Intermittent Transcription Dynamics
for the Rapid Production
of Long Transcripts of High Fidelity
Martin Depken,1Juan M.R. Parrondo,2,3and Stephan W. Grill3,4,*1Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, the Netherlands 2Departamento de Fisica Ato´mica, Molecular y Nuclear and GISC, Universidad Complutense de Madrid, 28040 Madrid, Spain
3Max Planck Institute for the Physics of Complex Systems, No¨thnitzerstrasse 38, 01187 Dresden, Germany 4Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstrasse 108, 01307 Dresden, Germany
*Correspondence:grill@mpi-cbg.de
http://dx.doi.org/10.1016/j.celrep.2013.09.007
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-No Derivative Works License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited.
SUMMARY
Normal cellular function relies on the efficient and
ac-curate readout of the genetic code. Single-molecule
experiments show that transcription and replication
are highly intermittent processes that are frequently
interrupted by polymerases pausing and reversing
directions. Although intermittent dynamics in
repli-cation are known to result from proofreading, their
origin and significance during transcription remain
controversial. Here, we theoretically investigate
tran-scriptional fidelity and show that the kinetic scheme
provided by the RNA-polymerase backtracking
and transcript-cleavage pathway can account for
measured error rates. Importantly, we find that
inter-mittent dynamics provide an enormous increase in
the rate of producing long transcripts of high fidelity.
Our results imply that intermittent dynamics during
transcription may have evolved as a way to mitigate
the competing demands of speed and fidelity in the
transcription of extended sequences.
INTRODUCTION
Central to fidelity in both replication and transcription is that the different nucleotides possess different affinities for base pairing with template nucleotides inside the respective polymerase. Although substantial (Blank et al., 1986), this selectivity is ulti-mately limited by early and immutable evolutionary choices pertaining to nucleotide chemistry. As organisms evolved and diversified, longer genes and bigger genomes needed to be processed (Xu et al., 2006). With a concurrent increase in de-mand on fidelity, both DNA and RNA polymerases (RNAPs) have evolved proofreading mechanisms capable of removing errors that have already been incorporated into their growing polymer product. Utilizing such mechanisms, replication rea-ches an error ratio (number of incorrect bases divided by the number of correct bases in the final transcript) of the order
of 1/108 (Kunkel and Bebenek, 2000), whereas transcription
achieves an error ratio of at least 1/105 (Sydow and Cramer, 2009).
The theoretical underpinning of kinetic proofreading was established by Hopfield almost 40 yeas ago (Hopfield, 1974). However, the standard treatment assumes the bases to be repeatedly checked before being permanently incorporated into the growing transcript, and this preincorporation selection (PIS) results in an ever-growing transcript. With the event of single-molecule techniques, it is now well established that both RNA and DNA polymerases elongate their product in a highly intermittent manner: repeatedly pausing, moving backward, and cleaving off bases from the growing molecule (Erie et al., 1993; Donlin et al., 1991; Thomas et al., 1998; Orlova et al., 1995; Wang et al., 2009; Galburt et al., 2007; Wuite et al., 2000; Ibarra et al., 2009; Kireeva and Kashlev, 2009). In fact, postincor-poration proofreading (PIP) through transcript cleavage has long been recognized as playing a vital role in error suppression (Erie et al., 1993; Thomas et al., 1998; Sydow and Cramer, 2009; Ibarra et al., 2009; Kunkel and Bebenek, 2000; Jeon and Agarwal, 1996) but has received little attention at a quantitative theoretical level (Voliotis et al., 2009, 2012).
Using stochastic modeling, we here explore the downstream effects of PIP in transcription and elucidate the connection among proofreading, intermittent transcription dynamics, and overall elongation performance. Our stochastic hopping model (Greive and von Hippel, 2005) is built using structurally well-characterized states, with transition rates measured in physio-logically relevant settings. Importantly, the model introduces a coupling between chain elongation and the observed transcript shortening induced by proofreading (Erie et al., 1993; Thomas et al., 1998)—a connection not accounted for in Hopfield’s original development (Hopfield, 1974). In fact, the highest error suppression calculated within the standard scheme corre-sponds to a pathological situation with a net shortening of the transcript over time. This highlights the importance of performing a detailed analysis of both fidelity and elongation dynamics.
Our analysis shows that PIP can account for an error reduction of over two orders of magnitude under ideal conditions. It can thus in principle account for the difference between what is
observed in fidelity experiments and what can be expected from the relative base-pairing energies of matched and mismatched base pairs (bp) (Sugimoto et al., 1995; Blank et al., 1986). By considering the transcription of extended sequences, we further suggest evolutionary arguments for why the dynamics of an efficiently transcribing RNAP should be expected to be inter-mittent when adapted to transcribing long sequences at high fidelity and speed.
