Tuning and switching a DNA polymerase motor with mechanical tension

July 17, 2003
100 (17) 9699-9704


Recent single-molecule experiments reveal that mechanical tension on DNA can control both the speed and direction of the DNA polymerase motor. We present a theoretical description of this tension-induced “tuning” and “switching.” The internal conformational states of the enzyme motor are represented as nodes, and the allowed transitions between states as links, of a biochemical network. The motor moves along the DNA by cycling through a given sequence of internal states. Tension and other external control parameters, particularly the ambient concentrations of enzyme, nucleotides, and pyrophosphates, couple into the internal conformational dynamics of the motor, thereby regulating the steady-state flux through the network. The network links are specified by bulk-phase kinetic data (in the absence of tension), and rudimentary models are used to describe the dependence on tension of key links. We find that this network analysis simulates well the chief results from single-molecule experiments including the tension-induced attenuation of polymerase activity, the onset of exonucleolysis at high tension, and insensitivity to large changes in concentration of the enzyme. A major dependence of the switching tension on the nucleotide concentration is also predicted.
Single-molecule experiments (13) enable the velocity of a polymerase motor to be monitored as it catalyzes the replication of a mechanically stretched DNA template. Fig. 1 shows results for the widest range of tension currently available (2). Applying piconewton forces to the DNA template markedly alters (“tunes”) the rate of replication catalyzed by T7 DNA polymerase (DNAp). Stretching the DNA with tension greater than ≈35 pN induces the enzyme motor to reverse (“switch”) directions and begin depolymerizing the double-stranded (ds) DNA. Previous theoretical models (26) have related the tension dependence of the replication rate to free-energy changes but have not accounted for the exonucleolysis process that occurs above 35 pN. This process is of particular interest, because its mechanism may prove to be akin to the proofreading or editing function of the enzyme, which is triggered by a mismatch in DNA base pairing.
Fig. 1.
Experimental schematic (Inset), data (points), and model results (red curve). (Inset) A single DNA molecule is stretched between plastic beads, permitting micromanipulation by optical tweezers or micropipettes. The speed of ds DNA polymerization (nn + 1) is denoted by vpoly, and that of ds DNA excision (nn – 1) is denoted by vexo. In each forward step, the enzyme motor incorporates one nucleotide (dNTP) into the DNA and releases one molecule of PPi into the surrounding solution. The concentration of PPi typically is too low to permit direct reversal of dNTP hydrolysis (i.e., pyrophosphorolysis) to occur at an appreciable rate. Experimental data (5) for tension dependence of net replication rate catalyzed by T7 DNAp are compared with net flux, Jnet, derived from the network model shown in Fig. 2B with rate constants from Table 1 with [DNAp] = 800 nM, [dNTP] = 0.6 mM, and [PPi] = 1 μM.
This article treats both the polymerase and exonucleolysis modes by means of a pragmatic modeling procedure based on the analysis of reaction cycles extensively developed by Hill (7). This offers a systematic way to incorporate structural, thermodynamic, and kinetic data from bulk as well as single-molecule experiments and to examine assumptions pertaining to the effect of tension or other control variables on enzyme action. Aspects of the results are chiefly heuristic but serve both to suggest experimental tests and to aid the design of molecular dynamics simulations (6).

