this rate-limiting step. A Thermodynamic Analysis That Predicts the Positions of. Intrinsic Termination. By treating the reaction described in. Eq. 2 as a "virtual" ...
Proc. Nati. Acad. Sci. USA
Vol. 88, pp. 2307-2311, March 1991 Biochemistry
Transcript elongation and termination are competitive kinetic processes (transition state theory/RNA synthesis/termination efficiency)
PETER H. VON HIPPEL AND THOMAS D. YAGER Institute of Molecular Biology and Department of Chemistry, University of Oregon, Eugene OR 97403-1229
Contributed by Peter H. von Hippel, December 5, 1990
In this paper, we develop a kinetic approach ABSTRACT to predict the efficiency of termination at intrinsic (factor independent) terminators of Escherichia coli and related organisms. In general, our predictions agree well with experimental results. Our analysis also suggests that termination efficiency can readily be modulated by protein factors and environmental variables that shift the kinetic competition toward either elongation or termination. A quantitative framework for the consideration of such regulatory effects is developed and the strengths and limitations of the approach are discussed.
Each step of transcription by Escherichia coli RNA polymerase can be thought of as a kinetic competition between elongation and termination. Elongation. The polymerase can extend the 3' OH end of the RNA transcript by adding a nucleotide that is complementary to the DNA residue at template position I + 1:
RNA, + NTPI+1 RNA,+1 + PP1. [1] Here kforard is the overall rate constant for nucleotide addition, NTPI+1 is the nucleotide triphosphate to be added, and RNA, and RNA,+1 are transcripts of lengths I and I + 1 with 3' ends located at template positions I and I + 1, respectively. This nucleotide addition reaction must involve at least the two elementary steps of chemical bond formation and translocation of the complex by one position along the template. The actual forward rate under any particular set of conditions will depend on the detailed enzymology of the reaction, on the DNA sequence around template positions I and I + 1, on environmental conditions (pH, and the types and concentrations of different salts), and on the concentrations of the participating reactants. Termination. At each position there is also the possibility that the polymerase may dissociate from the template DNA and release the nascent transcript (RNA,): (ternary
complex), -m EDNA + RNA, + polymerase. [2]
Termination can be experimentally detected on a gel by the appearance of an RNA band of length I that cannot be "chased" into a longer product. Terminators that do not require additional protein factors have been called "intrinsic" (1). Within the limits of experimental resolution (currently -1 sec), termination at an intrinsic terminator appears to occur in an "all-or-none" fashion (2). However, some evidence for a rate-limiting elementary step within the RNA release reaction has been obtained (3); the apparent rate constant for the overall process (krelease) must then apply to this rate-limiting step. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
A Thermodynamic Analysis That Predicts the Positions of Intrinsic Termination. By treating the reaction described in Eq. 2 as a "virtual" process governed by equilibrium thermodynamics, we may write: DNA +
RNA, + polymerase = (ternary complex),. [3]
At each template position I, we equate the thermodynamic stability of the ternary complex with its standard free energy of formation (AG'compiex). To calculate AG'compiex we assume the "transcription bubble" model for the structure of the elongation complex (see refs. 4 and 5; Fig. 1). AG'fcomplex is calculated as the following sum:
fGcomplex = AGCDNA bubble +
AGf,RNA-DNA hybrid + AGfpol binding'