RESULTS
Transcriptional Error Suppression
Much of previously published theoretical work on RNA tran-scription has focused on recovering the sequence-dependent part of pausing patterns captured in transcription gels (Bai et al., 2004; Tadigotla et al., 2006; Maoile´idigh et al., 2011) and more recently through deep sequencing of nascent transcripts (Churchman and Weissman, 2011). However, proofreading is a process that should be efficient on a wide variety of genes. To identify mechanisms that function on generic sequences, we adopt a sequence-averaged view throughout this work. This is not to say that there is no sequence dependence in tran-scriptional pausing, which is clearly evidenced by strong pause sites (Nechaev et al., 2010), but is rather to focus on the high level of stochastic backtracking observed throughout, for example, highly transcribed genes (Churchman and Weissman, 2011).
To capture the effect of thermal fluctuations, we describe tran-scription as a stochastic hopping process between well-defined states, with transition rates set by the intervening free energy barriers (Risken, 1989). Although there is a wealth of biochemical studies of transcription and transcriptional fidelity (Kellinger et al., 2012; Thomas et al., 1998), the kinetic rates extracted from these studies are often effective rates that depend on implicit model assumptions. On the other hand, the more directly interpreted single-molecule data are incomplete for any partic-ular polymerase. Therefore, we here rely on the great structural homology present among bacterial, eukaryotic, and archaeal polymerases (Ebright, 2000; Hirata et al., 2008) to infer order of magnitude estimates of transition rates between microscopic states for an imagined generic RNAP. Although there will be quantitative differences for any particular real polymerase, the physical mechanisms we consider will remain applicable.
FollowingHopfield (1974), we take the error suppression to be achieved through a sequence of serially connected energy-consuming, molecular-scale, and error-correcting checkpoints. The quality of a checkpointi is judged by the fraction riof errors
it lets through, and the quality of several sequential checkpoints is given by the product of individual error fractionsr1r2r3. (see
Extended Results).
In transcription, both PIS and PIP are controlled by the same multifunctional active region inside the RNAP (Kettenberger et al., 2003; Opalka et al., 2003). The PIS process likely involves several steps before the incoming nucleotide is catalyzed onto the growing RNA molecule (Sydow and Cramer, 2009). The relative occupation of the catalysis-competent state is dictated by the free energy cost of aligning different nucleotides in the active site. A lower bound on the energetic penalty for aligning a mismatch rather than a match can be estimated
from the free energy cost of nucleotide mismatches in RNA-DNA duplexes in solution:DGactz6 kBT (Sugimoto et al., 1995;
Blank et al., 1986), givingrPISzexpðDGact=kBTÞz1=400. By
utilizing differences in base-pairing energies alone, and assum-ing that the polymerase operates close to the lower energetic bound on mismatch penalties, we would therefore expect no more than a few hundred bp to be reliably transcribed without errors.
To correctly transcribe longer sequences, further error sup-pression is needed. One way of increasing fidelity beyond what is given by base-pairing energies is through additional discriminatory interactions in the active site. Accounting for the full fidelity of the polymerization process in this way (Kellinger et al., 2012) would require the total barrier to catalysis to differ by a fullln ð105Þk
BTz12 kBT between the correct and incorrect
base. Although there is likely a change in the barrier to transcrip-tion as an error is incorporated, the extent of this change is unknown. Here, we investigate the theoretical bounds on fidelity gains offered by the experimentally established alter-native to direct energetic penalties: proofreading through PIP (Erie et al., 1993; Thomas et al., 1998; Sydow and Cramer, 2009; Ibarra et al., 2009; Kunkel and Bebenek, 2000; Jeon and Agarwal, 1996). In the next section, we show that, by the inclu-sion of PIP, the total error suppresinclu-sion can reach physiological levels without assuming large energy penalties beyond that offered by base pairing alone.
Proofreading through Backtracking
PIP-mediated fidelity increases are controlled by the RNAP entering a backtracked state followed by error removal through transcript cleavage. The backtracked state is an off-pathway state where the whole polymerase is displaced rearwards on the template (Shaevitz et al., 2003; Galburt et al., 2007) and from which the nascent transcript can be cleaved upstream from the 30end (seeFigures 1A and 1B). Although the physio-logically relevant scenario is one with multiple positions to backtrack to and cleave from, it is instructive to first consider the hypothetical case of a polymerase restricted to only one backtracked state where the polymerase is displaced by a single bp.
Single-State Backtracking
As the polymerase backtracks, the internal 8–9 bp RNA-DNA hybrid remains in register by breaking the last-formed base pair and reforming a previous base pair at the opposite end of the hybrid (Shaevitz et al., 2003; Galburt et al., 2007; Wang et al., 2009) (see Figures 1B and 1C). This exposes already incorporated bases to the active site, blocking further elongation but enabling cleavage of the most recently added base (cata-lyzed by the transcription factor IIS in eukaryotes and GreA and GreB in prokaryotes) (Fish and Kane, 2002; Borukhov et al., 2005; Jeon and Agarwal, 1996; Awrey et al., 1998; Opalka et al., 2003; Kettenberger et al., 2003; Sosunov et al., 2003; Thomas et al., 1998). If cleaved, a potential error is removed, the active site is cleared, and elongation can resume. The cleav-age process competes with the spontaneous recovery from the backtrack (Galburt et al., 2007), by which the polymerase returns to the elongation-competent state without removing the potential error (seeFigure 1C).