Kinetic Network Analysis

The kinetic pathway (810) for T7 DNAp shown in Fig. 2A can be described by the cyclic network of Fig. 2B; the nodes (i = 1–7) of the network represent known states of the enzyme–DNA complex, and the links specify transitions with the rate constants (αij) given in Table 1. Crystal structures (1113) show that the enzyme has a “right-hand” shape with finger, thumb, and palm domains and two distinct active sites, where the polymerase (red steps) and exonuclease (green steps) catalysis occurs. The relative prevalence of these modes of enzyme activity thus is influenced by a three-way competition (14) among binding of the enzyme to the polymerase (poly) and exonucleolysis (exo) sites and shuttling between the sites (black steps). Completion of one poly cycle corresponds to movement of the enzyme motor one step forward along the DNA template, and completion of one exo cyle corresponds to one step backward. These movements actually occur as spirals: The poly cycle converts EpDNAn to EpDNAn+1 (state 3 to 3′), and the exo cycle converts ExDNAn to ExDNAn1 (state 2 to 2′).
Fig. 2.
Kinetic network. (A) Pathways and intermediates determined from bulk kinetic studies (1113). As the enzyme motor visits the sequence of states 3 → 4 → 5 → 6 → 7 → 3′, it completes one polymerase cycle (red); a switch via 3 → 2 enables the sequence 2 → 2′, completing one exonuclease cycle (green). E, enzyme in solution; Ep,xDNAn, complex with DNA bound in poly (p) or exo (x) active sites; *, closed conformation at polymerase active site; n, number of base pairs in duplex portion of DNA; dNTP, deoxynucleotide. (B) Cyclic representation of reaction pathways. Distinct cycles for polymerase (red) and exonuclease (green) pathways are linked by a cycle (black) involving binding or unbinding of DNA to the poly or exo active sites. The poly cycle converts EpDNAn to EpDNAn+1 in five steps, with the enzyme motor recovering its original state at the end of each cycle. The exo cycle converts ExDNAn to ExDNAn1 via unknown intermediate states (indicated by “?”). Jpoly and Jexo denote the number of cycles completed per unit time. Arrows indicate reaction steps with rates proportional to ambient concentrations of enzyme, nucleotide, or pyrophosphate, or assumed to depend on tension f applied to the DNA.
Table 1.
Rate constants (s1) at zero tension
α12250 [DNAp]/μMα211,022
α13170 [DNAp]/μMα310.2
α22′ ≡ αexo900
α34>50 [dNTP]/μMα43>1,000
α73′>1,000α3′7>0.5 [PPi]/μM
All rate constants (8-10) are first-order (or pseudo first-order) and are given in units of s-1 (when multiplied by concentrations of DNAp, dNTP, or PPi in μM). Typical concentrations are given in the Fig. 1 legend.
The probabilities, pi, for the known states of the network to be occupied are determined by solving a system of coupled differential equations,
\[ \begin{equation*} \frac{dp_{i}}{dt}={{\sum^{7}_{j=1}}}({\alpha}_{ji}p_{j}-{\alpha}_{ij}p_{i}),\;\;\end{equation*}\]
subject to the normalization constraint, ∑ pi = 1, and steady-state condition, dpi/dt = 0. The transition flux from state ij is given by Jij = αijpi – αjipj. At steady state, these Jij values are the same for each link within a cycle and thus specify the overall cycle flux, Jpoly or Jexo, and the net replication flux, Jnet = JpolyJexo. This cycle formalism (7) is a multistate generalization of the Michaelis–Menten mechanism.
For zero tension, most of the input rate constants come from bulk kinetic studies (810). Although for three links of the Jpoly cycle only equilibrium constants are available (Fig. 2 A), this was not a limitation. In those cases, the kinetic data also provide lower bounds on the rate constants (indicated in Table 1). These bounds show that the three links are so facile as to have only a minor influence on Jpoly (found to be nearly unchanged upon increasing the rate constants 10-fold, with ratios fixed by the equilibrium constants). Jpoly is determined chiefly by the rate-limiting 4 → 5 step; therein the enzyme undergoes a conformational change at its active site, with the fingers flexing from open to closed (13, 15, 16). For the Jexo cycle, only the rate constant for the overall process, 2 → 2′, is available; intermediate states likely exist (indicated by “?” in Fig. 2B), but as yet their structural and kinetic properties remain unknown. The shuttling cycle, linking the states 1, 2, and 3 with the enzyme either free in solution or bound to the poly or exo site, has a key role. Labeled arrows indicate reaction steps with rates proportional to ambient concentrations of enzyme, nucleotide, or pyrophosphate or expected to depend on tension exerted on the DNA template.

