GLOSSARY

Top Abstract Glossary Introduction Results Discussion Appendix References

f				load force from the laser trap (pN)
kBT				4.1 pN-nm
1				polymerization rate of RNA by the nucleotides hydrolyzed on RNAP (1/s)
2				dissociation rate of PPi (1/s)
3				polymerization rate of RNA by the nucleotides hydrolyzed in solution when a PPi is bound to the RNAP (1/s)
4				polymerization rate of RNA by the nucleotides hydrolyzed in solution when no PPi is bound to the RNAP (1/s)
1				depolymerization rate of RNA (the reverse of 1) (1/s)
2				association rate of PPi (1/s)
3				depolymerization rate of RNA (the reverse of 3) (1/s)
4				depolymerization rate of RNA (the reverse of 4) (1/s)
				length of a base pair (0.34 nm)
1(n, t)				the probability density that the RNAP is in state Pn at time t (i.e., with no PPi bound to the RNAP)
2(n, t)				the probability density that the RNAP is in state P'n at time t (i.e., a PPi is bound on the RNAP)
n				Length of the RNA transcript

	INTRODUCTION

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RNA polymerase (RNAP) processes along a DNA strand and transcribes the information coded in the DNA's base pair sequence into RNA (Erie et al., 1992; Polyakov et al., 1995; Yager and von Hippel, 1987). During this procession, RNAP polymerizes an RNA transcript via a sequence of reactions taking place on its surface. First the incoming DNA chain is separated into single strands, one of which will serve as the template for the RNA. The two strands reanneal before they exit the posterior end of the enzyme; in between they form a "transcription bubble" ~15 bp (1 bp = 0.34 nm) long (Fig. 1 a). The entire enzyme is ~30 ± 5 bp long. The growing RNA chain is synthesized at a catalytic site within the transcription bubble by the addition of nucleotides complementary to the sequence of the template strand. Viewed as a molecular machine, RNAP processes along the DNA by converting the free energy of nucleotide binding and hydrolysis into a force directed along the DNA axis. However, the molecular mechanism by which these chemical bond energies are transduced into the force that drives the RNAP along the DNA strand remains a mystery.

View larger version (60K): [in this window] [in a new window]   FIGURE 1   (a) Schematic diagram of polymerase. The E. coli RNAP consists of four subunits (denoted 2') with overall dimensions ~9 × 11 × 16 nm. Approximately 30 bp fits into the major DNA groove (1 bp = 0.34 nm). During transcription, the entering DNA strand is separated into a template strand and a nontemplate strand to form a "transcription bubble" that is ~15 bp long. Within this domain is the catalytic site where nucleotides are added to the growing RNA chain. The DNA-RNA hybrid region is 8-12 bp long. (b) The catalytic locus may consist of two binding sites (Erie et al., 1992). A substrate binding site on the surface of the enzyme binds solution NTP weakly. If the incoming base is complementary to the template base at the substrate site, they form a hydrogen bond. This triggers hydrolysis of the pyrophosphate group, and a phosphodiester link is forged with the preceding base (redrawn from Erie et al., 1992).

The velocity and force of procession can be measured by using laser trap technology to produce a force-velocity curve (M. D. Wang, personal communication). Measurements on RNA polymerase show that its mechanical properties differ from those measured for other molecular motors in two important respects. The load-velocity curve is concave, rather than convex or linear, as characteristic of other molecular motors (Berg and Turner, 1993; Finer et al., 1994; Hunt et al., 1994; Molloy et al., 1995; Svoboda and Block, 1994). Thus the velocity is nearly constant up to loads above 20 pN, whereupon it falls off to a stall load between 25 and 30 pN (M. D. Wang, personal communication). This stall force is 5-6 times larger than that of myosin or kinesin.