In order for cleavage from the backtracked state to lower the error content, the net effect of backtracking and cleavage must be to selectively remove erroneous bases rather than correct ones. The inability of incorrectly matched bases to form proper Watson-Crick base paring within the RNA-DNA hybrid induces this selectivity. If an error has been catalyzed onto the 30 end of the nascent RNA molecule, the total energy of the transcrip-tion complex is lowered if the RNAP moves into a backtrack (seeFigure 1D). Doing this, the RNAP extrudes the unmatched bp from the hybrid and, so, returns to the low energy state of a perfect Watson-Crick base pairing within the entire hybrid (seeFigure 1C).
Because the backtracking process needs not to equilibrate, the total fidelity enhancements might depend on the free energy of transition barriers as well as on the free energy of the states themselves. Specifically, the manner in which misin-corporations affect the transition state to backtracking deter-mines if fidelity is increased through a selectively increased entrance rate into the backtrack (no shift of transition state) or by a decreased exit rate out of the backtrack (transition state shifts with the hybrid energy). For the latter case to have an appreciable proofreading capability, every single base must be extruded from the hybrid through backtracking, or else only a fraction of the incorporation errors will be proofread by selectively prolonged exposure to cleavage. The required high backtracking frequency would render the polymerization pro-cess inefficient, possibly even reverse it (see below). This is
clearly not what is observed in experiments (Galburt et al., 2007; Abbondanzieri et al., 2005). We thus take the selectivity to reside in the entrance step of the backtrack (seeFigure 1D), which ensures that all bases are checked, that backtracks are predominantly induced by incorrect bases, and consequently, that transcription can be both fast and accurate. For rates as illustrated in Figures 1B and 1C, this corresponds to
krec=kIrec=kbt and kbtI =kbtexpðDGact=kBTÞ (rates for incorrect
bases are denoted with the superscript I). We will simply refer tokbt as the backtracking rate, and we illustrate the resulting
form of the free energy landscape inFigure 1D. For our generic RNAP, we use kcat= 10=s (Neuman et al., 2003;
Tolic-Nu˛rrelykke et al., 2004) (prokaryotic) (Galburt et al., 2007) (eukaryotic), backtracking rate kbt= 1=s (Depken et al., 2009) (prokaryotic), and cleavage rate kclv= 0:1=s (Galburt et al.,
2007) (eukaryotic).
It is worth pointing out that there is evidence for an intermedi-ate stintermedi-ate between elongation and backtracking (Herbert et al., 2010). Based on the average diffusion constant over several backtracks (Depken et al., 2009), as well as direct observation of backtracking in individual traces (Abbondanzieri et al., 2005), we here assume a backtracking rate of the order one step a second. Because this rate is close to the rates expected for traversing the intermediate state, we simply absorb such a possible state into the first state of the backtrack.
In a development largely parallel to the theory of kinetic proof-reading through PIS (Hopfield, 1974), these rates are used to
A B C D E 2,000 1,000 -1,000
Figure 1. Single-Step Backtracking
(A) The basic hopping model coupling one-step backtracking to elongation is presented. The re-petitive unit is highlighted, with the off-pathway backtracked state indicated as BT. After entering a backtrack, elongation can resume either through cleavage out to a previous state of the chain ðNMPÞn1or by recovery without cleavage to the
entrance stateðNMPÞn.
(B) Schematic illustration shows the repeat unit with a correct base incorporated last. The tem-plate strand, the nascent transcript, and the hybrid region of the polymerase are shown. The poly-merase can enter a backtrack with ratekbtor add a base to the transcript with rate kcat. From the backtracked state, recovery by cleavage occurs with ratekclv, whereas realigning without cleavage occurs at a ratekrec.
(C) is the same as (B) but with an incorrect base at the growing 30 end of the transcript. The corresponding rates are indicated with the superscript I.
(D) Sketch shows the free energy landscape corresponding to (B) and (C). Solid black line corresponds to the last base correct; dashed red line corresponds to the last base incorrect. DGact refers to the free energy increase at the active site when the last incorporated base is wrong, whereasDGcat denotes the correspond-ing increase in the barrier to catalysis (cat). Recovery without cleavage occurs at a ratekbt, which places all selectivity in the entrance step to the backtrack (see text).
(E) Three traces simulated with a Gillespie algorithm are illustrated: a generic polymerizing RNAP with (kcat= 10=s, kbt= 1=s, kclv= 0:1=s; see main text); a stalled polymerase (kcat= 1=s, kbt= 10=9=s, kclv= 10=s); and a depolymerase (kcat= 1=s, kbt= 10=s, kclv= 10=s). Traces are black when the polymerase is elongating and red when backtracked.
calculate the error suppression of PIP (seeExtended Results; Figure S1): rz kcat kcat+kclvexp ðDG act+DGcatÞ kBT : (Equation 1)
Here,DGcatdenotes the change in barrier height for the
tran-sition to catalysis when trying to incorporate a base directly after an error (see Figure 1D). We can get an estimate of DGcat from published experiments that use the nucleotide
inosine to simulate the effect of a misincorporation on the addi-tion of the next nucleotide to the nascent strand (Thomas et al., 1998). From this, we estimate DGcatz2kBT. Assuming that
DGact is governed by the energy cost of mismatches, we
esti-mateDGactz6 kBT (see above). For our generic polymerase,
this implies proofreading capabilities for PIP that amount to a modestrPIPz1=30.