Models for Tension-Dependent Steps

Tension applied to the DNA template couples into the conformational dynamics of the enzyme motor by perturbing the free energy of intermediate states or transition-states of the enzyme–DNA complex, stabilizing some states and destabilizing others. Structural data (1113, 1719) on pertinent conformations of the complex (nodes in Fig. 2B) suggest that three reaction steps are likely to respond to tension: (i) the 4 → 5 transition, involving change from the open to closed polymerase conformation; (ii) the 3 → 2 transition, whereby the DNA template switches from the poly to exo active site of the enzyme; and (iii) the 2 → 2′ transition, in which a base is excised from the DNA primer at the exo site. We specify simple models for the tension dependence of these steps and exhibit the important role of detailed balance constraints. These constraints require that if any step within a cycle depends on tension, one or more other steps must have a compensating dependence on tension. Accordingly, the cycle analysis shows that the dependence on tension observable in single-molecule experiments actually pertains not simply to the rate constant for an individual reaction step but to the net flux of the coupled reaction network.
Open to Closed Conformational Change of DNAp. Models for the 4 → 5 step (26) use transition-state theory to relate α45(f) to the rate constant in the absence of tension by
\[ \begin{equation*}\;\;{\alpha}_{45}(f)={\alpha}_{45}(0)F_{45}(f)={\alpha}_{45}(0){\mathrm{exp}}[-{\Delta}g^{{\dagger}}(f)/k_{B}T],\;\;\end{equation*}\]
where F45(0) = 1 and Δg(f) is the tension-dependent portion of the free energy of activation associated with change from the open-fingers (reactant state) to the closed-fingers conformation of the DNA–enzyme complex (considered the transition state). At present, there is no experimental means to determine Δg(f). Recent molecular dynamics simulations of an enzyme–DNA complex (6) show that previous models (25) focused on distance changes are unrealistic. In the range f < 30 pN, pertinent for Jpoly, the simulations indicate that Δg(f) arises chiefly from angular reorientations of DNA segments nearest the active site. Fig. 3A plots the Δg(f) function we have adopted in accord with the molecular dynamics simulations, and Fig. 3B shows the corresponding tension dependence of the rate constant. This model corresponds to a “late” transition state, the free-energy of which shifts concomitantly with that of the closed conformation.
Fig. 3.
Tension dependence. (A) Free energy of activation Δg(f) for α45 (red curve), estimated from molecular dynamics simulations (9) (points); free-energy change, Δg(f) = gds(f) – gss(f), for converting one ss residue to a ds residue of DNA (dashed black curve), derived from experimental force-extension curves for DNA uncomplexed with enzyme (8, 24); corresponding Δg*(f) for enzyme–DNA complex (solid black curve) at poly site estimated from flux data of Fig. 1; and nominal activation free-energy Δgx(f) for excision at exo site (green curve), also estimated from flux data. (B) Tension-dependent factors used in the model shown in Fig. 2B: F45(f), red curve from Eq. 2; F32(f), dashed black curve; \( \begin{equation*}F_{32}^{*}(f)\end{equation*}\), solid black curve from Eq. 3 with m = 6; and Fexo(f), green curve from Eq. 4.
Switching Between Poly and Exo Sites. Structural studies (1719) of polymerases related to T7 DNAp show that the 3 → 2 switch from the poly- to the exo-binding site involves major conformational changes. It appears that a β-hairpin separates the two catalytic domains, located ≈35 Å apart. DNA immigrating from the poly site must rotate appreciably to bind to the exo site. Moreover, the exo site binds several single-stranded (ss) residues of DNA, formed by “melting” terminal base pairs of the duplex DNA. An array of protein residues at the exo site aids the melting and holds the ss DNA segments kinked within the exonuclease-binding pocket. We assume that the switching process is limited by DNA melting, which is fostered by tension, and model the tension dependence of the equilibrium constant, K32 = α3223, as K32(f)=K32(0)F32(f), where F32(0) = 1 and the tension dependence has the form,
\[ \begin{equation*}\;\;F_{32}(f)=\frac{[1+{\mathrm{exp}}(m{\Delta}G(0)/k_{B}T)]}{[1+{\mathrm{exp}}(m{\Delta}G(f)/k_{B}T)]}.\;\;\end{equation*}\]
Here m denotes the number of base pairs that must be melted to enable the template DNA to switch from the poly to the exo site and ΔG(f) is the free energy required to melt one base pair. This is given by ΔG(f) = ΔG(0) – Δg(f), where –Δg(f) = gss(f) – gds(f) pertains to converting one ds residue into a ss residue during the melting. For free DNA, thermal data (20) give ΔG(0) ∼ 2.3 kBT, and Δg(f) can be obtained from measurements of the stretching of DNA strands subjected to tension (5, 21, 22). It would be more appropriate to use a free energy for melting in the presence of the enzyme, but lacking knowledge of that, we tried ΔG(f) for free DNA. Fig. 3 shows Δg(f) and the corresponding F32(f) (dashed black curve), evaluated with m = 6.
In our calculations of Jnet(f), we found the exonuclease flux at saturation (f > 45 pN in Fig. 1) to be sensitive to the value adopted for m in Eq. 4, and the use of ΔG(f) for free DNA required the choice of m = 6 to match the experimental data at saturation (f > 45 pN). Structural data for the T7 DNAp-editing complex are not yet available, but that for kindred polymerases (RB69 and Klenow) indicate that 3 or 4 bp are melted in the exo-binding pocket (1719). Bulk kinetic data on T7 DNAp were considered (15) to suggest m as large as 8 or 9 and thus are compatible with m = 6 but do not provide conclusive evidence for or against that assignment.