where A GfDNA bubble is the (unfavorable) standard free energy of formation of the open DNA bubble from intact duplex DNA; AGfRNA.DNA hybid is the (favorable) standard free energy of formation of the RNADNA hybrid (within the open DNA bubble) from single-stranded DNA and RNA chains; and AG',01 bindin is the (favorable) standard free energy of binding of the polymerase to the nucleic acid components of the ternary complex. Eq. 4 is defined for standard state concentrations (1 M) of all components. Thus, it omits the concentration-dependent portions of the free energy terms for the mixing entropy that is lost as the separate components (on the left side of Eq. 3) come together to form the ternary complex. Certain aspects of the definition of the standard state are arbitrary (5). The AGfDNA bubble and AG?,RNA.DNA hybrid terms'of Eq. 4 can be computed (5) by using thermodynamic reference data (6, 7) derived from the thermal melting of short nucleic acid duplexes. The AG'poi binding term is evaluated by assuming that AGOcomplex 0 kcal/mol (1 cal = 4.184 J) at sites of efficient intrinsic termination (5). This yields an estimate of -30 ± 3 kcal/mol for AGO, I binding under standard in vitro transcription conditions. The average value of AG'complex is found to be about -18 kcal/mol at nontermination positions along the DNA template (S). This represents the average free energy that must be added to the ternary complex to reach the state in which AG'.complex 0 kcal/mol. To release the nascent transcript, an activation free energy must also be added to reach the transition state for termination. Thus, at nontermination positions (which probably include >99.9% of all sites on E. coli DNA; see ref. 5), the ternary (elongation) complex should move along the DNA template in a deep potential energy well, elongation should be highly processive, and the probability of spontaneous dissociation of the elongation complex should be very low. These predictions are confirmed by the finding that elongation complexes can be "stalled" (at nontermination positions) by depletion of NTP substrates Abbreviation: TE, termination efficiency.
2307
[4]
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Biochemistry: von Hippel and Yager
Proc. Natl. Acad. Sci. USA 88 (1991)
TE as a Function of AG'nopex. Fig. 2A presents some measured values of TE for variants of the trp and thr attenuators of E. coli. The experimental measurements were performed under comparable in vitro conditions [60-150 mM KCl, 4-10 mM MgCl2 (pH 7-8)]. For each terminator, TE is plotted as a function of the maximum value of AGfcomplex, which is calculated from Eq. 4. Most ofthe data points cluster at values of (AGf,compiex)max between -4 and +4 kcal/mol and TE appears to increase monotonically with this parameter. In Fig. 2B, the open squares present analogous TE data for a number of other terminators and attenuators ofclasses I, II,
DNA Bubble (17±1 bp) A,
state E
1.0
state T
FS
o mr attenuator variants
c
"I,5>>g
0)
'DNA-RNA Hybrid
and can be reactivated days later by adding back NTPs (8-10). Thus, we can use this approach to define DNA template positions at which termination cannot occur because it is "thermodynamically forbidden" (5). As an intrinsic termination sequence is encountered by the polymerase, the magnitude of AGOcomplex changes abruptly from about -18 kcal/mol to about 0 kcal/mol over one or two nucleotide positions along the template (5). This gives AGOcomplex the properties of a "binary switch" and makes the thermodynamic prediction of sites of intrinsic termination a robust function of DNA sequence. We (and others) have proposed that this discontinuous behavior of AGfcomplex is caused by a major conformational transition within the transcription bubble. In this transition, part of the RNADNA hybrid is competitively displaced by a "terminator hairpin" that forms in the nascent RNA as an intrinsic termination site is traversed (see the lower structure in Fig. 1 and refs. 4 and 5). To apply the above model, not just to a virtual process but to an actual transcription event in real time, we must assume that the competitive displacement of the DNARNA hybrid by the RNA terminator hairpin reaches equilibrium prior to the rate-limiting RNA release step of Eq. 2. That is, krelease must be assumed independent of the rate of the nucleic acid displacement reaction. The general validity of this assumption is borne out by the considerable success of our thermodynamic model in predicting the positions of actual intrinsic termination events.
RESULTS Definition of Termination Efficiency (TE). In vitro transcription reactions are typically performed with a DNA fragment that contains a single promoter and a single terminator. Of all the RNA chains that initiate and "clear" the promoter, some fraction is released at the terminator at position I and is detected as a single band on a gel (RNA,). The remaining fraction is extended to the end of the DNA fragment and is also detected as a single ("runoff") band (RNArnoff). The TE at position I (TEX) is defined as: =
RNA,/(RNA, + RNArunoff).