A comparison of the regular traces of Figure 1E with those from single-molecule experiments (Galburt et al., 2007; Depken et al., 2009; Ibarra et al., 2009) also demonstrates that the one-step backtracking does not adequately capture the observed intermittent transcription dynamics (see also below). Although much of the intermittent dynamics seen in experiments has been attributed to structural heterogeneity through sequence-specific pauses (Herbert et al., 2006; Neuman et al., 2003; Ibarra et al., 2009; Bai et al., 2004; Tadigotla et al., 2006; Maoile´idigh et al., 2011), we will argue for another functional explanation for a large subpopulation of these pauses: they are essential for improving fidelity beyond what is allowed by
Equation 1.
Multistep Backtracking
It is clear from Equation 1 that, apart from increasing the energy penalty for a mismatch, a low error ratio can be achieved through having a high transcript-cleavage rate compared to the elongation rate. Given their reverse arrangement (kcat= 10=s, kclv= 0:1=s), it seems reasonable to consider the case where
the evolution of these rates has been strongly limited by external constraints pertaining to nucleotide chemistry and the inter-cellular environment. To mediate these external constraints, the polymerase would have to evolve alternative internal paths to increase error suppression.
One such internal path could be to reduce the free energy of the backtracked state. This would suppress spontaneous reversal of the backtrack and increase the probability of cleav-age and error removal. Because a substantial part of the free energy relates to the energetics of base matching within the hybrid, the energy level of the backtracked state is likely con-strained by the structure of the hybrid—again presumably fixed by early evolutionary choices. However, there exists another possibility: an effective entropic reduction in the free energy level of the backtracked state can be achieved by extending the number of accessible states. RNAP is indeed able to back-track by more than just one base and thermally move between the different backtracking states that are available (Nudler et al., 1997; Komissarova and Kashlev, 1997a, 1997b; Shaevitz et al., 2003; Galburt et al., 2007) (seeFigures 2A and 2B). With
N off-pathway and backtracked proofreading states, the free
energy associated with the backtracked state would in an equilibrium setting be reduced by the entropic termkBT lnðNÞ.
This mechanism delays spontaneous recovery and raises the
A B
C
second first
Figure 2. Multistep Backtracking
(A) The basic repeat unit of multistate backtracking in a nested scheme is presented. For visual clarity, only the backtracked states in the highlighted repeat unit are drawn.
(B) Sketch shows the free energy landscape of a multistate backtrack. Solid black line corresponds to the last base correct; dashed red line corre-sponds to the last base incorrect. Also illustrated are the multiple backtracked states and the effect of cleavage. See legend to Figure 1B for a description of the rates.
(C) Three traces simulated with a Gillespie algo-rithm are illustrated: a generic polymerizing RNAP with (kcat= 10=s, kbt= 1=s, kclv= 0:1=s); a stalled complex (kcat= 10=s, kbt= 10=s, kclv= 0:1=s); and a depolymerizing one (kcat= 1=s, kbt= 10=s,
kclv= 0:1=s)—all in accordance to the theoretical predictions derived in the Extended Results. A section of the trace for our generic polymerase has been magnified, showing two backtracks: one rescued to elongation by cleavage, and one by diffusion. Only the backtrack reentering elongation through cleavage would have corrected an error at the end of the transcript. Traces are black when the polymerase is elongating and red when backtracked.
chance of error removal also in our out-of-equilibrium setting (seeExtended Results).
When acting through extended backtracked states, the error suppression of PIP can be calculated as (seeExtended Results
andFigure S1) r1:PIPz k cat kcat+ ffiffiffiffiffiffiffiffiffiffiffiffi kclvkbt p exp ðDG act+DGcatÞ kBT = kcat kcat+kbtexp DG 1:PIP kBT ; (Equation 2)
DG1:PIP=DGact+DGcat
1 2kBT ln k bt kclv : (Equation 3)
ComparingEquation 2toEquation 1, we see that fidelity is increased by extending the space available for backtracking: the low cleavage rateffiffiffiffiffiffiffiffiffiffiffiffiffi kclv is replaced by the geometric mean
kclvkbt
p
. This increases the fidelity by a factor of about three for our generic polymerases and provides an error reduction of
r1:PIPz1=100.