Here we propose a mechanism that can account for both of these features of the load-velocity curve, and which makes definite predictions about how the load-velocity curve changes when the concentrations of pyrophosphate and nucleotide are varied. The model also allows us to make predictions about the statistical behavior of the enzyme. In the next section we formulate the mathematical model. In the third section we present an analytic expression for the load-velocity curve, and show in the fourth section how the model parameters are computed from experimental data. In the fifth section we show how the load-velocity curve and the stall force should vary as pyrophosphate and nucleotide concentrations are changed. We also discuss how the model can be extended to incorporate DNA sequence-dependent effects, such as pausing and backsliding (Landick, 1997), and the appearance of "inchworming" motion in DNA footprinting experiments (Chamberlin, 1994; Krummel and Chamberlin, 1992). We present only the key results in the body of the paper and place the mathematical derivations of these results in the Appendices.

	A MODEL FOR FORCE GENERATION IN RNAP

We shall discuss our model in the context of the experimental situation of Yin et al., shown schematically in Fig. 2 a (Yin et al., 1995; M. D. Wang, personal communication). In these experiments the RNAP is affixed to the substratum, and a 0.5-µm bead attached to the end of the DNA is held in a laser trap. When nucleotide is added to the solution, the RNAP exerts tension on the DNA strand that pulls the bead off the center of the laser trap. Because the force developed by the laser trap is calibrated to piconewton accuracy, and the location of the bead can be tracked with nanometer precision, the force exerted on the bead by the RNAP can be accurately measured. In this fashion a load-velocity curve can be constructed. In practice, this is made difficult by the propensity of RNAP to pause intermittently in a sequence-dependent fashion.

View larger version (51K): [in this window] [in a new window]   FIGURE 2   (a) Schematic diagram of the experimental setup (Yin et al., 1995). The RNAP is attached to a coverslip, and a 0.5-µm bead is attached to the DNA strand. The position of the bead can be monitored optically. Procession of the RNAP reels in the DNA, pulling the bead out of the trap center. The trap is closely approximated by a linear elastic element, and so the force exerted on the DNA strand can be computed from the bead displacement. (b) Mechanical equivalent of the experimental setup used in formulating the model. Stress trajectories must be continuous; the stress flow is laser trap  DNA  RNA tip  RNAP barrier  substratum  laser trap. The barrier F represents the interaction between the RNAP and the DNA-RNA hybrid, through which tension in the DNA strand is transferred to the RNAP, while the template DNA strand is allowed to pass freely. The clamp on the front part of RNAP is redrawn from Landick (1997). The protein elasticity is in series with the DNA and trap elasticities to form a composite elastic system. Intercalation of a hydrolyzed nucleotide between the growing tip of the RNA chain and the front end of the RNAP can occur if Brownian fluctuations in the system produce a gap larger than the size of a nucleotide. Until pyrophosphate is released, a new nucleotide cannot enter the RNAP binding site; thus PPi release is the rate-limiting step for RNAP progression. The spring connecting the front and rear parts of the RNAP indicates that the two may move somewhat independently to produce the appearance of "inchworming" in DNA footprinting experiments.

Mechanical assumptions

The geometry of the model is shown in Fig. 2 b. The DNA strand is attached at one end to the bead, which is held in the laser trap. The other end within the RNAP is annealed to the growing RNA strand via the 8-12-bp hybrid in the transcription bubble. The RNA strand, in turn, can make contact with the RNAP at a site at the front of the catalytic site (labeled F in Fig. 2 b). This site acts as a "barrier" against which the tip of the transcript can fluctuate. In the model, F is the site where tension in the DNA strand is transferred to the RNAP, while allowing the template DNA strand to pass freely. Here the "barrier" F is meant to represent the "clamp" on RNAP (Landick, 1997), but it does not have to be a discrete site. Rather, it represents the overall interaction between the RNAP and the DNA-RNA hybrid, which prevents the DNA-RNA hybrid from being pulled out of the RNAP by the load force. The RNA transcript also may be attached to the RNAP at hybrid position posterior to the catalytic site and further back in an RNA before it exits the enzyme (Nudler et al., 1997). Both of these sites can transmit stress from DNA to RNAP. In the transcription process, RNAP needs to work against the load force and overcome the attachments of RNAP to DNA and RNA. It does this by rectifying the Brownian diffusion against the "barrier" F using the free energy of nucleotide triphosphate (NTP) binding and hydrolysis. Thus a dominant portion of the load force should flow through the "barrier" F to the substratum and then back to the laser trap.

