and forth between statesfive and six 23 times before it finally passes to state four. While it is possible togenerate this kind of trap without cooperativity at one end of the chain or the otherby tuning k b , cooperative interactions are needed to simultaneously achieve trapsat both ends. Finally, it is important to note that phenomenon is not limited toactivator binding: cooperative interactions in fast molecular reactions elsewhere inthe transcriptional cycle, such as in the dynamics of pre-initiation complex assembly,could, in principle, also induce slow fluctuations.Inspired by the MWC model of protein allostery [44, 50], a second way to bridgethe timescale gap between activator binding and transcriptional bursting is to posittwo distinct system configurations: an ON configuration where binding is favored( k b (cid:29) k u ) and an OFF configuration that is less conducive to binding ( k b (cid:28) k u ).From any of the seven binding states, this system can transition from OFF to ONby traversing M on slow steps, each with rate k ion (cid:28) k u , where i is the step number(Figure 3G). Similarly, transitions from ON to OFF are mediated by M off steps withrates given by k ioff . Stochastic simulations indicate that this system yields bimodaltranscription that fluctuates between high and low activity regimes on timescalesset by the rate-limiting molecular steps (Figure 3H). Thus, as long as these steps10nduce a sufficiently large shift in activator binding ( k b ), the rate-limiting step modelreconciles rapid activator binding with transcriptional bursting.Figure 1B suggests candidates for these slow molecular steps. For example, theON state in Figure 3G could correspond to an open chromatin state that favorsbinding while the OFF state could indicate that a nucleosome attenuates bindingsuch that M on = M off = 1. Our model also allows multiple distinct rate-limitingsteps. For instance, chromatin opening could require multiple histone modifications( M on ≥ M off = 1), or chromatin opening may need to be followed by enhancer-promoter co-localization to achieve a high rate of transcription ( M on = 2, M off = 1).Although they are not the only possible models, the cooperativity and rate-limiting step scenarios discussed above represent two distinct frameworks for thinkingabout how slow processes like bursting can coincide with, and even arise from, rapidprocesses like activator binding. The next challenge in identifying the molecularprocesses that drive transcriptional bursting is to establish whether these modelsmake experimentally distinguishable predictions.
4. Using bursting dynamics to probe different models of transcription
While we cannot yet directly observe the microscopic reactions responsible forbursting in real time, these processes leave signatures in transcriptional dynamicsthat may distinguish molecular realizations of bursting such as those of our coop-erative binding (Figure 3E) and rate-limiting step (Figure 3G) models. Inspiredby [27, 51–53], we examine whether the distribution of observed burst separationtimes (Figure 4A) distinguishes between these two models. In keeping with litera-ture convention, we refer to these separation times as first-passage times from OFFto ON.The variability in reactivation times provides clues into the number of hiddensteps in a molecular pathway. For instance, suppose that bursts are separated by anaverage time τ off = 1 /k on , as defined in the two-state model in Figure 2A and B.If there is only a single rate-limiting molecular step in the reactivation pathway( M on = 1 in Figure 3G), then the first-passage times will follow an exponentialdistribution (Figure 4B) such that the variability, defined as the standard deviation( σ off ), will simply be equal to the mean ( τ off ). Now, consider the case where twodistinct molecular steps, each taking an average τ off /
2, connect the OFF and ONstates ( M on = 2). To calculate the variability in the time to complete both steps andreactivate, we need to add the variability of each step in quadrature: σ off = (cid:114)(cid:16) τ off (cid:17) + (cid:16) τ off (cid:17) = τ off √ . (8)11ore generally, in the simple case in which each step has the same rate, given anaverage first-passage time of τ off , the variability in the distribution of measuredfirst-passage times will decrease as the number of rate-limiting steps, M on , increasesfollowing σ off ( M on ) = τ off √ M on . (9)As predicted by Equation 9, increasing the rate-limiting step number reduces thewidth of the distribution for the rate-limiting step model obtained from stochasticsimulations, shifting passage times from an exponential