With an extended backtracking space, it is now clear from simulated traces (Figure 2C) that the intermittent dynamics of our generic polymerase qualitatively match the intermittent dynamics observed in single-molecule experiments (Herbert et al., 2006; Neuman et al., 2003) (see below for a quantitative assessment). It is worth pointing out that even with infinite room for backtracking, our generic polymerase would only take aroundkbt=kclv= 10 diffusive backtracking steps and reach
a typical depth of aroundN = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikbt=kclv
p
z3 before being cleaved off. This depth is below the lower estimates for the distance to RNA hairpin barriers in the trailing RNA strand (Klopper et al., 2010). We should thus be justified in assuming the available backtracking distance to be effectively infinite even when ac-counting for possible barriers to backtracking (seeFigures 2A and 2B).
By comparing the experimental effects of cleavage-stimu-lating factors with the effect of increasing cleavage rates in simulated traces, we provide further support of our kinetic scheme in the Extended Results (Figure S2). We also show that our model can capture the stalling dynamics of a polymer-ase as it transcribes against an increasing force (Galburt et al., 2007; Depken et al., 2009). It is interesting to note that because
of these dependencies of backtracking dynamics on the load, one would also expect the fidelity of transcription to generally depend on load (Voliotis et al., 2012). Although the extension of the backtracking space provides for an increased fidelity already on this level, we now show that it gives additional proofreading benefits by supplying the polymerase with further PIP checkpoints.
Further Backtracking Checkpoints
Even when additional bases have been added to the transcript after an erroneous incorporation, the error can in principle still be corrected through an extended backtrack and cleavage (Voliotis et al., 2009). For this to lead to an appreciably increased likelihood of error removal, the RNAP must still be biased toward entering into the backtrack. With an error at the penultimate 30 position of the transcript, the polymerase experiences such a bias: moving into a backtrack will elimi-nate a bad bp stacking within the hybrid (seeFigure 3). This is followed by another heavily biased step to completely extrude the error from the hybrid and, so, exposing it to the cleavage reaction. We know of no direct measurement of the penultimate biasDG2:PIP, but because the typical stacking
energy in a nucleic acid complex is 1:5 4:5 kBT (Sugimoto
et al., 1995), we assume DG2:PIPz3 kBT. This second PIP
checkpoint provides an error ratio (see Extended Results;
Figure S3) of r2:PIPz kcat kcat+kbtexp DG 2:PIP kBT : (Equation 4)
For our generic polymerase r2:PIPz1=3, and the total
PIP-induced error reduction is rPIP=r1:PIPr2:PIPz1=300. Combined
with the PIS error correction expected from base-pairing en-ergies alone, the PIP error correction brings the error fraction down torPISrPIPz1=400,1=300z1=105; thus, our scheme
quan-titatively accounts for the typically observed error suppression (Rosenberger and Hilton, 1983; Blank et al., 1986; de Mercoyrol et al., 1992).
There could in principle be additional inherent PIP check-points that would enable the polymerase to reach an even lower error ratio. A free energy penaltyDGn:PIP for moving the
error further into the hybrid would incur the necessary long-range bias and additional fidelity gains according to (see
Extended Results)
rPIP=r1:PIPr2:PIP/rn:PIP/
rn:PIPz kcat kcat+kbtexp DG n:PIP kBT : (Equation 5)
It is interesting to note that for the considered range of rates, the error suppression for multistate backtracks depends on the backtracking rate (Equations 4and5), whereas for single-state backtracks, it did not (Equation 1). The backtracking rate is a parameter not directly dependent on nucleotide chemistry and the external environment. This could render the backtrack-ing rate susceptible to selective pressures pertainbacktrack-ing to fidelity requirements.
Figure 3. A Second PIP Checkpoint
The polymerase is expected to be sensitive to errors incorporated also next to last. The magnitude of the rates is illustrated by relative thickness of the transition arrows; bad base stackings are indicated in red.G indicates the free energy of the complex with respect to the elongation-competent state.
Dynamic Effects of Proofreading
Before we address the problem of how much proofreading is appropriate, we here illustrate the consequences of the proof-reading states on pause duration and frequency, as they might be observed in single-molecule experiments. To this end, we simulate our generic RNAP transcribing a long homogeneous sequence and compare it to a simulation of an otherwise identical polymerase, but which has PIP turned offðkbt= 0=sÞ.
In Figure 4A, we show a particular stochastic realization of incorporation errors, i.e., those errors that pass through the PIS checkpoint and get incorporated (red), together with the errors left after the section has been exposed to PIP (black).
The fidelity enhancements are clearly visible, but they come at the cost of both a decreased velocity as well as an increased intermittency. These effects are qualitatively visible already at the level of individual traces but are quantitatively best seen in the changes of the dwell time distribution (seeFigure 4B) or in the transition rate (inverse dwell time) distribution (see Fig-ure 4C). In the dwell time distribution, proofreading introduces a power law regime, throughout which the probability of a long pause falls off with duration t as t3=2 (Galburt et al., 2007; Depken et al., 2009), until it drops off exponentially beyond
t = 1=kclv. With no cleavage, the power law would extend to
indefinitely long times (Depken et al., 2009). In Figure 4B, we see a clear exponential behavior of the dwell time distribution for both processes at around t = 1=kel= 0:1 s, whereas the
proofreading polymerase also has a broad regime consistent with the above-mentioned power law decay extending out tot = 1=kclv= 10 s. Similarly, considering the transition rate
dis-tributions, we see a narrow but significant low-velocity peak
develop around the transition ratekclv= 0:1=s, diminishing the
bare elongation peak situated around the rate kcat= 10=s (see
Figure 4C; note the log scale).
