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Proc Natl Acad Sci U S A. Feb 24, 2009; 106(8): 2583–2588.
Published online Feb 5, 2009. doi:  10.1073/pnas.0803507106
PMCID: PMC2650307
Applied Mathematics, Biophysics

Elongation dynamics shape bursty transcription and translation

Abstract

Cells in isogenic populations may differ substantially in their molecular make up because of the stochastic nature of molecular processes. Stochastic bursts in process activity have a great potential for generating molecular noise. They are characterized by (short) periods of high process activity followed by (long) periods of process silence causing different cells to experience activity periods varying in size, duration, and timing. We present an analytically solvable model of bursts in molecular networks, originally developed for the analysis of telecommunication networks. We define general measures for model-independent characterization of bursts (burst size, significance, and duration) from stochastic time series. Inspired by the discovery of bursts in mRNA and protein production by others, we use those indices to investigate the role of stochastic motion of motor proteins along biopolymer chains in determining burst properties. Collisions between neighboring motor proteins can attenuate bursts introduced at the initiation site on the chain. Pausing of motor proteins can give rise to bursts. We investigate how these effects are modulated by the length of the biopolymer chain and the kinetic properties of motion. We discuss the consequences of those results for transcription and translation.

Keywords: bursts, interrupted Poisson process, transcriptional pausing, waiting times

The stochasticity of molecular processes contributes to heterogeneity in populations of isogenic cells. Cellular heterogeneity is manifested by differences in the copy numbers of molecules and in the timing and duration of processes. Recent advances in single-cell measurement have facilitated the quantification of stochastic phenomena (14) (reviewed in refs. 5 and 6). Together with models and theory much insight has been obtained into the sources of noise and how particular network designs contribute to noise suppression and amplification (710).

Stochasticity of gene expression has been described by distributions of macromolecules in a population of cells (1, 2). Whether averaging over a population captures the entire spectrum of molecular fluctuations a particular cell experiences over one generation, depends on magnitudes and rates of fluctuations. If these are slow but high in amplitude, the required averaging duration may extend over a generation span (11). Then, a single cell may not even be able to reach protein states accessible to other members, thus rendering cell–cell protein level distributions uninformative with respect to behavior of genetic circuits (12). In such cases, waiting times for individual birth and death events need to be monitored to assess physiological constraints on a single-cell level. This stochastic nature of waiting times will be our focus. Little analytical theory has been developed to deal with this phenomenon despite its relevance for single-cell behavior.

The waiting times in a first-order process with rate constant k follow an exponential distribution; the mean waiting time between events and its standard deviation are equal to 1/k. The waiting time for an event is no longer exponentially distributed if it is regulated by another process. This mechanism underlies bursts in synthetic activity. The interrupted Poisson process (IPP) was introduced to study bursts in queuing and telecommunication theory (13). In an IPP, a stochastic switch modulates a process with exponentially distributed waiting times. Depending on the time scale separation between the process and the switch, multiple timescales may appear in waiting times for production events.

Bursts have received ample attention in the biophysics literature (5, 10, 1417). These studies tend to focus predominantly on the protein number distributions, but do not analyze the distributions for waiting times in much depth. We show that such statistics are relevant for burst characterization and the molecular mechanisms giving rise to bursts.

Bursts have been experimentally observed for synthesis of mRNA and protein (3, 4, 1822). They are characterized by rapid productions of a number of mRNA or protein molecules during short time intervals. Periods of synthetic silence occur between bursts. Bursts may give rise to significant disturbances of cellular physiology depending on burst size and the duration of synthetic silence and activity. Even though the benefit of bursts needs to be analyzed further, they could be beneficial for cells living in rapidly fluctuating environments (23). Bursts may give rise to a bimodal distribution of protein expression across cell populations (24). Thereby, 2 subpopulations could emerge having different adaptive potentials.

We apply the analytical theory of IPPs to a molecular mechanism for bursts. To identify and characterize bursts we derive 3 new indices: burst size, duration, and significance. We demonstrate how motor protein trafficking along biopolymer chains (such as mRNA polymerase and DNA polymerase along DNA, ribosomes along mRNA, and cargo-carrying dynein along microtubuli) can generate bursts depending on the length of the biopolymer and stochasticity of initiation and motion. We show that motor proteins can generate bursts by pausing or by memory of initiation bursts.

