doi:10.1016/j.jmb.2004.11.075
J. Mol. Biol. (2005) 346, 399–409
A Mathematical Model for Transcriptional Interference by RNA Polymerase Traffic in Escherichia coli Kim Sneppen1*, Ian B. Dodd2*, Keith E. Shearwin2, Adam C. Palmer2 Rachel A. Schubert2, Benjamin P. Callen2 and J. Barry Egan2 1
NORDITA, Nordic Institute for Theoretical Physics, Niels Bohr Institute, Blegdamsvej 17 DK-2100 Copenhagen Denmark 2
Discipline of Biochemistry School of Molecular and Biomedical Science, University of Adelaide, SA 5005 Australia
Interactions between RNA polymerases (RNAP) resulting from tandem or convergent arrangements of promoters can cause transcriptional interference, often with important consequences for gene expression. However, it is not known what factors determine the magnitude of interference and which mechanisms are likely to predominate in any situation. We therefore developed a mathematical model incorporating three mechanisms of transcriptional interference in bacteria: occlusion (in which passing RNAPs block access to the promoter), collisions between elongating RNAPs, and “sitting duck” interference (in which RNAP complexes waiting to fire at the promoter are removed by passing RNAP). The predictions of the model are in good agreement with a recent quantitative in vivo study of convergent promoters in E. coli. Our analysis predicts that strong occlusion requires the interfering promoter to be very strong. Collisions can also produce strong interference but only if the interfering promoter is very strong or if the convergent promoters are far apart (O200 bp). For moderate strength interfering promoters and short inter-promoter distances, strong interference is dependent on the sitting duck mechanism. Sitting duck interference is dependent on the relative strengths of the two promoters. However, it is also dependent on the “aspect ratio” (the relative rates of RNAP binding and firing) of the sensitive promoter, allowing promoters of equal strength to have very different sensitivities to transcriptional interference. The model provides a framework for using transcriptional interference to investigate various dynamic processes on DNA in vivo. q 2004 Elsevier Ltd. All rights reserved.
*Corresponding author
Keywords: transcriptional interference; RNA polymerase; occlusion; sitting duck; collision
Introduction It may seem to an outside observer that a cell is faced with an immense traffic problem because so many proteins must occupy specific sites on the DNA while many other proteins, such as polymerases, must traverse the DNA, often in both directions. We still do not know the traffic rules on this busy one-lane two-way street. Not only have cells somehow solved this problem, but one imagines that they have also taken full advantage of such traffic interactions. An example is the Abbreviations used: pA, aggressive promoter; pS, sensitive promoter; EC, elongating complex; SDC, sitting duck complex; RNAP, RNA polymerase(s). E-mail address of the corresponding author:
[email protected]
RNAP–RNAP interactions that result from convergent (face-to-face) or tandem arrangements of promoters. Such arrangements are not uncommon, both in prokaryotes and eukaryotes.1 In such cases transcription from one promoter can have a significant inhibitory effect on transcription from the other promoter, often with important regulatory consequences.1–8 This phenomenon, termed transcriptional interference, is poorly understood in most cases and then only qualitatively, and for any situation it is not clear what factors determine the strength of interference and what mechanisms are most significant. A recent quantitative study of transcriptional interference by face-to-face bacteriophage promoters in Escherichia coli1 provided an opportunity to develop and test a general mathematical model for transcriptional interference by RNAP–RNAP interactions.
0022-2836/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.
400 Callen et al.1 examined interference in vivo when a weak promoter (the “sensitive” promoter) and a strong promoter (the “aggressive” promoter) were opposed over short DNA distances, from 62 bp to 208 bp, within the E. coli chromosome. Interference with the sensitive promoter was measured as the ratio of its activity (measured by a lacZ reporter gene) in the absence of the aggressive promoter (mutationally inactivated) to its activity in the presence of the aggressive promoter. On the basis of various rearrangements of the promoters, they were able to exclude mechanisms involving RNA– RNA hybridization and promoter competition and showed that most interference was due to the passage of RNA polymerase (RNAP) across the sensitive promoter. Three mechanisms were suggested to explain the interference1 (Figure 1).
Modelling Transcriptional Interference
(a) Occlusion, where the polymerase passing over the opposing promoter temporarily prevents binding of RNAP, was originally proposed by Adhya & Gottesman5 to explain interference by an upstream promoter upon a downstream promoter in tandem. (b) Sitting duck interference was the term introduced for interference due to removal of promoterbound complexes (sitting ducks) by the passage of RNAP from the opposing promoter. This mechanism should also work with tandem promoters, as long as pre-initiation complexes at the downstream promoter are removed by RNAP coming from upstream. (c) Collision between elongating polymerases moving in opposite directions, causing one or both polymerases to terminate, has been invoked by a number of authors to explain interference by face-to-face promoters.3,4,8 Interference through collisions is not expected to apply to tandem promoters. Here we mathematically model these mechanisms of transcriptional interference by stochastic simulation, by a numerical (mean field) method and by an approximate analytical approach. All three methods were able to successfully reproduce the interference data,1 while requiring the fitting of few unknown parameters. The model identifies the factors that determine the strength of the interference mechanisms in a wide range of situations. Many of the critical parameters for any given combination of promoters are either known or easily measurable. However, our analysis confirms and quantifies the proposal that knowledge of in vivo kinetic parameters of the sensitive promoter, and to a lesser extent the aggressive promoter, is necessary to predict interference.1 The converse of this is that interference measurements can provide information about promoter kinetics in vivo and can also be used to measure other dynamic processes on the DNA, such as RNAP elongation speed.