Aside from uncertainty regarding the value of m, the nominal F32(f) function evaluated with ΔG(f) for free DNA did not simulate well the experimental results. As seen in Fig. 1, the observed net velocity of the motor switches from the poly to exo mode near 35 pN and reaches the saturation level at ≈45 pN; however, the net flux obtained by using the nominal F32(f) does not switch until 45 pN and does not reach saturation until >60 pN. This deviation is in the direction expected. For DNA complexed with the exo site of the enzyme, melting would likely be enhanced appreciably. By iterating the flux calculations, we obtained a switching function, \( \begin{equation*}F_{32}^{*}(f)\end{equation*}\), and a melting free energy, Δg*(f), shown in Fig. 3 (solid black curves) for which the corresponding net flux (red curve in Fig. 1) switches and saturates in accord with the experimental results. A simple relationship holds to a good approximation: Δg*(f) = 1.5Δg(f). Although only of heuristic value, finding that Δg*(f) is ≈50% larger than Δg(f) serves to illustrate how the presence of the enzyme is likely to foster the tension-induced melting of DNA base pairs.
Whether Eq. 3 is evaluated with \( \begin{equation*}F_{32}^{*}(f)\end{equation*}\) or F32(f), if m = 6 the tension dependence of the equilibrium between the Poly- And exo-binding sites is very strong. For example, tension of 50 pN applied to the DNA is predicted to increase K32(f) above K32(0) by ≈6 orders of magnitude, thus markedly favoring the exo site. The tension dependence of the equilibrium constant, K32 = α3223, might arise from the rate constant for 3 → 2, from its reverse, or both. We therefore evaluated Jnet(f) for the two limiting cases, with the tension dependence wholly contained in either α32 or α23. The tension dependence would be expected to reside chiefly in α32 or in α23, depending on whether the transition state is “late,” in which tension shifts the free energy of activation concomitantly with state 2 (exo site), or “early,” in which tension changes the free energy of activation in parallel with state 3 (poly site). At steady state, we found that the net flux was virtually the same whether the tension dependence was attributed to α32 or α23, indicating that the contribution to Jnet(f) from switching between the poly and exo sites is governed by the equilibrium constant rather than the rate constants.
Excision of DNA at Exo Site. Structural data for editing complexes (18, 19) indicate that the melted portion of the DNA primer strand is held apart from the template strand by interaction with protein residues within the exonuclease active site, wherein one or more bases of the DNA primer strand can be excised. Tension applied to the template is likely to perturb the protein–DNA interactions within the exo-binding pocket. Thus, we expect that template tension may destabilize the transition state for excision within the exonuclease–DNA complex in a fashion qualitatively similar to polymerization in Eq. 2.
There is a striking contrast between the rate constant, αexo(0) = 900 s1 for the overall excision process (step 2 → 2′) at zero tension and the saturation flux Jexo(fsat) ≈ 30 s1 needed to fit the observed data for f > 45 pN (2). In the network shown in Fig. 2B, the exo flux is essentially determined just by the product of αexo(f) and p2(f), the steady-state occupation probability of the exo site.** The large disparity between αexo(0) and Jexo(fsat) therefore suggests considering two limits for the tension dependence: (i) αexo(f) remains near αexo(0), but tension markedly reduces p2(f) to a low saturation level such that αexo(0)p2(fsat) ≈ 30 s1; (ii), tension acts to bring p2(f) up to a saturation level near unity but drastically reduces the rate constant to αexo(fsat) ≈ 30 s1. For i, the saturation flux, Jexo(fsat), would be very sensitive to p2(f), and much less sensitive for ii. The dependence of Jexo(fsat) on the nucleotide concentration, [dNTP], provides a means to determine which of these limits is more realistic. At very low [dNTP], the population that otherwise would be in the poly cycle is shuttled into the exonuclease cycle (state 2). Thus, we expect that p2(f) would be strongly influenced by [dNTP] for limit i but not for ii. The experimental observation that the exo saturation flux remains nearly the same regardless of whether dNTPs are present (2) thus offers strong evidence that the tension dependence resembles limit ii. To model that, we used αexo(f) = αexo(0)Fexo(f), with
\[ \begin{equation*}\;\;F_{{\mathrm{exo}}}(f)=a+(1-a){\mathrm{exp}}[-{\Delta}g_{x}(f)/k_{B}T]\;\;\end{equation*}\]
and a = αsatexo(0) = 30/900. Here Δgx(f) represents a nominal activation free energy for the tension dependence of excision, chosen such that Δgx(0) = 0 and αexo(f) descends smoothly from αexo(0) to αsat. Fig. 3 includes (green curves) Δgx(f) and the corresponding factor Fexo(f). For simplicity, we took Δgx(f) as a linear and steep function of f to make Fexo(f) descend rapidly at low tension, which is not required by the available experimental data but ensures that the contributions of F32 and Fexo to the net flux are approximately separable. Although convenient in our exploratory calculations, such separability seems unlikely to hold when fuller information about tension dependence is attained from experiments or molecular dynamics simulations.