The pause density, or the total time a polymerase spends at each position along the DNA molecule, is another important experimentally observable parameter. In Figure 4D, we show pause density plots along a sequence of 500 bp, with darker bands indicating longer total time spent at that position during the transcription process. Comparing transcription with and without PIP, it is clear that PIP leads to greater heterogeneity, ex-hibiting distinct regions of markedly increased occupation den-sity even where there are no incorporation errors. Thus, our model can partially account for both the observed intermittency as well as the broad pause time distributions (Neuman et al., 2003; Galburt et al., 2007)—without the need to introduce any sequence-specific effects (Depken et al., 2009; Galburt et al., 2007).
Having shown that external constraints can be mediated through accessing an internal extended backtracked space—re-sulting in intermittent transcription dynamics—we now turn our attention to the specific level of backtracking observed in exper-iments. The induced intermittency is tuned by the backtracking rate and because increasing it would render all proofreading checkpoints more effective (seeEquation 5), one might wonder why the backtracking rate is kept moderate and not made much larger (Depken et al., 2009; Abbondanzieri et al., 2005).
Evolutionary Optimization
In order to elucidate a possible underlying reason for why the backtracking rate is kept moderate, we now consider the
A
B C
D
20,000 40,000 60,000 80,000 100,000
Figure 4. The Effects of Proofreading
(A) On top, we show a realization of incorporation errors according to our free energy estimates (only PIS in red), and below, we show the errors that survive (or, possibly, are inserted by) the proofreading mechanisms (PIS and PIP in black). (B) The dwell time distribution for adding one nucleotide to the nascent transcript, for processes with (black-filled circles) and without (red filled squares) proofreading (p.r.), is shown. Proof-reading gives rise to a power law regime, signifi-cantly increasing the fraction of long pauses. (C) The transition rate (inverse dwell time) distri-bution for the same processes as in (B) is shown, where the effects of proofreading can be seen through a shift from a unimodal to a bimodal distribution because many excessively slow transitions involving backtracks start influencing the kinetics.
(D) The pause density, or the total occupation time plotted for a 500 bp sequence transcribed by the same two polymerases as used in (B) and (C), is presented. The darker the bands, the longer the total occupation time at that position. The scales are individually normalized to cover the range of occupation times for each polymerase (reaching about 1 s without proofreading, and about 100 s with proofreading). Two incorporation errors are indicated with red markers.
phenotypic space made available through extended backtrack-ing and variations in the backtrackbacktrack-ing rate. The various quantities used to access polymerase performance—as it varies with the level of PIP—are calculated in theExtended Results(see also
Figure S2) using continuous-time random-walk theory (Montroll and Lebowitz, 1987). We start with considering three instanta-neous transcriptional efficiency measures pertaining to the level of the individual bp and then move on to consider the efficiency on extended sequences or genes.
Performance on the Base Pair Level
We are interested in quantifying how much longer it takes to tran-scribe with and without proofreading. Based on our estimate ofrPIS, we expect that there is only about one error passing
through the PIS checkpoint every 400 bases. Therefore, we can ignore the effect of errors on the overall elongation dynamics. Without proofreading, the average elongation time is 1=kcat. In the Extended Results we calculate the average
elongation time with proofreading tel. Taking these together,
we define the elongation efficiency measurehel,
hel= 1=kcat tel = ð1 kbtÞ=kcat 1=2+ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=4Þ + ðkbt=kclvÞ p ; (Equation 6)
which describes the relative slowdown due to PIP. With no PIP ðkbt= 0Þ, the efficiency is appropriately hel= 1, whereas it
vanishes at the transition between net polymerization and net depolymerizationkcat=kbt. At this point, elongation stops
pro-ceeding with a well-defined velocity and behaves diffusively on large-length scales. Net depolymerization sets in for kcat<kbt.
This situation is pathological and shows that backtracking cannot dominate the dynamics even though this would be judged optimal in terms of fidelity. The transition to nonfunctional polymerases can be seen directly in the experimental transcrip-tion traces presented in Figure 3D inGalburt et al. (2007), where an opposing force was used to increase the entrance rate into the backtrack—bringing the system to stall around 14pN. The same stalling effect due to excessive backtracking and cleavage can be seen in the simulated traces presented in Figure 2C (see alsoFigure S2). Also note that the overall elongation rate increases with increasing cleavage rate, as observed
experi-mentally (Herbert et al., 2010; Fish and Kane, 2002; Proshkin et al., 2010).