Approach Parameters
Figure 1. The three mechanisms of transcriptional interference for convergent promoters. The sensitive promoter (pS) is on the right ((a), (c)–(e)) and the aggressive promoter (pA) is on the left ((b)–(e)). Some nomenclature is given in Table 1. kSon is the rate of formation of sitting duck complexes at pS (the superscripts S and A denote pS and pA parameters), kSf is the rate of firing of those complexes (see the text). qS is the fractional occupancy of the pS promoter by sitting duck complexes. The boxed equations are steady-state balance equations for the gain and loss of sitting duck complexes at the promoter. c is the probability of pS being free of passing RNAP from pA. N 0 ZNK40 is the distance between front ends of RNAPs at two promoters separated by distance N bp.
The mechanics of the model are outlined in Figure 1, while Table 1 shows the parameter values used. The values of rZ70 bp for the length of promoter-bound RNAP and lZ35 bp for the length of elongating RNAP are estimates from footprinting studies on open complexes and stalled elongating complexes.9,10 For each specific promoter combination, five additional parameters were needed: the DNA distance between the promoter start sites, N; the intrinsic strengths, KA and KS, of the aggressive promoter (pA) and sensitive promoter (pS), respectively; and the “aspect ratios”, aA and aS, which are the ratios of the on rate and firing rates of the aggressive and sensitive promoters (see below). The four promoters considered were bacteriophage 186 pR and pL and bacteriophage P2 pe and pc.1 The relative strengths of the promoters were obtained from the lacZ reporter data of Callen et al.1 Separate
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Modelling Transcriptional Interference
Table 1. Model parameters Parameter
Meaning
Values
References
l r N KA
Length of elongating RNAP (bp) Length of promoter for RNAP binding (bp) Distance between transcription start points (bp) Rate of RNAP production from pA (sK1)
35 (C20 to K15) 70 (C20 to K50)
10 9 1
KS
Rate of RNAP production from pS (sK1)
v aS
Speed of transcription (bp/s) Aspect ratio of pS
aA
Aspect ratio of pA
pR: 0.056a pE: 0.028 pL: 0.0056 pC: 0.0028 40 pL: 1 pC: 0.08 pR: 1 pE: 1
1 Fitted Fitted Fitted Fitted Fitted
a Promoters are bacteriophage 186 pR and pL and bacteriophage P2 pe and pc, with relative promoter strengths for pR:pE:pL:pC of 20 : 10 : 2 : 1.1 By our reporter assays, pR is w25% as strong as the bacteriophage l PL promoter, which has a production rate of 0.22 sK1.11
reporter assays comparing 186 pR and the bacteriophage lPL promoter (data not shown), allowed us to estimate K values for all of the promoters from an estimate of 0.22 sK1 for l PL in vivo.11 The value of vZ40 bp/s for the velocity of elongating RNAP in E. coli and the aspect ratios of the four promoters were obtained by fitting to the interference data. Our estimate of v is comparable to literature reported values for our conditions (25–65 bp/s).5,12–14 We used three methods to model the mechanisms of interference shown in Figure 1. We found that the approximate analytical approach was most useful for understanding the critical parameters and their contribution to interference. This approach can be applied when the sensitive promoter is weak, which is likely to be the most common natural situation. The equations in Figure 1 and in the text refer to this approach. These equations become increasingly inaccurate as the strength of pS approaches pA or when the inter-promoter distance becomes large. The more general mean field approach and the computer simulation approach are briefly described below; further description of mean field and analytical approaches is given in the on-line Supplementary Material. The isolated promoter We first modelled an isolated weak, sensitive promoter, pS (Figure 1(a); for a weak promoter, “self-occlusion” is negligible, see below). The production of elongating RNAP from a promoter is a complicated multi-step process.15–17 However, for the purposes of describing transcriptional interference we can use a simplified kinetic description. Production of elongating RNAP from pS, occurring with an overall rate KS, is divided into two steps. Firstly, free RNAP binds to the free promoter and forms a “sitting duck” complex (SDC) with a rate kSon (sK1). Secondly, this SDC can fire to form an elongating complex (EC) with a rate kSf (sK1) and leave the promoter. A sitting duck complex is defined as one that (1) can take a significant time to form at the promoter,