Relations Imposed by Detailed Balance

In the cycle analysis, detailed balance imposes constraints on ratios of rate constants and their dependence on tension or other variables. The ratio of the product of the clockwise to the product of the counterclockwise rate constants (Π+) equals an equilibrium constant given by exp(–ΔGcycle/kBT), where ΔGcycle is the net free-energy change in the cycle and may depend on tension. In practice, this constraint is not useful for cycles such as poly and exo, where the tension dependence of ΔGcycle is not known. However, because the cycle 1 → 3 → 2 → 1 involves no net change in free energy, detailed balance requires the ratio Π+ = α13α32α2112α23α31 = 1 and remains unaffected by tension.†† Accordingly, the tension dependence must satisfy
\[ \begin{equation*}\;\;K_{13}(f)K_{32}(f)K_{21}(f)=K_{13}(0)K_{32}(0)K_{21}(0).\;\;\end{equation*}\]
The variation with tension of K32(f), shown in Fig. 3B, must be offset by the tension dependence of one or both of the other equilibrium constants in this cycle: K21 = α2112 or K13 = α1331, involving two pairs of rate constants. There are many ways that the requisite tension dependence might be apportioned among these rate constants. We examined limiting cases in which the compensating factor, given by \( \begin{equation*}F_{32}^{*}(f)\end{equation*}\) or its reciprocal, was assigned entirely to one or another of the four rate constants. As with the poly-exo switching, we found that the steady-state flux Jnet(f) is determined almost solely by the equilibrium constants K21(f)or K13(f) and is insensitive to how the tension dependence was partitioned between each corresponding pair of rate constants. As shown in Fig. 4, markedly different results are obtained if the compensating tension dependence resides in K21 (case A) rather than K13 (case B). Experimental observations (2) definitely rule out case B, because it exhibits strong variations with [DNAp] that are not seen. Case A indicates that high tension acts to stabilize the enzyme–DNA complex in its exo site (state 2) with respect to both its poly site (state 3) and the dissociated complex (state 1).
Fig. 4.
Jnet versus tension curves as DNAp concentration is varied from 8,000 to 0.008 nM, with [dNTP] = 0.6 mM and [PPi] = 1 μM. (A) For case A. (B) For case B.