distribution when M on = 1to increasingly peaked gamma distributions when M on > k u in our case) cause first-passage time distributions to exhibit approximatelyexponential behavior [54].The coefficient of variation ( CV = σ off /τ off ) provides a succinct way to sum-marize the shape of passage time distributions for a wide range of parameter values.Figure 4D plots σ off against τ off for each of the model architectures considered inFigure 4B and C for a range of different τ off values. Points representing distributionswith CV = 1 will fall on the line with slope one and points for distributions with CV < CV values of approximately one for a widerange of τ off values, consistent with exponential behavior. Conversely, all modelswith multiple rate-limiting steps have slopes that are significantly less than one.Thus, by moving beyond experimentally measuring average first-passage time fora given gene and examining its distribution , it is possible to rule out certain molecularmechanisms. For example, a non-exponential distribution would be evidence againstthe cooperative binding and single rate-limiting step models (see Appendix C.1and Appendix C.4 for details about stochastic simulations and first-passage timecalculations). While these conclusions are specific to the models considered here,the general approach of invoking the distributions rather than means and usingstochastic simulations to derive expectations for different models can be employed todiscriminate between molecular hypotheses in a wide variety of contexts. Indeed, theexamination of distributions has been revolutionary throughout biology by making itpossible to, for example, reveal the nature of mutations [55], uncover mechanisms of12 C) (D)(A) (B) 0 5 10 15 20 25first-pasage time (min)00.020.040.060.080.10.12 p r o b a b ili t y t r a n s c r i p t i o n r a t e p r o b a b ili t y first passagetimes M on , number ofrate-limiting stepstwo-statemodel fitsimulation ONOFF OFF ON0 2 4 6 8 10mean first passage time, τ off (min)0246810 s t a n d a r d d e v i a t i o n , σ o ff ( m i n ) C V = 1 ( e x p o n e n t i a l ) CV<1 (gamma)CV>1
Figure 4:
Using first-passage time distributions to discriminate between models of tran-scriptional bursting . (A) The outcome of stochastic simulations like those in Figure 3D, F,and G (purple) is fit to a two-state model (black) and the first-passage times out of the OFFstate are measured. (B)
First-passage times for the rate-limiting step model as a function ofthe number of rate-limiting steps M on calculated using stochastic simulations. A single step re-sults in an exponential distribution, but distributions break from this behavior when more stepsare added, yielding increasingly peaked gamma distributions. (C) In contrast, first-passage timesfor the cooperative binding model follow an exponential distribution. (D)
Standard deviation asa function of mean first-passage time for various parameters choices of the cooperative binding(blue) and the rate-limiting step models (red, with color shading indicating the M on values con-sidered in (B)). Distributions with CV = 1, such as the exponential distribution, fall on the lineof slope one while gamma distributions, with CV <
1, fall in the region below this line. (Param-eters: B, k uon = k uoff = 0 . − , k boff = 0 .
01 s − , k bon = 21 s − , M off = 1, k off = 0 . − , k ion = M on . − ; C, k b = 0 . − , k u = k off , and ω = 6 .
7; D, see simulation scripts onGitHub for exact parameter values.) qualitatively estimating the order of magnitude ofbursting timescales, raw fluorescence measurements from MS2 and PP7 experimentssuch as those in Figure 1A-C do not directly report on the promoter state. Rather,the signal from these experiments is a convolution of the promoter state and the dwelltime of each nascent RNA molecule on the gene body [16]. As a result, inferencetechniques like those developed in [16, 26] are often required to infer underlyingburst parameters and promoter states that can be used to estimate first-passagetime distributions. Other techniques, such as measuring the short-lived luminescentsignal from reporters [27], have also successfully estimated first-passage times.The first-passage time analyses discussed here are just one of an expansive setof approaches to determining the best model to describe experimental data. Forinstance, direct fits of models to experimental time traces could be used to identifythe most appropriate model (see, e.g. [26, 61]). A discussion of this and otherapproaches falls beyond the scope of this work, but we direct the reader to severalexcellent introductions to elements of this field [61–64].