Because stress trajectories must be continuous, we shall assume that the stress follows the path

(1)

We treat the enzyme as an elastic body attached firmly to the substratum. DNA, RNA, and laser traps are all modeled as springs. However, treating these objects as elastic bodies does not prevent us from focusing our attention only at the catalytic site. In Appendix B we demonstrate that the composite elastic forces can be very well approximated by a constant force, because the protein elasticity is in series with the DNA and trap elasticities.

We shall take advantage of the fact that there are three time scales inherent in the problem:

t_M is the relaxation time to mechanical equilibrium between the RNA tip (the last subunit added) and the front of the catalytic site (labeled F in Fig. 2) after a polymerization event. t_M  ²/2D_R, where D_R is the diffusion constant of the RNA tip relative to F, and  is the size of a nucleotide (0.34 nm).

t_P is the time scale for NTP to enter the substrate site. This time scale depends on two coincidental processes: 1) diffusion of the NTP to the substrate site, and 2) the gap between the transcript tip and the barrier, F, being greater than .

t_R is the time scale of pyrophosphate release; this is the rate-limiting step in the progression of the RNAP.

When t_M  t_P, i.e., /t_P  2 D_R/, the system is always at thermodynamic equilibrium with respect to the tip of the transcript. (The time scale t_R does not disturb the relaxation to the thermodynamic equilibrium of the system. That is, even if PP_i release were very fast (t_R  t_M), the system is still at thermodynamic equilibrium as long as the condition /t_p  2D/ is satisfied. Note that 2D/ is the "perfect ratchet velocity" (Peskin et al., 1993).) In Fig. 8 in Appendix A, we demonstrate that this condition is satisfied, even if we use the much smaller diffusion coefficient of the bead in the laser trap (~10⁵ nm²/s). Therefore, we can safely assume an equilibrium distribution of the gap between the RNA tip and the RNAP "ratchet barrier," F.

The key result of these calculations is that the rate of relaxing to the thermodynamic equilibrium after the addition of a nucleotide is much faster than the nucleotide insertion rate. So the distribution of the gap between the RNAP barrier and the transcript tip is time independent. This allows us to consider the motion of the RNAP to occur in discrete steps of length = 0.34 nm, the length of a single base pair. Thus we can formulate the model as a discrete state Markov chain.

Kinetic assumptions

Transcript elongation takes place at the catalytic site shown schematically in Fig. 1 b. There is some ambiguity about the exact sequence of events taking place at the catalytic site (Erie et al., 1992). One possible sequence is

(2)

In this scheme the RNAP states are

P_n: The transcript length is n.

B_n+1: The transcript length is n + 1 with NTP bound in the substrate site.

R_n+1: The transcript length is n + 1 with NTP bound in the substrate site and hydrogen bonded to the complementary site on the template DNA strand.

H_n+1: The transcript length is n + 1, with the nucleotide bound in the substrate site and to the template DNA strand after the hydrolysis of nucleotide.

This scheme is illustrated graphically in Fig. 3 a, where we have plotted the spatial displacement, n, of the RNAP along the horizontal axis and the reaction coordinate along the vertical axis. All transition rates with a horizontal (n) component depend on the load force, f, from the laser trap. We have placed the origin of our coordinate system at a particular nucleotide (e.g., the first) added to the growing RNA chain, and denoted by n the position of the RNA transcript tip. The subscript refers to the length of the transcript (or equivalently, the position of the RNAP from the beginning of the transcript).