The network model developed here employs, in addition to the rate constants for zero tension (Table 1), three tension-dependent factors (F45, \( \begin{equation*}F_{32}^{*}\end{equation*}\), and Fexo; Eqs. 2, 3, and 4). At present, the assumptions adopted to specify these factors are necessarily rudimentary and tentative, especially for \( \begin{equation*}F_{32}^{*}\end{equation*}\) and Fexo, because we lack adequate kinetic and structural data for the exo mode. The network analysis nonetheless facilitates identifying how the dependences on tension or concentration of various links in the network and detailed balance constraints influence properties observable in single-molecule experiments. Molecular dynamics simulations based on structural data for enzyme–DNA complexes offer prospects for improvements, as recently accomplished for the F45 factor (6). Meanwhile, our present model demonstrates generic features arising from the three coupled cycles shown in Fig. 2 and provides specific estimates for testable effects including some not yet observed. Here we outline such aspects and note significant differences from previous interpretations of the single-molecule experimental results for the poly and exo activity of T7 DNAp (2).
Generic Aspects of the Coupled Cycles. From the perspective of a kinetic network analysis, the response of a motor enzyme to tension applied to a DNA template actually pertains not simply to the rate constant for a rate-limiting individual step, as usually assumed, but to the net flux of the coupled-cycle system. Although the tension dependence of the rate-limiting step has a major role, the net flux involves also the occupation probabilities of states of the enzyme–DNA complex. Those probabilities can be influenced strongly by the topology of the network (7). A key feature of the T7 DNAp system is that the entrance nodes for the poly and exo cycles (states 3 and 2 in Fig. 2) are each linked to a common node (state 1). The triangular cycle comprised of those three nodes is subject to a rigorous constraint (Eq. 5) imposed by detailed balance. If the cycle were isolated, at steady state each of its three links would also be at equilibrium (24), and thus the transition flux Jij would vanish for each link. To a very good approximation, we found that this holds even when the triangular cycle is connected to the neighboring poly and exo cycles: At steady state, J13 = J32 = J21 remained nearly zero (≈105/s or less) for million-fold variations in concentrations ([DNAp], [dNTP], and [PPi] each varied by factors of 10±3 from those shown in Fig. 1) and for tensions ranging from f = 0to65pN. Over this wide range, the ratios of the occupation probabilities of the three nodes (states 1–3) therefore are fixed by equilibrium constants regardless of what occurs within the adjacent poly and exo cycles.
Of prime interest is the ratio of the entrance probabilities for the exo and poly cycles, given by p2/p3 = K32(f). This ratio is a thermodynamic rather than kinetic property, because it is determined by the free-energy difference between the exo and poly active sites of the enzyme–DNA complex. Recognizing that aspect is important in designing and assessing either interpretive models or molecular dynamics simulations. In particular, the inference that template tension stabilizes the exo site (state 2) relative to both the poly site (state 3) and the dissociated complex (state 1) is likely to be reliable, because this pertains to a thermodynamic property despite being drawn from comparing a simplistic model with kinetic data (Fig. 4).
Specific Properties Amenable to Experimental Tests. The primary quantity obtained from our network model is the net steady-state flux, Jnet = JpolyJexo. This compares well with experimental results (2) over the entire range of tension studied (f = 0–65 pN), as seen in Fig. 1. It is significant that at zero tension, Jnet(0) = 130 bases/s was obtained without normalization to the experimental data. The contrast with α45 = 300/s illustrates that the network substantially modulates the effect of the rate-limiting step. Except for the region below 25 pN, where F45 is dominant, the agreement stems largely from choosing parameters in \( \begin{equation*}F_{32}^{*}\end{equation*}\) and Fexo to fit key features of the experimental data, fswitch and Jexo(fsat), and from imposing the detailed balance constraint via K21 rather than K13 (case A of Fig. 4). However, several consequent properties of the model invite experimental tests.
(i) The Jnet(f) curve should remain nearly the same for wide variations in the concentration of DNAp (0.8–8,000 nM) or PPi (1 nM to 1 mM). This insensitivity arises because over these ranges the steps involving [DNAp] and [PPi] are much more facile than those determining Jpoly and Jexo. Further, the detailed balance constraint imposed in case A has the effect of reducing the enzyme concentration required to reach the range in which the net flux becomes insensitive to [DNAp].
(ii) For [DNAp] below ≈0.8 nM or [PPi] above 1 mM the exo saturation flux level should decrease in magnitude and the poly flux should also drop, but the tension (fswitch) at which the enzyme motor reverses direction should remain about the same. These trends occur because the population of the poly entrance node, p3, drops when low [DNAp] inhibits loading that site (via 1 → 3) or high [PPi] opens an additional drain (via 3 → 7); the population of the exo entrance node, p2p3K32(f), then drops in parallel. The poly and exo fluxes likewise drop in parallel when ffswitch, where they become equal (with p2p3), and thus fswitch remains unchanged (Fig. 4A).
(iii) In contrast, as [dNTP] is decreased, the exo saturation level should change little (only a factor of 2 increase for 106-fold decrease in dNTP), but fswitch should move steadily lower. As shown in Fig. 5, the net flux at low tension should also drop markedly. Because lowering [dNTP] inhibits the first step (3 → 4) in the poly cycle, it causes p3 to build up and thereby shifts flux from the poly to the exo cycle. The tension, fswitch, at which Jpoly = Jexo moves downward as [dNTP] decreases, because the poly flux drops rapidly while the exo flux changes only modestly. Fig. 6 displays the shifts of steady-state populations of network nodes with tension for a typical concentration of dNTP (Fig. 6 Upper, corresponding to red curves of Figs. 1 and 5), and a concentration 103-fold lower (Fig. 6 Lower, corresponding to brown curve of Fig. 5).
Fig. 5.
Jnet versus tension curves (for case A of Fig. 4) as [dNTP] is decreased over a104-fold range from 6 mM to 0.6 nM, with [DNAp] = 800 nM and [PPi] = 1 μM. In this range, the tension (fswitch) at which the enzyme motor reverses direction decreases from ≈40 to 20 pN. The red curve is the same as in Figs. 1 and 4A.
Fig. 6.
Variation of steady-state occupation probabilities for nodes of Fig. 2B with tension on DNA template for case A of Fig. 4, with [DNAp] = 800 nM and [PPi] = 1 μM. (A) For [dNTP] = 600 μM, corresponding to red curves of Figs. 1 and 5. (B) For [dNTP] = 600 nM, corresponding to brown curve of Fig. 5.
As yet, experiments varying [PPi] or testing the properties noted in iii have not been reported. Examining the effect of decreasing [dNTP] on the switching tension is particularly desirable. The model definitely predicts that reducing the available chemical energy, supplied by dNTP, facilitates exonucleation.
Differences from Previous Interpretations. The rate data of Fig. 1 in the region f < 30 pN, where polymerization is dominant, were originally interpreted by Wuite et al. (2) using a “global” model. This used Eq. 2 but with the free energy of activation for the rate-limiting open to closed conformational change taken as nΔg(f), where n denotes the number of adjacent template nucleotides converted from ss to ds geometry in the closed conformation, n – 1 of which revert to ss geometry before onset of the next replication step. The term Δg(f), determined from experimental force-extension curves for ss and ds DNA, is the same as that appearing in our Eq. 4. Aside from n, the model does not depend on the enzyme at all, because the force-extension curves pertain to bare DNA. Comparison with the data indicated n = 2 for catalysis by T7 DNAp (2); a similar study found n = 2 for sequenase and n = 4 for Klenow (3). These results were incompatible with crystal structures of enzyme–DNA complexes, which show n = 1 in the closed conformation of the poly site. Molecular dynamics simulations (6) have now confirmed that n = 1 and shown (Fig. 3A) that the free energy of activation, Δg(f), differs greatly from Δg(f) as a consequence of steric constraints on the DNA imposed at the poly active site of the enzyme.
Previous interpretations of rate data for the region f > 40 pN, where exo becomes dominant, were qualitative. From comparison with the tension dependence observed for Escherichia Coli exonuclease I, it was suggested that melting or fraying of the primer strand was not rate-limiting during tension-induced exo by T7 DNAp, which instead might result from deformation of the 3′ end of the primer (2). The inference remains uncertain, however. Melting occurs chiefly during the switching transition from the poly active site (which is lacking in E. Coli exonuclease I) to the exo site (15) and likely would be fostered by tension-induced deformation of the 3′ end. Still more open is the process responsible for saturation of the exo rate at tensions between 45 and 65 pN. This was considered to suggest that T7 DNAp imposes a tension-independent mediation on the melting rate (2). Our model assumes that melting governs the poly → exo switch, but is not necessarily rate-limiting and attributes the exo saturation to a combined effect of melting and the subsequent excision process rather than solely a cap on the melting rate.
The rate data also exhibit “bursts and gaps” in the progress of the enzyme motor as it moves along the DNA strand (2). These spurts and pauses of the motor correspond to an enzyme molecule loading via its poly or exo site onto the stretched template strand, traveling some ways forward or backward, and then falling off. Our model calculations reported in this article deal only with steady-state fluxes. Although bursts and gaps contain additional information, qualitative interpretations seem inadequate and thus indicate the need for a more comprehensive analysis.