(2) has the potential to form an EC, and (3) can be removed from the DNA by a passing elongating complex. An important property of SDCs is that their removal by passing ECs interferes with the activity of the promoter. Thus, we expect that any RNAP bound to double-stranded promoter DNA and in rapid equilibrium with free RNAP, for example initial closed complexes,15 are not SDCs; their removal is not detrimental since they can be re-formed rapidly. Formation of the various open complexes, including early initiation complexes, involves major structural rearrangements of RNAP and the DNA.17 We therefore expect these complexes to qualify as SDCs, since they are no longer in rapid equilibrium with free RNAP, their formation may be rate limiting for a promoter,15 and they have not yet formed the tight binding conformation characteristic of ECs that results when the synthesized RNA enters the exit channel.17,18 The rate constant for formation of SDCs, kSon (given a fixed free [RNAP] in the cell) is intrinsic to the promoter and subsumes the many kinetic processes leading to SDCs. Similarly, the firing rate constant, kSf , is intrinsic to a promoter and subsumes a number of kinetic processes in the transition from an SDC to an EC. As we shall show, the ratio of kSon and kSf , which we term the aspect ratio of the promoter, aS, is critical in interference:
aS Z
kSon kSf
(1)
At steady state of the uninhibited promoter, the rate of formation of sitting ducks by binding to the free promoter balances the rate of loss of sitting ducks through firing and there is a fixed fractional occupancy, qS, of the promoter by SDCs. This gives the “balance” equation (boxed in Figure 1(a)), which can be rearranged to express qS for the uninhibited promoter in terms of the promoter’s intrinsic rate constants:
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Modelling Transcriptional Interference
qS Z
kSon aS Z aS C 1 kSon C kSf
(2)
KS , the overall activity of the uninhibited promoter, can also be expressed in terms of these constants (Figure 1(a)). We next modelled an isolated aggressive promoter, pA (Figure 1(b)). The important property of the aggressive promoter is its rate of production of elongating RNAPs, K A. The average time between RNAPs produced from pA is 1/KA seconds (measured front to front). This time cannot be zero because there is a finite time needed for RNAP to clear and regenerate the promoter before the next RNAP can bind. We assume that, as soon as it begins elongating, the RNAP obtains a length lZ 35 bp, and a speed v, therefore this “self-occlusion” time is l/v seconds. This effect can significantly reduce the firing rate for very strong promoters, and sets an upper limit on KA of v/l (w1.1 sK1). However, the effect is negligible for weak promoters and is ignored in the above treatment of pS. The average time between the back end of one RNAP and the front end of the next RNAP from pA, the “gap time”, is 1/KAKl/v and we term the inverse of this average gap time, KA (sK1; Figure 1(b)). KA is important in all the mechanisms of interference. Note that KA zKA as long as pA is not a very strong promoter (KA%0.1 sK1). Interference by occlusion Occlusion interferes with the formation of SDCs (Figure 1(c)). The RNAP attempting to bind pS must search for promoter DNA of length rZ70 bp (K50 to C20 positions of the promoter), exposed in the gaps between ECs arriving from the aggressive promoter pA. Each EC is expected to occupy lZ 35 bp of DNA and thus occlude the sensitive promoter for the time it takes it to traverse rClZ 105 bp at a speed of v bp/s. (It should be noted that these footprinting-derived values for r and l may not truly reflect the extent of DNA occupation as felt by another RNAP.) The term c is the probability of pS being free of ECs from pA, and is calculated by multiplying the probability that the K50 position of pS is free (probZ ð1=KA Þ=ð1=KA Þ; Figure 1(b)) by the probability that the gap to the next arriving polymerase from pA is sufficiently long to allow binding (probZ1/exp[(time gap needed)/(average time gap)]Z1/exp[(r/v)/(1/KA * )]; Figure 1(c)). Thus, c ranges from 1 (no occlusion) to 0 (complete occlusion) and the effective on rate kSon then becomes ckSon . Introducing c into the balance equation allows S , the activity of pS in the derivation of a term for Kocc presence of occlusion (Figure 1(c)). The interference due to occlusion, Iocc, is thus the ratio of KS S (uninhibited) to Kocc : Iocc Z
KS ckSon C kSf aS C 1=c Z Z S Kocc ckSon C ckSf aS C 1
(3)
Sitting duck interference In sitting duck interference, the SDC at pS is removed by an EC from pA (Figure 1(d)), and we make the initial assumption that no SDCs survive such an encounter. Thus, we need to introduce a term into the loss side of the SDC balance equation to account for the rate at which ECs arrive from pA once an SDC has formed at pS. KA is the rate at which front ends of ECs arrive at pS after the previous EC from pA has cleared pS. However, the rate at which front ends of ECs arrive at pS once an SDC has formed at pS can be slightly higher than KA . This is because KA is an average of (1) steady state production of ECs from pA and (2) lower rates of production during the approach to steady state after the previous EC has cleared pA. Thus, the steady state rate of production is slightly higher than KA . This difference is greater the longer the time taken to reach the steady state, which in turn is dependent on the aspect ratio of pA, aA, and is greatest when aAZ1. Because the formation of an SDC at pS can often take longer than the time it takes for pA to reach steady state, the pS SDC often encounters ECs from pA at this higher rate. We A approximate this effect by introducing the term K A instead of K in the balance equation (Figure 1(d)). A is dependent on the aspect ratio of The value of K A A A pA ðK zK ða C 1=aA C 2Þ=ðaA C 1=aA C 1ÞÞ and is approximately 33% higher than KA when aAZ1 but tends towards KA as aA deviates from 1. This adjustment is further discussed in the Supplementary Material. The loss of SDCs through encounters with ECs from pA reduces the SDC occupancy of pS (qS), reducing the rate of production of ECs from pS, S (Figure 1(d)). The combined qS kSf , to KoccCsd interference by occlusion and sitting duck removal is then: IoccCsd Z