View larger version (28K): [in this window] [in a new window]   FIGURE 3   (a) The chemical kinetic states for the model shown in Fig. 2. The horizontal axis marks distance in units of base pairs (1 bp = 0.34 nm), and the vertical axis is the reaction coordinate. All transitions with a horizontal component depend on the load force, f. With the RNA strand in polymerization state Pn, the catalytic site expands by a distance   = 0.34 nm to accommodate the binding of a nucleotide: Pn  Bn+1. If the nucleotide is complementary to the template, it binds to form the recognition complex, Rn+1, which triggers rapid hydrolysis to state Hn+1. Release of pyrophosphate carries the system to state Pn+1. If PPi release is the rate-limiting step, we can combine states Bn+1, Rn+1, and Hn+1 into a composite state, P'n+1, shown shaded. An alternative pathway is shown by the dashed transitions. Here binding does not require an expansion of the substrate site, but recognition does. In this case, the state Pn is the composite state shown enclosed by dashes, and P'n+1 contains only Rn+1 and Hn+1. (b) The Markov chain model for the transition diagram in a, where we assume that the rate-limiting step is PPi release. Thus the transition rates between states Bn+1, Rn+1, and Hn+1 are fast, so that these states can be combined into the single shaded state P'n+1. All transitions with a horizontal component depend on the load force, f.

In this scheme, NTP binding to the substrate site requires that the site expand by a distance   = 0.34 nm to accommodate the incoming nucleotide. Once docked at the substrate site, a recognition step takes place wherein if the incorrect nucleotide has bound, it is quickly released and another binds. If the nucleotide matches the template, hydrogen bonds are formed between the incoming nucleotide and the template strand, which triggers hydrolysis and subsequent release of pyrophosphate.

Alternatively, binding to the substrate site could occur before intercalation:

(3)

In this scheme the catalytic site expands by  to permit recognition only after NTP has bound to the substrate site. This is indicated by the dashed transitions in Fig. 3 a. Other kinetic sequences are also possible. However, we shall sidestep these ambiguities by assuming that under no-load conditions, the rate-limiting step for transcript elongation is the release of pyrophosphate from the catalytic site (Erie et al., 1992; Yin et al., 1995). This enables us to collapse RNAP states connected by fast transitions that are load independent into one composite state, shown enclosed in Fig. 3 a (Scheme 2 by solid lines, Scheme 3 by dashed lines).

With this key assumption about the rate-limiting step, we can represent the state transition diagram in Fig. 3 a by the Markov chain shown in Fig. 3 b. In this simplified model of the polymerization kinetics, an RNAP containing a transcript of length n can exist in two polymerization states:

P_n (containing a transcript of length n with no PP_i bound)

P'_n (containing a transcript of length n with PP_i bound).

We use in the diagram and the subsequent analysis the notation listed in the Glossary above.

The governing equations for the Markov chain in Fig. 3 b are

(4)

where ₁(n, t) + ₂(n, t) = 1. The solution to these equations will provide the force-velocity curve we seek.

	RESULTS

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In Appendix C we solve Eq. 4 corresponding to the model in Fig. 3 b to obtain the following expression for the load-velocity curve:

(5)

Independent information about the motor function is contained in the statistical variance of the motor's motion about its mean velocity. This variance can be characterized by an "effective" diffusion constant, D_eff, given by (cf. Appendix D)

(6)

A plot of many trajectories should show that the variance of the trajectories increases with time at a rate

(7)

(In Appendix D we show that D_eff contains the same information as the "randomness parameter" defined by Schnitzer and Block to investigate the step size of kinesin (Schnitzer and Block, 1995, 1997).)

	DETERMINATION OF RATE PARAMETERS

In Eqs. 5 and 6 the transition rates with subscripts 1, 3, and 4 depend on the load force f. The constraint that the model obey detailed balance at chemical equilibrium (which is required for consistency with thermodynamics) requires that (Hill, 1977, 1989)

(8)

where G_i is the free energy drop of the transition process i. There are two approaches to determining the force dependence of the transition rates. If the potential wells holding the nucleotide in position on the substrate and product sites were known, then one could model these rates by the Kramers rate law (Hanggi et al., 1990; Risken, 1989), and detailed balance is ensured. However, to use this approach, one must know the free energy changes in the whole transition process, not just at the two end points. Alternatively, if only the total free energy drop in the transition process is known, one can calculate one of the parameters, say , using experimental results, and obtain the other parameter from Eq. 8. In Appendix E we use the principle of detailed balance and the empirical measurements listed in Table 3, augmented by two physically reasonable assumptions, to compute the transition rates. The results are shown in Table 1.