Analytical Expression of the Waiting Time Distribution

In this section, we study a small network to gain insight into burst-generating mechanisms. This will allow us to derive general indices for the characterization of burst properties. These indices will be applied to characterize biological mechanisms.

The network consists of a source switching between an inactive OFF and an active ON state according to a Poisson process (Fig. 1A). OFF and ON periods are defined on the level of the switch (see refs. 15 and 17 for the discussion of mechanisms giving rise to genetic switches). In the active state, production of P, e.g., mRNA or protein, occurs at exponentially distributed intervals of length τini = 1/kini. The average ON period lasts for τon = 1/ksw. Production periods are interrupted by transitions to the OFF state. Each rate constant corresponds to the inverse of the mean first passage time for a complex kinetic mechanism. We assume that it follows an exponential waiting time distribution.

Fig. 1.
Bursts generated by the minimal model. (A) The network consists of a switching source and a Poissonian product generator. Full arrows denote reactions. Product P is synthesized only in the ON state. ksw+, ksw, kini, and kdeg denote the ON switching, ...

Product generation is bursty if it occurs many times during one ON state. In addition, the duration of the OFF state (τoff = 1/ksw+) should be longer than or comparable to the ON period. Under these conditions, waiting times display 2 timescales (Fig. 1B). Because our interest is the statistics of intervals between production, the degradation of P does not play a role.

The mechanism discussed here specifies an interrupted Poisson process (IPP) investigated in the field of queuing theory (13). An IPP is a Poisson process for event occurrence (arrivals) modulated by a random switch. In this framework a gene that switches between an ON and OFF state as function of a transcription factor would be considered the source. Arrivals would, for instance, correspond to initiations of transcription giving rise to elongating RNA polymerases. The IPP theory provides the probability density function (PDF) for waiting times between production events, fX(t), with the stochastic variable X as the waiting time with value t. It is instructive to realize that the PDF can become >1 (it is not a probability) and that ∫0 fX(t)dt = 1. The probability of an interval between consecutive events being within [t, t + dt] equals fX(t) dt. The PDF of an IPP is a weighted sum of 2 exponential distributions,

equation image

equation image

where K = ksw+ + ksw + kini, and the weight factors w1 = 1 − w2 = (kinir2)/(r1r2) [set membership] (0, 1); derived in the supporting information (SI) Appendix. The PDF can reveal the presence of 2 times scales in a stochastic time series (Fig. 2Upper).

Fig. 2.
Theoretical analysis of the minimal burst model (Fig. 1). Columns correspond to different parameterizations. (Upper) The waiting time PDF for a pure Poisson process and for 2 IPPs with different burst characteristics (Eq. 1). The vertical dashed lines ...

The length of intervals between production events follows from the superposition of 2 independent processes: production during a single ON state, and periods of synthetic inactivity. The latter may result from multiple switches between ON and OFF states without producing any P. This is the case if the mean number of productions per ON state is small, i.e., kiniksw (Fig. 1C). Accordingly, synthetic activity and silence periods (at the level of P production) do not strictly overlap with ON and OFF states of the switch.

Characteristic timescales of the fast and the slow process appear in Eq. 1 as rates r1 and r2. Weight factors w1 and w2 are the probabilities to observe the short period (mean duration 1/r1) and the long period (mean 1/r2) between P productions, respectively. For large timescale separation, i.e., when ksw+ and ksw are much smaller than kini (Fig. 2 Right), the rates become r1kini and r2ksw+. In this regime, the expected burst size, βe, equals the number of initiations per ON state, i.e., kini/ksw.

The point of timescale separation we refer to as τX (Fig. 2). At this interval the probability to observe a waiting time resulting from the fast and the slow process is identical (two terms of Eq. 1 are equal). Two timescales are observable if kini > ksw+ksw; only then τX > 0 (SI Appendix).

The mean waiting time determined from Eq. 1 corresponds to the macroscopic estimate (SI Appendix),

equation image

The inverse of this equation equals the mean arrival rate, kini left angle bracketONright angle bracket. It has the interpretation of the expected burst size divided by the duration of a single switch cycle. The noise in the waiting time is given by

equation image

The second term expresses the deviation of IPP from a Poisson process. It is small if either the ON state is short-lived (βe decreases) or silence periods are negligible.