5. Conclusions
The rapid development of live-imaging technologies has opened unprecedentedwindows into in vivo transcriptional dynamics and the kinetics of the underlyingmolecular processes. We increasingly see that transcription is complex, emergent,and—above all—highly dynamic, but experiments alone still fail to reveal how indi-vidual molecular players come together to realize processes that span a wide rangeof temporal scales, such as transcriptional bursting.Here we have argued that theoretical models can help bridge this crucial dis-connect between single-molecule dynamics and emergent transcriptional dynamics.By committing to mathematical formulations rather than qualitative cartoon mod-els, theoretical models make concrete quantitative predictions that can be used togenerate and test hypotheses about the molecular underpinnings of transcriptionalcontrol. We have also shown how, although different models of biological phenomenamight be indistinguishable in their averaged behavior, these same models often makediscernible predictions at the level of the distribution of such behaviors.Moving forward, it will be critical to continue developing models that are explicitabout the kinetics of their constituent molecular pieces, as well as statistical meth-ods for connecting these models to in vivo measurements in an iterative dialoguebetween theory and experiment. In particular, robust model selection frameworks14re needed to navigate the enormous space of possible molecular models for transcrip-tional control. Such theoretical advancements will be key if we are to synthesize theremarkable experimental findings from recent years into a truly mechanistic under-standing of how transcriptional control emerges from the joint action of its molecularcomponents.
6. Acknowledgements
We are grateful to Simon Alamos, Lacramioara Bintu, Xavier Darzacq, JonathanDesponds, Hinrich Boeger, Michael Eisen, Julia Falo-Sanjuan, Anders Hansen, JaneKondev, Jonathan Liu, Mustafa Mir, Rob Phillips, Alvaro Sanchez, Brandon Schlo-mann, Mike Stadler, Meghan Turner and Aleksandra Walczak for useful discussionsand comments on the manuscript. However, any errors and omissions are our own.HGG was supported by the Burroughs Wellcome Fund Career Award at the ScientificInterface, the Sloan Research Foundation, the Human Frontiers Science Program, theSearle Scholars Program, the Shurl & Kay Curci Foundation, the Hellman Founda-tion, the NIH Director’s New Innovator Award (DP2 OD024541-01), and an NSFCAREER Award (1652236). 15 ppendix A. Literature summary of timescales of transcriptional burstingand associated molecular processes
In this section, we present a survey of timescales observed for transcriptional burstingacross a broad swath of organisms (Appendix Table A.1). Further, we review in vivo and in vitro measurements that have revealed the timescales of the molecular transactionsunderlying transcription and its control.Recent technological advances such as single-molecule tracking, live-cell imaging, anda variety of high-throughput sequencing methods, have revealed how eukaryotic tran-scription is driven by a dizzying array of molecular processes that span a wide range oftimescales. The overview of these timescales presented in Figure 1E show how many ofthese processes are significantly faster than transcriptional bursting.Chromatin accessibility is a central control point for regulating transcription in eu-karyotes [44, 65]. DNA wrapped around nucleosome restricts transcription factor access[65, 66]. Multiple studies have determined the timescales of spontaneous DNA unwrap-ping and rewrapping to be around 0.01-5 s [67–69]. While unwrapping and rewrapping areprobably too fast to directly lead to long transcriptional bursts, DNA unwrapping mightrepresent a “foothold” by which factors transiently bind DNA and enact larger-scale,sustained chromatin modifications [65].Interestingly, nucleosome turnover occurs over a longer timescale compatible withbursting, with multiple studies suggesting timescales of several minutes to hours [70–74]. Recent genome-wide studies have measured average nucleosome turnover time to beapproximately 1 hour in the fly and in yeast [70, 71]. Further, histone modifications maymodulate nucleosomal occupancy [65, 75, 76], and the half-life as well as addition of thesemodifications can also span a broad range of timescales compatible with bursting, fromseveral minutes to days [77–83].Once the chromatin is open, enhancers, DNA stretches containing transcription factorbinding sites and capable of contacting promoters to control gene expression, become ac-cessible. Transcription factor binding recruits co-factors and general transcription factorsto the promoter, triggering the biochemical cascade that ultimately initiates transcrip-tion [20]. While the resulting bursts of RNAP initiation last from a few minutes to hours(Figure 1A-D), single-molecule live imaging has shown that transcription factor bindingis a highly transient process, with residence times of 0.5-15 s [45, 46, 84–89]. The vastmajority of transcription factors bind DNA for seconds, but it is worth noting that sometranscription factors and chromatin proteins can bind DNA for minutes [90].However, the binding of transcription factors, the general transcriptional machinery,and RNAP to the DNA might be more complex than the simple cartoon picture of individ-ual molecules engaging and disengaging from the DNA. For example, recent experimentshave revealed that both mediator and RNAP form transient clusters with relatively short16ifetimes in mammalian nuclei of 5-13 s 10 s, respectively [91–94]. In addition, it is demon-strated that transcription factors can also form clusters in vivo [45, 46]. However, howthese cluster dynamics relate to transcriptional activity remains unclear.Further, enhancers and promoters are often separated by kbp to even Mbp. Themechanism by which enhancers find their target loci from such a large distance, and howthis contact triggers transcription, remain uncertain and are reviewed in [95].