	RELATIONSHIP OF THE MODEL TO EXPERIMENTS AND PREDICTIONS

Using the transition rates calculated in Appendix E and listed in Table 1, the model makes the following predictions about the mechanical behavior of RNAP:

1. Fig. 4, a and b, shows the load-velocity curves for various concentrations of NTP and PP_i. The model predicts that the load velocity is concave. This experimental result (M. D. Wang, personal communication) is not used in our calculation of the transition rates in Appendix E; i.e., we did not fit our model to generate this concave load-velocity curve. Rather, the model predicts this concave load-velocity curve based on the parameters obtained from other experimental data.

View larger version (17K): [in this window] [in a new window]   FIGURE 4   (a) Load-velocity curves computed from Eq. 5 for 1 µM PPi and different concentrations of NTP. At 1 mM NTP and 1 µM PPi, the motor initially proceeds almost independently of the resisting force at low loads, then falls sharply to zero at the stall force. As the concentration of NTP decreases, the load-velocity curve becomes less concave. A Java code for the model can be run from within Netscape at http://teddy.berkeley.edu:1024/Java/ratchet_java.html. (b) Load-velocity curves computed for 1 mM NTP and different concentrations of PPi. Note that as the concentration of PPi is increased from 1 µM to 1 mM, the transcription velocity decreases by a factor of 2, whereas the stall force is virtually unchanged.

2. Fig. 5, a and b, predicts how the stall force and the maximum velocity should respond to changes in the solution concentrations of NTP and PP_i. When the concentration of PP_i is increased from 1 µM to 1 mM, the stall force is virtually unchanged, whereas the maximum velocity is reduced by half. This is another experimental result (Yin et al., 1995) that we did not use in our calculation of the transition rates, but is a prediction borne out by experiments. It is also important to point out our predictions for situations where experimental results are currently not available. In particular, our model predicts that, as the concentration of NTP decreases, the stall force decreases by roughly the same percentage as the maximum velocity.

View larger version (14K): [in this window] [in a new window]   FIGURE 5   (a) The maximum velocity (vm) and the stall force (fs) as functions of NTP concentration at 1 µM PPi. fs is computed from Eq. E.9. At low NTP concentrations both vm and fs rise logarithmically with [NTP]; however, at higher concentrations, fs continues to rise, whereas vm levels off and is nearly constant. (b) The maximum velocity (vm) and the stall force (fs) as functions of PPi concentration at 1 mM NTP. At low pyrophosphate concentrations, vm and fs are practically constant; at moderate pyrophosphate concentrations, fs remains constant, but vm starts to fall off, first logarithmically, then at a slower pace; at higher concentrations, fs decreases logarithmically and converges to zero along with vm.

3. In principle, variance measurements on RNAP can be performed in a manner similar to that of measurements made on kinesin (Schnitzer and Block, 1995; Svoboda et al., 1994). In the previous section we showed that the variance can be characterized by an effective diffusion constant: var(x(t)) = x(t)²  x(t)² = 2D_efft. Fig. 6 shows how D_eff varies with solution concentrations of NTP and pyrophosphate.

View larger version (16K): [in this window] [in a new window]   FIGURE 6   The effective diffusion constant (Deff) computed from Eq. 6 as a function of NTP concentration at 1 µM PPi, and Deff as a function of PPi concentration at 1 mM NTP.

All curves in Figs. 4-6 are computed from Eqs. 5 and 6, using the transition rates given in Table 1.

In Fig. 4 a, at 1 mM NTP and 1 µM PP_i, the motor initially proceeds almost independently of the resisting force at low loads, then falls sharply to zero at the stall force, given by

(9)

where Q is a steady-state constant given by Eq. E.10 in Appendix E.