Measures for Characterization of Bursts

The size of a burst and the burst duration are relevant burst properties. For biological applications they need to be determined on the basis of a stochastic time series as the mechanism underlying bursts is typically unknown.

The burst size β is defined as the mean number of production events not interrupted by a long inactivity period. Inactivity periods (interruptions) occur as often as bursts. The total number of production events divided by the number of interruptions yields the burst size. To determine the timescale of interruptions and hence their number, we define a sequence size function Φ([theta]),

equation image

where FX([theta]) is the cumulated distribution function (CDF), i.e., FX([theta]) = ∫0[theta] fX(t) dt. For a given threshold [theta], the function yields the sequence size such that events are grouped into sequences interrupted by intervals longer than [theta]. Because of the timescale separation, there is a specific interval [theta]b for which Φ([theta]) equals the burst size β. The value of [theta]b can be determined on the basis of the functional dependence of Φ([theta]) as illustrated in Fig. 3.

Fig. 3.
Diagram of a sequence size function Φ. (Upper) Time series of production events. Horizontal bars denote intervals longer than thresholds [theta]A–D. (Bottom) For a given [theta], Φ is constructed by dividing the total number ...

As a measure for burst size β we evaluate the sequence size function at the interval of timescale separation, τX, which has a straightforward interpretation for the minimal burster from Fig. 1. This point lies in the middle of 2 intervals τ1 and τ21 < τ2); they correspond to the change of Φ([theta]) from convex to concave to convex as function of [theta], respectively (SI Appendix).

The burst size β is greater than the expected burst size βe because the βe excludes the possibility of an ON state without a production event. Both measures are approximately equal for a large timescale separation: β = βe + O(log kini), if kini [dbl greater-than sign] ksw+ and kswksw+. Fig. 2 illustrates the PDF and Φ([theta]) for different parameter regimes of the minimal model. Bursts become more pronounced for high kini over ksw ratios (increased timescale separation). The behavior of the sequence size function for nonexponential waiting times is explored in the SI Appendix. In short, a gamma-distributed waiting time for the OFF to ON transition increases the timescale separation. The applicability of the indices is not affected.

Once the burst size is known, the duration of a burst, τβ, can be obtained by multiplying β by the mean waiting time within a burst (1/r1 in the minimal model). Whether the interval is part of a burst can be deduced by using the threshold of timescale separation, τX (determined on the basis of the waiting time PDF). In addition to burst size, burst significance is important. Bursts lose significance if the interruption period becomes comparable to intervals within a burst. To quantify this we introduce a dimensionless significance coefficient ξ = 1 − τ12, ξ [set membership] (0, 1).

We use all 3 measures, β, τβ, and ξ, to analyze stochastic time series for more complex schemes. The advantage is that the indices are mechanism-independent. In addition to their unbiased nature, they have a clear mechanistic interpretation for the minimal burster. This property facilitates interpretation of yet unidentified mechanisms giving rise to bursts.

Motor-Protein Traffic Jams along Biopolymer Chains

Bursts have been observed experimentally for single-cell synthesis of mRNA and protein (3, 4, 1820) (review, ref. 6). For such cases, free-energy-driven motion of a catalytic motor protein along a biopolymer template is required. Here, we investigate the role of the stochasticity in the initiation of motion and in the motion itself for the observation of bursts at the end of the chain. We will consider different lengths of the polymer and kinetics of initiation and transport.

Fig. 4 shows a canonical 1D macromolecular trafficking model. It contains the switching source, as described in the previous section, followed by sites on the polymer. Each site can be occupied by a single motor, moving forward only, with the elongation rate constant kel (sites per time). Evidently, motors cannot pass each other and shall collide. We also consider the motor occupying more than one site (SI Appendix). Previously we discussed conditions for bursts to emerge at the start of the polymer. Whether bursts are preserved at the end of the chain depends on the characteristics of the stochastic motion.

Fig. 4.
Canonical model of macromolecular trafficking along a biopolymer. In the ON state of the switch, proteins initiate elongation with a rate constant kini. Elongation occurs with a rate constant kel. “O” and “U” denote occupied ...