Z
KS S KoccCsd
Z
A ckSon C kSf C K ckSon C c kSf
A aS C 1=c aS =c K C aS C 1 ðaS C 1Þ2 KS
(4)
The interference due to the sitting duck effect alone, Isd, can be obtained from these equations by setting c to 1 (no occlusion): Isd Z
A A KS K aS K Z 1 C Z 1 C (5) S ðaS C 1Þ2 KS Ksd kSon C kSf
Interference by collisions Once an SDC at pS fires and becomes an EC, it can still be interfered with by colliding with an EC from pA. Given the structural and catalytic differences between open complexes and ECs, the possible outcomes of a collision between two ECs are likely to be quite different from those after an encounter of
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Modelling Transcriptional Interference
Table 2. Observed and predicted interference Aggressive promoter (pA)
Interpromoter distance (N)
Sensitive promoter (pS)
pR
62 bp
pL
pR pR pR pR pE pE pL pC
164 208 123 t 85c 62 62 62 62 62
pL pL pL pC pC pL pR pE
a b c
Observed interferencea (Iobserved)
5.6G0.2 6.1G0.4b 7.2G0.3 9.8G0.8 3.3G0.1 3.5G0.2 2.1G0.2 3.0G0.1 1.1 1.1
Predicted interference Stochastic simulation
Mean field
Analytical
5.7
5.4
5.6
8.2 10.5 2.8 3.6 2.1 3.0 1.0 1.0
8.0 9.4 2.9 3.6 2.1 2.9 1.0 1.0
8.3 9.9 2.9 3.7 2.2 3.0 1.1 1.0
Data from Callen et al.1; G indicates 95% confidence interval. These constructs had slightly different inter-promoter sequences. t indicates the 73% efficient trpa terminator, with distances to and from the end of the T-tract.
an EC with an SDC. We made the initial assumption that both ECs are removed from the DNA immediately. The probability that the pS EC will escape collision is a function of (1) the time that the pS EC takes to reach pA, N 0 /v, (where N 0 ZNK 40 bp, the distance between fronts of the RNAPs when bound at pA and pS) and requires also that pA has not fired in the previous N 0 /v seconds, and (2) the adjusted rate at which ECs are fired from pA, A . Thus, the probability of a pS EC escaping K collision is 1/exp[(time gap needed)/(average time gap)]Z1/exp[(2N 0 /v)/(1/K**A)] (Figure 1(e)). Total interference is then: Itotal Z
KS KA 2N 0 K v
S KoccCsd e
Z IoccCsd e
A 2N 0 K v
(6)
Mean field treatment and stochastic simulation The above analytical approach is a simplified, approximate solution because it ignores selfocclusion at pS and also any effect that pS has on pA when the promoters are convergent. As pS becomes stronger, its ability to partially “defend” itself by interfering with pA activity becomes significant. This effect is magnified with increasing N due to the increase in collisions. Thus the analytical approach tends to increasingly overestimate the interference felt by pS as KS and N increase. To provide a more general description that allowed us to extend the parameter regime for calculating interference (e.g. when pS is strong or N is large), a mean field treatment was developed. This allowed us to take into account the ability of pS to defend itself against pA, by computing the average activity of pA in the face of RNAPs from pS. Essentially, initial activities for both promoters are calculated and their mutual interaction is iterated numerically until a steady state is reached
(Supplementary Material). We found that the selfdefence effect becomes significant only when KS is above about 0.02 sK1 when the promoters are separated by less than 300 bp. This agrees with the observation that the weak pL and pC promoters did not interfere with the pR and pE promoters (Table 2). However, the mean field approach also is an approximation because it does not take into account fluctuations and the consequent correlations. These become more significant as the DNA between the promoters becomes occupied by multiple ECs, that is as N becomes greater than v/KA. This is discussed in the Supplementary Material. Thus the final benchmark is the stochastic simulation. In the stochastic simulation, the dynamics are implemented by updating at any time step [t,tC dt]Z[t,tC1bp/v] the presence of any RNAP according to the processes shown in Figure 1. That is, in time steps dt, a promoter forms an SDC with probability kSon dt unless it is occupied or occluded by other RNAPs. An SDC at a promoter initiates elongation with a probability kSf dt except when an EC from the opposing promoter is positioned such that it will collide with the SDC in the time step dt, in which case the SDC is removed. Any EC is moved a vdt step in the direction of elongation, except when it collides with an EC moving in the opposite direction, in which case both RNAPs are removed from the system. The interference is found by dividing the number of ECs which pass the opposing promoter when it is assumed to be silent by the number obtained when it is active. Each value in Table 2 was the result of simulating the promoter pair for the equivalent of 700 hours. The program, written in FORTRAN, is available on request. The stochastic model is also available for direct simulation in the form of a Java applet†. † http://www.nordita.dk/research/complex/ models/DNA/rnap.html