In vivo measurements of enhancer-promoter separation in the
Drosophila embryo have shownthat this distance fluctuates with a timescale of tens of seconds to several minutes [96–98]—timescales strikingly similar to those of bursting. However, recent work has cast doubton the simple “lock and key” model of enhancer association (stable, direct contact betweenenhancers and promoters triggers transcription), suggesting instead that enhancers mayactivate cognate loci from afar and, in some cases, may activate multiple target locisimultaneously [32, 95, 96, 98–101]. Many important questions remain about the natureof enhancer-driven activation and it remains to be seen whether enhancer associationdynamics are generic aspect of eukaryotic transcriptional regulation, or whether theyonly pertain to a subset of organisms and genes.A single transcriptional burst generally consists of multiple RNAP initiation events( ∼ Drosophila , for instance)[11, 16]. The transcriptional bursting cycle thus encompasses a smaller, faster biochem-ical cycle in which RNAP molecules are repeatedly loaded and released by the generaltranscription machinery. One interesting hypothesis for the molecular origin of transcrip-tional bursting is that the OFF state between bursts is enacted by an RNAP moleculethat becomes paused at the promoter, effectively creating a traffic jam [102]. Live imagingand genome-wide studies have shown that RNAP pausing before initiation is common ineukaryotes [102–105] and that its half-life of up to 20 min can be consistent with tran-scriptional bursting [106–113].Although the dynamics of some of the molecular processes outlined above are com-patible with the long timescales of transcriptional bursting, we still lack a holistic pictureof how these kinetics are integrated to realize transcriptional bursts and, ultimately, tofacilitate the regulation of gene expression by transcription factors.We must also acknowledge that we still lie at the very beginning of a reckoning withthe dynamics of transcriptional processes as measurements for some molecular processesresults in a range of timescales that are difficult to reconcile. In particular, we still lacksolid dynamic measurements regarding the assembly of the transcription preinitiationcomplex. Yet, perhaps more egregious than the lack of any individual dynamical mea-surement is the lack of a comprehensive, quantitative, and predictive understanding ofhow these molecular processes interact with one another in time and space to give rise totranscriptional bursting. 17 able A.1:
Literature summary of transcriptional bursting.
We attempted to summarize theduration of a single transcriptional burst from various organisms and genes. In the cases where thesingle-cell data is not available, such as in data stemming from smFISH experiments, we used populationaveraged T ON and/or T OF F values instead to give a sense on the timescales.
System Method Bursting Timescale Reference
Bacteria in vitro single-molecule as-say 5-8 minutes [37]
Tet system MS2 T ON ≈ T OFF ≈
37 minutes [8]
Fruit fly embryo even-skipped stripe 2 MS2 few minutes [11, 16] even-skipped
MS2 few minutes [15]
Notch signaling MS2 5-20 minutes [36] snail, Kr¨upple
MS2 5 minutes [32]gap genes: hunchback, gi-ant, Kr¨upple, knirps smFISH T ON ≈ T OFF ≈ hunchback MS2 few minutes [34] even-skipped stripe 2 MS2 few minutes [11]
C. elegans
Notch signaling MS2 10-70 minutes [17]
Human, Mouse
TGF- β signaling luciferase assay few hours [114] TFF-1 signaling MS2 few hours [18]
HIV-1 viral gene MS2 few minutes [115]liver genes smFISH T ON ≈
30 minutes - 2hours [116]mammalian genes luciferase assay few hours [19]
Amoeba actin gene family RNA-seq few hours [117]actin gene family MS2 10-15 minutes [26] able A.2: Summary of measured timescales of underlying molecular processes associated withtranscription.