Abbreviations: DNAp, DNA polymerase; ds, double-stranded; ss, single-stranded; exo, exonucleolysis site or cycle; poly, polymerase site or cycle.
In each exo cycle, the free energy released by hydrolysis of a DNA base pair is very large, approximately –10 kBT, and thus the exo cycle operates far from equilibrium (23). Thus, we assume that even at high tension the reverse reaction (2′ → 2) does not contribute significantly to Jexo(f).
For this cycle, the reported rate constants at zero tension (810) gave Π+ ≈ 0.1 rather than unity. To satisfy detailed balance, we therefore adjusted the reported values of α12 and α13 (in Fig. 2A) by factors (0.5 and 4.25, respectively) well within the experimental uncertainty (S. Patel, personal communication); the adjusted values are shown in Table 1. These adjustments to the enzyme loading rates from solution at zero tension were found to make almost no difference to Jnet(f) for case A over a wide range of enzyme concentrations.
The importance of enforcing the detailed balance constraint of Eq. 5 was assessed by calculations omitting any tension dependence of K21 or K13, which gave Jnet(f) curves that behaved qualitatively similar to those of Fig. 4B, with the exo saturation flux typically too small (by a factor or 5 or more) and much too strong variation with [DNAp]. We were unable to avoid these discrepancies by altering the F32(f) and Fexo(f) factors without incorporating detailed balance.