For the molecular motors for which a load-velocity curve has been obtained, the relationship is either linear or convex, in contrast to the load-velocity curves for RNAP in Fig. 4 a, which are concave down. At low loads, the RNAP transcription velocity does not decrease significantly as the load force increases.

In general, such a concave shape will arise when there is a rate-limiting chemical step that is not affected by the load force. This can be seen by considering a process consisting of cycles of two sequential steps. Suppose that the rate of the first step decreases exponentially with the load, but the rate of the second step is load independent and is much smaller than that of the first step at zero load. At low loads, the second step is the rate-limiting step; that is, the rate of the process cycle is roughly the same as the second step, no matter how fast the first step is. When the load is increased sufficiently, the rate of the first step eventually decreases to values comparable to that of the second step, whereupon the rate of the process cycle drops sharply to zero. This leads to a concave load velocity curve.

For RNAP during normal progression, the rate-limiting chemical event is pyrophosphate release, which we have assumed is independent of the load. In a typical transcription cycle, polymerization is followed by the release of pyrophosphate. At low loads, the polymerization step is much faster than the release of PP_i, and so the transcription rate is approximately the rate of PP_i release. At high loads the polymerization rate eventually falls below the rate of PP_i release, whereupon the transcription rate is roughly given by the polymerization rate, which falls off significantly with load.

The stall force f_s  28 pN is well below the thermodynamic limit of f_S^TD = G/  145 pN, where G  12k_BT = 50 pN-nm is the mean free energy of NTP hydrolysis, and  = 0.34 nm is the step size (Yin et al., 1995; M. D. Wang, personal communication). Because the thermodynamic maximum stall force f_S^TD  1/(step size), the large stall force of RNAP may be attributed to the small step size of the RNAP. The stall force of RNAP is 5-6 times larger than that of myosin or kinesin. However, if we measure the efficiency of energy conversion near stall of each forward step by the ratio f_S/G, then we find that RNAP is significantly less efficient than myosin or kinesin. These three molecular motors are compared in Table 2. Note that at stall the RNAP is still hydrolyzing NTP at a steady rate, so energy consumption continues, although no work is being performed. Thus the hydrolysis of NTP and the transcript elongation are not tightly coupled.

Dependence on pyrophosphate and NTP concentrations

The two quantities that are easiest to manipulate experimentally are the concentrations of pyrophosphate and nucleotide. Fig. 5 shows how the stall force (f_s) and the maximum velocity (v_m) depend on these variables:

At low NTP concentrations, both v_m and f_s rise logarithmically with [NTP]; however, at higher concentrations, f_s continues to rise, whereas v_m levels off and is nearly constant.

At low pyrophosphate concentrations, v_m and f_s are practically constant. At moderate pyrophosphate concentrations, f_s remains constant, but v_m starts to fall off, first logarithmically, then at a slower pace; at higher concentrations, f_s decreases logarithmically and converges to zero along with v_m.

Sequence dependence

The model developed so far treats DNA as a homopolymer with but one nucleotide type. There are several ways to generalize the model to include sequence-specific effects. The most important mechanical parameter is the strength of the bonds holding the terminal nucleotide onto the end of the transcript. This is embodied in the "horizontal" dissociation rate constants, _i, in Fig. 3. If all other factors are equal, the 's can take on one of two values: A-U and A-T pairs are joined by two hydrogen bonds (denote by "), and G-C pairs are held together by a triplet of hydrogen bonds (denote by ). RNAP processing along a homopolymeric DNA consisting of all G's will have a higher stall force than one consisting of all A's, because the breaking rate, 's, will be smaller for the former than for the latter (cf. Appendix E). The stall force of a DNA strand consisting of a random sequence of bases will have a mixture of two values of  = (", ). The average stall force will lie between two extremes: f_S(")  f_S  f_S().