While the vast majority of transcription factors bind DNA for seconds, it is worth noting thatsome transcription factors (e.g. TATA-binding protein) and chromatin proteins (e.g. CTCF, Cohesin) can bindDNA for minutes. These outliers are not included in Figure 1
System Organism Experimental method Timescale Reference
Nucleosomal DNA Wrapping/Unwrapping
Mononucleosomes
In vitro reconstitution FRET 0.1-5 s [69]Mononucleosomes
In vitro reconstitution FRET 10-250 ms [68]Mononucleosomes
In vitro reconstitution Photochemical crosslink-ing < Nucleosome Turnover
Histone H3.3
Drosophila cell Genome-wide profiling 1-1.5 h [70]Histone H3 Yeast Genomic tiling arrays ∼ Histone Modification dCas9 inducible recruit-ment Mammalian cell Single-cell imaging several hours to days [82]rTetR inducible recruit-ment Mammalian cell Single-cell imaging several hours to days [81]Chemical-mediated re-cruitment Mammalian cell Chromatin in vivo assay several days [83]Histone H3 Human cell Liquid chromatography,mass spectrometer andheavy methyl-SILAClabeling several hours to days(half-maximal time ofmethylation) [80]Targeted recruitment Yeast ChIP 5-8min (reversal of tar-geted deacetylation)1.5 min(reversal oftargeted acetylation) [79]Histone H2a, H2b, H3, andH4 Mammalian cell Isotope labeling <
15 min (acetylationhalf-life) [77]Histone H2, H2a and H2b Mammalian cell Isotope labeling ∼ ranscription Factor Binding Bicoid
Drosophila embryo SMT ∼ Drosophila embryo SMT ∼ Drosophila embryo SMT ∼ ∼ ∼ ∼ ∼ ∼ ∼ In vitro reconstitution SMT ∼ Chromatin Protein Binding
CTCF Mammalian cell SMT ∼ ∼
22 min [118]
Enhancer-Promoter Interaction snail shadow enhancer
Drosophila embryo MS2, PP7 labeling ∼ snail enhancer Drosophila embryo MS2, PP7 labeling several minutes [98]endogenous even-skipped locus with homie insulator
Drosophila embryo MS2, PP7 labeling several minutes [96]
Transcription Initiation even-skipped stripe 2 en-hancer
Drosophila embryo MS2 labeling ∼ ∼ hb P2 enhancer
Drosophila embryo MS2 labeling ∼ RNAP Cluster Dynamics
RNAP tagged with Den-dra2 Mammalian cell tcPALM ∼ ∼ NAP tagged with Den-dra2 Human cell tcPALM ∼ Mediator Cluster Dynamics
Mediator tagged with Den-dra2 Mammalian cell tcPALM ∼ Promoter-Proximal Pausing
RNAP tagged with GFP Human cell FRAP ∼
40 s [106]RNAP (genome-wide)
Drosophila cell RNA sequencing ∼ Drosophila cell ChIP-nexus ∼ Drosophila cell Genome-wide footprinting ∼ Drosophila salivaryglands ∼ ∼ Drosophila cell scRNA-seq 15-20 min (at geneswith low activity) [112]LacO-tagged minimalCMV promoter Human cell MS2 labeling, FRAP ∼ ppendix B. Two-state model calculations As noted in the main text, the average initiation rate is equal to r times thefraction of time the promoter spends in this ON state p on , (cid:68) initiation rate (cid:69) = r p on . (B.1)To predict the effect of bursting on transcription initiation, it is necessary to deter-mine how p on depends on the bursting parameters. In the mathematical realizationof the two-state model shown in Figure 2A, the temporal evolution of p off , theprobability of being in the OFF state, and p on is given by dp off dt = − k on p off + k off p on , (B.2)and dp on dt = k on p off − k off p on . (B.3)To solve these equations, we make the simplifying assumption that our system is insteady state such that p on and p off are constant in time. In this scenario, we canset the rates dp off /dt and dp on /dt to zero. We can then solve for k off in terms of k on resulting in k off = k on p off p on . (B.4)Plugging in p on + p off = 1 gives us k off = k on (1 − p on ) p on , (B.5)which can be solved in terms of k on , k off p on = k on k on + k off . (B.6) Appendix C. Molecular model calculations
Here we provide a brief overview of the calculations relating to the three theoret-ical models of transcription presented in Section 3: the independent binding model(Figure 3C), the cooperative binding model (Figure 3E) and the rate-limiting stepmodel (Figure 3G). We also provide resources relating to the calculation of first-passage time distributions discussed in section 4.22 ppendix C.1. Stochastic simulations
We make heavy use of stochastic simulations throughout this work. A custom-written implementation of the Gillespie Algorithm [47] was used to simulate trajecto-ries for the various models discussed in the main text. These simulated trajectorieswere used to generate the activity trace plots in Figure 3D, F, and G, as well asthe first-passage time distributions in Figure 4B-D. All code related to this project(including the Gillespie Algorithm implementation for stochastic activity trace gen-eration) can be accessed on GitHub.