In Fig. 5A we consider a polymer of length 100 sites, preceded by a bursty switch with 100 initiations per ON state, on average. We plot the mean occupancy of sites at the beginning and at the end of the polymer as function of a dimensionless ratio, kel/kini. If a motor protein travels many sites between initiations during a single ON state (kel/kini [dbl greater-than sign] 1), its progression is not hampered by collisions. A traffic jam arises at the beginning of the chain if the number of traversed sites during consecutive initiations is small (kel/kini [double less-than sign] 1). This results in high mean site occupancy at the initial segment of the polymer. Congestion weakens at the end of the polymer because motors have fewer partners ahead of them. As shown in Fig. 5B, the net occupancy gradient along the chain increases with the polymer length.

Fig. 5.
Mean site occupancy. (A) At the beginning (solid line) and at the end (dashed line) of a 100-site polymer (ksw+ = ksw = 1 [1/T], kini = 100 ksw); error bars, standard deviation; (B) along 1- to 100-site polymers (dotted), numbers indicate ...

If collisions are significant, the timescale separation generated at the initiation stage is disrupted. The frequency of P production becomes exclusively determined by the motor protein progression at the end of the chain. The effect intensifies as the length of the polymer increases as illustrated in Fig. 6A.

Fig. 6.
Analysis of the canonical model of macromolecular trafficking. Gillespie simulations of at least 1E-05 events, ksw+ = ksw = 1 [1/T], kel = kini = 100 ksw. (A–D) The dashed line marks the minimal burst model (no elongation). ( ...

Because of collisions, the rate at which motors leave the polymer becomes smaller than kini. The mean interval between productions (the inverse of the macroscopic flux) becomes longer for longer chains (Fig. 6B). Additionally, the standard deviation becomes comparable to the mean waiting time. The process becomes exponentially distributed and the memory of the state of the switch is lost entirely.

Motor protein collisions disrupt inactivity periods of initiation bursts. As a result, intervals between production events become comparable. In this regime, the burst size β measured at the end of a long polymer is seemingly larger than for a short one (Fig. 6C). Therefore, it is necessary to aid the measurement of bursts with the significance index, ξ. As the length of the polymer increases, the significance of bursts diminishes (Fig. 6D). Below, we introduce a mechanism that can recover bursts at the output even if no bursts occur at the input.

Pausing of Motor Proteins Can Generate Bursts

We will consider a polymer without a switching source that modulates initiation. Initiation takes place at a fixed rate constant kini. Such a model has only one timescale, thereby no bursts at initiation can occur. We consider motor protein pausing along the chain as a potential burst-generating mechanism. At every site, a motor can switch at a rate kp+ to a paused state that lasts 1/kp (Fig. 7). This mechanism is known to occur for RNA polymerase (2529) and ribosomes (3034).

Fig. 7.
Model of macromolecular trafficking with pausing motor proteins. Proteins initiate motion with a fixed rate constant kini. A site can be unoccupied (U), occupied (O) by a motor, or occupied by a motor in a paused state (S). Motors can pause at a rate ...

Pausing of a single motor causes congestion due to its collision with consecutive motors during its dwell time. This allows for the buildup of a burst packet. The packet can survive until the end of the chain only if the pausing frequency is low for a given chain length L. If this is not the case, there is a high probability that proteins within the potential burst will also pause and thus divide the packet (Fig. 8A, curves for kp+ = 100 [1/T]). Another requirement for bursting concerns the lifetime of the paused state. If too short, compared with the initiation rate kini and the elongation rate kel, the consecutive proteins will not catch up (Fig. 8A, curves for kp = 100 [1/T]). Timescale separation in the waiting times at the end of the chain, and hence bursts, appear only when motors do not pause too frequently during the elongation, and when the paused state is sufficiently long-lived (Fig. 8A, solid curve for kp+ = kp = 1 [1/T]).

Fig. 8.
The effect of chain length and pausing parameters on waiting time statistics for the model with pausing. Data obtained from Gillespie simulations of 1E-06 events. Common parameters, kini = 100 [1/T], kel = kter = kini. (A) The appearance of the timescale ...