For a given DNA sequence, the Markov chain in Fig. 3 b can be simulated by replicating each Markov unit with a  (" or ) corresponding to the bond type at that location. In this way, a distribution of stall loads will be computed for many replications of the numerical "experiment." No data are currently available for sequence-dependent stall loads; however, the model makes definite predictions of how such measurements should go.

Backsliding

We have viewed RNAP as a processive "sliding clamp" with the 3' terminus of the RNA transcript always aligned with the catalytic site (Landick, 1997). However, there is evidence that the RNA transcript can slip out of the catalytic site, allowing the RNAP transcription bubble to backslide along the RNA and DNA, a phenomenon that is probably sequence dependent (Landick, 1997). Moreover, in the laser trap force measurements, RNAP frequently paused for variable amounts of time before resuming transcription and force generation (M. D. Wang, personal communication). This also may be due to slipping of the leading RNA nucleotide out of the catalytic site. Such effects can be incorporated into the model, as shown in Fig. 7, by introducing a parallel sequence of states that allow backsliding of the RNAP along the RNA and DNA, which is probably force dependent.

View larger version (44K): [in this window] [in a new window]   FIGURE 7   (a) Schematic diagram of RNAP backsliding. The RNA transcript slips out of the catalytic site, allowing the RNAP transcription bubble to backslide along the RNA and DNA. The catalytic site at location n slips one or more nucleotides back along the transcript, so that the 3' terminus of the transcript is still at position n, but the catalytic site is now at position n  m. (b) Markov chain model for simulating RNAP backsliding. The top portion of the diagram is the same as in Fig. 3 b. The RNAP starts backsliding at position n. The dashed box represents states where the 3' terminus of the RNA transcript is not aligned with the catalytic site. These states are numbered by m = 1, 2, ... M, where M is the number of base pairs that have slipped out of the catalytic site.

The appearance of "inchworming" in DNA footprinting

According to the model as formulated, the motion of the RNAP proceeds "smoothly," one step at a time, which is in accord with some recent observations (Nudler et al., 1997). However, there is also evidence from DNA footprinting studies that RNAP "inchworms" along the DNA in a saltatory motion (Chamberlin, 1994; Krummel and Chamberlin, 1992). Such motion can be accommodated into the model by allowing the posterior subunit of the RNAP to be elastically tethered to the front subunit. as shown schematically in Fig. 2. This can produce an apparent variation in the RNAP footprint by two different mechanisms.

First, suppose that the attachment of the transcript to the posterior subunit can be modeled as a series of potential wells representing the attachments (e.g.. hydrogen bonds) of the transcript to the RNAP. As the front end moves steadily forward, the spring is stretched and the force on the rear increases until the threshold for breaking the bonds is reached. Thereafter, the rear end will process at the same rate as the front end, but the overall dimension of the RNAP will be dilated. When the progress of the front end is terminated during the footprinting assay, the RNAP will relax back to its equilibrium length, and the assay will give the impression of a variable-length protected region of the DNA. This length will be sequence dependent, because the strength of the attachment of the transcript to the RNAP is sequence dependent.

Alternatively, if the binding of the transcript to the rear end has the character of a "sliding friction," then the rear end will be stationary until a threshold force is exceeded, whereupon it will lurch forward to its equilibrium position. The front end will then extend the spring once again. Thus inchworming of the rear end of the RNAP will accompany the smooth motion of the front end.

	DISCUSSION

Top Abstract Glossary Introduction Results Discussion Appendix References

As RNA polymerase carries out transcription under cellular conditions, it creates downstream supercoiling that generates an opposing load force of ~6 pN (Yin et al., 1995). However, laser trap experiments have shown that it can move against a load several times greater than the maximum force developed by kinesin (M. D. Wang, personal communication; Svoboda and Block, 1994). Moreover, the load-velocity curve appears to differ qualitatively from that measured for kinesin and myosin. Rather than falling off almost linearly with load, RNAP moves at nearly constant velocity until loads above ~20 pN are applied, whereupon it falls off rapidly to a stall force of 25-30 pN.