Appendix C.2. Independent binding model
All calculations in this section pertain to the independent binding model pre-sented in Figure 3C.
Appendix C.2.1. Calculating state probabilities
Calculating the probability of each activity state is central to determining a sys-tem’s overall transcriptional behavior. Because our mathematical model is a linearchain with no cycles (see Figure 3B), we can make progress towards calculating thestate probabilities, p i , by imposing detailed balance, which gives p n k + ( n ) = p n +1 k − ( n + 1) , (C.1)where k + and k − are the effective rates of adding and subtracting a single activatormolecule that we define in Figure 3B. Plugging in Equation 5 and Equation 6 fromthe main text results in p n ( N − n ) k n,n +1 = p n +1 ( n + 1) k n +1 ,n , (C.2)where, the rates k n,n +1 and k n +1 ,n are the microscopic binding and unbinding ratesdefined in Figure 3A, respectively. Now we make use of the fact that there are onlytwo unique microscopic rates in independent binding system: activator moleculesbind at a rate k n,n +1 = k b = k b [ A ], with [ A ] being the activator concentration and k b the binding rate constant, and unbind at a rate k n,n − = k u . Plugging these valuesinto Equation C.2 and rearranging leads to p n +1 = (cid:16) N − nn + 1 (cid:17)(cid:16) k b k u (cid:17) p n . (C.3)To further simplify the expression in Equation C.3, we write k u k b as a dissociationconstant ( K d ), resulting in p n +1 = (cid:16) N − nn + 1 (cid:17) p n K d , (C.4)23hich has the form of a recursive formula for calculating state probabilities fromtheir predecessors. For instance, for the case where n = 0 we have p = N p K d . (C.5)We can extend this logic to calculate the probability of any state, n , as a function of p , leading to p n = N !( N − n )! n ! p K nd = W ( n ) p K nd , (C.6)where the factorial terms captured by W ( n ) on the far right-hand side can be thoughtof as accounting for the fact that a given number of activators bound, n , may cor-respond to multiple microscopic binding configurations (compare Figure 3A and B).Note that W (0) = 1, which means that Equation C.6 is valid even when n = 0. Fi-nally, we impose the normalization condition that the sum of the state probabilitiesshould be equal to 1, which leads to p n = p W ( n ) K − nd p (cid:80) Ni =0 W ( i ) K − id . (C.7)Canceling out the factors of p gives us our final expression for p n , namely p n = W ( n ) K − nd (cid:80) Ni =0 W ( i ) K − id = W ( n ) K − nd Z , (C.8)where Z on the far righ-hand side indicates the sum of all state weights. Thus, givenvalues of the rates k b and k u , which define K d , we can calculate the probability of thesystem being in each binding state n . This probability is shown diagrammatically inthe shading of the different states in Figure 3C. Appendix C.2.2. Independent binding cannot produce bimodal transcriptional output
A basic requirement for bimodal transcriptionl behavior is that p > p and p N > p N − , where N is the total number of binding sites. Couching this in terms ofEquation C.8 leads to p p = 1 N K d > , (C.9)which simplifies to K d > N (C.10)for the low activity regime and p N p N − = 1 N K d > , (C.11)24eading to K d < N (C.12)for the high activity regime. Since K d is set by the ratio k u k b , which is constant for allstates in the independent binding model, it is not possible for it to be simultaneouslylarger (Equation C.10) and smaller (Equation C.12) than the number of bindingsites N . We thus conclude that independent binding is incompatible with bimodaltranscription, regardless of the number of binding sites N . Appendix C.2.3. Diffusion-limited binding
In the main text we state that we set k b = [ A ] k b to 0 . − for the simulationsshown in Figure 3C. This is convenient because it leads to a model where half theavailable sites are bound, on average. This choice is also physically reasonable. Bicoidconcentrations in the embryo in the region of hunchback expression are on the orderof 10 nM ([ A ] ≈
10 nM) [42], so that k b ≈ . − thus implies a k b of approximately0 .