The length of the biopolymer chain affects the statistics of bursts as well. The addition of sites increases the probability that a single motor pauses a number of times during its progression. Thereby, such a motor destroys the burst it was part of. The effect is equivalent to an increase in kp+ at a fixed L. As a result, the waiting times lack the short timescale originating from frequent product initiation. Instead, they are dominated by the lifetime of the paused state. The mean waiting time left angle brackettright angle bracket and its standard deviation increase with increasing L, and the timescale separation becomes less pronounced (Fig. 8 B and C). Because burst size β decreases, bursts tend to disappear for longer chains. They can always be recovered by decreasing the probability of a single motor protein to pause many times during its progression. As an illustration we will change the pausing rate for the longest chain considered, L = 500. If we set kp+ 10 times smaller than the value used for L = 50, the timescale separation is recovered and β increases (Fig. 8 B and D, dashed line). This indicates that pausing may prevent or promote bursts depending on the properties of the biopolymer.

Aggregative Behavior of Multiple Burst Generators

Statistics of bursts change when they arise from the simultaneous activity of a number of independent burst-generating mechanisms. In biological terms, this superposition may describe the transcription of an mRNA from independent copies of a gene or the translation of protein from a number of mRNAs. Here, we focus on the extension of the simple model of bursts to a superposition of many independent interrupted Poisson processes.

We choose parameters such that a single burster initiates 1 product per ON state (ksw+ = 0.1 [1/T], ksw = 1 [1/T], kini = ksw). The resulting burst size is small, β ≈ 2.3. The waiting time PDF almost completely loses its double-exponential character for >8 sources (Fig. 9A). Although the burst size increases for many sources (Fig. 9B), their significance ξ diminishes until the sequence size function Φ becomes always convex and ξ can no longer be evaluated (Fig. 9B Inset). This indicates that the clustering of events into bursts no longer occurs; waiting times follow a single-exponential distribution. Analytical results for the pooled mechanism, the PDF, and the Poissonian behavior in the limit of many independent IPPs, can be found in the SI Appendix. These results are particularly insightful for the conditions for bursts in the protein level. A burst of a few mRNA transcripts does not necessarily cause a protein burst. It depends on temporal correlation between translation that occurs independently on each transcript.

Fig. 9.
Superposition of independent bursters (ksw+ = 0.1 [1/T], ksw = 1 [1/T], kini = ksw). (A) Analytical waiting time PDF as a function of the number of burst sources. The curve becomes almost exponential for 8 sources. (B) Analytical sequence ...

Discussion

For most applications the exact burst-generating mechanism is not known or too complex to handle analytically. To overcome this problem we defined 3 measures for the characterization of bursts: burst size, duration, and significance. We stress the usefulness of the significance measure, because large bursts can arise at a negligible timescale separation. The indices have a transparent interpretation for the system in Fig. 1 and allow for model-independent analysis of more complicated mechanisms. Additionally, we offer a rigorous method to obtain the indices from stochastic time series. We applied those measures to investigate the influence of the stochastic motion of motor proteins along a biopolymer on bursts of product release at the end of the chain, e.g., of protein, cargo-vesicles, or mRNA. Our study was inspired by the experimental discovery of bursts in transcription and translation (3, 4, 1820). We found that bursts at the input of the chain tend to be smoothed out by longer chains because of the congestion of motor proteins. Because of collisions, timescales within and between bursts become comparable causing the burst size to be larger but of less significance. Hence, the stationary output flux decreases with the length of the chain. At a fixed initiation rate, bursts can emerge because of the pausing of motor proteins.

We discussed 2 mechanisms that could give rise to bursts in production. Bursty transcription initiation was the first one. The second considered motor protein pausing as a source of bursts. How can these 2 mechanisms be distinguished experimentally and how can existing data be interpreted in this light? We will consider this in more detail for transcription. Promising mechanisms for initiation-induced bursts in mRNA production are genes controlled by strong repressors occasionally leaving the promotor to allow a few RNA polymerases to initiate transcription as suggested in refs. 3 and 4. On average, every τini minutes an RNA polymerase initiates elongation if the gene is in the ON state, i.e., in the presence of an activating transcription factor or in the absence of a repressor. During this time, polymerase traverses kel τini nucleotides. If no significant congestion occurs along the DNA, the mean waiting time left angle brackettright angle bracket for polymerases at the end of the chain is proportional to τini (Eq. 2); the timescale of initiation bursts (if present) is preserved at the end of the chain. If motors collide during their progression this relationship is destroyed. Fig. 10 illustrates this for switch parameters corresponding to measurements by Golding et al. (3); τon = 6, τoff = 37 min. This figure displays the dependency of left angle brackettright angle bracket on the initiation time τini for the minimal, canonical trafficking, and detailed model (with initiation switch and pausing). Assuming that the experimental system studied by Golding et al. (3) is in the regime where collisions do not disrupt the proportionality between left angle brackettright angle bracket and τini, their measured waiting time within a burst corresponds to τini = 2.5 min. For these parameters, Fig. 10 indicates that most of the control on the waiting time is exerted by the initiation; pausing properties do not affect left angle brackettright angle bracket. The calculated burst size (3.6) is close to the experimental result (≈2.2).