404
Results and Discussion All three modelling approaches, with the parameter values of Table 1, showed good consistency with the experimental interference data of Callen et al.1 (Table 2). The fit of the predictions to the data is also reasonably robust to variation in the parameters a, v, l and r. A twofold change in aS for pL affects I by less than 10%, while a twofold decrease or increase in aS for pC decreases or increases I by up to 30% or 50%, respectively. An uncertainty of 10 bp/s in v gives no more than a 10% uncertainty in I (for NZ62 bp). Changing l by 10 bp or r by 20 bp has no more than a 4% effect on I. The fit is more sensitive to K; increasing KA or decreasing KS produces roughly proportional increases in I (as long as KA%0.1). However, the effect on I is much less if KA and KS are changed in the same direction, and thus the predictions are reasonably robust to systematic error in the estimates of promoter strengths. The response of the sitting duck, occlusion and collision mechanisms to aS, K and N, and the fitting of v are examined in more detail in the subsequent sections. Sitting duck interference The sitting duck mechanism is determined by the strength ratio, KA/KS, of pA and pS (more precisely A =KS ) and the aspect ratio of pS (see equation (5)). K In Figure 2(a), the interference due to the sitting duck effect alone is plotted against pS aspect ratio for the range of pA/pS strength ratios examined by Callen et al.1 (Note that aspect ratios of 1 for the aggressive promoters have been used here and in A and sitting subsequent Figures, maximizing K duck and collision interference.) It can be seen that large interference values are possible with this
Modelling Transcriptional Interference
mechanism. The striking feature of sitting duck interference is that it falls off steeply as the pS aspect ratio deviates from 1 (equation (5); Figure 2(a)). This is because when aS!1, kSf is high relative to kSon (equation (1)) and RNAP does not wait long on the promoter, so SDCs are rarely present to be removed. Conversely, when aSO1, kSon is high relative to kSf and there is a high fraction of SDCs but removal of these complexes has little effect on promoter activity because SDCs quickly reform. This property of sitting duck interference means that the sensitivity of a promoter to transcriptional interference can be tuned by varying kSon and kSf without necessarily affecting the intrinsic strength of the promoter. Figure 2(b) shows the effect of pS aspect ratio on the total interference (including all effects) predicted for the four different strong-weak promoter pairings (NZ62 bp) studied.1 The graphs show that the two observed interference values for pL require its aspect ratio to be close to 1 (and also for the aspect ratios of pR and pE to be close to 1). Thus for pL, kSon Z kSf Z 0:011 sK1 (kSon Z KS ðaC 1Þ and kSf Z KS ðaC 1Þ=a). In contrast, the low interference seen with pC requires its aspect ratio to be far from 1, close to either 0.08 or 12.5. We can choose between these possibilities in this case because the rate at which open complexes initiated transcription in vitro was 6.5-fold faster at pC than at pL,1 so we expect kSf for pC to be higher than for pL. Given that pC is half as strong as pL in vivo, we thus expect that kSon for pC must be much lower than for pL and therefore that the lower aspect ratio for pC, aSZ 0.08, is correct (aSZ0.04 is expected if the ratio of kSf values is 6.5). Thus, for pC we estimate kSon Z 0:0030 sK1 and kSf Z0.038 s K1 . So, despite being weaker than pL, pC is much less sensitive to sitting duck interference. A comparison of Figure 2(a) and (b) shows that for pL, the model predicts the sitting duck mechanism to be the major component of the interference at NZ62 bp (giving IZ4.4 of the IobservedZ5.6 to 6.1 for pR–pL and IZ 2.6 of the IobservedZ3.0 for pE–pL), supporting the conclusion of Callen et al.1 Occlusion
Figure 2. Sitting duck interference. (a) Interference due to the sitting duck mechanism only, as a function of aspect ratio of pS, for three different strength ratios: KA/KSZ20 (pR–pC), ten (pR–pL and pE–pC) and five (pE–pL). (b) Total interference versus aspect ratio for the four different promoter combinations with NZ62. The horizontal lines show the measured interference values for each combination. Calculations were made using the mean field approach, with aAZ1.