We thank Gijs Wuite, Carlos Bustamante, Antoine van Oijen, Sunney Xie, Peter von Hippel, Myron Goodman, and Ken Johnson for helpful discussions. We are grateful for support of this work by the National Science Foundation (Grant PHY 0210437). A.G. is also grateful for support from the Klivingston Fellowship for Biodynamics provided by the Fetzer Institute.


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Published in

Go to Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
Vol. 100 | No. 17
August 19, 2003
PubMed: 12869690


Submission history

Published online: July 17, 2003
Published in issue: August 19, 2003


We thank Gijs Wuite, Carlos Bustamante, Antoine van Oijen, Sunney Xie, Peter von Hippel, Myron Goodman, and Ken Johnson for helpful discussions. We are grateful for support of this work by the National Science Foundation (Grant PHY 0210437). A.G. is also grateful for support from the Klivingston Fellowship for Biodynamics provided by the Fetzer Institute.



Anita Goel
Department of Physics, Harvard University and Harvard–Massachusetts Institute of Technology Division of Health Sciences and Technology, Cambridge, MA 02138; Department of Physics and Astronomy, University of Maine, Orono, ME 04469; and Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138
R. Dean Astumian
Department of Physics, Harvard University and Harvard–Massachusetts Institute of Technology Division of Health Sciences and Technology, Cambridge, MA 02138; Department of Physics and Astronomy, University of Maine, Orono, ME 04469; and Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138
Dudley Herschbach
Department of Physics, Harvard University and Harvard–Massachusetts Institute of Technology Division of Health Sciences and Technology, Cambridge, MA 02138; Department of Physics and Astronomy, University of Maine, Orono, ME 04469; and Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138


To whom correspondence may be addressed. E-mail: [email protected] or [email protected].
Contributed by Dudley Herschbach, May 23, 2003

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