Fig. 10.
Detailed model of elongation: mean waiting time left angle brackettright angle bracket for mRNA production as function of initiation intervals τini = 1/kini. The gene consists of 1,000 nt, RNA polymerase occupies 50 nt (35). Parameters of the initiation switch are ...

How would pausing of RNA polymerases and altered initiation rates (different genes) change this picture? In Fig. 10, we investigate this question by using realistic pausing parameters. The panel for burst size shows that initiation bursts are reduced by pausing. Spontaneous collisions and hence deviation of left angle brackettright angle bracket from linear dependence (Eq. 3) occur only at high initiation rates. The discrepancy is larger if pausing is considered. Therefore, pausing is a more plausible mechanism for such deviations in real biological systems where τini exceeds 0.1 min. This makes pausing a potent target for regulation, e.g., by NusA and NusG, in accordance to recent experiments (2629). Voliotis et al. (35) have postulated collision-induced bursts caused by backtracking of RNA polymerases. Fig. 10 offers a convenient method to determine whether collisions induced by pausing or spontaneous are additional controlling processes besides initiation.

Which genes are likely to generate bursts? Strong repressors could induce bursts according to bursty initiation as described above. In experimental studies this mechanism yielded a small burst size (3, 4). If the lifetime of mRNAs and proteins is shorter than the OFF period, transient bursts (“puffs”) are produced. Savageau's demand principle (36) predicts that infrequently used genes are regulated by repressors to prevent them from accumulating mutations. Such systems should be susceptible to bursts, e.g., repressor-regulated operons of prokaryotic signaling networks.

Highly activated genes under the control of an activating transcription factor can give rise to bursts by the pausing mechanism (Fig. 8). Such systems are predominantly in the ON state (e.g., by way of enhancer dependence) and have an approximately constant and high initiation rate. This should make them prone to bursts induced by pausing. Such a mechanism has not been experimentally observed so far. Escherichia coli's rrn genes (coding for ribosomal RNA) would be likely candidates, for they are among the genes with the highest expression activity (37). In mammalian systems, poised and paused polymerases occur often and could underlie the experimentally observed bursts (20, 38, 39).

Typically, products are not synthesized in a single macromolecular process. We showed that if bursts are generated independently, the resultant burst tends to lose significance with an increasing number of sources. Hence, mRNA bursts are more likely to occur than protein bursts, because protein is typically produced from a few transcripts simultaneously. Depending on the significance and size of the burst generated by a single catalytic process, this rules out significance of bursts in, for instance, metabolism where the copy number of catalytic proteins is thousands.

Bursts are a powerful mechanism to generate cellular heterogeneity. Key processes such as transcription and translation are particularly prone to generate bursts. How biological systems manage to function reliably in the presence of bursts, whether they actively suppress them or control their characteristics, remains to be experimentally shown. Exact burst properties and their functional consequences depend on the relative timescales of initiation, elongation, and termination processes. The analytical theory we presented gives insight into generic burst properties. The proposed burst indices allow for quantitative studies of specific systems and their comparison.

Supplementary Material

Supporting Information:

Acknowledgments.

We thank J. G. Blom, J. Vidal Rodríguez, H. Westerhoff, and P. R. ten Wolde for helpful input and the anonymous reviewers for their very insightful comments. This work was supported by Netherlands Organisation for Scientific Research (NWO) Computational Life Sciences Project NWO-CLS-635.100.007 and by the Netherlands Institute for Systems Biology and NWO (F.J.B.).

Footnotes

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0803507106/DCSupplemental.

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