Because the details of nucleotide binding, hydrolysis, and polymerization are not known exactly, we have formulated a kinetic model based on energetics that allows us to understand certain qualitative and quantitative aspects of RNAP's mechanochemistry. Each step of the transcription process consists of several kinetic steps whose transition rates are calculated based on experimentally determined free energy differences. The energy driving RNAP motion derives ultimately from the hydrolysis and subsequent binding of the nucleotide triphosphates it uses to build the RNA transcript. However, it is not clear how this free energy is transduced into so large a processive force. This transduction process cannot be too efficient, for the measured stall force is much less than the thermodynamically maximum force obtained by dividing the free energy of hydrolysis by the length of a single base: ~12k_BT (at 1 µM PP_i and 1 mM NTP)/0.34 nm  145 pN (Yin et al., 1995).

The model presented here provides an explanation for this energy transduction; it is essentially an extension of the Brownian ratchet polymerization models developed earlier (Mogilner and Oster, 1996; Peskin et al., 1993). A related idea for RNAP procession was proposed by Yager and von Hippel (1987); however, we are not aware of any quantitative model for RNAP force generation. The model demonstrates that a Brownian ratchet mechanism with a step size of one nucleotide is sufficient to account for the sizable stall force measured in the laser trap experiments. No conformational changes need be invoked, although the model does not rule them out. Indeed, in a previous model for kinesin force generation, it was shown that a combination of biased diffusion and a conformation change driven by binding free energy was necessary to reproduce the observed mechanical measurements (Peskin and Oster, 1995).

In addition to accounting for the large observed stall force, the model accounts for the atypical concave shape of the load-velocity curve. This shape arises because of the multiple chemical steps involved in the ratchet mechanism. In particular, this concave shape is essentially due to the slow (relative to the mechanical motions) rate of pyrophosphate release, which is the rate-limiting step in transcription. Because the time scale for mechanical relaxation to equilibrium is so much faster than the rate of the fastest reactions involved, the progression of RNAP can be treated as a sequence of mechanical equilibrium states and modeled as a Markov chain with thermally excited transition rates. A similar viewpoint was taken by Erie et al. in discussing the kinetics of transcription (Erie et al., 1992).

We have restricted ourselves to the simplest case of two kinetic steps: polymerization and the rate-limiting step of pyrophosphate release. However, there is no difficulty in generalizing the model to include other kinetic schemes. It is also relatively straightforward to include sequence-dependent rates once such data become available.

Experimentally determining the load-velocity curve for RNAP has proved difficult because of the occurrence of confounding effects. For example, RNAP occasionally slides backward, resulting in the leading nucleotide temporarily slipping out of the catalytic region. This backsliding is much slower than a normal transcription step, and so, when it occurs, backsliding becomes the rate-limiting step in the overall transcription. Between the backsliding events, the RNAP moves with roughly constant velocity. Thus the velocity of RNAP is constant when it is "on the track," but drops to zero when it slides "off the track." When it slides back on the track, the velocity jumps up again to its normal rate. Thus the "net" velocity of transcription when the RNAP is on the track is given by the envelope of the RNAP velocity (M. D. Wang, personal communication). In each step of transcription between backslidings, the release of pyrophosphate is the rate-limiting step. Although our model was developed for the normal transcription where the RNAP is on the track, it can be extended to take the backslidings into consideration (see Fig. 7 and the accompanying discussion).

We have avoided explicit treatment of the controversial issue of inchworm motion by RNAP (Chamberlin, 1994; Nudler et al., 1997), although we have indicated how this feature can be included by modeling RNAP as two elastically joined subunits and taking into account the binding of the transcript to the posterior section. This addition also accounts for pausing by upstream RNA hairpin formation.

Finally, the model makes definite predictions about how the stall force depends on pyrophosphate and nucleotide triphosphate concentrations, how the various kinetic steps reflect on the load-velocity behavior, and how measuring the statistics of progression can provide information about the kinetics of transcription. Thus the model can serve as a basis for a unified view of the kinetics, thermodynamics, and mechanics of transcription by RNAP.