0.8 H
0 0
_0. w.
FIG. 1. Structural basis of the thermodynamic calculations described in the text. The elongation complex is shown in the upper diagram and the intrinsic termination complex is shown in the lower diagram. bp, Base pairs.
TE1
k
* t attenuatovarants
[5]
This definition excludes RNA chains that are "abortively" released prior to the initial formation of the stable elongation complex (11, 12).
C. 0
*-
H
:3
0.2
To
0.0 _ -10
-20
0
10
(Af, complex max (kcal/mol) 1.0 B 0 C. c
0.8
._
C
0.6
o classes l, II, III
terminators * class IV
terminators
.
C
._
0.4
0)
H-
I
0.2
0.0 -40
-20
0
20
(AGf complex max (kcal/mol) FIG. 2. The observed efficiency of intrinsic terminators (TE) plotted as a function of the maximum value of AGtcomp ex which is calculated by using Eq. 4. All TE values are from in vitro determinations unless otherwise indicated. The fitted lines are (krelease + kfor0wamd) values calculated by using transition state theory (Eiqs. 6-8). (A) e, Wild type and variants of the E. coli'thr attenuator (13, 14); n, wild-type trp attenuators of E. coli (3, 15-20), Serratia marsecens (21), and Salmonella typhimurium (22, 23) and the following variants of the E. coli trp attenuator: L77, L78, L80, L81, L82, L150, L151 (16); L115 (19); L126 (19, 20); L153 (16, 20); a135 (18); A trp Lc 1419 (3, 15, 18, 20). (B) n, Wild-type E. coli leu, thr, trp, his, ilvBN, ilvGMEDA, pheST, pheA, and pyrBI attenuators (5), S. marsecens and S. typhimurium trp attenuators (20-22), S. typhimurium his attenuator (5), E.' coli supBE, tonB, his operon, rrnB tl, deoCABD, trp t terminators (5), and the M13 "central" and A"oop" terminators (5). Data are presented for both orientations ofthe E. coli tonB and his operon terminators. Data for the E. coli his operon and deoCABD terminators are from in vivo determinations. *, Wild-type E. coi rpsO-pnp, frd-ampC, rpIKAJL-rpoBC A, nusA-infl, and lacI-iacZ intergenic attenuators, and the TnWO tetA-orfL intergenic attenuator (5). The datum for the TnWO tetA-orfL intergenic attenuator is from an in vivo determination.
Proc. Natl. Acad. Sci. USA 88 (1991)
Biochemistry: von Hippel and Yager and III, as defined in ref. 5. These structures are attenuators near the 3' ends of amino acid or nucleotide biosynthesis operons or are intrinsic terminators of a wide variety of simple operons. These in vitro measurements of Fig. 2B have been made under roughly the same conditions as those of Fig. 2A. The trend and general conclusions are the same as for Fig. 2A, although the apparent scatter of the data is greater. (The solid squares in Fig. 2B represent data from another class of terminators and will be discussed below.) The experimental TE values of Fig. 2 are a roughly monotonic function of (AG~,complex)max, which suggests that our simple thermodynamic model might serve as a "firstapproximation" predictor ofthe efficiencies, as well as of the template positions, of intrinsic terminators. However, the scatter of the data points precludes a quantitative analysis of TE values by this approach for two reasons. First, the efficiency of a terminator is expected to be much more sensitive to the exact value of AG complex than is its position. Second, at loci where termination is thermodynamically allowed, this process must compete with elongation, which can also remove the transcription complex from template position I. Thus, a competitive kinetic approach must be taken to analyze TE. Analysis of TE with Transition State Theory. To explain the distribution of TE values in Fig. 2, we must consider the relative rates of the competing processes of termination and elongation. We can define TE in terms of rate constants as follows: TE = krelease/(krelease + kforward)
[6]
We assume that the actual rates of the release and forward reactions are accurately represented by the respective rate constants-i.e., that the concentrations of NTPs are not rate-limiting, that release is not prevented by exogenous protein factors, etc. (If this is not true under certain experimental conditions, then TE can be defined instead as a ratio of rates.) We interpret the rate constants of Eq. 6 in terms of Eyring transition state theory, which is used generally to predict the rates of chemical reactions. The Eyring theory (24, 25) is based on the concept of a "transition state," which, by definition, is the state of highest free energy along the reaction coordinate that joins reactant and product. Eyring and coworkers defined an equilibrium constant K* for the formation of the transition state from the reactant state and proposed that the rate constant (k) for the conversion of reactants to products could be related to the free energy difference between the transition state and the reactant state. This free energy difference is called the "activation free energy barrier" for the reaction. We may write the following expressions for the rate constants of the forward and release reactions:
kforward = Kf(kT/h)e
forwad/T
[7a]
and
krelease Kr(kT/h)e-Ac relex/RT,
[7b] where Kf and Kr are the "transmission coefficients" for passage of the transcription complex through the two transition states, k is Boltzmann's constant, T is the absolute temperature of the reaction (Kelvin), h is Planck's constant, and AGIorw d and AGfeieaae are, respectively, the activation free energy barers to elongation and termination. We as-
2309
been reported to lie in the range of 10-100 msec (8). We assume a median value of 30 msec. From this, we estimate kfod to be -30 sec-1 at such a site. Using Eq. 7a, we then calculate the height of the average activation barrier to elongation to be about +16 kcal/mol. For the moment, we for assume comparable values for kfoad and AGf "read-through" at a termination site; this assumption will be relaxed below. The Termination Barrier at Sites at Which Termination Is Allowed. We use the model of Yager and von Hippel (5) to establish that AGOcomplex 0 kcal/mol at an efficient intrinsic terminator. We next set the probabilities of elongation and termination to be approximately equal at such a site (TE 0.5). This leads to the assignment AG*release A G forward +16 kcal/mol, as depicted in Fig. 3 (Right). The Termination Barrier at a Site at Which Termination Is Not Allowed. In contrast, at template positions where termination is thermodynamically forbidden, the transcription complex resides in a deep potential energy well with AGOcomplex -18 kcal/mol. At such a site (assuming as above that AG~release fGorard +16 kcal/mol) the total height of the barrier to termination must be about +34 kcal/mol. This is -18 kcal/mol higher than the elongation barrier at the same position, as depicted in Fig. 3 (Left).* Several features of Fig. 3 require more explanation. (i) In each panel, the end of the reaction coordinate shows state I + 1 differing from state I by AG'r ,d. This corresponds to the standard free energy change (on a per mol basis) for extending the RNA transcript by one nucleotide residue. (ii) The stippled boxes at the peaks of the activation barriers in Fig. 3 (Right) correspond to changes of ± 1.4 kcal/mol in the heights of these barriers. This indicates the variations in these heights that might be expected from experimental uncertainties, changes in environmental conditions, physiological regulation, etc. A change of 1.4 kcal/mol in the height of such a barrier corresponds to a 10-fold change in polymerase dwell time at the corresponding template position (see below). (iii) The final termination state (after transcript release is complete) is not shown in Fig. 3. This state is rather arbitrarily defined (5). Furthermore, its absolute free energy is not relevant to the present analysis, since it will not affect the heights of the activation barriers as measured from the reactant state. (iv) Finally, we note that depiction of the elongation and release processes as one-step reactions is obviously an oversimplification. In reality, each of these processes consists of two or more elementary steps and thus the transition state diagrams of Fig. 3 should contain multipeaked barriers. The height ofeach of the single-peak barriers shown in Fig. 3 must therefore be understood to represent the rate-limiting step of each process. TE versus A(AG*). At a particular terminator, the relative probabilities of elongation and termination will depend only on the difference in the activation barrier heights for the two processes. This can be shown by inserting Eqs. 7a and 7b into Eq. 6 to obtain:
=
sume the same transmission coefficients for the forward and release reactions. The Elongation Barrier. The "dwell time" (t1/2) of the elongation complex at an average elongation position has
TE = [1 + e-&(AG')/RTI-1
[8]
*An independent lower limit estimate of about +26 kcal/mol can also be obtained directly for the total height of the termination barrier by using Eq. 7b and an assumed average lifetime (01/2 1 week) for a typical "stalled" elongation complex (8-10). This represents a lower limit value because we do not know whether the observed lifetimes of these very stable stalled complexes reflect a "real" dissociation process as represented by Eq. 2. Alternatively, the observed lifetimes may be limited by artifacts such as aggregation of the complexes, adsorption to the walls of the vessel, degradation of the RNA, etc.