We next introduce an efficiency parameter for the fidelity of the transcription process, hPIP= 1 rPIP, which is zero (0) in
the absence of PIP and one (1) for perfect PIP. Finally, we parameterize the nucleotide efficiency of the transcription process by the ratio of final transcript length and the average number of nucleotides consumed in its production. This ratio is given by the simple expression (see Extended Results) hNTP= 1 kbt=kcat. This measure is unity without PIP and
vanishes at stall, where transcript is continuously cleaved and polymerized at equal overall rates. Figure 5 shows the three efficiency measures hel, hPIP, and hNTP as functions of
the backtracking rate kbt (within the operational range
0=s%kbt%kcat= 10=s) for an otherwise generic RNAP. We see
that, whereas transcription velocity and nucleotide efficiency correlate positively, they both correlate negatively with fidelity, directly illustrating the cost of ensuring fidelity. This hints at an underlying competition, which we now explore by considering transcription of extended sequences.
Performance on the Gene Level
Here, we demonstrate that a moderate rate of backtracking is necessary for rapidly generating transcripts with few mistakes from extended sequences. The suitable amount of proofreading needed varies with the length of the sequence that is to be tran-scribed: the longer the sequence, the harder it is to transcribe it without errors. It is thus instructive to introduce the probabilityPl of producing a long error-free sequence. It is important to note that this sequence lengthl should not necessarily be interpreted as the complete gene length. Instead,l should be seen as the typical error-free lengthl = 1=rfuncthat is required when
transcrip-tion needs to reach error ratesrfuncto produce acceptable
tran-scripts. For each attempt, the probability of transcribing a sequence of length l without an error is given by
PlðrÞ = ð1 rÞlzexpðlrÞ, with r = rPIPrPISrepresenting the total
error fraction.
We define the production rate gaincelon extended sequences
by comparing the rate at which error-free transcripts are produced with and without PIP, cel=helPlðrPISrPIPÞ=PlðrPISÞ =
helexpðlrPIShPIPÞ. Similarly, we introduce the NTP efficiency
gain on extended genes cNTP by comparing the number of
error-free transcripts produced per nucleotide used, with and without PIP, cNTP=hNTPexpðlrPIShPIPÞ. These simple forms
justify the otherwise arbitrary-seeming local efficiency measures for fidelity introduced above. From both these quantities, it is clear that even moderate PIP provides enormous gains in the rate of perfectly transcribing longðl>1=rPISÞ sequences.
With the two sequence-wide measures that we have intro-duced, it is now possible to address transcriptional efficiencies on the level of transcription of whole genes. As an example, we consider a sequence length comparable to that of the typical human gene, l = 104bp. InFigure 6A, we plot the efficiencies
cel andcNTP as a function of the backtracking ratekbt. Each
measure has a definite optimal value, and we see that the gains in both rate of perfect transcript production and nucleotide efficiency can be enormous, here reaching more than 13 orders of magnitude.
Figure 5. Polymerase Performance
Proofreading efficiencyhPIP(red dot-dashed line), elongation efficiencyhel (black solid line), and nucleotide efficiencyhNTP(blue dashed line) as a function of the backtracking rate, for an otherwise generic polymerase withkcat= 10=s andkclv= 0:1=s, is illustrated. Values indicated by diamonds were obtained numerically, through Gillespie simulations.
If our generic RNAP was optimized to transcribe this particular sequence length, then we would expect the true value of the backtracking rate to lie somewhere in the intermediate region between the peaks: representing a compromise between NTP efficiency and production rate. For the intermediate value of
kbt= 1=s —coinciding with our estimate of the physiologically
relevant backtracking rate—it would take a polymerase about 1 hr to produce an error-free transcript. This should be compared to the time needed to complete the same task without PIP, which for our simple PIS efficiency estimate amounts to approximately 1 billion years.
Intermittency Increases with Gene Length
Finally, it is interesting to ask how the region of optimal back-tracking rate changes as the transcribed sequence length varies.Figure 6B shows the backtracking rate that optimizes cel (black solid line) and cNTP (red dashed line) as a function
of sequence length l. The inset in Figure 6B highlights the backtracking rate for our generic polymeraseðkbt= 1Þ and the
implied sequence lengths ð104 4,104bpÞ for which this
backtracking rate would be optimal. A complete discussion would need to account for relaxed fidelity constraints due to, for example, codon redundancy (Alberts et al., 2003). Even so, considering that the average gene length in eukaryotes lies in the range 104–105 bp (Xu et al., 2006), it is thought provoking that the moderately observed backtracking rates of
around 1/s are the result of an evolutionary optimization for rapidly and efficiently producing functional transcripts from genes in the tens of kbp range.
DISCUSSION
Through an analytical study of the dynamic coupling of back-tracking, chain elongation, and cleavage, we have shown that intermittent transcription dynamics are to be expected for an efficient transcription process. Our work shows that base-pairing energetics, together with transcriptional proof-reading through backtracking and cleavage, can account for observed error rates. This proofreading relies on a conglom-erate of checkpoints, all contributing to fidelity while depend-ing on the occurrence of extended backtracks. Through backtracking and cleavage, an incorporated error is proofread at least twice but could in principle be proofread as many times as there are bases in the RNA-DNA hybrid within the elongation complex. Investigating if there are additional proof-reading checkpoints is an interesting line of future research, potentially providing a link between the structure of the elonga-tion complex and overall transcripelonga-tional efficiency and fidelity. Such work might offer clues as to why the RNA-DNA hybrid has a length of about 8–9 bp (Kent et al., 2009; Nudler et al., 1997).