Occlusion increases with increasing strength of pA, KA, but is suppressed if pS has a high aspect ratio (equation (3)). This is because when aS is large, kSon is high relative to kSf , and the occlusion-inhibited step is not limiting for the activity of pS. Figure 3(a) shows interference due solely to occlusion versus KA for a range of aS values. For pR (KAZ0.056) interfering with pL (aSZ1), occlusion is expected to cause only 1.1-fold interference. Even with its lower aspect ratio, pC is only occluded 1.2-fold by pR. Clearly, strong occlusion requires a very strong aggressive promoter. Figure 3(b) shows the predicted combined effect of occlusion and sitting duck interference for a sensitive promoter with the same strength as pL. Since collisions are ignored in Figure 3, these predictions should be equally
405
Modelling Transcriptional Interference
Figure 3. Occlusion. (a) Interference due to occlusion only, as a function of strength of pA, KA, for three different aspect ratios of pS. The calculation is done with pSZpL, but is unaffected by the strength of pS for tandem arrangements and, as long as pS is weak, for convergent arrangements. (b) As (a) but showing the combined effect of occlusion and sitting duck interference (collisions ignored), which is the total interference predicted for tandem promoters. Calculations were made using the mean field approach, with aAZ1.
applicable to tandem promoters. Thus, unless the aggressive promoter is very strong, the sitting duck effect is likely to be the dominant interference mechanism with tandem promoter arrangements. Adhya & Gottesman5 proposed the occlusion mechanism to explain an w30-fold reduction in the activity of the gal promoter when the strong l PL promoter was located 6.5 kb upstream. By our model, however, occlusion by l PL (KAZ0.22 sK1 (11)) should give at most only wtwofold interference (Figure 3(a)). Therefore, it is difficult to see how l PL transcription could cause 30-fold interference by occlusion alone. One possibility is that RNAP from l PL may pause substantially over Pgal, increasing the occlusion time. Another possibility is to invoke the sitting duck mechanism, which should also apply to tandem promoters. Figure 3(b) shows the predicted combined effect of occlusion and sitting duck interference for face-to-face promoters where pS is weak and therefore is the same as expected for total interference with tandem promoters (no collision). The strength ratio of l PL and pgal in these experiments is not clear but if pgal is tenfold weaker than l PL, then IoccCsd ranges from wfour to wten (depending on aS); if pgal is 30-fold weaker than l PL, then IoccCsd ranges from w9 to w25. Thus, a combination of sitting duck interference and occlusion explains the Adhya & Gottesman data much better than occlusion alone.
Figure 4. Collisions. (a) Interference due to collisions only, as a function of the inter-promoter distance, N. (Note that promoters are expected to overlap when separated by less than 40 bp.) The different curves show the effect of different strength ratios, KA/KS, with KA as for pR. KA/KS was varied from 10 to 1, as indicated. The continuous lines show interference at the weaker promoter (pS) and the broken lines show the interference felt by the stronger promoter (pA) for the same promoter combinations. (b) Total interference versus N for the pR–pL combination, showing the effect of varying the speed of RNAP, v, from 25 bp/s to 50 bp/s. The points are the observed interference values (Table 2). Calculations were made using the mean field approach, with aAZ1.
Collisions Figure 4(a) shows the effect of N on interference due to collisions alone, examining the case where pA has a KA equal to that of pR and varying the strength of pS (continuous curves). It can be seen that very little interference by collision alone is predicted for the pR-62-pL arrangement (KA/KSZ10 curve). When pS is weak relative to pA, interference due to collisions rises exponentially with increasing N. Increases in the strength of pS initially have little effect on interference but, as the strength of pS gets close to pA, interference is dramatically reduced due to self-defence. This is because RNAPs from pS can clear a path for themselves. The dotted curves in Figure 4(a) show the reciprocal effect, that is the interference that pS has on pA. For example, at a 1000 bp separation, two opposing pR promoters should interfere with each other wfivefold by the collision mechanism (Figure 4(a); KA/KSZ1 curve). If the activity of one of the promoters is reduced by half (for example, by repression), then I rises sharply for this promoter and falls sharply for the other promoter (KA/KSZ2 curves), producing a large change in relative activities of the two promoters. Thus for comparable promoters at large distances, the self-defence effect can magnify small differences in promoter strengths.