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Biochemistry: von Hippel and Yager
Proc. Natl. Acad. Sci. USA 88 (1991)
At Elongation Sites
At Termination Sites
Termination Barrier
+30 LL Elongation
Ietmrnation Barr'.ie r
E onga|io Barn er
Barrie,
w
l
To CZ
-4--
/
I
+10 rE
.
+w1
~11
a.
-
Reaction Coordinate FIG. 3. Schematic diagram of the relative activation barrier heights for elongation and termination at an elongation site (Left) and at an intrinsic termination site (Right). For clarity, the zero-point on the y axis is set equal to the free energy of the reactant state of the transcription complex at template position I. The total height of each barrier is the sum of two components: (i) a thermodynamic component (AGfcompiex) that must first be input to make AGf°,cx (AGL AG1*1C that must next be added .d to reach the appropriate transition state. Barrier heights corresponding to 10-fold changes in dwell time ("pausing") at elongation or termination positions are shown as alternative heavy lines at the tops of the peaks. The stippled areas correspond to peak height variations of ±1.4 kcal/mol; a total variation of 2.8 kcal/mol in the relative heights of the elongation or termination barriers should change the TE from -99% to '1% (or vice versa) at a given termination site. or
where A(AGt) = (AGt d- AG*icase) In Fig. 2A, we superimpose theoretical curves of TE versus A(AGt), calculated by using Eq. 8, on the experimental TE data plotted against (AGocom lex)max. The horizontal axes of the two plots have been aligned by setting A(AGt) = 0 kcal/mol at TE = 0.5 and by setting (AGfcompex)nax = -2 kcal/mol at TE = 0.5. This brings into coincidence the midpoints of the experimental data distribution and the central theoretical curve. It also suggests that the best average value of AGF,pol binding for this set of terminators may actually be -32 kcal/mol, rather than -30 kcal/mol as estimated previously (5). We note that the centers of the distributions of the thr and trp data points in Fig. 2A appear offset from one another by =3 kcal/mol along the horizontal axis. This may reflect a real difference in the value of AGfpol binding for these two terminators. The outlying theoretical curves of Fig. 2A have the same form as the central curve but are shifted horizontally by either +2 or -2 kcal/mol. The resulting 4 kcal/mol-wide "envelope" oftheoretical curves contains most ofthe experimental data points for the thr and trp attenuator variants shown in this figure. This suggests that the experimental data on termination efficiencies (at least for these two attenuators) might be "explained" adequately by the above kinetic competition model. The data of Fig. 2B (c) are from a collection of 21 different intrinsic terminators and attenuators (see figure legend). More scatter in the data is evident here than in Fig. 2A. However, the trend of these data points in Fig. 2B is clearly the same as that of Fig. 2A. DISCUSSION The good fit of the kinetic competition model to the experimental data (Fig. 2) appears to justify the use of the Eyring
formalism to analyze the efficiencies of intrinsic E. coli terminators. There is ample recent precedent for this approach in studies of macromolecular transconformation reactions (25). We emphasize that most of the assumptions of our thermodynamic model of elongation and termination (5) are retained in this kinetic competition model. Fig. 3 shows, at an average (nontermination) template position, that the barrier height to termination is about +34 kcal/mol, while that to elongation is only about +16 kcal/ mol. Thus, the spontaneous release of the RNA transcript is highly improbable at nontermination sites. In contrast, at a "canonical" intrinsic termination site the