In contrast to physiological settings, where the DNA can be covered by various DNA-binding proteins, we have in this study considered transcription along bare DNA. Additional enhance-ments of the proofreading capabilities are expected for genes crowded by proteins because the physical hindrance induced by such proteins should bias the RNAP away from catalysis, toward backtracking and increased proofreading.
Considering the effects of proofreading both on NTP con-sumption and the production rate of extended functional transcripts, we show how the internal hopping rate in the back-tracked state is not optimized for fidelity. Instead, our analysis suggests that it is kept moderate in order to enable rapid pro-duction of extended transcripts of high quality. Importantly, there will be many more backtracks (about one in every ten base pairs) than there are errors to remove. This apparent inefficiency can be understood as a direct consequence of an enormous asymmetry in cost: undetected errors have the poten-tial to render the whole transcript dysfunctional, whereas removing a correct base carries only a moderate cost in energy and time. A substantial level of paranoia is thus desirable on part of the polymerase and could be the reason behind why backtracking is observed in vivo at levels comparable to what we predict (Churchman and Weissman, 2011). Although such frequent backtracking decreases the instantaneous average transcription rate, the observed level of backtracking—perhaps counterintuitively—drastically increases the rate at which high-fidelity transcripts are produced.
Interestingly, our work shows that the frequency of back-tracks for different RNAPs should be expected to correlate positively with the sequence length that has induced the highest evolutionary pressures on each transcription process (see Fig-ure 6B). For example, genomes with genes of increasing length would be expected to be transcribed with increasingly
A
B
10,000 30,000 50,000
Figure 6. High-Fidelity Transcript Production
(A) On the left vertical axis, we mark the production rate gain on extended sequencescelas a function of the backtracking rate (black solid line). On the right vertical axis, we mark the NTP efficiency gaincNTP as a function of the backtracking rate (red dashed line), all for a sequence of lengthl = 104bp. The region between the two peaks is where one might expect the optimal value ofkbtto lie.
(B) The backtracking rate that optimizes the production rate gain (black solid line) or the energy efficiency gain (red dashed line) as a function of sequence length is shown. The red arrow indicates the sequence length used in (A), and the dashed horizontal line marks the backtracking rateðkbt= 1=sÞ of our generic RNAP. The gray area marks the sequence lengths that could be optimized forkbt= 1=s. Inset is a magnification showing that PIP is optimal for gene lengths between 104and 43 104bp (gray shaded region) atkbt= 1=s.
intermittent dynamics in order to maintain transcriptional effi-ciency. Considering that, for example, in eukaryotes, Pol I, Pol II, and Pol III transcribe drastically different sequence lengths (Watson et al., 2013), it would be interesting to determine if the predicted increase in backtracking rate (Galburt et al., 2007; Depken et al., 2009)—with the concurrent increase in the intermittency of the dynamics—could be found for polymer-ases optimized for increasing sequence lengths.
Although the mechanistic details are very different, the asym-metry that governs intermittency of RNAP dynamics is even larger for DNA polymerases: whereas transcription errors affect the production of transient proteins, errors in replication are propagated to descendant generations. The general concept of cost asymmetry thus applies also to replication, and it would be interesting to extend our analysis to DNA polymerases to see if they also remove many more bases than there are errors incorporated into the growing product.
Understanding the evolutionary pressures that have formed the different polymerization machines processing genomes remains a fascinating experimental and theoretical challenge. We hope our model can serve as a benchmark for more detailed descriptions. Generally, our work highlights that PIP offers enor-mous fidelity gains and how this fundamental mechanism has become so vital for normal cellular function.
SUPPLEMENTAL INFORMATION
Supplemental Information includes Extended Results and four figures and can be found with this article online athttp://dx.doi.org/10.1016/j.celrep. 2013.09.007.
ACKNOWLEDGMENTS
We thank Eric Galburt, Justin Bois, and Abigal Klopper for fruitful discussions and suggestions. J.M.R.P. acknowledges financial support from grants ENFA-SIS (FIS2011-22644, Spanish government) and MODELICO (Comunidad de Madrid). S.W.G. acknowledges funding by the EMBO Young Investigator Pro-gram and the Paul Ehrlich Foundation. M.D. acknowledges partial support from FOM (which is financially supported by the ‘‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek’’) as well as the hospitality of both the Max Planck Institute for the Physics of Complex Systems in Dresden and the Vrije Universiteit in Amsterdam, where part of this research was performed. This research was supported in part by the National Science Foundation under grant NSF PHY05-51164. Received: May 16, 2012 Revised: February 1, 2013 Accepted: September 5, 2013 Published: October 10, 2013 REFERENCES
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