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Modelling Transcriptional Interference
Estimating the speed of elongation Previous estimates of 25–65 bp/s for the speed of RNAP elongation in vivo (v) have been based on the rate of RNA appearance,12,19 the timing of protein expression5 or the rate of RNAP-mediated entry of T7 DNA into the cell,13 requiring use of relatively long DNA templates. Interference measurements provide an alternative method to estimate v that, though indirect, is potentially sensitive enough to be used with short DNA lengths. The method relies on collision being the only interference mechanism to be affected by N (Figure 1), thus any change in interference resulting solely from a change in spacing is due solely to differences in collisions. We can use the equation (6) to obtain: A v Z 2K
DN D ln Iobserved
(7)
allowing an estimation of v independently of the other mechanisms and all parameters except for A A . K is dependent primarily on KA but is also K slightly dependent on l, aA (and v itself). However, aA has at most a 33% effect, while twofold changes A . Using in l produce changes lower than 10% in K the data for the three different pR–pL spacings (NZ 62, 164, 208; Table 1) in equation (7) gives vZ 49.6 bp/s. Figure 4(b) shows how the predicted total interference for the pR–pL combinations varies with N over a range of v values. It can be seen that v has little effect on the predicted interference when NZ62 (also true for other promoter combinations), while the predictions for the NZ164 and 208 cases are more sensitive to v. As a whole, the data are fitted best when vZ40 bp/s. Thus, this limited data set provides an estimate that is similar to those from other methods, uses a considerably shorter DNA length, and which is dependent almost solely on our estimate of the strength of pR. (Note that data from which we indirectly estimate pR strength,11 are not themselves dependent on any estimate of v.) However, this estimate of v is based on the assumption that every pL EC that collides with a pR EC terminates. If a fraction of elongating RNAPs escape termination after a collision, then the estimate of v must be reduced proportionately. Also, this does not take into account the possibility that ECs from pR may have an average v different from ECs from pL, which may result if consecutive ECs from pR can cooperate in overcoming pause sites14 (see below). The nature of sitting duck complexes Two observations support our expectation that there is a correspondence between SDCs and open complexes. Firstly, the rate of transition from open complexes to productive elongating complexes in vitro was 6.5-fold faster for pC than for pL.1 A similar difference (3.4-fold) in the rate of transition from SDCs to ECs ðkSf Þ for pC compared with pL can be estimated from the in vivo interference data,
suggesting that there is some overlap between the two processes. Secondly, in vivo KMnO4 footprinting of open complexes at pL showed a 3.6-fold reduction in these complexes in the presence of pR.1 This is reasonably similar to the 5.2-fold reduction in sitting duck occupancy (q) predicted by the model. Transcriptional interference by roadblock Although we have made simple assumptions about what happens after collisions between an EC and an SDC and after collisions between two ECs, and these assumptions allow predictions that fit the available data, we cannot exclude other scenarios. Some DNA binding proteins are known to be able to form a barrier or roadblock to elongating RNAP in vitro and in vivo.19–22 It is possible that an SDC can be such a roadblock and thus can interfere with transcription from a face-to-face or upstream promoter. When an EC encounters a roadblock or pause site its progress can be arrested and it can backtrack, where the RNAP moves backwards along the DNA such that the RNA 3 0 end is displaced from the catalytic site and can even exit the RNAP via the secondary channel within RNAP.23 The cellular factors GreA and GreB can reactivate the EC by causing cleavage of the RNA to give a 3 0 end in the catalytic site.22,24 The Mfd protein can also cause reactivation or release of the backtracked complex, by pushing it forward.25 Reactivation can also occur by RNAP “cooperation” in which a second EC transcribing the same DNA strand arrives behind the complex, pushing it forward and helping it pass through the roadblock.14,19 It is not clear how often stalled or backtracked ECs are removed from the DNA in vivo (or how they are affected by ECs transcribing the other DNA strand). Since pL should be well occupied by RNAP, its lack of interference with pR suggests that RNAP complexes at pL are not a significant roadblock for ECs from pR. With NZ 62 bp, it is unlikely that a second RNAP could be loaded at pR to cooperate with an EC roadblocked at pL. Thus it seems that SDCs at pL are too weakly bound to block a single EC. However, an SDC at pR or pE may be more tightly bound and therefore may be a roadblock to ECs from pL or pC. If half of the pS ECs which reach pA are terminated by a sitting duck roadblock at pA, a reasonable fit to the data can be retained if a of pL and pC are decreased by about threefold. Even higher roadblock efficiencies are consistent with the data if the aspect ratios of pR and pE are decreased, since this reduces the occupancy of pR and pE and reduces the magnitude of the roadblock effect on interference. Our analysis supports the idea that a high proportion of SDCs are removed (or at least, inactivated) after an encounter with an EC. In particular, the high sensitivity of the gal promoter to activity of the upstream tandem l PL promoter5 indicates that most SDCs struck “from behind” are inactivated.