barrier heights for elongation and termination are each about +16 kcal/mol. Under these conditions, termination is predicted to occur with -50% efficiency (Eqs. 7 and 8). Regulation. As the theoretical curves of Fig. 3 show, a change of only 2-4 kcal/mol in the relative heights of the elongation and termination barriers should suffice to drive virtually the entire population of transcribing RNA polymerase molecules down one or the other kinetic pathway. Thus, the efficiency of an intrinsic terminator is predicted to be highly sensitive to minor DNA (and RNA) sequence effects and environmental factors. How might the relative heights of the two barriers be perturbed in vivo to achieve regulation at intrinsic terminators? One possibility could involve changes in the transition states, and thus in the barrier heights, for either elongation or termination. A second possibility could involve changes in the thermodynamic stability of the transcription complex at termination sites, as reflected in the AGf complex parameter of Eq. 4. We refer to these two classes of effects as "kinetic" and "thermodynamic," respectively. Regulation by Kinetic Effects. An example of a kinetic effect may be provided by a recently characterized RNA polymer-
Biochemistry: von Hippel and Yager ase mutant. This mutant displays a decreased elongation rate in vivo and in vitro and also an increased TE at both intrinsic and rho-dependent sites (D. J. Jin and C. A. Gross, personal communication). This finding can be interpreted in terms of an increase in the activation barrier height for elongation, which would both slow down the elongation reaction and also favor the termination branch of the kinetic pathway. Regulation by Thermodynamic Effects. The value of one of the right-hand terms of Eq. 4 could be altered at a termination site in a manner not specified in our "canonical" model for intrinsic termination (lower diagram of Fig. 1). For example AGOpol binding might shift to a value that is less favorable for elongation. Fig. 2B shows some data that are consistent with such an effect. The solid squares represent measurements of TE for six intrinsic "intergenic attenuators" (class IV terminators as defined in ref. 5). Each experimental TE value is plotted as a function of (AG',complex)., which we have calculated by using Eq. 4 and assuming a value of AG',poi binding of -30 kcal/mol (5). In this figure, the solid squares appear to fall on a curve that is offset by about 9.5 kcal/mol from that of the canonical attenuators and terminators of classes I, II, and III (open squares; see above). This offset might be explained if the AGf, I binding term of Eq. 4 were to take on a value of about -20 Vcal/mol at these termination sites, rather than the value of about -30 kcal/mol that appears to be typical for intrinsic terminators of classes I, II, and III. This sudden change in the AGOp01 binding term would cause the overall stability of the transcription complex at such sites to be -10 kcal/mol less than expected (see Eq. 4); consequently, the TE would be anomalously high. Factor-Dependent Termination. Our model can also be used to analyze factor-dependent termination events. A termination factor (e.g., rho) could alter the relative heights of the elongation and termination barriers by other means than the displacement ofthe DNARNA hybrid by an RNA terminator hairpin. For example, such a factor might bind to the polymerase to alter directly the transition state for elongation or termination. Alternatively, factor binding could perturb AGfpoj binding. Figs. 2 and 3 suggest that an alteration of =4 kcal/mol in relative barrier heights by either of these mechanisms could switch TE for a particular terminator from 99% or vice versa.