05 nM − s − . This is comfortably below the upper bound for k b set by diffusion(0.1–10 nM − s − ), which we take to be the speed limit for independent binding [120]. Appendix C.3. Cooperative binding
All calculations in this section pertain to the independent binding model pre-sented in Figure 3E.
Appendix C.3.1. Deriving cooperativity weights
In Equation 7 we incorporated cooperative binding by adding multiplicativeweights, ω , giving k i,i +1 = k b ω i . (C.13)This functional form follows from the assumption that each bound activator increases k b by a constant factor ω ≥
1. This leads the expression for k + ( n ) k coop + ( n ) = ( N − n ) ω n k n,n +1 , (C.14)which is a nonlinear function of n . Now, in analogy to the calculations presented inAppendix C.2.1, let’s re-derive our expressions for p n . To start, we have p n +1 = (cid:16) N − nn + 1 (cid:17)(cid:16) k b k u (cid:17) ω n p n . (C.15)Again expressing k u k b as a dissociation constant ( K d ), we obtain p n +1 = (cid:16) N − nn + 1 (cid:17) ω n p n K d . (C.16)25e can also extend this logic to calculate the probability of any state, n , as a functionof p , leading to p n = N !( N − n )! n ! ω n ( n − p K nd = W ( n ) ω n ( n − p K nd . (C.17)Finally, by requiring that all state probabilities sum to one, we obtain p n = W ( n ) ω n ( n − K − nd Z , (C.18)where Z again denotes the sum of all state weights as in Equation C.8. We have usedthese expressions to calculate the probability of each state shown using the shadingin Figure 3E. Appendix C.3.2. Cooperativity permits bimodal expression
Now, let’s use Equation C.17 to examine how the addition of the cooperativityfactor ω makes bimodal bursting possible. Recall that bimodal expression requiresthat p > p and p N > p N − . For the low activity regime, cooperativity is notrelevant and so the form of the requirement remains the same, namely p p = 1 N K d > . (C.19)However, things change in the high activity regime. Here, we have p N p N − = 1 N ω N − K d > . (C.20)In stark contrast to the independent binding case, we see that the addition of ω makesit possible to realize both conditions simultaneously, opening the door to bimodalburst behaviors. Specifically, bimodality demands K d > N, (C.21)and ω > K N − d (C.22)to be true. Not only do these requirements demonstrate that cooperativity is requiredto achieve bimodal bursting, they also indicate that K d must be greater than thenumber of binding sites in the model, which corresponds to a system where, absentcooperative effects, activator binding is highly disfavored.26 ppendix C.3.3. Cooperativity is necessary to simultaneously achieve kinetic trap-ping at both ends of the chain The reasoning here closely mirrors the discussion from the previous section. Toachieve kinetic trapping at both the high and low ends of the binding chain modelsimultaneously, we require (at a minimum) that k − (1) > k + (1) and k − ( N −
1) = ω N − ( N − K d > . (C.24)We can simplify these requirements to obtain upper and lower bounds on ω , namely (cid:104) K d ( N − (cid:105) N − < ω < K d N − . (C.25)We see that Equation C.25 implies restrictions on the relationship between K d and N . Specifically, there must be a gap between the upper and lower bounds in Equa-tion C.25 such that there exist viable ω values. This means that (cid:104) K d ( N − (cid:105) N − < K d N − , (C.26)must hold. Upon simplification, this gives K d > ( N − . (C.27)Equation C.27 tells us that the dissociation constant must be larger than one(indeed, it must be larger than 25 for a 6 binding site system). This implies that theexpression for the lower ω bound on the left-hand side of Equation C.25 is guaranteedto be greater than one as well, which indicates that cooperative interactions arenecessary to realize kinetic traps on both ends of the chain. Appendix C.4. First-passage time calculations
In this Review we used stochastic simulations (briefly outlined in Appendix C.1)to arrive at expectations for the form of first-passage time distributions for the co-operative binding and rate-limiting step models. All relevant scripts are available at27itHub. We also note that the functional forms for waiting time distributions can beusing analytical methods such as Laplace Transforms. We do not provide the detailsfor this approach here, but point the reader to [54, 121], as well as the sources citedtherein, for more information.
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