407
Modelling Transcriptional Interference
Survival of collisions between ECs Many uncertainties exist about the outcome of collisions between ECs transcribing in opposite directions. Do one or both ECs stall, or does one EC force the other backwards, maybe backtracking all the way back to the promoter? Are stalled ECs reactivated for repeated collisions? How are ECs removed from the DNA? The increase in interference with increasing N can be explained if vZ 50 bp/s and all ECs from the sensitive promoter are terminated by such collisions. However, if a fraction of these ECs are allowed to survive, then the fit to the data can be maintained by decreasing v proportionately. In addition, the data do not indicate whether all or none of the colliding ECs from the aggressive promoter survive, because only a very small proportion of the ECs from pR and pE should meet ECs from the much weaker pL and pC. It is possible that ECs from the stronger promoters prevail after a collision because of cooperation from additional ECs arriving behind them.14 Thus, when N is large enough to accommodate trailing ECs behind a collision (not the case for NZ62), the fraction of ECs from the weaker promoter that survive the collision may decrease. Promoter arrangements in E. coli It is hard to assess how often transcriptional interference is likely to be important in gene expression because it is easy for convergent and even tandem promoters to escape detection in the
Figure 5. Frequency of promoter arrangements in E. coli. The list of 4462 known and predicted promoters from version 4.0 of the RegulonDB database26 was analysed for the occurrence of tandem, convergent and divergent promoter pairs. Tandem promoters were divided into forward or reverse orientations. The cumulative frequency of each type of promoter pair was plotted as a function of the distance between the transcription start points, for inter-promoter distances of up to 800 bp.
usual processes for analysing the transcription of genes. We looked for convergent, divergent and tandem promoter pairs in the RegulonDB database (v.4) of 4462 known or predicted promoters in the E. coli genome.26 We counted the numbers of each type of pair as a function of distance between their start sites (Figure 5). In these cumulative plots we see that there are 166 non-overlapping tandem promoters (starts O70 bp apart) and 54 nonoverlapping convergent promoters (starts O40 bp apart) that are separated by !200 bp. Thus, there appears to be a significant fraction of promoters for which sitting duck interference and occlusion may be important. There appear to be very few convergent promoters between 200 bp and 800 bp apart, suggesting that “untunable” and costly collisional interference is used less often as a control mechanism. Conclusions and outlook Our analysis shows that three mechanisms for transcriptional interference in E. coli can account for quantitative data for face-to-face and tandem promoters. These mechanisms are dependent on four key properties of each promoter pair: their strengths, KA and KS, the aspect ratio of the sensitive promoter, and when the promoters are convergent, the distance between them. Interestingly, the aspect ratio of the aggressive promoter also has an effect, though this is relatively small. Interference due to collisions can be severe when face-to-face promoters are separated by long distances but is minimal at short distances, unless the aggressive promoter is very strong. Occlusion is also capable of causing high levels of interference, for tandem as well as face-to-face promoters, but again only with very strong aggressive promoters. With promoters of moderate strengths and at short spacings, whether convergent or in tandem, only the sitting duck effect is capable of causing a high level of interference. However, because the aspect ratio of the sensitive promoter is critical in setting the strength of sitting duck interference (and, to a smaller degree, occlusion), it is not possible to predict the degree of interference from the promoter strengths and distances alone. On the other hand, this feature provides regulatory flexibility, allowing the strength of interference to be tuned by evolution by alteration of the kinetic properties of the sensitive promoter without necessarily altering its strength or location. Our model provides a basis for the development of tools to examine dynamic processes on DNA in vivo. Interference measurements can provide information about the interaction between RNAP complexes on the DNA and on the kinetics of RNAP action at promoters in vivo. Our estimates of kon and kf for pL and pC (konZ0.011 sK1 and 0.0030 sK1, and kfZ0.011 sK1 and 0.038 sK1, respectively) show that these in vivo parameters can vary substantially between promoters. It would be interesting to know the range of kon and kf values that can be attained. It
408 may be possible to use interference studies to test the effect of activators on these rates in vivo. Interference methods also provide a way to determine in vivo RNAP elongation speed over relatively short DNA distances, and could be used to measure the effects of DNA or RNA sequence features (e.g. pause sites) or DNA-bound proteins on elongation. At present the available interference data do not allow us to ascertain whether a sitting duck complex can act as a roadblock or to determine the frequency of termination after collisions between elongating complexes. However, we hope that further modelling will allow the design of interference experiments, such as using identical strong promoters arranged face-to-face and separated by a greater range of distances, that may be able to distinguish alternative physical models of RNAP interactions in vivo. We also plan to adapt the model to account for promoter competition, where the promoters are so close that their SDCs overlap.27,28 We expect our model to be directly transferable to other eubacterial systems. Some aspects, particularly our treatment of interference by collisions, might also be applied to eukaryotes. However, most eukaryotic promoters seem to require the assembly of large, ill-defined multiprotein complexes spanning very large DNA regions. Thus, although disruption of these complexes by elongating RNAP6,7 shares similarities to sitting duck interference, mathematical modelling of this interference will require knowledge of the rates of formation of these multiprotein complexes.
Acknowledgements K.S. is supported by the Swedish Research Council through Grants No. 621 2002 4135 and 639 2002 6258. The Egan laboratory is supported by grant GM62976 from the USA National Institutes of Health.
Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jmb.2004.11.075
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Edited by M. Gottesman (Received 22 September 2004; received in revised form 25 November 2004; accepted 29 November 2004)