CONCLUSIONS The approach presented here and in ref. 5 allows us to predict the positions and efficiencies of intrinsic termination events in E. coli and related organisms. Because this approach is based directly on a structural and thermodynamic model (the transcription bubble paradigm), it leads to specific hypotheses about the various free energy (Eq. 4) and activation free energy (Eqs. 6-8) terms that define the outcome of the elongation-termination decision at each template position. This approach thus provides a conceptual framework that we
Proc. Natl. Acad. Sci. USA 88 (1991)
2311
hope will be useful in guiding and interpreting future experiments. We thank many colleagues (especially J. A. Schellman, J. Geiselmann, 0. G. Berg, and M. J. Chamberlin) for useful discussions and insights. This research was supported, in part, by U.S. Public Health Service Research Grants GM-15792 and GM-29158 (to P.H.v.H.) and by a grant to the University of Oregon from the Lucille P. Markey Charitable Trust. P.H.v.H. is an American Cancer Society Research Professor of Chemistry. 1. Manley, J. L., Proudfoot, N. J. & Platt, T. (1989) Genes Dev. 3, 2218-2222. 2. Andrews, C. & Richardson, J. P. (1985) J. Biol. Chem. 260, 5826-5831. 3. Farnham, P. J. & Platt, T. (1981) Nucleic Acids Res. 9, 563577. 4. Yager, T. D. & von Hippel, P. H. (1987) in E. coli and S. typhimurium: Cellular and Molecular Biology, ed. Neidhardt, F. C. (Am. Soc. Microbiol., Washington, DC), pp. 1241-1275. 5. Yager, T. D. & von Hippel, P. H. (1991) Biochemistry 29, in press. 6. Breslauer, K. J., Frank, R., Blocker, H. & Marky, L. A. (1986) Proc. Natl. Acad. Sci. USA 83, 3746-3750. 7. Freier, S. M., Kierzek, R., Jaeger, J. A., Sugimoto, N., Caruthers, M. H., Nielson, T. & Turner, D. H. (1986) Proc. Nati. Acad. Sci. USA 83, 9373-9377. 8. Rhodes, G. & Chamberlin, M. J. (1974) J. Biol. Chem. 249, 6675-6683. 9. Arndt, K. M. & Chamberlin, M. J. (1990) J. Mol. Biol. 213, 79-108. 10. Morgan, W. D., Bear, D. G. & von Hippel, P. H. (1984) J. Biol.
Chem. 259, 8664-8671. 11. Gill, S. C., Yager, T. D. & von Hippel, P. H. (1990) Biophys.
Chem. 37, 239-250. 12. Carpousis, A. J. & Gralla, J. D. (1980) Biochemistry 19, 32453253. 13. Lynn, S. P., Bauer, C. E., Chapman, K. & Gardner, J. F. (1985) J. Mol. Biol. 183, 529-541. 14. Lynn, S. P., Kasper, L. M. & Gardner, J. F. (1988) J. Biol.
Chem. 263, 472-479. 15. Bertrand, K., Korn, K., Lee, F. & Yanofsky, C. (1977) J. Mol. Biol. 117, 227-247. 16. Stauffer, G. V., Zurawski, G. & Yanofsky, C. (1978) Proc. Natl. Acad. Sci. USA 75, 4833-4837. 17. Zurawski, G. & Yanofsky, C. (1980) J. Mol. Biol. 142, 123-129. 18. Christie, G. E., Farnham, P. J. & Platt, T. (1981) Proc. Nail. Acad. Sci. USA 78, 4180-4184. 19. Kolter, R. & Yanofsky, C. (1984) J. Mol. Biol. 175, 299-312. 20. Reynolds, R. A. (1988) Ph.D. thesis (Univ. of California, Berkeley). 21. Miozzari, G. F. & Yanofsky, C. (1978) Nature (London) 276, 684-689. 22. Yanofsky, C. & van Cleemput, M. (1982) J. Mol. Biol. 155, 235-246. 23. Winkler, M. E., Mullis, K., Barnett, J., Stroynowski, I. & Yanofsky, C. (1982) Proc. Natl. Acad. Sci. USA 79, 2181-2185. 24. Glasstone, S., Laidler, K. J. & Eyring, H. (1941) The Theory of Rate Processes (McGraw-Hill, New York). 25. McCammon, J. A. & Harvey, S. C. (1987) Dynamics of Proteins and Nucleic Acids (Cambridge Univ. Press, Cambridge, U.K.).
